Je Remercie Infiniment M Costas

Kiparissides Ainsi Que Mme

Jovita Moreno Vozmediano D'

Guiseppe Storti

Frank, Muriel Olivier

Elodie Pierre-Yves Christophe

Lewalle L Fouarge

A Auwera

M V D & Brande

Keywords: polyethylene, Induced Condensing Agents (ICA), Sanchez-Lacombe equation of state (SL EoS), solubility, diffusivity, Random-Pore Polymer Flow model (RPPFM), quaternary systems polyéthylène, agents de condensation induits (ICA), équation d'état de Sanchez-Lacombe

Je souhaiterai en premier lieu remercier le laboratoire CP2M de m'avoir accueillie ainsi

En 2020, les polyoléfines sont la famille de polymères la plus utilisée au monde puisqu'ils représentent environ 45% de la production de matières plastiques dans le monde. Le marché mondial du polyéthylène devrait augmenter substantiellement à environ 150 millions de tonnes d'ici 2027 1,2 . En effet, le polyéthylène, objet de la thèse actuelle, est le polymère le plus produit sur la planète en raison de sa large gamme de propriétés et d'utilisations, de son faible coût de production et d'un impact environnemental relativement faible. Les polyéthylènes sont utilisés dans de nombreux domaines essentiels, comme par exemple l'électricité et l'électronique, l'emballage, l'automobile, les produits pharmaceutiques, le bâtiment et la construction 3 .

Le polyéthylène peut être fabriqué dans des procédés en solution, en suspension et en phase gazeuse en utilisant de l'éthylène et très souvent un comonomère α-oléfine 4 . La polymérisation de l'éthylène sur un catalyseur supporté dans des réacteurs à lit fluidisé (FBRs) en phase gazeuse est le procédé prédominant pour la production d'environ 75% de polyéthylène linéaire basse densité (LLDPE) et 25% de polyéthylène haute densité (HDPE) dans le monde. Cependant, en raison de la nature hautement exothermique de la polymérisation, la vitesse maximale à laquelle cette chaleur peut être éliminée du réacteur est clairement une étape limitante dans la vitesse de production du polymère dans les réacteurs en phase gazeuse 5 . Le procédé en mode condensé est utilisé afin d'augmenter la capacité d'évacuation de la chaleur, permettant ainsi d'optimiser la productivité. En effet, en mode de fonctionnement condensé, en plus d'injecter de l'éthylène, de l'hydrogène, de l'azote, et éventuellement un comonomère; des solvants inertes, qui sont principalement des alcanes comme le propane, l'isobutane, le n-pentane ou le n-hexane, sont utilisés comme agents de condensation induits (ICAs) 6 . Ces alcanes permettent principalement d'évacuer la chaleur générée à l'intérieur du réacteur puisqu'ils sont partiellement liquéfiés dans un échangeur de chaleur externe puis pulvérisés à l'intérieur du réacteur. Toutes ces espèces présentes dans la phase gazeuse, inertes ou non, diffuseront dans la phase amorphe du polymère.

Ils modifieront donc la structure physique et éventuellement chimique du polymère produit, qui à son tour peut influencer des paramètres importants tels que la vitesse de la réaction, par le biais du changement de la solubilité et de la diffusivité de ces espèces 7,8 . Afin de mieux contrôler et comprendre le processus de polymérisation, il est évidemment assez important de connaître avec précision la solubilité ainsi que la diffusivité des espèces individuelles dans la phase amorphe du polymère.

Jusqu'à présent, la plupart des études expérimentales de solubilité sont réduites aux systèmes binaires (pénétrant -PE), avec un nombre limité de systèmes ternaires (pénétrant 1 -pénétrant 2 -PE). Cependant, la plupart des études ternaires mesurent la solubilité globale du mélange gazeux dans la phase amorphe du polymère. Les modèles thermodynamiques, permettant de calculer la solubilité totale mais aussi partielle des différentes molécules composant le mélange gazeux, sont donc très important à développer. Évidemment, cela signifie que ces modèles thermodynamiques doivent être appropriés, c'est-à-dire prendre en considération les interactions entre les différentes molécules dans le mélange. Il est clair que l'on ne peut pas utiliser des modèles de solubilité binaires pour prédire les solubilités individuelles des molécules de pénétrant dans les systèmes à plusieurs composants, puis estimer la solubilité globale en additionnant simplement les solubilités partielles 9 .

L'équation d'état de Sanchez-Lacombe (SL EoS) est l'un des modèles thermodynamiques les plus couramment utilisés dans le domaine des polyoléfines 10 . Par contre, cette équation d'état doit être ajustée aux données de solubilité expérimentales afin d'identifier des paramètres d'interactions relatifs à SL, et ces données font cruellement défaut pour les systèmes à plusieurs composants dans des conditions similaires à celles utilisées dans un processus de polymérisation industriel. Actuellement, les expériences basées sur la méthode gravimétrique utilisant une balance à suspension magnétique (MSB) sont peut-être la technique la plus utilisée pour ce type de mesure de solubilité en raison de leur relative simplicité et de leur haute précision. Un grand nombre d'études ont utilisé cette méthode pour étudier l'impact des pressions partielles et de la température sur la solubilité et la diffusivité des systèmes (principalement) binaires et (très occasionnellement) ternaires. Cependant, ces expériences n'aboutissent souvent qu'à l'obtention de données thermodynamiques globales d'un mélange donné dans un polymère, et des étapes supplémentaires sont nécessaires afin d'estimer les solubilités partielles. Afin de surmonter le manque de données de solubilités individuelles dans les systèmes à plusieurs composants, une nouvelle approche permettant l'estimation des paramètres d'interaction de Sanchez-Lacombe est proposée. En effet, cette méthode est basée sur la combinaison de mesures expérimentales de solubilité totale avec des estimations du facteur de compressibilité des mélanges gazeux à travers l'équation d'état de Peng-Robinson 11 . Ces deux données seront alors ajustées à SL EoS afin de déterminer les paramètres d'interactions spécifiques à un système donné. Cette nouvelle approche permet donc de calculer les solubilités individuelles des deux pénétrants dans un système ternaire non idéal où la troisième phase est le polyéthylène.

En raison du manque important de données de solubilité dans des conditions industrielles, la méthode expérimentale de perte de charge a été employée afin de mesurer expérimentalement XIII les solubilités globales et individuelles des systèmes à plusieurs composants. Les solubilités individuelles et globales ternaires et quaternaires ont été mesurées grâce à cette méthode et les solubilités globales ont été validées avec la technique gravimétrique. Ensuite, ces systèmes ont été modélisés à partir de l'équation d'état de Sanchez-Lacombe. En effet, pour la première fois à notre connaissance, l'application de l'équation d'état de Sanchez-Lacombe est étendue du système ternaire au système quaternaire, comprenant donc trois pénétrants et un polymère.

L'effet des ICAs ainsi que du comonomère sur la solubilité de l'éthylène dans la phase amorphe du polymère a été étudié. Les effets de co-solubilité, entrainant une augmentation de la solubilité de l'éthylène en présence de composés plus lourds, ainsi que les effets anti-solvants, provoquant une baisse de la solubilité des composés lourds en présence d'éthylène, ont été mis en évidence.

De plus, un manque significatif de données concernant la diffusivité des mélanges à plusieurs composants en phase gazeuse est également observé dans la littérature. Cette information est cruciale car il a été montré que la prise en compte de la co-solubilité ainsi que des effets de codiffusion ont un impact notable sur le transfert de masse et de chaleur et donc sur la vitesse de polymérisation. Il est donc important d'obtenir d'avantage d'études expérimentales et théoriques sur la diffusion des mélanges de pénétrants dans la phase amorphe du polymère. La diffusivité globale des systèmes binaires, ternaires et quaternaires sera donc mesurée par la méthode gravimétrique, et analysée à partir de la solution de Crank à la loi de Fick en utilisant une représentation des particules à deux niveaux 12 . En effet, ce modèle suppose que la particule de polymère est constituée de deux tailles de granulés de polymère compact, avec des fractions différentes. Les effets de co-diffusion ont été démontré dans les systèmes ternaires par l'augmentation de la diffusivité globale en présence de composés plus lourds (i.e. ICAs, comonomère). Les diffusivités quaternaires sont quant à elles assez proches des diffusivités ternaires dans la gamme de pression analysée.

La connaissance de ces données thermodynamiques est essentielle pour décrire la polymérisation de l'éthylène en phase gazeuse puisque les propriétés physiques et chimiques du polymère seront fortement influencées par la variation de la composition de la phase gazeuse.

En effet, lors de la polymérisation de l'éthylène, la particule de catalyseur est d'abord injectée dans le réacteur. Le monomère diffuse de la phase continue du réacteur, et dans les pores de la particule supportée, où se trouvent les sites actifs du catalyseur. Lorsque le monomère entre en contact avec les sites actifs de la particule de catalyseur, la formation de polymère a lieu. A mesure que le polymère s'accumule au niveau des sites actifs, la structure initiale de la particule XIV subit une accumulation de contraintes conduisant au processus de fragmentation. La formation continue de polymère, qui se déposera sur la surface et dans les pores du catalyseur, entraine une adsorption et une diffusion du monomère à travers cette couche de polymère pour atteindre les sites actifs, où la polymérisation a lieu. Au fur et à mesure que la réaction se déroule, la particule continue de se dilater et le volume de polymère continue de croître 13,14 . Cela se traduira par des gradients de température et de concentration à travers la particule de polymère en croissance conduisant à des variations des vitesses de réaction locales et des propriétés du polymère. Ainsi, pour décrire la réaction de polymérisation, il est impératif de pouvoir prédire les dynamiques de transfert de matière et de chaleur dans la particule en croissance.

Un modèle de croissance des particules, tenant compte des limitations de masse et de transfert de chaleur se produisant pendant la croissance d'un catalyseur Ziegler-Natta lors de réaction de polymérisation en phase gazeuse, est développé. Le "Random-Pore Polymer Flow Model" (RPPFM) 15 sera adapté afin d'estimer les vitesses de polymérisation ainsi que l'évolution temporo-spatiale des profils de concentration et de température dans la particule de polymère et la distribution de masse molaire. Evidemment, la détermination précise à la fois de la concentration des pénétrants dans la phase amorphe du polymère et du coefficient de diffusion des pénétrants est cruciale dans toute étude de modélisation de la croissance des particules. En incorporant un modèle thermodynamique précis, le RPPFM est donc capable de prédire l'impact de la composition de la phase gazeuse (ICA, hydrogène) sur la polymérisation de l'éthylène en phase gazeuse. Il a été conclu que le comportement inverse a été observé pour l'ICA et l'hydrogène. En effet, une augmentation de la pression partielle de l'hydrogène entraine une diminution de la vitesse de polymérisation ainsi que de la distribution de la masse molaire alors qu'une augmentation de la pression partielle de l'ICA entraine le comportement inverse.

Enfin, il convient de rappeler que les procédés en phase gazeuse ne sont pas les seuls utilisés pour la production du polyéthylène. Etant donné l'impact des différentes espèces utilisées dans la phase gazeuse, il est raisonnable de supposer que les particules de polymère gonfleront aussi dans les réacteurs en suspension. La façon dont ils gonflent dépendra ainsi du diluant utilisé, de la température du réacteur et des propriétés du polymère. Comme pour les systèmes en phase gazeuse, les données thermodynamiques concernant les systèmes binaires diluants liquides/PE liés à la polymérisation en suspension ne sont pas suffisamment courantes dans la littérature 16 .

La connaissance de ces données thermodynamiques est cruciale si l'on a besoin de contrôler la masse molaire du polymère produit ou d'optimiser le processus de dégazage. En outre, ceci est d'autant plus important que ces diluants liquides modifieront les propriétés physiques du polymère produit en provoquant son gonflement et son ramollissement en raison de leur forte solubilité dans la phase amorphe du polymère. Une étude expérimentale a donc été réalisée afin de mesurer la solubilité et le gonflement de la phase amorphe du polymère en présence de différents diluants liquides dans des conditions industrielles avec différents grades de polyéthylène. Nous avons conclu que la solubilité ainsi que le gonflement augmentent lorsque l'on augmente la température ou que l'on diminue la longueur de la chaine carbonée des diluants liquides et la cristallinité du polymère. Il est aussi important de noter que le taux de gonflement du polymère en présence de diluants liquides légers (i.e. n-pentane, n-hexane) peut atteindre 57 %vol à 90°C. Cette augmentation drastique du gonflement est une propriété très importante à contrôler dans les réacteurs industriels afin d'éviter un encrassement du réacteur dû à la plastification du polymère 17 .

Toutes ces données thermodynamiques, aussi bien en phase gazeuse qu'en suspension, doivent être soigneusement prises en compte, car elles sont un outil important pour comprendre l'évolution de la particule de polymère en croissance au cours de la réaction de polymérisation de l'éthylène permettant ainsi une modélisation précise des réacteurs industriels. 𝐶 𝑝,ov overall heat capacity of the particle (J.kg -1 .K -1 )

𝐶 𝑝g,in heat capacity of the inlet gas stream (J.K -1 .mol -1 )

[𝐶 𝑖 ] eq equilibrium concentration of component i in the particle (mol.m -3 )

[𝐶 𝑖 ] concentration of component i (mol.m -3 )
𝐷 𝑖,ov overall (or effective) diffusivity of component i (m 2 . s -1 )

𝐷 𝑟 concentration of dead polymer chains (mol.m -3 ) D diffusion coefficient (m 2 .s -1 )

𝐷 0 pre-exponential factor of Vrentas and Duda (m 2 .s -1 )

𝐸 a diffusion activation energy (J.mol -1 ) h heat transfer coefficient (J.m -2 .s -1 .K -1 )

[𝐻 2 ]
hydrogen concentration in the bed (mol.m -3 ) 𝑘 c,p overall thermal diffusivity of the particle (J.m -1 .s -1 .K -1 ) 𝑄 vap total enthalpy change due to evaporation of liquid in the reactor (J.mol -1 ) R universal gas constant (J.mol -1 .K -1 )

𝑘
𝑅 p polymerization rate (mol.m -3 reactor bed.s -1 )

𝑅 v polymerization rate (mol.m -3 cat.s -1 ) 𝑆 𝑖 solubility of component i (g/g.am.polymer) 𝑇 g,in inlet gas stream temperature (K)

𝑇 r reduced temperature (K)

𝑇 w temperature of the exchanger (K) 𝑇 b bulk temperature of the reactor (K)

T temperature (K) U overall heat transfer coefficient (J.m -2 .s -1 .K -1 )

𝑉 ̂𝑖 * specific critical hole free volume required for the i th component (cm 3 .g -1 )

𝑉 R volume of reactor bed (m 3 )

𝜔 𝑖 mass fraction of the i th component (-)

Greek letters

𝜉 13 ratio of molar volumes for the solvent and polymer jumping units (-)

𝜌 𝑜𝑣 overall particle density (kg.m -3 )

-𝛥𝐻 p overall enthalpy of polymerization (J.mol -1 ) 𝜐 molar volume (m 3 .mol -1 )

𝜙 𝑖 volume fraction of the i th component (-)

𝜒 solvent-polymer interaction parameter (-) w acentric factor (-)

General Introduction

In 2020, polyolefins represent approximately 45% of the production of plastic materials worldwide, which makes them the most widely used family of polymers in the world. The worldwide market for polyethylene alone is expected to rise substantially to about 150 million tons by 2027, making it the most extensively produced polymer on the planet because of its broad range of properties and end-uses, low production cost, and relatively low environmental impact. PE products are used in many important fields, for example electrical and electronics, packaging, automotive, pharmaceuticals, building and construction.

Polyethylene can be made in solution, slurry and gas phase processes using ethylene, and very often an α-olefin comonomer. The polymerization of ethylene on supported catalysts in gas phase fluidized bed reactors (FBRs) continues to be the predominant process for production of approximately 75% of linear low density polyethylene (LLDPE) and 25% of high density polyethylene (HDPE) worldwide. However, due to the highly exothermic nature of the polymerization, the maximum rate of heat removal from the reactor is clearly a limiting step in the polymer production rate in gas phase reactors. Therefore, in order to optimize the productivity, condensed mode cooling is employed in order to increase the capacity of heat removal. In condensed mode operation, in addition to injecting ethylene, hydrogen, nitrogen, and eventually a comonomer, inert compounds, which are mainly alkanes as propane, isobutane, n-pentane or n-hexane, are used as induced condensing agents (ICAs). These alkanes can be partially liquefied in an external heat exchanger and then sprayed inside the reactor mainly to serve as a dispersing heat medium for removing the heat generated within the reactor.

All of the species present in the gas phase, inert or not, will diffuse into the amorphous phase of the polymer. They will therefore change the physical, and possibly the chemical structure of the final polymer, which in turn can influence important parameters such as the reaction rate, through the change in the individual species solubility and diffusivity. In order to better control and understand the polymerization process, it is obviously important to know the solubility as well as the diffusivity of the individual species in gas mixtures in the amorphous phase of the polymer with precision.

Up to now, most of the experimental solubility studies were restricted to binary systems (penetrant -PE), with a limited number looking at ternary systems (penetrant 1penetrant 2 -PE). However, most of the ternary studies only measure the overall solubility of the gas mixture in the amorphous phase of the polymer, making thermodynamic models crucial when looking at the effect of the individual solubilities of the species inside the polymer. Obviously, this means that if one wishes to develop models to quantify these effects, appropriate thermodynamic models are needed. It is clear that one cannot always use binary sorption models in order to predict the individual solubilities of penetrant molecules in multicomponent systems, and then estimate the overall solubility by simply adding the partial solubilities. It is necessary to account for the way that they interact with the polymer in a ternary (or quaternary) system. This is of course possible using equations of state (EoS), such as the Sanchez-Lacombe EoS.

The Sanchez-Lacombe EoS (SL EoS) is one of the most commonly used thermodynamic models in the field of polymer reaction engineering. However, the main drawback of this model is that it requires empirical, adjustable parameters that are specific to a given system. These adjustable interaction parameters in the SL EoS need to be estimated experimentally as there is no way to estimate them à priori. To do so, this equation of state has to be fitted to experimental solubility data, and such data is sorely lacking for multicomponent systems under conditions similar to those used in a polymerization process. Currently, gravimetry based on a magnetic suspension balance (MSB) is perhaps the most widely used technique for this type of solubility measurement because of its relative simplicity and high precision. A large number of studies have used this method to investigate the impact of partial pressures and temperatures on the solubility of (mostly) binary and (very occasionally) ternary systems (and as far as we have been able to determine, never quaternary systems). However, MSB experiments give only the total solubility of a given mixture in a polymer, and additional measurements or steps need to be taken to estimate the individual solubilities. Besides, the advantage of this method is that it can also give the global diffusivity of multicomponent systems, which is an important information.

In Chapter 2, the gravimetric technique was used to measure the total solubility of ternary systems that are not available in the literature, such as ethylene/propane/LLDPE, ethylene/isobutane/LLDPE and ethylene/1-butene/LLDPE. Only global solubilities of the mixtures are measured with the magnetic suspension balance. In order to overcome the lack of individual solubility date in multicomponent systems, a novel approach for the estimation of the Sanchez-Lacombe interaction parameters, is proposed. This method is based on combining total solubility measurements with estimates for the compressibility factor of gas mixtures using the Peng-Robinson Equation of State (PR EoS). This novel approach makes it possible to calculate individual components solubilities for the two penetrants in non-ideal ternary systems where the third phase is polyethylene.

General Introduction

The important lack of solubility data in polyolefin systems, especially in ternary and quaternary systems at industrial conditions, inspired the work presented in Chapter 3. As an alternative of the gravimetric experiments, an accurate and reliable procedure based on pressure decay was developed. The advantage of this method is that it can give the overall solubility, as gravimetric technique, but also the partial solubilities of each component in the gas mixture. Ternary and quaternary individual and overall solubilities were measured using the pressure decay method and validated by the gravimetric technique. The effect of ICAs as well as comonomer on the solubility of ethylene in the amorphous phase of the polymer were studied in this chapter. The Sanchez-Lacombe EoS was expanded to include four components, therefore the pressure decay method was used to estimate the total and individual solubilities. Total solubilities were validated with experimental data through gravimetric method. This appears to be the first time that the application of the Sanchez-Lacombe EoS is extended from the ternary system to the quaternary system, including therefore three penetrants and a polymer.

A significant lack of data concerning the diffusivity of multicomponent systems is also observed in the literature. This information is crucial since it has been shown that accounting for the co-solubility as well as the co-diffusion effects have a noticeable impact on mass and heat transfer and therefore on the polymerization rate. The diffusivity of different penetrants in polyethylene will be studied in Chapter 4. The diffusion in multicomponent polymeric systems is so far not completely understood, especially in ternary and more importantly in quaternary systems. It is therefore crucial to have more experimental and theoretical studies about the diffusion of mixtures of penetrants in the amorphous phase of the polymer since it has a great importance in ethylene polymerization reaction in gas phase. The overall diffusivity of binary, ternary and quaternary systems will therefore be measured through the gravimetric method, based on the magnetic suspension balance, and analyzed with Crank's solution to Fick's law using a two-level particle representation. This particle model assumes that the polymer particle is constituted of two sizes of compact polymer granules, with different fractions.

Knowledge of this thermodynamic data is key when describing ethylene polymerization in gasphase since the physical and chemical properties of the produced polymer will change when varying the gas phase composition. During ethylene polymerization, the catalyst particle is first injected into the reactor. Monomer diffuses from the continuous phase of the reactor, and into the pores of the supported particle, where the active sites of the catalyst are found. When monomer contacts the active sites of the catalyst particle, polymer formation takes place and as polymer accumulates at the active sites, the initial structure of the particle suffers a build-up of stress leading to the fragmentation process. The continuous formation of polymer, which will deposit on the catalyst surface and pores, causes the monomer to absorb and diffuse through this polymer layer to reach the active sites, where polymerization takes place. As the reaction proceeds, the particle continues to expand and the volume of polymer keep growing. This will result in gradients of temperature and concentration through the growing polymer particle leading to variations in local reaction rates and polymer properties. Thus, to describe the polymerization reaction, it is imperative to be able to predict the dynamics of mass and heat transfer in the growing particle.

In Chapter 5, a comprehensive single particle growth model, accounting for mass and heat transfer limitations appearing during the early growth of a Ziegler-Natta catalyst in gas phase olefin polymerization, is developed. The Random-Pore Polymer Flow Model (RPPFM) will be adapted in order to estimate the overall particle polymerization rate as well as mass and energy balances, in order to predict the temporal-spatial evolution of concentration and temperature profiles in the polymer particle as well as molecular weight distributions. This chapter is based on the thermodynamic and diffusivity models developed in the first chapters, as the accurate determination of both the penetrants concentration inside the amorphous phase of the polymer and the effective penetrants diffusion coefficient is crucial in any particle growth modeling study. Therefore, the SL EoS will be used in order to describe the solubility of the different species inside the amorphous phase of the polymer at equilibrium (used as a boundary condition). And Vrentas and Duda diffusivity model is applied to calculate the diffusivity of penetrant molecules in the semicrystalline polymer. By incorporating an accurate thermodynamic model, the RPPFM is therefore able to predict the impact of the gas phase composition (i.e. ICA, hydrogen) on ethylene polymerization.

Finally, it should be recalled that gas phase processes are not the only ones used for PE production. The different species in a liquid phase cause the polymerizing particles to swell in a slurry process. The way they swell will depend on the used diluent, the reactor temperature and polymer properties. As in gas phase systems, very little information on the thermodynamic data of PE/liquid diluents related to slurry polymerization is available in the literature. Very few studies have investigated the solubility of liquid diluents in the amorphous phase of the polymer and the way the diluents swell the polymers. Knowledge of this thermodynamic data

is crucial if one needs to control the molecular weight of the produced polymer or optimize the degassing process. Besides, this is all the more important since these liquid diluents will change the physical properties of the produced polymer as they cause its swelling and softening due to their high solubility in the amorphous phase of the polymer. In Chapter 6, an experimental study has been performed in order to measure the sorption and swelling of the amorphous phase of the polymer in presence of different liquid diluents under industrial conditions (i.e. diluents, temperatures) with different grades of polyethylene. The effect of temperature, diluent chain length, crystallinity and partial dissolution of the polymer have been studied.

All these thermodynamic data, both in gas and slurry phase, represent important tools for understanding the evolution of the growing polymer particle during ethylene polymerization reaction, and are necessary for accurate modeling of industrial reactors.

Chapter 1

Literature Review

Introduction to polyolefins

The term polyolefins is a general one that usually refers to polymers of ethylene (polyethylene, PE) or propylene (polypropylene, PP), and their copolymers with each other and other -olefins such as 1-butene or 1-hexene. Polyolefins exhibit a huge range of properties despite their chemical simplicity (they are composed uniquely of carbon and hydrogen atoms, and nothing else!). Indeed, they range from rigid materials, which are used for car parts, to soft materials such as flexible fibres. Some are as clear as glass; others are completely opaque 1 . Some, such as microwave food containers, have high heat resistance while others melt easily. Indeed, the wide-spread use of polyolefins is due to their extensive range of properties and end-uses, low production cost and relatively low environmental impact. Polyethylene, the focus of the current thesis, is a macromolecule containing simple repeating units of monomers composed of covalently linked carbon atoms in which two hydrogen atoms are attached. Figure 3 shows polyethylene molecular structure, where n is the number of ethylene monomer units in the α-olefin.

Figure 3. Polyethylene molecular structure

Polyethylene is used in the manufacture of a wide variety of consumer products, such as films, pressure pipes, insulation of wires and cables, and many other products 4 . These products are used in many important fields, as for example electrical & electronic, packaging, automotive, building & construction, and pharmaceuticals.

Polyethylene is divided into at least three main families, usually depending on its density: highdensity polyethylene (HDPE), linear low-density polyethylene (LLDPE), and low-density polyethylene (LDPE).

Low-density polyethylene (LDPE) is made using free radical process and contains short chain branches (SCB) and long chain branches (LCB). These resins are produced under high pressures in autoclave or tubular reactors. High-density polyethylene (HDPE) and linear low-density polyethylene (LLDPE) are made with coordination catalysts and contain only SCBs. Table 1 summarizes the main properties of the different types of polyethylene resins.

These resins can be obtained by slurry, solution or gas-phase processes, using different types of coordination catalysts. HDPE can be made with Ziegler-Natta, metallocene or chromium catalysts, but the latter cannot be used effectively for LLDPE as they do not incorporate comonomers very well. Special grades of LLDPE resins are made in solution processes, but most of this family of polymer is produced in gas-phase processes since the amorphous phase tends to be soluble in the diluents used in slurry reactors 5 . Polyethylene is often classified according to its melt flow index (MFI) and density. The overall density of the polymer is related to its crystallinity; more branches lead to lower density of the final polymer and therefore lower crystallinity. Indeed, this density can be controlled by adding a comonomer that will lead to higher branches inside the produced polymer. The MFI depends on the molecular weight distribution (MWD) and the LCB content of the polymer. It is a measure of the above molecular and structural chain characteristics which in turn affect the processability of the produced polymer; a lower polymer molecular weight leads to a higher MFI. These are part of the important properties of the polymer that have to be taken into account when producing special grades of polyethylene.

1.2.Catalysts for ethylene polymerization

As we just mentioned, there are three major types of catalyst used for the polymerization of olefins: Ziegler-Natta, Philips and metallocene catalysts.

Ziegler-Natta (ZN) catalyst are the most widely used catalyst due to their high activity and selectivity, and low cost. The fifth generation of Ziegler-Natta catalyst can lead to a productivity of 100-130 kg of polymer / gram of catalyst compared to 2-4 kg of polymer / gram of catalyst for the 1st generation of Z-N catalyst. These catalysts are composed of a transition metal salt of metals from group IV to VII, which is the (pre)catalyst itself, and a metal alkyl from group I to III; known as the co-catalyst or activator 6 . When the transition metal is exposed to the co-catalyst, the polymerization active sites are created by alkylation then reduction of the transition metal centers. When Ziegler-Natta catalysts are used in supported form, they are usually found in the form of titanium tri-or tetrachloride, and are supported on magnesium dichloride (MgCl2), or on silica (SiO2) coated with MgCl2. The presence of more than one type of active site in heterogeneous Ziegler-Natta catalyst can lead to a wide molecular weight distribution (MWD), with a polydispersity index (PI) between 4-10, and an inhomogeneous chemical composition distribution (CCD) 7 . Ziegler Natta catalysts can also be used in solution processes, most often for elastomers, but in this case, they are usually based on vanadium or vanadium oxide.

Phillips chromium catalysts are composed of a chromium oxide (CrOx) impregnated on a silica support. The particularity of this catalyst is that there is no need for a co-catalyst because it is activated in situ by the monomer (i.e. ethylene) during the polymerization. The exact mechanism by which the actives species are being created still remains unknown. However, several authors revealed that the metal forming the active site can exist simultaneously in different oxidation states 8 . This will lead to products having (very) broad molecular weight distribution with a PI that can be as high as 15 to 30 (as opposed to 4-10 for supported Ziegler-Natta catalysts). However, the high sensitivity of this catalyst to impurities can lead to some difficulties when using it. Phillips catalyst are only used in the supported form.

Finally, metallocenes, similarly to ZN catalysts, can be used in both molecular and supported form. Most metallocene catalysts require an activator like methylaluminoxane (MAO).

However, recent studies revealed that it is possible to obtain commercially viable metallocene catalysts without MAO 9 . The main difference with the other catalysts is that only one degree of oxidation of the metal atom during the activation state is reached, leading to the formation of only one type of active site, even when supported. This is a very important property if one needs very fine control over comonomer incorporation or over MWD. However, metallocenes are much more expensive than Phillips or ZN catalysts. Free radical polymerization (FRP) at high pressures is only used to produce LDPE homopolymer and copolymer (e.g. EVA grades). In the FRP process, the polymerization reaction is carried out in a pressure range of 1200-3500 bars and temperature range of 150-330 °C. These processes are inherently different from the gas phase and slurry processes that are the focus of the current thesis. In particular, they are run under supercritical conditions, and do not employ diluents or inert compounds to control heat transfer. This means that the thermodynamic aspects relevant to the solubility of small molecules in the polymer, and to the polymer swelling are unrelated to the thermodynamics of lower pressure processes.

As described above, olefins can be polymerized using a catalyst at lower pressures and temperatures. In this case, the catalytic polymerization of olefins is carried out in three main types of processes depending on the form of the catalyst (molecular or supported) and phase of the continuous medium in which the reaction takes place: solution, slurry, and gas phase.

In solution reactors, both the catalyst and the polymer are soluble in the reaction medium. The polymerization reaction is carried out in a temperature range of 100-250°C and a pressure from 40-100 bars, depending on the solvent used and on the desired polymer properties. These reactions are usually performed in a continuous stirred tank reactor (CSTR) or in a loop reactor.

We will not consider this type of process in the current thesis.

Slurry processes can be classified into diluent and bulk. In diluent processes, used exclusively for PE, an inert diluent (C3 -C6 alkanes) is used to suspend the growing polymer particles. Bulk slurry processes are used only to make isotactic PP, and use liquefied monomer as the continuous medium. These processes can be run in autoclaves or loop reactors in a temperature between 80-110°C and a pressure from 5-65 bars.

Finally, in gas-phase polymerization processes, the polymer is formed on the supported catalyst and the growing particles are dispersed in a continuous gas phase. Gas-phase reactors are used to polymerize ethylene, propylene and higher α-olefins. Gas-phase processes can be used for both PE and PP. If PE is made in a gas phase process, one finds only fluidized bed reactors (FBRs) because they are the only type of reactor to provide sufficient heat removal capacity.

PP can be made in both FBRs and stirred bed reactors because heat removal, while still an important challenge, is not quite as limiting in terms of production for PP 5 . In FBRs, the polymer particles are fluidized by a gaseous mixture of monomer, hydrogen and other compounds, while in stirred bed reactors, polymer particles are dispersed in the gas-phase due to a mechanical stirring. Gas phase processes typically operate at temperatures from 70-120°C and pressures of 5-30 bars.

In the present thesis we will focus our attention on ethylene polymerization processes, essentially in gas phase systems, but we will also look at certain aspects related to diluent slurry processes. For this reason, the rest of the discussion on the state-of-the-art and literature review will focus essentially on ethylene polymerization processes. Nevertheless, the reader should be aware that, from a fundamental point of view of thermodynamics and reactor operation, there are significant similarities between PP and PE processes.

Slurry Phase Ethylene Polymerization

As mentioned in the previous section, slurry processes for olefin polymerization can be divided into two main categories: diluent and bulk. Bulk slurry processes are used only to make isotactic PP, and use liquefied monomer as the continuous medium. Diluent processes use a chemically inert diluent (C3 -C6 alkanes) to suspend the growing polymer particles. In Phillips processes, isobutane is preferred over n-butane as it is a poorer solvent and swells the polymer less 5 .

However, stirred tank slurry processes use mostly n-hexane as diluent 11 .

Slurry processes utilize either stirred tank reactors or loop reactors. The first generation of slurry reactors for olefin polymerization were autoclave reactors, except for Phillips, which was the first company to introduce a loop reactor for olefin polymerization. Both configurations have residence time distributions that can be approximated as Continuous Stirred Tank Reactors (CSTRs).

Slurry loop reactors consist of 1 to 2 loop reactors in series. Typically, industrial slurry-phase loop reactors operate at temperatures of 70-120°C, pressures of 30-90 bar and a polymer-solids concentration of approximately 45% w/w [12][13][14] . The reaction mixture, consisting of monomer, diluent, catalyst, hydrogen and polymer, flows in the loop reactor by means of an axial centrifugal pump placed at the bottom of the reactor. Indeed, this pump is designated to maintain efficient suspension of the solid polymer particles in the liquid diluent 15 and therefore provides high flow velocities of the reaction mixture (i.e. 5-7 m.s -1 ) that results in very welldefined mixing conditions and high heat transfer coefficients through the reactor wall 16 . Indeed, as the volume of the reactor and the solid concentration of the fluid slurry increase, the demands on the pump also increase 17 . Loop reactors are usually vertically installed (Borstar, Philips or Spheripol processes). The loop reactor is continuously fed with monomer and comonomer (i.e. ethylene and 1-hexene or 1-butene) and diluent (i.e. supercritical propane, isobutane, n-pentane, n-hexane) 18 .

Table 3 shows the different industrial slurry reactors as well as the different diluents and comonomers used. With the exception of a small third phase region near the monomer injection part, two phases coexist in the major fraction of the volume of a slurry-loop reactor, namely, a liquid phase (i.e. diluent, monomer, comonomer and hydrogen) and a polymer phase (i.e. polymer and sorbed quantities of diluent, monomer, comonomer and hydrogen) 19 . During the polymerization reaction, polymer slurry is regularly discharged from the loop reactor by means of the settling legs placed at the lower part of the loop reactor 15,20 . These settling legs periodically open to remove the highly concentrated slurry which consists of polymer solids and a fraction of the liquid dissolved in the latter 21 . Then, the product stream leaving the first reactor is either injected inside a second loop reactor with fresh monomer, diluent and hydrogen or removed from the reactor. If so, the highly concentrated slurry leaving the loop reactor is fed to a hot-flash separator where most of the diluent and unreacted monomers are flashed off and recycled, with high monomer conversion rates of 95-98%. The polymer is then dried and pelletized. The major advantages of a slurry process include mild operating conditions, high monomer conversion, ease of heat removal, and relative ease of processing. Indeed, slurry processes have better heat transfer capacity when the polymer particles are suspended in a liquid, with respect to the gas phase processes. Better heat transfer means higher space-time yield in slurry reactors, leading to higher volumes of produced polymer in shorter times. As a result, grade changes can be faster in slurry. However, slurry processes require the purchase, purification, removal and recycling of the solvent, which can impose extra costs and unit operations.

Control of temperature in slurry reactors is important since the operating temperature has a complex impact on the chemical and physical phenomena in PE slurry reactors. Indeed, increasing temperature can lead to higher catalyst productivity but also enhances the risk of 1 Literature Review polymer swelling since the latter tends to become more soluble in the continuous phase at higher temperatures 21 . If the temperature reaches a critical point, particle swelling can occur and can lead to the production of more viscous or adhesive polymer slurry 14,22 causing the formation of relatively low densities (lower than 940 kg.m -3 ). These issues of reactor fouling by polymer adhesion are consequently linked to the swelling of the amorphous phase of the polymer due to the sorption of these industrial liquid diluents.

Nevertheless, very few studies of phase equilibria for PE/liquid diluent systems can be found in the literature. Knowledge of this thermodynamic data is crucial if one needs to control the molecular weight of the produced polymer or to optimize the degassing process. Schnell et al. 23 experimentally and theoretically (Sanchez-Lacombe Equation of State) studied cloud point temperatures and pressures as a function of n-hexane/PE mixture at pressures up to 150 bar and temperatures up to 227°C. De Loos et al. 24 evaluated experimentally the lower solution temperature (LST) in n-hexane/LLDPE system as a function of pressure at temperatures of 127-327°C and pressures up to 130 bar. They showed that cyclohexane is a better solvent than nheptane which is a better solvent than n-hexane. Pochivalov 25 also studied the LCST for LDPE in n-decane and n-tridecane in a wide temperature-concentration range at 290°C using gravimetric method. They concluded that n-decane is slightly a better solvent for LDPE than ntridecane. Hamada et al. 26 determined the molecular weight dependence of the lower critical solution temperature (LCST) for linear PE fractions in n-pentane, n-hexane, n-heptane and noctane. They found out that LCST increases with increasing the number of carbon atoms in the n-alkane. In the same manner, Strathmann et al. 27 studied n-hexane, n-octane, n-decane and nhexadecane in polyethylene and found out that LCST increases as the chain length of the alkane increases. Besides, Nakajima et al. 28 showed that the amount of polyethylene dissolved in liquid solvents is function of the temperature and alkanes chain length. Aminabhavi et al. 29 showed the sorption of 14 organic liquids into HDPE geomembranes at 25, 50 and 70°C, as well as the swelling of the HDPE geomembrane. They showed that increasing the temperature increases the sorption and diffusion of the organic liquids through HDPE, hence increasing the swelling of HDPE. Finally, Krajakova et al. 30 studied the sorption and swelling of different PE films in presence of n-pentane, n-hexane and n-octane at temperatures in the range of 25-68°C. They showed that longer hydrocarbons diluents have a slightly higher solubility in all PE studied. Besides, they showed a significant increase of liquid n-alkanes solubility in function of PE crystallinity for crystallinities lower than 60 wt%, although almost constant solubility for sample crystallinity higher than 60 wt%.

1 Literature Review To conclude, it has been shown that -the heavier the liquid solvent, the higher its solubility in the polymer, and therefore the greater the swelling of the amorphous phase of the polymer -the higher the temperature, the greater the solubility of liquid solvent in the amorphous phase of the polymer -the lower the crystallinity, the higher the solubility of liquid solvents in the amorphous phase of the polymer.

Clearly, the thermodynamic data of PE/liquid diluents related to slurry polymerization is not enough available in the literature, contrarily to the thermodynamic of gas phase which was extensively studied and described in the literature. Indeed, gas phase polymerization reactors, such as FBRs, are the most commonly used reactors for the production of HDPE and LLDPE, but are limited by heat transfer issues due to the highly exothermic nature of ethylene polymerization. This process was commercialized for the first time in the late 1970s. Ethylene polymerization is a highly exothermic reaction. Heat needs to be removed continuously in order to keep the temperature in the reactor at the desired level; the more heat one removes, the faster the polymerization can be allowed to go and the higher the productivity will be. Because of this, the only type of reactor currently used to polymerize ethylene in gas phase reactions is the FBR (except for the new Hyperzone technology which should come on line for the first time this year). An example of this FBR is shown in the scheme in Figure 4. Due to the solubility limitations inherent in LLDPE processes, this type of polymer must be made in the gas phase so it is in fact produced only in FBRs. It should be noted that plants making LLDPE can also make HDPE, and about a quarter of all HDPE is also made in a gas phase processes.

Gas phase ethylene polymerization in fluidized bed reactors

Overview of FBRs

The reactor itself is simply a cylinder with an enlarged section at the top, and a distributor plate at the bottom. The role of the distributor plate is to ensure an even flow of the feed stream into the bottom of the reactor and to avoid growing polymer particles from falling into the feed zone.

The diameter of the main section of the reactor is on the order of 2-6 meters, and the height of the main cylindrical section is about 10-20 meters, without taking into account the upper section of the reactor [31][32][33] . Industrial FBRs typically operate at temperatures of 75-110°C, and at pressures of 15-40 bar [START_REF]Polyolefin Reaction Engineering[END_REF]35 . The feed stream contains a mixture of ethylene, hydrogen, nitrogen and various α-olefins (typically 1-butene or 1-hexene, and eventually 1-octene) that enter the reactor through the distributor plate. The catalyst particles (or occasionally pre-polymerized catalyst particles) are fed continuously to the reactor at a point above the gas distributor 36 . In addition to the reactive compounds injected to the reactor, inert compounds, usually alkanes such as propane, isobutane, isopentane or n-hexane, can be used as induced condensing agents (ICA) to enhance heat removal 37 and thus aid in increasing process productivity and safety 15- 17 .

As can be seen in Figure 5, an FBR is typically run with the gas (and eventually small amounts of liquid) flowing upwards, more or less in plug flow, and with the polymer particles flowing in a toroidal fashion, upwards through the middle of the bed, and downward along the reactor walls. Polymerization takes place as the catalyst and growing particles move through the bed, thus increasing in weight and size 38,[START_REF] Fernandes | Fluidized bed reactor for polyethylene production. The influence of polyethylene prepolymerization[END_REF] . The unreacted gas mixture is recovered at the top of the bed, compressed in an external compressor and cooled in a heat exchanger before being fed back to the bottom of the reactor. The cooling of the feed stream helps to remove the heat of the reaction and compensate pressure drop [START_REF] Mirzaei | Fluidized Bed Polyethylene Reactor Modeling in Condensed Mode Operation[END_REF][START_REF] Bernier | Gas phase polymerization process[END_REF][START_REF] Iii | Fluidized bed reaction systems[END_REF] . The polymer product is removed near the bottom of the FBR, and unreacted materials are separated from the product, cooled, liquefied and recycled. The gas stream inside the reactor provides the fluidization of the growing polymer particles.

Typically, the relative gas-particle velocity is on the order of 2-8 times the minimum fluidization rate; on the order of 0.5-1 m/s [START_REF] Dumain | Process for gas phase polymerization of olefins in a fluidized bed reactor[END_REF] . In order to get such high flow rates through the bed, low per pass ethylene conversions and high recycle ratios are common. Thus, even if the per pass conversion is between 5 and 20%, the overall conversion of this type of process can be quite high, on the order of 90-98% [START_REF] Mckenna | Condensed Mode Cooling of Ethylene Polymerization in Fluidized Bed Reactors[END_REF][START_REF] Mcauley | A comparison of two-phase and wellmixed models for fluidized-bed polyethylene reactors[END_REF] . Furthermore, the particle size distribution (PSD) in the bed can be quite broad; going from several tens of microns up to several hundreds of microns for the final product. This shows the challenge of maintaining proper fluidization in order to avoid the presence of too many fines that could be moved out of the reactor to the compressor 1 Literature Review while ensuring that the largest particles do not settle to the bottom and block the distributor plate. The disengagement zone of the FBR is designed to avoid the first problem. Here the diameter usually doubles, causing the linear velocity of the gas to drop by a factor of 4, leading to a decrease of the particle velocity below the value for minimum fluidization causing fines to fall back into the bed. This is why the minimum fluidization velocity of an FBR, which varies as the square of the particle diameter, has to be carefully estimated. Nevertheless, some particles are small enough so they do not fall and go to the disengagement zone. Therefore, the diameter of the reactor is enlarged by a factor of at least two, in order to minimize the fines from going out of the reactor by defluidizing them [START_REF] Di | Apparatus for improved fluidized beds and reactors containing same[END_REF] . This disengagement zone allows not decreasing the gas flow through the bed, which would have important limitation on the maximum reaction rate because of heat transfer limitations.

Ethylene polymerization in gas-phase FBRs is highly exothermic, producing approximately 3600 kJ.kg -1 of converted ethylene [START_REF] Mckenna | Condensed Mode Cooling of Ethylene Polymerization in Fluidized Bed Reactors[END_REF] , which can pose significant problems of heat removal.

One of the solutions to overcome these heat removal issues is the addition of induced condensing agents (ICAs) when injecting the gas phase mixture. This process is usually referred as condensed mode operation.

The use of alkanes to enhance heat removalsuper dry and condensing mode operation

While space time yields in slurry processes might be higher than in gas phase, the latter remains the only way we have to make products like LLDPE or other polyolefins with a high amorphous content. The polymerization of ethylene in gas-phase fluidized bed reactors (FBRs) remains an important process for the production of polyethylene (i.e. LLDPE, HDPE). Nevertheless, the highly exothermic nature of ethylene polymerization and the poor heat transfer characteristics of the gas phase can pose significant problems of heat removal. It is therefore important to continuously remove heat generated by providing means to cool the bed since the maximum rate at which the heat of the reaction can be removed from the reactor is clearly a limiting step in the polymer production rate in gas phase reactors 5,22,26,27,28,[START_REF] Herzog | Process for the continuous gas-phase (co-) polymerisation of olefins in a fluidized bed reactor[END_REF] . Indeed, increasing the polymer production rate without being able to remove the associated heat release would result in a drastic increase in the reactor temperature that can cause the polymer fusion and subsequently the reactor shut down [START_REF] Herzog | Process for the continuous gas-phase (co-) polymerisation of olefins in a fluidized bed reactor[END_REF] .

As mentioned above, the temperature of the FBR is controlled to an isothermal level through continuously removing the heat of polymerization by circulating the gas from the fluidized bed to a heat exchanger outside the reactor and recirculation of the cooled gas stream back into the reactor 36 .

In order to understand better how heat transfer can be improved, let us look at a simplified steady-state state energy balance around a FBR, given by Equation (1):

𝑚̇g ,in 𝐶 𝑝g,in (𝑇 g,in -𝑇 g ) -𝑈𝐴(𝑇 g -𝑇 w ) -𝑄 vap + 𝑅 p 𝑉 R (-𝛥𝐻 p ) = 0 (1)
where 𝑚̇g ,in is the inlet gas mass flow rate, 𝑇 g,in is the inlet gas stream temperature and 𝐶 𝑝g,in is the heat capacity of the inlet gas stream. U is the overall heat transfer coefficient, A is the surface area of contact between the exchanger and the gas, 𝑇 g is the temperature of the gas in the bed, 𝑇 w is the reactor wall temperature, 𝑄 vap is the total enthalpy change due to evaporation of liquid in the reactor, 𝑅 p is the rate of the polymerization per unit of volume of reactor bed 𝑉 R and -𝛥𝐻 p is the overall enthalpy of polymerization.

Equation ( 1) can then be rearranged as follow:

𝑅 𝑝 𝑉 𝑅 = 𝑚̇g ,in 𝐶 𝑝g,in (𝑇 R -𝑇 g,in ) + 𝑈𝐴(𝑇 g -𝑇 w ) + 𝑄 vap (-𝛥𝐻 p )
This equation shows that if one wishes to increase the productivity of the reactor (𝑅 𝑝 𝑉 𝑅 ), this can be done by increasing the heat capacity of the inlet gas stream and/or increasing the enthalpy change due to the evaporation of liquid in the reactor. Increasing the value of the overall heat transfer coefficient is extremely difficult since it is a function of the gas flow rate but the minimum fluidization velocity has to be carefully respected if we want to avoid fines to be blown out of the reactor. Furthermore, it is also difficult to change the value of the bed temperature 𝑇 𝑅 significantly as this is linked to product quality, and can lead to major operability problems if it goes too high. In order to increase the productivity, alkanes can be added to the reactor, either in liquid or gas phase. If the inlet fluid temperature is greater than its dew point, 𝑄 𝑣𝑎𝑝 will be equal to zero, which will leave us the heat capacity of the inlet gas stream as only way to increase the productivity. The heat capacity of common process components in gas phase ethylene polymerization reactors is given in Table 4.

Looking at these values, and considering an inlet gas stream containing 9 bar of ethylene, 1 bar of hydrogen and 6 bar of nitrogen, this will lead to a heat capacity of 8.8 cal.K -1 .mol -1 . If we replace 6 bar of nitrogen by 1 bar of nitrogen + 5 bar of propane, the heat capacity will be increased about 38% higher. Consequently, replacing nitrogen by ICAs will increase the heat 1 Literature Review capacity of the inlet gas stream, allowing us to generate and remove more heat, thereby leading to an increase of the productivity of the reactor. Of course, if the feed stream is cooled to below its dew point, then the feed to the reactor will contain a certain amount of liquid (depending on its composition and temperature). This liquid will vaporize at a certain point in the reactor, allowing us to remove even more heat.

We will refer to running the process with a vapor only feed stream that contains ICA as "super dry mode" (as opposed to dry mode where no additional inert compounds are added and nitrogen is used to regulate partial pressures). A process with a partially liquefied feed is said to be running in condensed mode [START_REF] Wu | Introducing a condensing agent into a fluidized bed polymerization reactor operated in the condensing mode[END_REF] .

Beyond Heat Transfer -Further impact of ICA

A recent review [START_REF]Polyolefin Reaction Engineering[END_REF] shows that using ICA, whether added in liquid form or only as vapors can have numerous effects beyond simply enhancing heat transfer. This is shown schematically in Figure 6. Of course heat transfer is the initial reason for adding ICA, but (much like -olefin comonomers) they will also influence many important aspects related to polymer thermodynamics (solubility, diffusion, etc.), the physical properties of the polymers, and certain aspects related to reactor operability (stickiness, pressure drop, etc.). Let us consider briefly each of these general points in turn.

Solubility and diffusion: thermodynamically-related quantities

ICAs that are added to the reactor are chemically inert in the sense that they do not have any influence on the behavior of the active sites. However, the impact of ICAs is more complex than simply increasing the heat removal rate from the reaction. Indeed, they are able to change the chemical and physical properties of the polymer.

The amount and type of ICA injected in a FBR is one of the most important factors that affects the overall quantity of the dissolved species in the polymer. Indeed, these ICAs will dissolve inside the amorphous phase of the polymer, as well as the other gaseous species injected into the reactor (i.e. monomer, hydrogen, comonomer), and will therefore influence many properties of the polymer, and in particular, the concentration of the different species in the amorphous phase of the polymer. It turns out that the presence of a heavier component (i.e., ICA or comonomer) in the gas phase enhances the solubility of the lighter component (i.e., ethylene) in the amorphous phase of polymer, and vice-versa 36,[START_REF] Yao | Modeling the solubility of ternary mixtures of ethylene,iso-pentane,n-hexane in semicrystalline polyethylene[END_REF][START_REF] Novak | Ethylene and 1-hexene sorption in LLDPE under typical gas-phase reactor conditions: Experiments[END_REF][START_REF] Sun | Solubility measurement of hydrogen, ethylene, and 1-hexene in polyethylene films through an intelligent gravimetric analyzer[END_REF][START_REF] Bashir | Partial Molar Volumes and Thermal Expansion Coefficients as an Explanation for Co-Solvent Effect of Penetrants in Multicomponent Polymer Mixtures[END_REF][START_REF] Namkajorn | Condensed-Mode Cooling for Ethylene Polymerization: The Influence of Inert Condensing Agent on the Polymerization Rate[END_REF][START_REF] Mckenna | Improved understanding of the impact of alkanes when using condensed mode cooling for PE production[END_REF][START_REF] Alizadeh | Modeling Condensed Mode Operation for Ethylene Polymerization: Part I. Thermodynamics of Sorption[END_REF][START_REF] Banaszak | Ethylene and 1-Hexene Sorption in LLDPE under Typical Gas Phase Reactor Conditions: A Priori Simulation and Modeling for Prediction of Experimental Observations[END_REF] . This phenomenon is often referred to as cosolubility effect, where heavy component acts as a co-solvent to ethylene and ethylene as an anti-solvent to the other.

This co-solubility effect will then enhance the local concentration of ethylene in the amorphous phase of the polymer causing an increase in the rate of polymerization, or decreases the 1 Literature Review solubility of a comonomer leading to changes in the crystallinity of the polymer [START_REF] Ishola | Condensed Mode Cooling for Ethylene Polymerization: Part VI. Impact of Induced Condensing Agents on Comonomer Incorporation[END_REF] . Alizadeh et al. 37,[START_REF] Namkajorn | Condensed-Mode Cooling for Ethylene Polymerization: The Influence of Inert Condensing Agent on the Polymerization Rate[END_REF] showed that adding 2.5 bars of n-pentane to 7 bars of ethylene leads to an increase in the initial polymerization rate by a factor 2.5 times with respect to that observed for 7 bars of ethylene alone. The same tendency is presented for n-hexane as ICA, where it has been shown that adding 0.8 bars of n-hexane leads to an increase of the initial activity of 1.8 times higher than with ethylene alone.

In addition to the impact of the different species on the solubility (i.e. at equilibrium) and the swelling of the amorphous phase of the polymer, changes in their diffusivity (i.e. before equilibrium) through the polymer to the active sites also occurs. For instance, Kanellopoulos et al. [START_REF] Kanellopoulos | An Experimental and Theoretical Investigation into the Diffusion of Olefins in Semi-Crystalline Polymers: The Influence of Swelling in Polymer-Penetrant Systems[END_REF] showed that polymer swelling increases the diffusion rate for binary systems. Alves et al. [START_REF] Alves | Estimation of diffusion coefficients for multiple penetrant/polyolefin systems based on sorption data[END_REF] developed a diffusion model for ternary systems based on the free volume of penetrant/polymer systems that accounts for polymer swelling, as well as co-solubility effect. They showed that the presence of ICA increases the overall solubility of the mixture leading to higher degree of swelling of the polymer resulting in an increase of the diffusion coefficient.

Obviously, different ICAs and other penetrants will induce different changes in a given system.

For instance, the swelling of the polymer particle by an ICA was studied in several papers [START_REF] Bashir | Partial Molar Volumes and Thermal Expansion Coefficients as an Explanation for Co-Solvent Effect of Penetrants in Multicomponent Polymer Mixtures[END_REF][START_REF] Kanellopoulos | An Experimental and Theoretical Investigation into the Diffusion of Olefins in Semi-Crystalline Polymers: The Influence of Swelling in Polymer-Penetrant Systems[END_REF][START_REF] Bashir | An Equation of State-Based Modeling Approach for Estimating the Partial Molar Volume of Penetrants and Polymers in Binary Mixtures[END_REF][START_REF] Alves | Modeling Condensed Mode Cooling for Ethylene Polymerization: Part II. Impact of Induced Condensing Agents on Ethylene Polymerization in an FBR Operating in Super-Dry Mode[END_REF] since this property is crucial when modeling fluidized bed reactors, and this will be discussed in detail in Section 3.3.3. Bashir et al. [START_REF] Bashir | Partial Molar Volumes and Thermal Expansion Coefficients as an Explanation for Co-Solvent Effect of Penetrants in Multicomponent Polymer Mixtures[END_REF][START_REF] Bashir | An Equation of State-Based Modeling Approach for Estimating the Partial Molar Volume of Penetrants and Polymers in Binary Mixtures[END_REF] used the Sanchez-Lacombe equation of state to model the swelling of the polymer in presence of different ICAs, and showed that heavier ICAs have higher solubility in the amorphous phase of the polymer and therefore leads to a higher degree of swelling of the polymer particle. Indeed, adding ICA increases the free volume of the amorphous phase of the polymer and promotes greater mobility of the chains, leading to the swelling of the particle. The ability to develop models to predict the contribution of these different phenomena is clearly very important if we wish to fine tune our control of polymer properties. This point will be discussed in more detail in Section 4.

Physical Properties

Since the sorption effects mentioned change the concentrations of reactive species in the polymer phase and thus at the active sites, they will have an impact on the polymerization rate and polymer properties. A series of recent articles from our research group studied theoretically and experimentally the solubility of ICAs in presence of ethylene inside the polymer 58,59,61,67- 1 Literature Review [START_REF] Andrade | Condensed Mode Cooling for Ethylene Polymerization: Part IV. The Effect of Temperature in the Presence of Induced Condensing Agents[END_REF] . They showed that the solubility of a given component in a polymer is function of the temperature, pressure, polymer crystallinity, and in the case of ternary systems, of the composition of the gas phase. Indeed, increasing the total pressure or decreasing the temperature leads to an increase in the solubility of the different species.

It has also been shown that the increase in ethylene concentration at the active sites due to the co-solvent effect of ICA, leads to an increase in the molecular weight of the produced polymer. Indeed, Namkajorn et al. [START_REF] Namkajorn | Condensed-Mode Cooling for Ethylene Polymerization: The Influence of Inert Condensing Agent on the Polymerization Rate[END_REF] observed a significant increase in the molecular weight distribution when adding ICAs such as isopentane or n-hexane. Other authors also showed that the crystallinity of the particles in the reactor was substantially higher when ICA was present in the reactor than in the case of dry mode (i.e. no ICA). They attributed this to the impact of ICA on the kinetics of crystallization, which was validated in a separate study [START_REF] Andrade | Condensed Mode Cooling for Ethylene Polymerization: Part V-Reduction of the Crystallization Rate of HDPE in the Presence of Induced Condensing Agents[END_REF] . Furthermore, in the case of copolymerization, it has been shown that changing levels of ICA concentration for fixed monomer and comonomer pressures can also lead to changes in polymer crystallinity (i.e. incorporation of comonomer) by changing the monomer to comonomer ratio at the active sites [START_REF] Ishola | Condensed Mode Cooling for Ethylene Polymerization: Part VI. Impact of Induced Condensing Agents on Comonomer Incorporation[END_REF] .

As will be discussed in the Section 4, changing the rates of reaction, diffusivity, and polymer properties can all have an influence on how the polymer particles grow and how their morphology can change in the reactor; these are all critical issues from many points of view.

Reactor Operation

ICAs can have direct consequences on the basic operation of FBRs for PE. For instance, one can encounter different operational challenges caused by the addition of ICAs in the reactor.

The introduction of liquid into a gas phase fluidized bed reactor could result in plugging of the distribution plate, non-uniformity of monomer concentrations inside the FBR and accumulation of liquid at the bottom of the reactor which can result in a complete reactor shut-down [START_REF] Iii | Fluidized bed reaction systems[END_REF][START_REF] Dumain | Process for gas phase polymerization of olefins in a fluidized bed reactor[END_REF][START_REF] Cai | Condensing mode operation of gas-phase polymerization reactor[END_REF][START_REF] Iii | Method for fluidized bed polymerization[END_REF] .

Other issues include a limit on the type and amount of ICA that can be introduced inside the reactor to avoid having the polymer becoming too sticky to discharge or to maintain a normal fluidization velocity [START_REF] Cai | Condensing mode operation of gas-phase polymerization reactor[END_REF] . While extremely important from an operational point of view, these and similar issues are outside of the scope of the current thesis. The reader is referred to references [START_REF]Polyolefin Reaction Engineering[END_REF][START_REF] Dumain | Process for gas phase polymerization of olefins in a fluidized bed reactor[END_REF][START_REF] Cai | Condensing mode operation of gas-phase polymerization reactor[END_REF][START_REF] Iii | Method for fluidized bed polymerization[END_REF] for more detailed discussion of these points. Adding ICA during ethylene polymerization in gas phase also changes the temperature inside the reactor. For instance, Andrade et al. [START_REF] Andrade | Condensed Mode Cooling for Ethylene Polymerization: Part IV. The Effect of Temperature in the Presence of Induced Condensing Agents[END_REF] showed that in dry mode (i.e. without ICA), increasing the temperature in the reactor should increase the polymerization rate due to the activation energy of the exothermic reaction. Besides, the heat capacity of the gas phase is much lower in dry mode than in presence of ICAs, so the difference between the particle temperature and the gas phase will be lower in presence of ICA than in dry mode. This effect was also shown by Alizadeh et al. 37 by developing a single particle model in order to show the gradients of temperature and concentration within the growing polymer particle. The model demonstrated that the temperature of the particle rapidly increases at the beginning of the reaction and then gradually decreases as the reaction proceeds, to a steady-state value around the bulk temperature. This observation was explained by the fact that the temperature evolves in the same manner as the reaction rate during the polymerization reaction. This was also explained by the fact that the polymer particle size and thus its external surface area grow very rapidly during the initial steps of the polymerization which facilitates the exchange of generated heat and causes the particle temperature to decrease. Furthermore, the importance of the cosolubility effect on the reactor operation was shown by Alves et al. [START_REF] Alves | Modeling Condensed Mode Cooling for Ethylene Polymerization: Part II. Impact of Induced Condensing Agents on Ethylene Polymerization in an FBR Operating in Super-Dry Mode[END_REF] , who developed a model demonstrating the impact of change in the polymer density and then in the polymer swelling on reactor modeling. Indeed, their model shows the impact of ICA on the production, reactor behavior and polymer PSD. They showed that the increase of ICA partial pressure increases the productivity, production, ethylene conversion and pressure drop. As an example, adding ICAs during ethylene polymerization in gas phase decreases the residence time inside the reactor of 22% in presence of propane and 42% in presence of isobutane with respect to the polymerization reaction in dry mode. This means that the catalyst needs a shorter time to produce polymer, leading therefore to an increase in the PE production rates.

Single particle modelling and kinetics

A growing polymer particle can be thought of as a microreactor, where heat and mass transport phenomena coupled to polymerization reactions and other processes take place [START_REF] Grof | Modelling of Morphogenesis of Growing Polyolefin Particles[END_REF] . Regardless of whether we are operating in a gas or slurry phase process, it can be said that the catalyst particle injected into a reactor will change rapidly, first by undergoing fragmentation, and then through the growth of the polymer particle. These processes, along with reactor operating conditions and the characteristics of the original catalyst support, are important steps in determining the final properties of the produced polymer. The properties of the final polymer product depend on the molecular properties such as comonomer content, molecular weight distribution, degree of crystallinity, but also on the morphology of polymer particles such as the growth and the shape of particles and the intra-particle distribution of pore and polymer phases. It is therefore crucial to understand the different phenomena that occurs during ethylene polymerization in gas phase.

4.1.Particle morphology and structure

Before describing the particle growth during ethylene polymerization as well as writing the mathematical expressions to quantify these phenomena, we begin our discussion by looking at the initial morphology of the catalyst particles and their evolution during the polymerization. Indeed, the role of a catalyst particle, and especially the role of the catalyst support, is mainly to transport the active sites into the reactor where they can be exposed to monomer to produce polymer. The advantage of starting and finishing with a particle is that it makes it simple to handle and to recover your final product. The knowledge of both the chemical nature of the support and its physical properties (i.e. high surface area, pore-size distribution) is essential in order to produce a catalyst and thus polymer particles with the desired properties. Supported catalyst particles are highly porous and have diameters varying from 10 to 100 µm, depending on the type of polymerization process that will be performed 5 . These supported catalyst particles, often referred to as macrograin or macroparticle, are composed of an association of smaller structures, reported as micrograins or microparticles [START_REF] Mckenna | Particle Growth During the Polymerisation of Olefins on Supported Catalysts, 1 -Nascent Polymer Structures[END_REF] . MgCl2 and SiO2 are the two most commonly used supports, and they have a high specific surface area so they maximize the number of accessible active sites per unit volume thereby allowing us to make as much product as possible with as little catalyst as possible. Most common catalyst supports have specific surface areas varying from 200 to 600 m 2 .g -1 . Besides, these supports have to be friable enough so that they can be fragmented during the polymerization in order to allow monomer access to the active sites (more on this below). However, they have to be strong enough to allow the fragmentation to proceed in a controlled manner by avoiding the formation of fines [START_REF] Alizadeh | Particle Growth during the Polymerization of Olefins on Supported Catalysts. Part 2: Current Experimental Understanding and Modeling Progresses on Particle Fragmentation, Growth, and Morphology Development[END_REF] . Supports used for Ziegler-Natta catalysts are both silica and MgCl2, whereas Phillips and metallocenes catalysts use mostly silica supports. MgCl2 is a crystalline, ionic solid into which the Ziegler-Natta active sites are inserted via isomorphic lattice substitution [START_REF] Soga | Ziegler-Natta catalysts for olefin polymerizations[END_REF][START_REF] Mckenna | Single particle modelling for oleÿn polymerization on supported catalysts: A review and proposals for future developments[END_REF] . Silica supports are amorphous solids, upon which either chromium oxide sites are fixed via calcination at high temperature, or metallocenes are adsorbed or immobilized on the silica surface [START_REF] Fisch | Investigation of silica particle structure containing metallocene immobilized by a sol-gel method[END_REF][START_REF] Pullukat | Silica-Based Ziegler-Natta Catalysts: A Patent Review[END_REF][START_REF] Mcdaniel | Fracturing silica-based catalysts during ethylene polymerization[END_REF][START_REF] Hamielec | Polymerization reaction engineering -Metallocene catalysts[END_REF][82] .

1 Literature Review
These catalytic systems are fundamentally different from a chemical point of view but present however a number of common characteristics. For this reason, the following modeling section can be applied for both MgCl2 and SiO2 supported particles. This discussion is intended to help the reader understand the importance of the thermodynamic aspects being studied in the current thesis, so we will not go into great details. The interested reader is referred to references [START_REF] Mckenna | Particle Growth During the Polymerisation of Olefins on Supported Catalysts, 1 -Nascent Polymer Structures[END_REF][START_REF] Alizadeh | Particle Growth during the Polymerization of Olefins on Supported Catalysts. Part 2: Current Experimental Understanding and Modeling Progresses on Particle Fragmentation, Growth, and Morphology Development[END_REF][START_REF] Hamielec | Polymerization reaction engineering -Metallocene catalysts[END_REF][START_REF] Mckenna | Growth and Evolution of Particle Morphology: An Experimental & Modelling Study[END_REF][START_REF] Grof | Modeling of morphogenesis of growing polyolefin particles[END_REF][START_REF] Grof | Modeling of morphogenesis of polyolefin particles: Catalyst fragmentation[END_REF] for more details.

4.2.Particle growth during ethylene polymerization

During ethylene polymerization, the catalyst particle is first injected into the reactor. Monomer diffuses from the continuous phase of the reactor into the pores of the supported particle, where the active sites of the catalyst are found. When monomer contacts the active sites of the catalyst particle, polymer formation takes place and the fast-forming chains will be growing on the catalyst surface and pores. As polymer accumulates in the pores of the catalyst particle, the initial structure of the particle suffers a build-up of stress at different points, leading to the fragmentation process [START_REF] Mckenna | Particle Growth During the Polymerisation of Olefins on Supported Catalysts, 1 -Nascent Polymer Structures[END_REF][START_REF] Mckenna | Growth and Evolution of Particle Morphology: An Experimental & Modelling Study[END_REF] . However, this fragmentation process lasts a very short period of time, on the order of 10 -1 to 10 -2 seconds, depending on the catalytic system [START_REF]Polyolefin Reaction Engineering[END_REF] . During that time, the initial structure of the catalyst fragments into smaller fragments, but because of the polymer formed (if this occurs correctly) the particle itself remains intact.

The fragmentation process is important for the polymerization to progress reasonably quickly by conserving a satisfactorily fast access of the monomer to the active sites through creating or maintaining a sufficient pore space inside the polymer particle. As the fragmentation proceeds, the active sites located on the fragments of the initial catalyst particle are completely surrounded by the semi-crystalline polymer. Then, the reaction continues as monomer diffuses through the boundary layer around the catalyst particle and through its pores to reach the active sites. The continuous formation of polymer, which will deposit on the catalyst surface and pores, causes the monomer to sorb and diffuse through this polymer layer to reach the active sites, where polymerization takes place. Since the polymerization reaction is exothermic, the heat produced at the active sites needs to be transferred on the opposite direction of the monomer diffusion; through the polymer layer to the bulk phase [START_REF]Polyolefin Reaction Engineering[END_REF] . As the reaction proceeds, the particle continues to expand and the volume of the polymer continues to grow [START_REF] Van Grieken | Effect of 1-Hexene Comonomer on Polyethylene Particle Growth and Kinetic Profiles[END_REF][START_REF] Estenoz | A structural model for the catalytic polymerization of ethylene using chromium catalysts. Part II: Thermal effects[END_REF] . This will result in gradients of temperature and concentration through the growing polymer particle as illustrated in Figure 7, leading to variations in local reaction rates and polymer properties. Thus, to describe the polymerization reaction, it is imperative to be able to predict the dynamics of mass and heat transfer in the growing particle. When modeling polymerization reactors, quantifying heat and mass transfer inside a growing polymer particle is essential. Besides, knowledge of the intraparticle temperature profile is important since it impacts the polymerization kinetics, as well as the concentration profiles of the different species during polymerization reaction which in turn will influence the polymerization rate and the molecular weight evolution. Up to now, a number of particle models have been proposed in order to solve mass and energy balances around a growing polymer particle in order to obtain temperature and concentration profiles of reactants at every point inside the particle. So far, the Particle Flow Model (PFM) [START_REF] Debling | Heat and Mass Transfer Effects in Multistage Polymerization Processes: Impact Polypropylene[END_REF][START_REF] Hutchinson | Polymerization of olefins through heterogeneous catalysis X: Modeling of particle growth and morphology[END_REF][START_REF] Hatzantonis | Recent developments in modeling gas-phase catalyzed olefin polymerization fluidized-bed reactors: The effect of bubble size variation on the reactor's performance[END_REF] and the multigrain model (MGM) [START_REF] Galvan | Molecular weight distribution predictions for heterogeneous Ziegler-Natta polymerization using a two-site model[END_REF][START_REF] Hoel | Effect of diffusion on heterogeneous ethylene propylene copolymerization[END_REF] are considered the closest approximations to describe the phenomena at the mesoscale level during polymerization reasonably well. These models are illustrated in Figure 8.

In the case of the PFM, the growing polymer particle is assumed to be pseudo-homogeneous.

Besides, the PFM assumes the polymerization reaction to be diffusion controlled, which makes this model able to predict the concentration and temperature gradients within the growing particle at each moment during the polymerization 37,[START_REF] Yiagopoulos | Heat and mass transfer phenomena during the early growth of a catalyst particle in gas-phase olefin polymerization : the effect of prepolymerization temperature and time[END_REF] . In the case of MGM, the growing polymer particle is assumed to be heterogeneous and is described as an agglomeration of identical spherical sub-particles. Each sub-particle consists of a fragment of the original catalyst particle with all actives sites on its external surface. The polymer particle is called macrograin 1 Literature Review or secondary particle and the sub-particles are called micrograins or primary particles [START_REF] Hutchinson | Polymerization of olefins through heterogeneous catalysis X: Modeling of particle growth and morphology[END_REF] . In brief, there are two ways of describing these two models 5 : the "structured" approach where the pores and the structure of the polymer particle are taken into account (MGM) and the "nonstructured" approach where the porous particle is considered to be pseudo-homogeneous (PFM). However, both PFM and MGM assume that the active sites are homogeneously dispersed throughout the growing particle at the beginning of the reaction. Besides, it is usually assumed that the fragmentation step is instantaneous and complete at the beginning of the reaction [START_REF]Polyolefin Reaction Engineering[END_REF] .

Figure 8. Representation of the morphology of a growing catalyst/polymer particle using PFM and MGM. Active sites are the black circles and the polymer matrix the yellow ones. Adapted from 37 In these models, though they consider different morphology of the growing porous polymer particle, mass transport is described with Fick's diffusion law, where it is assumed at one level for the PFM and two levels for the MGM. Both models can provide a detailed description of the gradients of concentration and temperature profiles within the growing polymer particle, and the overall particle polymerization rate. The PFM offers the advantage of a relatively simpler mathematical description via the hypothesis that the particle constitute a pseudohomogeneous medium that eases the numerical solution.

In the current study, a comprehensive mathematical particle growth model, accounting for mass and heat transfer limitations appearing during the early growth of a Ziegler-Natta or other supported catalysts in gas phase olefin polymerization is developed. PFM model will be adapted in order to estimate the overall particle polymerization rate as well as the spatial distribution of temperature, monomer, penetrant(s) and polymer molecular properties by solving a system of partial differential equations describing the conservation of energy and molar species in a growing polymer particle. However, the accurate determination of both the monomer concentration at the catalyst active site and the effective monomer diffusion coefficient is crucial in any particle growth modeling study. This study will therefore show the importance of having an accurate thermodynamic model when developing a particle growth model.

The PFM was developed in the early 1980s and has been applied for almost all types of olefin polymerization catalysts. In all of the following balance equations, t and r represent the time and radial position inside the particle, respectively and R is the radius of the pseudohomogeneous particle at each moment during its growth.

The mass balance for the monomer M can be written as

𝜕[𝐶 M ] 𝜕𝑡 = 1 𝑟 2 𝜕 𝜕𝑟 (𝐷 M,ov 𝑟 2 𝜕[𝐶 M ] 𝜕𝑟 ) -𝑅 v
Where [𝐶 M ] is the ethylene molar concentration, 𝐷 M,ov is the overall (or effective) diffusivity of ethylene, and 𝑅 𝑣 is the rate of ethylene consumption.

The mass balance for inert compounds ICA can be written as

𝜕[𝐶 ICA ] 𝜕𝑡 = 1 𝑟 2 𝜕 𝜕𝑟 (𝐷 ICA,ov 𝑟 2 𝜕[𝐶 ICA ] 𝜕𝑟 )
Where [𝐶 ICA ] and 𝐷 ICA,ov are the ICA molar concentration and overall diffusivity of the different ICAs, respectively.

Different initial and boundary conditions can be used for both mass balances but the most commonly used for monomer, comonomers or ICAs are

[𝐶] (𝑟,0) = 0 [𝐶] (𝑅,𝑡) = [𝐶] eq 𝜕[𝐶] (0,𝑡) 𝜕𝑟 = 0 [𝐶 M ]
eq is the overall equilibrium concentration of ethylene in the particle and [𝐶 ICA ] eq is the equilibrium concentration of the different reactants in the particle, both calculated with Sanchez-Lacombe EoS.

The energy balance equation can be written as Where T, ∆𝐻 p , 𝑘 c,p , 𝜌 𝑜𝑣 and 𝐶 p,ov represent the temperature, the heat of polymerization, the overall thermal conductivity of the particle, the overall particle density, and the heat capacity of the particle, respectively. In the first boundary condition, h, and 𝑇 b are the heat transfer coefficient and the bulk temperature of the continuous gas phase in the reactor, respectively.

The set of coupled partial differential equations can be solved in a number of ways in order to have a better understanding of the heat and mass transfer resistance inside the growing polymer particle. The polymerization rate at the active sites can be obtained by two levels of models:

𝑅 v = 𝑘 p 𝐶 * [𝐶 M ] or 𝑅 v = 𝑘 p 𝜇 0 [𝐶 M ]
Where 𝑘 p is the propagation rate constant described by Arrhenius law, 𝐶 * is the concentration of active sites per volume of catalyst and 𝜇 0 is the concentration of activated catalyst sites.

Indeed, the second model takes into account the kinetic model described in Table 5 where living and dead polymer chains are described, whereas the first model using 𝐶 * is a big approximation.

It is therefore more precise to use the second model. Floyd and Ray 94 developed a multigrain particle model in order to investigate the effects of intraparticle mass and heat transfer for a range of catalyst activities. They introduced correlations for diffusion coefficients and sorption, and showed the importance of having a good thermodynamic description of the system. However, they used a correlation from Stern et al. [START_REF] Stern | The effect of pressure on the permeation of gases and vapors through polyethylene. Usefulness of the corresponding states principle[END_REF] to estimate the Henry's law solubility coefficient, which is clearly deviating from realistic sorption data at higher pressures and from multicomponent systems. They also used a correlation given by Michaels and Bixler [START_REF] Michaels | Solubility of gases in polyethylene[END_REF] to estimate the diffusion coefficients of vapors through semicrystalline polyethylene. Similar approaches (with perhaps different correlations)

have been taken by most of the subsequent authors, but it is important to point out that they do not take into account mixture effects in multicomponent systems nor polymer swelling.

Floyd and Ray 97 used the multigrain particle model developed in their previous paper [START_REF] Floyd | Polymerization of olefins through heterogeneous catalysis. III. Polymer particle modelling with an analysis of intraparticle heat and mass transfer effects[END_REF] to analyze heat and mass transfer resistances in the external boundary layer of the polymer particles. They showed that external mass transfer is negligible, even for the most active catalysts, also demonstrated by McKenna et al. [START_REF] Mckenna | Modeling of transfer phenomena on heterogeneous Ziegler catalysts: Differences between theory and experiment in olefin polymerization (an introduction)[END_REF] and Chiovetta et al. [START_REF] Estenoz | A structural model for the catalytic polymerization of ethylene using chromium catalysts. Part II: Thermal effects[END_REF][START_REF] Estenoz | A structural model for the catalytic polymerization of ethylene using chromium catalysts. Part I: Description and solution[END_REF] . Nevertheless, they showed that external film temperature gradients can be large, especially for high-activity catalyst. Indeed, particle overheating due to external film resistance can result in gradients of 10s of K for large catalyst particles at the beginning of the polymerization. Besides, they also suggested that the heat transfer resistance in the external film is predicted to fall off rapidly during polymerization as polymer particles growth increases.

Hutchinson and Ray 100 refined the MGM of Floyd and Ray [START_REF] Floyd | Polymerization of olefins through heterogeneous catalysis. III. Polymer particle modelling with an analysis of intraparticle heat and mass transfer effects[END_REF][START_REF] Floyd | Polymerization of olefines through heterogeneous catalysis IV. Modeling of heat and mass transfer resistance in the polymer particle boundary layer[END_REF] and extended it to copolymerization systems. This model describes the effect of mass and heat transfer and predicts catalyst performance, polymerization rates and particle morphology. They showed the importance of mass transfer limitations inside the particle. They also demonstrated that the diffusion resistance could generally not explain the wide molecular weight distributions often observed in their studied systems, but did not account for compositional effects of the gas phase on the final results. stage. Besides, the overheating of polymer particles predicted by dynamic simulations is smaller in terms of maximum temperature rise than the overheating predicted by stationary models.

They reported that dynamic simulations are better suited for the analysis of small polymer particles in the early stages of their growth, than steady-state modeling of polymer particle overheating 102,103 .

Chiovetta and Estenoz 104 developed a geometrical model to reproduce the support structure in which they included diffusion and reaction phenomena to account for heat and mass transfer effects, and most importantly were the only authors to propose a model that allowed one to include the effect of particle fragmentation. This model was later extended by Chiovetta and Ferrero [105][106][107][108] , and finaly by Chiovetta and Estenoz [START_REF] Estenoz | A structural model for the catalytic polymerization of ethylene using chromium catalysts. Part II: Thermal effects[END_REF][START_REF] Estenoz | A structural model for the catalytic polymerization of ethylene using chromium catalysts. Part I: Description and solution[END_REF]104 . Indeed, they concentrated on the relationship between particle fragmentation and the development of concentration and temperature profiles in the growing polymer particle during the initial stages of polymerization.

However, most common models assume the fragmentation to be instantaneous and complete at the beginning of the reaction 37,[START_REF] Yiagopoulos | Heat and mass transfer phenomena during the early growth of a catalyst particle in gas-phase olefin polymerization : the effect of prepolymerization temperature and time[END_REF]101,109 . The particle model developed in the current thesis will also include this assumption.

Parasu Veera et al. 110 developed a multigrain particle model in which monomer transport occurs not only by diffusion, but also by convection through the pores. The convection is then measured experimentally by the pressure gradient created by the monomer consumption within the particles. This will therefore lead to a significant higher monomer transport rates than thus predicted. Kittilsen et al. 111 showed the importance of the convective transport for big catalyst particles, highly active catalyst particle and high fraction of the reactive species in the fluid (i.e. comonomer).

Accurate estimation of the monomer concentration at the catalyst active site as well as the effective monomer diffusion coefficient are two issues that need to be carefully applied in particle modeling study. McKenna et al. [START_REF] Mckenna | Modeling of transfer phenomena on heterogeneous Ziegler catalysts: Differences between theory and experiment in olefin polymerization (an introduction)[END_REF] and Weickert et al. 112 have shown that the high activities observed with the modern catalysts cannot be explained by the Fickian diffusion of monomer, nor by Henry's law correlation for sorption data. This is an important aspect since these thermodynamic effects have a significant impact on the early growth of the catalyst particle as they cause the variation of catalyst morphology and porosity. Yiagopoulos et al. [START_REF] Yiagopoulos | Heat and mass transfer phenomena during the early growth of a catalyst particle in gas-phase olefin polymerization : the effect of prepolymerization temperature and time[END_REF] developed a particle growth model based on the PFM, accounting for both external and internal mass and heat transfer limitations. They included diffusional and convectional mass and heat fluxes. They introduced Sanchez-Lacombe thermodynamic model to calculate the equilibrium monomer concentration in the amorphous polymer, but only for binary (one penetrant plus a polymer) systems. They investigated the effect of prepolymerization time and temperature on the overall polymerization rate and particle overheating. They showed that it is necessary to prepolymerize the highly active Ziegler-Natta catalyst before their injection into the reactor, which will reduce the internal and external mass and heat transfer resistance during the initial seconds of the polymerization. Wu et al. 113 presented similar results when investigating the effect of catalyst prepolymerization on the final polymer properties produced in gas phase ethylene/1-butene copolymerization Kanellopoulos et al. 109 developed a new particle model called the random pore polymeric flow model (RPPFM) based on the modified polymeric flow model (MPFM) and the random pore diffusion model of Wakao and Smith 114 . Indeed, they analyzed the effects of initial catalyst size, catalyst morphology (i.e. porosity), growth and overheating of highly active Ziegler-Natta catalyst particles in a gas phase olefin polymerization. Indeed, they used the Sanchez-Lacombe equation of state to calculate the equilibrium concentration of monomer in the amorphous phase of the polymer. They also introduced a new diffusion model that takes into account the change in the particle porosity, the crystallinity of the polymer, the size of the penetrant molecules and the temperature. They showed that increasing the initial catalyst size decreases the initial polymerization rate due to mass transfer limitations and increases particle overheating due to heat transfer limitations. Furthermore, they showed that as the degree of crystallinity increases, the polymerization rate decreases due to the decrease of ethylene diffusion coefficient, and increases particle overheating due to a decrease in the rate of particle swelling.

Alizadeh et al. 37 developed a particle model based on the PFM in order to predict the transient behavior of concentration and temperature profiles within the growing polymer particle during gas-phase ethylene polymerization. They included an accurate thermodynamic description through Sanchez-Lacombe EoS as sorption model and Vrentas and Duda 115,116 as diffusion model. Indeed, they were the first to include multicomponent thermodynamic effects and showed that the gas phase composition (i.e. ethylene, n-hexane) had a noticeable impact on the mass and heat transfer, as well as on the polymerization rate.

Hoel et al. [START_REF] Hoel | Effect of diffusion on heterogeneous ethylene propylene copolymerization[END_REF] also developed a Single Particle Model (SPM) to look at copolymer properties.

One of their most interesting contributions (in the context of the current work) was the fact that they postulated that the lack of fundamental data or correlations relating diffusivity and monomer sorption to copolymer composition or crystallinity increase the difficulty of modeling particle growth process, especially for copolymerization. The present thesis will then include a particle model based on the RPPFM developed by Kanellopoulos et al. 109 that will predict the concentration and temperature profiles within the growing polymer particle during gas-phase ethylene polymerization. This model is able to show the impact of different ICAs (i.e. n-hexane and n-pentane) on the mass and heat transfer, the polymerization rate but also on the molecular weight distribution of the produced polymer.

4.5.Kinetic modeling for the polymerization of olefins

Knowledge of the kinetics of polymerization is fundamental for the mathematical modeling of polyethylene particle growth. Understanding the kinetics on a microscale level is crucial in order to show the effects of multiple active sites on molecular weight distribution (MWD). Indeed, at this scale, multiple mechanisms occur such as site activation, polymer growth and catalyst deactivation. Several papers present a comprehensive detailed kinetic scheme for ethylene polymerization in gas-phase [START_REF] Yiagopoulos | Heat and mass transfer phenomena during the early growth of a catalyst particle in gas-phase olefin polymerization : the effect of prepolymerization temperature and time[END_REF]117 . Yiannoulakis et al. 118 employed a kinetic model for homopolymerization to look at the effect of reaction operation on molecular weight. Then, Kiparissides et al. 117 implemented a kinetic scheme into a MGM to investigate the effect of reactor operation conditions on particle growth in a catalytic ethylene/1-butene copolymerization gas-phase FBR. Both used the method of moments to describe the chain length, and offered efficient ways for estimating kinetics parameters from experimental data. P Population balances and the method of moments are the traditional ways of modeling the molecular weight distribution (MWD) for most polymerization reactions. Population balances are molar balances for the concentration of living and dead polymer chains of each length inside the reactor. Generally, only the zeroth, first and second moment equations are solved for both living and dead polymer chains. Besides, each active site type has generally a complete kinetic scheme involving activation, initiation, propagation, transfer and deactivation. The kinetic model in Table 5 is developed to account for any number of active site types. The kinetic mechanism described in this study is similar to those outlined by 10,117,118 .

The initiation of active sites consists of associating the active site to the monomer in order to form a living polymer chain of length 1. Then, the living polymer chains of length r grow by linking to monomer species, leading to a living polymer chain of length r+1 with terminal monomer attached to the active center, corresponding to the propagation. Most dead polymer chains are produced by chain transfer reactions. These transfer reactions occur with monomer (transfer to monomer) or with hydrogen (transfer to hydrogen), and both create living polymer chains of length one with different reactivities (i.e. 𝑃 H 𝑘 , 𝑃 1 𝑘 ) and generate a dead polymer segment of length r. The spontaneous chain transfer reaction regenerates new active sites that can be initiated. And finally, active sites can deactivate spontaneously to form dead sites and dead polymeric segments. Where C * is the concentration of vacant catalyst active sites, M is the monomer concentration, Pr is the concentration of "live" polymer chains, PH is the concentration of polymer chains containing hydrogen, Dr is the concentration of dead polymer chains, H2 denotes the hydrogen concentration, Cd is the concentration of deactivated catalyst site.

The single particle model can be numerically solved to calculate the temperature and concentration gradients inside the particle during ethylene polymerization in gas phase, but also to estimate the change in the reaction rate, the molecular weight distribution and the particle size when varying the gas phase composition inside the reactor. This model can be fitted to experimental data of both reaction rate and MWD in order to identify the kinetic parameters.

Indeed, knowledge of the heat and mass transfer resistance inside the growing polymer particle as well as the variation of the MWD is a very important tool when modeling gas phase 1 Literature Review polymerization reactors. MWD data can provide crucial details about polymer properties such as MFI which is related to the processability of the produced polymer.

Clearly, if we want to have confidence in model predictions, it is very important to know the exact values for the equilibrium concentrations of the different species present in the reactor in the amorphous phase of the polymer, as well as the diffusivity.

Penetrant sorption and diffusion in polyethylene

The discussion in the previous section demonstrates why it is so important to be able to model the sorption process, and to predict the impact of changing gas phase compositions on sorption, diffusion and related processes. In this section we will discuss the main models used for this purpose, show that in fact data for estimating model parameters is sorely lacking in the literature, and look at the different experimental studies that can help us to find useful data for process modelling.

5.1.Thermodynamic models describing sorption equilibrium

For the solubility studies of a single component in a polymer, the sorption of heavy components, especially at higher pressures, deviates considerably from the predictions of Henry's law.

Moreover, Henry's law cannot be used for the calculation of solubilities of individual species in multicomponent systems in a polymer. It is therefore essential to use a more sophisticated thermodynamic model in order to be able to have a more realistic thermodynamic description of the system 119 .

Advanced equilibrium thermodynamic models are classified into two main categories: activity coefficient models and equations of state (EoS). There are several drawbacks to the activity coefficient approach, including the fact that thermodynamic properties such as densities, enthalpies, entropies cannot be obtained from the same model because the excess Gibbs free energy is rarely known as a function of temperature and pressure 119 . This is why equations of state are favored for solvent-polymer systems, since they can capture the dependency of phase volume on pressure. This parameter is clearly crucial in the calculation of the solubility of solutes in the polymer phase and also the swelling degree of the polymer phase due to the sorption.

EoS are powerful tools for the investigation of thermodynamic properties and phase behavior of pure fluids and their mixtures. The most apparent progress in EoS for polymer systems has been made by applying statistical mechanics. Some models derived from statistical thermodynamics assumed molecules to have one or more segments, and the partition function of the system can be obtained by counting the possible configuration when these segments are arranged in hypothetical cells which are like the lattices in solid materials. Then the calculation of the thermodynamic quantities is made from the partition function on the basis of statistical mechanics. Although, many of the more recent theories describe molecules to be moving freely in continuous space 119 .

The Sanchez-Lacombe equation of state (SL EoS) 120 based on lattice models and the Perturbated Chain Statistical Associating Fluid Theory (PC SAFT) 121 based on the perturbation theory are two such models that are widely used in the field of polymer reaction engineering.

In this section, an overview of these two equations of state will be presented. Figure 9 shows the sorption equilibrium of multicomponent systems in polyethylene with PC-SAFT and Sanchez-Lacombe EoS. The red circles are assumed to be the active solid catalyst site, the blue line the monomer particles, and the grey circles the formed polymer particles. The PC-SAFT EoS is a continuum model where molecules, constituted of spherical segments of fixed diameter, are assumed to be moving freely in a continuous space [START_REF] Alizadeh | Modeling Condensed Mode Operation for Ethylene Polymerization: Part I. Thermodynamics of Sorption[END_REF] . This model is derived from the perturbation theory, where a reference system is considered to describe the repulsive interactions of the molecules. The perturbation terms are introduced as correction term that takes into account any deviation of real molecules from the reference system. In this model, the freely jointed molecules exhibit repulsive and attractive forces among them. The repulsive interactions are described by an expression of hard chain and the attractive interactions are divided into dispersion interactions (i.e. Van der Waals forces) and a contribution due to association (i.e. formation of hydrogen bonds). These contributions are assumed to be independent of each other and are then described by different perturbation terms.

In PC-SAFT EoS, a pure substance is characterized by three parameters: the temperatureindependent segment diameter σ, the number of segments per molecule m, and the energy corresponding to the interaction of two segments, ε/k. A detailed description of PC-SAFT development model can be found in elsewhere 121,126 . In order to solve this equation of state, kij interactions parameters between the penetrant molecules and the polymer need to be identified by fitting experimental solubility data to the model. Xiong et al. 127 compared the performance of SAFT with SL to model cloud point curves in polyethylene systems with n-butane and n-pentane. For all temperature-pressure-composition ranges, the SAFT model predictions were found to be closer to the experimental data.

Nevertheless, this comparison study has been reiterated in the past few years by the following authors.

Maity 128 used PC-SAFT to model the solubility of ethylene, carbon dioxide, nitrogen and methane in polyethylene below and above the melting point. He founds out that the adjustable solvent-solute binary interaction parameter kij is temperature-dependent, and developed a suitable correlation for all studied systems. Kanellopoulos and Krallis 129 studied the solubility of ethylene, propane, 1-hexene, 1-butene in homo-and co-polyethylene. They showed that PC-SAFT provides an accurate description of the solubilities of olefins in solvents while SL enables good predictions of olefins solubilities in polyolefins. Besides, they showed that PC-SAFT is recommended for ethylene homopolymerization, while SL gives better predictions for ethylene copolymerization.

Banaszak et al. [START_REF] Banaszak | Ethylene and 1-Hexene Sorption in LLDPE under Typical Gas Phase Reactor Conditions: A Priori Simulation and Modeling for Prediction of Experimental Observations[END_REF] used the molecular simulation method of united atom force field in order to parameterize the PC-SAFT EoS and then extended it to the ternary system of two solutes and a polymer. They predicted that introducing 1-hexene to the gas phase composition increases the solubility of ethylene in the amorphous phase of PE, whereas adding ethylene in the gas phase composition decreases 1-hexene solubility. This leads to a lower gas solubility compared to that predicted by summing the solubilities of the two binary systems. Their simulations were later experimentally validated by means of the gravimetric method conducted by Novak et al. [START_REF] Novak | Ethylene and 1-hexene sorption in LLDPE under typical gas-phase reactor conditions: Experiments[END_REF] .

More recently, Alizadeh et al. [START_REF] Alizadeh | Modeling Condensed Mode Operation for Ethylene Polymerization: Part I. Thermodynamics of Sorption[END_REF] studied the solubility of ethylene and n-hexane in polyethylene in a ternary system. They showed that SL EoS overestimates the solubility of both ethylene and n-hexane at all studied temperature, while PC-SAFT overestimates ethylene solubility and underestimates that of n-hexane for all measurement temperatures. Chmelar et al. 130 used the PC-SAFT equation for the modeling of the sorption of propane and 1-hexene in polyethylene at different temperatures and different crystallinities of the polymer.

They showed that the fitting error of the PC-SAFT model was about 5.7% in average with the experimental data.

Very recently, Greenhalgh et al. 131,132 have modeled the vapor solubility of ethylene, 1-butene, isobutane, isopentane, 1-hexene and n-hexane in a series of semicrystalline LLDPE as a function of temperature, pressure and vapor composition. They showed good agreement 1 Literature Review between the experimental solubility measured with a magnetic suspension balance and the PC-SAFT model.

Sanchez-Lacombe Equation of State (SL EoS)

Sanchez-Lacombe is a model based on a lattice theory, meaning that pure components are assumed to be broken into parts and placed into a lattice structure 120,133,134 . The SL EoS is basically an extension of the classic Flory-Huggins theory. The main advantage of this model is that it introduces vacant lattice sites or holes, and varying the fraction of holes varies the density and the compressibility. SL assumes a random mixture of holes and "mers", where pure components can interact with intermolecular potential. The energies of hole-hole and mer-hole interactions are zero, whereas the mer-mer interaction has a non-zero interaction energy. In the SL EoS, the properties of a pure component is described by three lattice parameters: the mermer interaction energy ε, the closed-packed molar volume of a mer ν and the number of sites (mers) a molecule occupies in the lattice r. Moreover, the knowledge of three characteristic parameters for both penetrants and polymers is required: the characteristic pressure 𝑃 * , the characteristric temperature 𝑇 * and the characteristic density 𝜌 * . Bashir et al. 135 performed a sensitivity analysis to understand the effect of these three parameters on the predictive capability of SL EoS. It was found that SL EoS solubility predictions are more sensitive to the changes made in characteristic parameters of monomers (especially the monomer characteristic temperature) than those made in the characteristic parameters of the polymer. On a positive note, it is possible to know these parameters a priori.

However, the SL EoS includes interaction parameters between the penetrant molecules and the polymer. These are adjustable parameters that need to be estimated in order to obtain accurate thermodynamic predictions. To do so, this equation of state has to be fitted to experimental solubility data (this is also true of PC-SAFT). Here is where the major difficulty comes in: there is very little meaningful data available for multicomponent systems that allow us to obtain the interaction parameters for use in modelling gas phase PE systems.

Bashir et al. also used the SL EoS to estimate the partial molar volumes (PMV) of gases and polymers in binary [START_REF] Bashir | An Equation of State-Based Modeling Approach for Estimating the Partial Molar Volume of Penetrants and Polymers in Binary Mixtures[END_REF] and ternary systems [START_REF] Bashir | Partial Molar Volumes and Thermal Expansion Coefficients as an Explanation for Co-Solvent Effect of Penetrants in Multicomponent Polymer Mixtures[END_REF] . They showed that the PMV of gases in the ternary mixtures are greater than their PMV values in the corresponding binary mixture, due to the enhanced thermodynamic interactions. Moreover, the PMV of penetrating molecules can be affected by their concentration in the gas phase, leading to a high average PMV of penetrant at low gas phase concentration of penetrant, and vice-versa. They attributed the cosolubility effect of a heavier component to its higher thermal expansion coefficient, which could create more free volume in the polymer phase required for the additional sorption of a lighter component. Furthermore, they extended the SL EoS to describe the solubility of ternary mixtures in gasphase [START_REF] Bashir | Modelling of multicomponent olefins solubility in polyolefins using Sanchez-Lacombe equation of state[END_REF] . They demonstrated that SL EoS is in good agreement with the experimental values presented by Novak et al. [START_REF] Novak | Ethylene and 1-hexene sorption in LLDPE under typical gas-phase reactor conditions: Experiments[END_REF] . This last group showed that the temperature, pressure and gas phase composition in the ethylene/1-hexene ternary mixture affect significantly the solubility behavior. In addition these same authors demonstrated that the co-solvent effect of 1-hexene dominates the anti-solvent effect of ethylene at high pressures and low temperatures, and viceversa. Besides, they postulated that the predictive capabilities of SL EoS depend on binary interactions parameters which are temperature-dependent for their studied systems.

More recently, Bashir et al. 136 showed the solubility of ethylene, propane, isobutane and nhexane and their mixtures in LLDPE at different temperatures and pressures, as well as the degree of polymer swelling due to the penetrant dissolution inside the polymer. They fitted SL EoS to binary experimental data in order to identify the interactions parameters kij, that they used for the ternary systems.

In the same direction, Alizadeh et al. [START_REF] Alizadeh | Modeling Condensed Mode Operation for Ethylene Polymerization: Part I. Thermodynamics of Sorption[END_REF] studied the solubility of ethylene and n-hexane in LLDPE and their mixture. They also used binary interaction parameters as input data to fit to ternary systems, and then rearranged k23 interaction parameter to have a better fitting of the ternary solubility.

Sun et al. [START_REF] Sun | Solubility measurement of hydrogen, ethylene, and 1-hexene in polyethylene films through an intelligent gravimetric analyzer[END_REF] compared the SL EoS estimations with experimental data for ethylene, hydrogen and 1-hexene and their mixture in PE. They showed good agreement with the prediction data from SL EoS. Other authors have used the SL EoS in order to describe the solubility of different penetrants in binary and ternary systems [START_REF] Bashir | Modelling of multicomponent olefins solubility in polyolefins using Sanchez-Lacombe equation of state[END_REF][137][138][139][140][141][142][143][144] and show reasonable results compared to the experimental ones.

Both of the SL and PC-SAFT EoS have been used successfully [START_REF] Bashir | Modelling of multicomponent olefins solubility in polyolefins using Sanchez-Lacombe equation of state[END_REF]127 , but the major drawback that they both share is the fact that they were developed for purely amorphous materials, so when applied to semi-crystalline materials, they cannot be used to predict the impact of the crystalline phase on penetrant solubility. Indeed, Andrade et al. [START_REF] Andrade | Condensed Mode Cooling for Ethylene Polymerization: Part V-Reduction of the Crystallization Rate of HDPE in the Presence of Induced Condensing Agents[END_REF] showed that when the crystalline fraction changes, part of the amorphous polymer may be affected by tie molecules that link crystalline lamella surrounding the amorphous region, causing the polymer to swell less. Nevertheless, we have chosen to use the SL EoS because of its mathematical simplicity, 1 Literature Review which allows an extension to multicomponent systems and use in single particle and reactor models 7,37 .

Table 6 contains a summary of most of the binary (one penetrant + PE) and ternary (two penetrants + PE) systems for which model parameters are available in the literature for the modelling of PE systems.

As we mentioned above, having a good estimation of the impact of the composition of the gas phase on the solubility of the monomer and the comonomer is important (in the very least; no information is available about the impact of heavy components on hydrogen) as this in turn will influence polymerization rates, polymer properties etc. In other words, if one wishes to model a realistic system, one cannot use binary sorption models to estimate the concentration of a single species in the polymer for multicomponent systems and simply add them to describe penetrant concentration in the system. It is necessary to account for the way that they interact with the polymer when mixed together. This means that interaction parameters between the different species and the polymer are needed in order to develop this kind of thermodynamic model. These interaction parameters are the only adjustable parameters in the SL EoS that need to be estimated in order to have a good thermodynamic representation. To do so, this equation of state has to be fitted to experimental solubility data, and such data is sorely lacking for multicomponent systems under conditions similar to those used in a polymerization process.

Peng-Robinson equation (PR)

Contrarily to modeling the thermodynamics of polymer-penetrant systems, which, as we just saw are highly non-ideal, calculating the compressibility factor of non-ideal gases and their mixture under mild temperatures and pressures is more straightforward. This can be done in different ways, one of which is using cubic equations of state. Two of the most popular cubic EoS include the Soave-Redlich-Kwong (SRK) and the Peng-Robinson (PR) equations. Ting et al. 151 evaluated the Peng-Robinson EoS for its ability to correlate and predict vapor-liquid equilibrium for asymmetric mixtures of alkanes, and found that the PR EoS is accurate for fitting such mixtures.

We will therefore use the PR EoS when this type of calculation is needed, in particular to calculate the compressibility factor of the different species in gas mixtures. The Peng-Robinson 152 equation is given by:

𝑃 = 𝑅 𝑇 𝜐-𝑏 m - 𝑎 m (𝑇) 𝜐(𝜐+𝑏 m )+𝑏 m (𝜐-𝑏 m ) (2)
Where 𝜐 is molar volume. The parameter 𝑎 m is a measure of the attractive forces between molecules in the mixture and the parameter 𝑏 m is the van der Waals co-volume occupied by these molecules in the mixture. These parameters are defined as follow:

𝑎 m = ∑ ∑ 𝑥 𝑖 𝑥 𝑗 (1 -𝑘 𝑖𝑗 ) √ 𝑎 𝑖 𝑎 𝑗 2 𝑗=1 2 𝑖=1 𝑏 m = ∑ 𝑥 𝑖 𝑏 𝑖 2 𝑖=1
𝑎 and 𝑏 are the respective PR parameters for pure substances. 𝑥 is the mole fraction of each component in the mixture and 𝑘 𝑖𝑗 is the binary interaction parameter.

Applying the first equation at the critical point, we have:

𝑎 𝑖 (𝑇 c,𝑖 ) = 0.45724 𝑅 2 𝑇 c,𝑖 1 Literature Review
A relationship between 𝛼 and the reduced temperature 𝑇 𝑟 can be made and linearized as follow

𝛼 𝑖 (𝑇 r,𝑖 , 𝑤 𝑖 ) = (1 + 𝑚 𝑖 (1 -𝑇 r,𝑖 1 2 ⁄ )) 2
The constant characteristic of each substance, 𝑚 𝑖 , is defined as follow:

𝑚 𝑖 = 0.37464 + 1.54226 𝑤 𝑖 -0.26992 𝑤 𝑖 2
Equation ( 2) can be rewritten as:

𝐴 = 𝑎 𝑃 𝑅 2 𝑇 2 𝐵 = 𝑏 𝑃 𝑅 𝑇 𝑍 = 𝑃 𝜐 𝑅 𝑇
Where Z is the compressibility factor.

5.2.Experimental methods for sorption measurement

As mentioned above, obtaining experimental data to fit the SL or PC-SAFT EoS is the key of being able to successfully model the impact of the composition of the gas phase, and in particular seemingly inert compounds on the polymerization and polymer properties.

Over the last decade, different experimental methods have been developed to measure the solubility and diffusivity of multicomponent systems in polymers. These experimental methods can be divided into four main classes:

pressure decay methods in which the amount of the sorbed species is calculated from pressure-volume-temperature (PVT) continuous measurements in an equilibrium cell containing the polymer sample and a known amount of gas species;

gravimetric methods in which the polymer sample is being saturated with a gas and the solubility is determined by continuously measuring the amount of gas that is sorbed or desorbed from the sample, until the sorption equilibrium is reached;

oscillating methods in which the resonance characteristics of a vibrating support estimates the increase in mass that is due to the sorption of solute species;

flow measurement methods like inverse gas chromatography, in which the gas phase composition is analyzed and from which the individual sorbed gases can be deduced.

Sorption studies for binary systems

Rogers et al. 153 (1959) were among the first investigators who measured using a quartz helix microbalance the equilibrium sorption of thirteen organic vapors in different polyethylene samples, having different crystallinities and densities at temperatures between 0-25°C. They revealed that the crystalline phase of polyethylene is physically impenetrable to the solute molecules, acting therefore like regions of cross-linkage. Another study from this research group showed that the solubility of these organic vapors follows Henry's law 154 . However, increasing the pressure of the solutes increases their solubility in the polymer, resulting in considerable plasticizing of polymer chains. The solubility of these vapors starts to increase exponentially and is then deviated from Henry's law.

Analogously, Michaels and Bixler 96 (1961) showed that the solubility of various gases like ethane, methane, ethylene and propylene in semicrystalline PE follows Henry's law throughout the range of their experimental investigation (5-55°C and low pressures < 1 atm). Besides, they confirmed that the crystalline phase of polyethylene does not sorb gas molecules, even small ones like Helium. These authors also found that the solubility coefficient was function of the Lennard-Jones force constant. This came from the fact that the more condensable the gas is, the higher is its solubility in the polymer.

Robeson and Smith 155 (1968) showed the solubility of ethane and butane in polyethylene in the temperature interval of 30-60°C. They also showed that increasing the pressure of the solute increases its solubility in the polymer, resulting in the plasticization of the polymer. They showed that the polymer film plasticizes ten times more with butane than with ethane in their temperature range of experiments.

Li and Long 156 (1969) investigated the solubility of ethylene, methane and nitrogen in polyethylene at 25°C using the quartz spring balance method. They showed that ethylene solubility increases exponentially with increasing the pressure, whereas those of methane and nitrogen increase asymptotically towards a saturation value. Besides, they found that the solubility of ethylene in polyethylene at high pressures (i.e. ethylene critical pressure) deviates from Henry's law.

Meyer and Blanks 157 (1983) measured the solubility of isobutane and propane in both HDPE and LLDPE at high temperatures (up to 260°C) and low pressures (< 3 bar) using a pseudogravimetric technique in a constant pressure system. They showed an increase of the solubility of both isobutane and propane when increasing their pressure and decreasing the temperature; isobutane solubility being higher than propane, as expected. Moreover, they showed the importance of correcting the solubility data for polymer crystallinity as it increases the solubility of isobutane depending on the type of polymer. Kulkarni and Stern 158 (1983) measured the equilibrium mass uptake of CO2, CH4, C2H4 and C3H8 by semicrystalline PE at gas pressures up to 40 atm and temperatures up to 35°C and found that the solubility coefficient of all penetrants was found to be independent of concentration, which indicates that the penetrant solubilities were within the limits of Henry's law limits.

Castro et al. 159 (1986) studied the solubility of n-butane, n-pentane, n-hexane and n-heptane in polyethylene membranes using gravimetric technique, at very low pressures (< 1 atm). They showed that the bigger the diluent, the higher is its solubility in the polymer. Moreover, they revealed a deviation of the experimental results with Henry's law, mostly at higher temperatures and pressures. This deviation pointed an important interaction between the solvent and the polymer.

Hutchinson and Ray 160 (1990) confirmed that Henry's law is not applicable for heavier hydrocarbon vapors that tend to swell and plasticize the polymer to a much greater extend.

Besides, they showed that the crystallinity plays an important role in the sorption process since the crystallites tie the amorphous region together and limit the amount of polymer swelling, and thus the possible diffusion of species in the polymer. Sato et al. 161 (1999) measured the solubility of CO2 and N2 in molten PP and HDPE at pressures up to 17 MPa by using a pressure decay method. They also showed a linear increase of the solubilities with pressure, which corresponds to Henry's law.

Yoon et al. 162,163 (1993) studied the effect of copolymer composition on the solubility of ethylene, propylene, 1-butene and 1-hexene in PE using a quartz spring balance. They showed that Henry's constant is almost independent of the copolymer composition for ethylene and propylene. However, they showed a strong dependency of the solubility of 1-butene and 1hexene on copolymer composition.

Moore and Wanke 164 (2001) used the gravimetric method to show the solubility of ethylene, 1butene and1-hexene in LLDPE having different crystallinities. They showed that the solubility of ethylene follows Henry's law for a wide range of temperatures and pressure. Moreover, they showed that Henry's law constant decreases with increasing the temperature and the 1 Literature Review crystallinity of the polymer. Nevertheless, Henry's law is no longer followed for larger diluents, such as 1-butene or 1-hexene, since the bigger the diluent, the higher its solubility in the polymer, as also pointed out by Hutchinson and Ray 160 . Kiparissides et al. 141 (2003) used the magnetic suspension microbalance to show the solubility of ethylene in semicrystalline polyethylene at temperatures up to 80°C and pressures up to 66 atm. They also showed that the measured solubilities decrease with increasing temperature and increase with increasing ethylene pressure.

Chmelar et al. 130 (2004) showed the sorption of ethylene and 1-hexene in three polyethylene copolymers samples using the gravimetric method at temperatures up to 150°C and pressures up to 30 bar. Similarly to the studies done previously, they showed that the solubility of lighter components like ethylene follows Henry's law, and that the sorption isotherms of heavier components like 1-hexene are non-linear and increase exponentially at elevated pressures. The same research group measured after few years the solubility of propane and 1-hexene in polyethylene at temperatures and pressures relevant to industrial catalytic gas-phase polymerization of ethylene 165 . They showed that both propane and 1-hexene solubilities increased with decreasing temperature, as expected, and with decreasing PE crystallinity.

Furthemore, the solubility of the heavier component, 1-hexene, is significantly larger than that of the lighter component, propane.

Reichert et al. 166 (2005) studied the sorption of propylene in PP particles formed by different polymerization processes, by means of a gravimetric method. They showed that the solubility of propylene in PP can be described by Henry's law at their experimental conditions. Nagy et al. 167,168 (2007) showed the solubility of iso-hexane and n-hexane in LLDPE using the Cailletet apparatus, from which dew points and bubble points of mixture of known composition can be determined visually. They showed that iso-hexane is a poorer solvent than n-hexane for LLDPE, having a cloud point temperature of 25K less than for n-hexane/LLDPE mixture.

Yao et al. 137 (2011) measured the solubility of propylene in semicrystalline isotactic propylene (iPP) by means of a pressure-decay method. The results showed that the solubility of propylene in iPP increases at elevated pressure and decreases at elevated temperature. The same research group showed the solubility of propylene in HDPE, LDPE and LLDPE using the same pressuredecay approach 138 . They confirmed that the crystallinity has an important impact on propylene solubility. Besides, they showed that propylene solubility depends on the chain microstructure of the polymer, since the solubility is higher in LDPE than in LLDPE, although these two samples have the same crystallinity. Sun et al. 57 (2016) showed the solubility of hydrogen, ethylene and 1-hexene in semicrystalline polyethylene films using an intelligent gravimetric analyzer (IGA). It was found that the solubility of ethylene and 1-hexene decreases with the temperature, while that of hydrogen exhibits an opposite variation tendency in the experimental temperature range.

Greenhalgh et al. 131,132 (2018) presented the solubility of ethylene, 1-butene, isobutane, Isopentane, 1-hexene and n-hexane in a series of semicrystalline LLDPE using a magnetic suspension balance (MSB). They showed that the heavier the alkane is, the higher is its solubility in the amorphous phase of the polymer, meaning that ethylene solubility is the lowest and 1-hexene solubility is the highest for their experiments.

A summary of the data we found in the literature for binary systems is given in Table 7. A wide range of data is available, so it is possible to fit the interaction parameters for a wide range of compositions. However this is not enough because higher order systems show more complex interactions.

To conclude on the binary studies in the literature, it was shown that:

• Penetrants are soluble only in the amorphous phase of the polymer.

• Vapor solubility increases with pressure and decreases with temperature;

• Henry's law coefficient increases with temperature; Deviation from Henry's law is observed for heavy hydrocarbons at all temperatures and pressures.

• The solubility in the amorphous fraction of the polymer decreases if the overall crystallinity of the polymer increases (at constant temperature and pressure). Indeed, the crystallinity may affect diffusion and swelling of the amorphous phase of the polymer due to tie molecules that are crystalline lamellar that link the amorphous regions of the polymer;

• Heavier alkanes and alkenes are more soluble than their lighter counterparts for a given partial pressure and temperature; Li and Long 156 (1969) determined the overall solubilities of ethylene-methane, ethylenenitrogen and methane-nitrogen in polyethylene at 25°C using the quartz spring balance method.

The data showed a very high plasticizing effect of the most permeating specie in the mixture at higher pressure. Indeed, they explain this effect by the co-solubility effect, where the solubility of the more soluble gas increases the solubility of the less soluble gas and therefore enhance the total solubility of the mixture.

Yoon et al. 162 (1993) studied the overall solubility of ethylene and propylene and their mixture in PE using a quartz spring balance. They found that the cosolubility effect is higher when increasing partial pressure of propylene while keeping the total pressure constant. They also showed that increasing the partial pressure of ethylene while keeping constant the partial pressure of propylene does not affect the solubility of propylene.

McKenna 170 (1997) showed the overall solubility of ethylene/butene in both HDPE and LDPE using a pressure-decay method at temperatures up to 94°C. No cosolubility effect was evident from the data, but it is possible that this is because of the very low partial pressures of butene (i.e. 0.65 bar) compared to the total pressure of 21 bar.

Moore and Wanke 164 (2001) showed the solubility of ethylene, 1-hexene, 1-butene and their mixtures in LLDPE using a gravimetric method at temperatures of 30-90°C and pressures up to 30 bar. They showed that there is a significant interaction between ethylene and 1-hexene during co-sorption since the amount of 1-hexene sorbed in the presence of ethylene is much lower than the amount sorbed by the polymer with pure 1-hexene. Besides, the solubility of these species, normalized with respect to the mass of the amorphous polyethylene, decreased with increasing polyethylene crystallinity.

Novak et al. [START_REF] Novak | Ethylene and 1-hexene sorption in LLDPE under typical gas-phase reactor conditions: Experiments[END_REF] (2004) also studied the individual solubility of ethylene and 1-hexene mixture in LLDPE using the gravimetric technique at different temperatures up to 150°C and pressure conditions up to 30 bars. They showed that the solubility of the mixture of ethylene and 1hexene is smaller than the sum of the solubilities of individual species at their respective partial pressures. They showed that adding 1-hexene enhances the solubility of ethylene, acting as a 1 Literature Review co-solvent; this phenomenon is often referred to as the cosolubility effect. They also showed that increasing the ethylene pressure decreases the solubility of 1-hexene with respect to the binary 1-hexene/PE system. In this case the lighter component is an anti-solvent.

Chen et al. 139 (2004) estimated experimentally the individual solubility of ethylene and nhexane mixture in LLDPE. They showed that the cloud point pressures are directly proportional to ethylene weight fraction. Moreover, supercritical ethylene increases the cloud point pressure of n-hexane/PE system. Yao et al. 55 (2006) measured the individual solubility of ethylene with two types of ICAs (i.e. isopentane and n-hexane) in a semicrystalline polyethylene using a pressure decay method at several temperatures and a total pressure of 20 bars. They also showed the co-solubility effect, corresponding to an enhancement of ethylene solubility in polyethylene in presence of both ICAs, and a slight decrease of the solubility of ICAs in presence of ethylene.

Raharjo et al. 171 (2007) measured the solubility of pure and mixed CH4 and n-C4H10 on poly(dimethylsiloxane) using dilation-sorption experiments and densimetery. They showed that the solubility of CH4 and n-C4H10 increases in presence of high concentration of n-C4H10 in PDMS. In contrast, the presence of CH4 does not measurably affects n-C4H10 solubility. Sun et al. 57 (2016) showed the individual solubility of ethylene and 1-hexene mixture in semicrystalline polyethylene using an intelligent gravimetric analyser. They confirmed the cosolubility effect, where 1-hexene is acting as a co-solvent by increasing ethylene solubility, whereas ethylene is acting as an anti-solvent by decreasing 1-hexene solubility in the polymer.

Moreover, it was found that the co-solubility of ethylene/1-hexene mixture is 10% smaller than the sum of the pure component solubilities.

Cancelas et al. 140 (2018) measured the overall solubility of ethylene, propylene and their mixture in isotactic polypropylene (iPP) with a high pressure sorption balance. They confirmed that the total solubility of the gas mixture is higher than the sum of gas solubilities of individual components. This is because the co-solvent effect of propylene on ethylene is higher that the anti-solvent effect of ethylene on propylene. The lower the temperature and the higher the pressure are, the higher is the solubility of ethylene and the lower is the solubility of propylene.

Finally, Greenhalgh et al. 131,132 (2018) presented the individual solubilities of mixed isobutane/isopentane, ethylene/isopentane and ethylene/n-hexane in a series of semicrystalline LLDPE using a magnetic suspension balance. They showed that increasing the weight fraction of the most soluble component in the mixture increases the solubility of the less soluble component. Besides, for a constant partial pressure, the solubility of the more soluble pure component is higher than the solubility of the mixture, whereas the solubility of the less soluble pure component is smaller than in binary system. All these phenomenon confirm the presence of co-solubility effect during the sorption of a mixture of species in a polymer.

A summary of the data we found in the literature for ternary systems is given in Table 8. Less data is available for ternary systems than binary. To conclude on the ternary studies in the literature, it was shown that:

• Heavy components of a given family of compounds increase the solubility of lighter ones (co-solvent effect);

• Light components of a given family of compounds decrease the solubility of heavier ones (anti-solvent effect);

• Most studies used a gravimetric approach;

• The solubility of the mixture of species in ternary systems is smaller than the sum of the solubilities of individual species at their respective partial pressures in binary system

• Most studies only measure overall solubility, which makes the fitting to thermodynamic models more complicated

If we look at Table 7 and Table 8 showing the set of binary and ternary experimental systems we found, it can be seen that most studies employ some version of gravimetric experiments based on a magnetic suspension balance because of their relative simplicity and high precision.

A large number of studies have used this method to investigate the impact of partial pressures and temperature on the solubility of (mostly) binary and (less occasionally) ternary systems. In terms of comonomer solubility, a large amount of data is available for 1-hexene and 1-butene solubility for different temperatures, pressures and PE crystallinity in mostly binary systems 130,131,141,[162][163][164] . A more limited amount of data is available for 1-octene 142,162 . In the case of ICAs, the solubilities of n-hexane, isopentane have been studied for binary and ternary systems for a large range of temperature and polymer type [START_REF] Yao | Modeling the solubility of ternary mixtures of ethylene,iso-pentane,n-hexane in semicrystalline polyethylene[END_REF]131,132,159,167,168 , but very limited data can be found for the sorption of isobutane and propane in PE 131,132,157,158 ; important diluents for a number of processes. Finally, it is not always an easy task to measure individual concentrations (and thus individual solubilities) in this type of apparatus, so very often it only provides total solubilities for a given mixture. The important lack of solubility data on polyolefin systems, especially ternary and quaternary systems at industrial conditions, has inspired this thesis. Indeed, we will focus in the current thesis on building an experimental apparatus able to measure individual solubilities of multicomponent polyolefin systems at industrial conditions. These experimental data will therefore be crucial for thermodynamic modeling as well as for reactor modeling.

5.3.Experimental and theoretical studies for diffusivity

The diffusion of gases and vapors in polymers has been the subject of a great number of studies, especially in binary systems 162,166,172-177 . However, most of these binary systems concern the diffusion of gases in polypropylene 140,166,172,177 , and very few studies looked at the effect of the diffusion of gases in polyethylene. Furthermore, the diffusion of multicomponent mixtures of penetrants in polymeric system, e.g. in ternary and more importantly in quaternary systems, has not been considered to any great length.

Pino et al. 178 studied the diffusion of oxygen, carbon dioxide and nitrogen, and their mixture through polyethylene films with different degrees and types of chain branching, using a mass spectrometer technique at 22-25°C. They showed that mixing the different gases changes the solubility and diffusivity of each other.

Freeman et al. 171 investigated the diffusion of n-butane/methane mixture in poly(dimethylsiloxane) (PDMS) at temperatures up to 50°C and found that methane has a higher solubility and diffusivity in PDMS in presence of n-butane, with respect to the binary methane/PDMS.

Patzlaff et al. 166 compared the diffusivity of propylene in polypropylene polymerized in gas phase and liquid phase using the same Ziegler-Natta catalyst system, through gravimetric method at temperatures between 25-70°C. They showed that the effective diffusion coefficient increases with increasing the size of polymer particles, and that the effective diffusion coefficient is higher for particles produced in slurry phase polymerization, with respect to gas phase process.

Palamara et al. 177 extended the static sorption technique in order to study the diffusion of propylene and ethylene in atactic polypropylene (aPP) at temperatures of 25-70°C. The static sorption method is based on measuring the mass uptake of the polymer sample in a capsule in presence of different gases, as for the gravimetric method. The only exception is that the static sorption method uses several capsule experiments in parallel and stop them at different times before equilibrium in order to measure the fractional mass uptake as a function of time. It was found that the diffusivity of ethylene is higher than that of propylene in aPP at the same temperature and concentration, as one would expect for the smaller molecule. They validated their experimental diffusivity results by employing the Vrentas and Duda free-volume theory.

Cancelas et al. 140 studied the overall diffusivity of ternary ethylene/propylene/iPP using the same magnetic suspension balance (MSB) presented in Chapter 2 at temperature between 50-85°C. They showed that the diffusivity of the ternary mixture is higher than the diffusivity of the pure components.

In the same manner, Alves et al. [START_REF] Alves | Estimation of diffusion coefficients for multiple penetrant/polyolefin systems based on sorption data[END_REF] looked at the diffusion of ethylene/propylene/iPP, ethylene/nhexane/PE and ethylene/n-pentane/PE by employing the free volume theory of Vrentas and Duda. They showed that, as expected, the ternary diffusivity of ethylene in presence of ICAs is higher than its binary diffusivity due to the co-solubility effect increasing ethylene concentration in the amorphous phase of the polymer. They also showed that the ternary 1 Literature Review diffusivity of the different ICAs increases compared to the binary one, which is not an expected result due to the anti-solvent effect of ethylene on ICA, decreasing ICA concentration in the amorphous phase of the polymer. This might be due to a change in the free volume of the system, which will be discussed in more details in Chapter 4.

Clearly, it is desirable to be able to integrate such information into process models in order to help us better understand the impact of the process conditions on the final polymer product. It is therefore crucial to have more experimental and theoretical studies about the diffusion of mixtures of penetrants in the amorphous phase of the polymer since it has a great importance in ethylene polymerization reaction in gas phase. As explained previously, accounting for the co-solubility effect can help explain the increase in productivity 37,[START_REF] Alves | Modeling Condensed Mode Cooling for Ethylene Polymerization: Part II. Impact of Induced Condensing Agents on Ethylene Polymerization in an FBR Operating in Super-Dry Mode[END_REF] . The solubility of the different species will have an impact on the free volume of the amorphous phase of the polymer, which in turn can influence the diffusivities of the penetrants in the system. Taking into account the interaction between the different species when calculating diffusion coefficients improves the capacity of single particle models to explain high initial rates of polymerization in presence of ICAs 37 .

The most widely used model that is able to take into account the interactions between the different penetrants when calculating the diffusivity of multicomponent system is the free volume theory. The free volume theory was developed by Fujita 179 and extensively modified by Vrentas and Duda 115,116,[180][181][182] . The free volume is commonly referred to as the empty space between molecules in solution. Molecular transport is therefore governed by: (1) the existence of a hole (or free volume) of sufficient size adjacent to a penetrant molecule and (2) that same molecule possesses enough energy to overcome intermolecular interactions to jump into the hole. From this definition, Duda et al. 182,183 developed the well-known correlations for the binary self-diffusion coefficient, 𝐷 1 , and the binary mutual coefficient, 𝐷, as follow:

𝐷 1 = 𝐷 0 exp ( 𝐸 a 𝑅 𝑇 ) exp ( -(𝜔 1 𝑉 ̂1 * + 𝜔 3 𝑉 ̂3 * 𝜉 13 𝜔 1 ( 𝐾 11 𝛾 ) (𝐾 21 + 𝑇 -𝑇 g1 ) + 𝜔 3 ( 𝐾 13 𝛾 ) (𝐾 23 + 𝑇 -𝑇 g3 ) ) 𝐷 = 𝐷 1 (1 -𝜙 1 ) 2 (1 -2 𝜒 𝜙 1 )
In these equations, 𝐸 a is the diffusion activation energy, 𝐷 0 is a pre-exponential factor that is only function of the penetrant molecules, 𝑉 ̂𝑖 * is the specific critical hole free volume required for the i th component to make a jump, 𝜔 𝑖 is the mass fraction of the i th component, 𝜉 13 is the ratio of molar volumes for the solvent and polymer jumping units, 𝛾 is an overlap factor It is clear that this equation requires a great number of parameters, that are not always available in the literature for all polyolefins systems, such as the pre-exponential factor 𝐷 0 as well as

𝐾 1𝑖 𝛾
⁄ and 𝐾 2𝑖 . Furthermore, these parameters will be even more numerous for ternary systems, and higher orders systems. More details about the development procedure of Vrentas and Duda equation for ternary systems will be found in Chapters 4 and 5. In order to overcome these issues, Alves et al. [START_REF] Alves | Estimation of diffusion coefficients for multiple penetrant/polyolefin systems based on sorption data[END_REF] replaced the denominator of Vrentas and Duda's equation which represents the hole free volume by 𝑉 × 𝑓 FV where V is the specific volume (m 3 kg -1 ) and 𝑓 FV is the free volume fraction measured by SL EoS. The detailed procedure on implementing the SL EoS into

Vrentas and Duda free volume equation can be found elsewhere [START_REF] Alves | Estimation of diffusion coefficients for multiple penetrant/polyolefin systems based on sorption data[END_REF] . It is highlighted here that experimental and theoretical studies about solubility of multicomponent systems are also needed for diffusivity calculation, and are therefore crucial tools.

1 Literature Review

Conclusion

In the current chapter, an introduction to polyolefins, and in particular polyethylene which is the focus of the current thesis, is provided in Section 1. The different commonly used catalysts for ethylene polymerization were briefly reviewed, as well as the different industrial processes for the production of polyethylene (i.e. gas phase, slurry and solution).

Gas phase ethylene polymerization processes have been presented in detail in Section 3.1, as this will be the focus of a large part of the thesis. The process of ethylene polymerization in gas phase FBRs has been described to better understand the phenomena occurring in an industrial point of view. Indeed, this process offers a wide range of advantages in terms of high overall conversion as well as broad particle morphology. However, the highly exothermic nature of ethylene polymerization in gas phase poses significant problems of heat removal. The rate of the polymer production in these reactors is limited by the rate at which the heat is removed from the reactor. In order to optimize the productivity, condensed mode cooling is employed in order to increase the capacity of heat removal. Condensed mode operation is described in Section 3.2, and its importance in gas phase FBRs is highlighted. This operation mode uses alkanes that are partially liquefied in an external heat exchanger and then sprayed inside the reactor mainly to serve as a dispersing heat medium. These alkanes are usually called induced condensing agents (ICAs) and will change the thermodynamic and physical properties as well as reactor operation of ethylene polymerization in gas phase.

However, the thermodynamics of ethylene polymerization in gas phase is yet not completely understood, and more experimental and theoretical studies are needed. The lack of experimental and theoretical studies on the solubility and the diffusivity of multicomponent systems in polyolefins is highlighted in Section 5.

It was revealed in Section 5.2. that most of the experimental solubility studies were restricted to binary systems (penetrant -PE), with a limited number looking at ternary systems (penetrant 1penetrant 2 -PE). These experimental studies demonstrate that the interactions between the different components in a gas phase mixture exist and must be taken into account. Thus, when modeling ternary (or higher orders systems), it is necessary to account for the way the different penetrants interact with the polymer. This can be achieved by using equations of state (EoS).

The Sanchez-Lacombe equation of state (SL EoS) based on lattice models and the Perturbated Chain Statistical Associating Fluid Theory (PC SAFT) based on the perturbation theory, which are the two widely used models in the field of polymer reaction engineering, were described in details in Section 5.1. The Sanchez-Lacombe EoS will be used in the current thesis. These experimental and theoretical studies demonstrated the importance of the co-solvent effect that leads to an increase of ethylene solubility in presence of heavier penetrant, and the anti-solvent effect decreasing the penetrant solubility in presence of lighter penetrants like ethylene. It would therefore be interesting to look at higher orders systems (i.e. quaternary) in order to analyze the interactions between the different penetrants with the polymer. Would the co-solvent and antisolvent effect remain the same? Would the interactions between the small molecules remain negligible? Can we assume ternary systems when describing quaternary systems? All these questions will be explained through experimental data using the gravimetric and pressure decay method and then validated with an extension of the SL EoS for quaternary systems.

A significant lack of data concerning diffusivity of multicomponent gas phase mixtures is also observed in the literature and was discussed in Section 5.3. Most experimental diffusivity studies are related to binary systems in polyolefins. Diffusion of multicomponent mixtures in polymeric systems is not extensively studied, especially in ternary and more importantly in quaternary systems. Some of the few ternary studied systems revealed a co-diffusion effect, where the ternary diffusivity of ethylene in presence of ICA is higher than its binary diffusivity. This was explained by the co-solvent effect increasing ethylene concentration in the amorphous phase of the polymer. However, studies about a possible anti-diffusion effect of the penetrants are not completely discerned and must be studied in more details. We can therefore ask if the co-diffusion effect is higher than the co-solvent effect? Does the anti-solvent effect affect the diffusion in the amorphous phase of the polymer? Would these anti and co-diffusion effects remain the same for higher order systems? All these questions will be investigated through experimental data using the gravimetric technique and validated with Vrentas and Duda diffusion model. Knowledge of this thermodynamic data is key when describing ethylene polymerization in gasphase since it will change the physical and possible the chemical properties of the produced polymer. The evolution of the particle morphology and growth during the polymerization is described in Section 4.1. and Section 4.2. A number of models are able to quantitatively describe these phenomena, and the Particle Flow Model (PFM) [START_REF] Debling | Heat and Mass Transfer Effects in Multistage Polymerization Processes: Impact Polypropylene[END_REF][START_REF] Hutchinson | Polymerization of olefins through heterogeneous catalysis X: Modeling of particle growth and morphology[END_REF][START_REF] Hatzantonis | Recent developments in modeling gas-phase catalyzed olefin polymerization fluidized-bed reactors: The effect of bubble size variation on the reactor's performance[END_REF] and the multigrain model (MGM) [START_REF] Galvan | Molecular weight distribution predictions for heterogeneous Ziegler-Natta polymerization using a two-site model[END_REF][START_REF] Hoel | Effect of diffusion on heterogeneous ethylene propylene copolymerization[END_REF] are considered the closest approximations to describe this phenomena at the mesoscale level during polymerization reasonably well. These models are described and analyzed in Section 4.3. The Random-Pore PFM (RPPFM) will be used in the current thesis. We could therefore ask what is the effect of using an accurate thermodynamic model in the particle model? Is it important to have both solubility and diffusivity data for ternary systems when describing homopolymerization? Will the gas phase composition have a great impact on the productivity as well as on the morphology of the produced polymer? All these questions will be solved through a particle model including a kinetic and thermodynamic models.

Gas phase processes are clearly not the only ones used for polyethylene production. The slurry phase processes discussed in Section 2 will be the focus of Chapter 6. The different phenomenon occurring in slurry phase have been highlighted, as well as process issues such as the control of temperature, the high amount of swelling of the amorphous phase of the polymer, the risk of reactor fouling due to plasticization of polymer slurry. Although slurry processes have better heat transfer capacity with respect to gas phase processes, the thermodynamics of slurry processes has to be carefully taken into account such as the solubility of liquid diluents in the amorphous phase of the polymer and the swelling of the later. However, very few experimental or theoretical studies related to slurry polymerization are available in the literature. A number of questions remain open in the field of slurry processes. How will the solubility and swelling vary with respect to the temperature, the type of diluent and the density of the polymer? What are the limits of temperatures for each solvent before the plasticization of the polymer? Are these properties important to take into account when modeling slurry reactors? All of these questions will be examined through experimental studies of the solubility, swelling and dissolution of the amorphous phase of the polymer in function of temperature and type of solvent.

Introduction

As discussed in Chapter 1, the sorption of ethylene, comonomers and alkanes (used as induced condensing agents, ICA, for instance) is non-ideal [1][2][3] . In other words the solubility of a mixture of gases in the amorphous phase of a polymer is not the same as the sum of the solubilities of the pure species. Over the last decade, different experimental methods have been developed to measure the solubility of gases in polymers. However, most of these experimental solubility studies are restricted to binary systems (penetrant -PE), with a limited number looking at ternary systems (penetrant 1penetrant 2 -PE) [4][5][6][7][8][9][10][11][12][13] . These studies showed that the solubility of a given component in a polymer is function of the structure of the penetrant, the temperature, the pressure and the polymer density and crystallinity. In the case of ternary systems, the composition of the gas-phase has been shown to be very important 14 . For instance, Yao et al. 4 measured ethylene-alkane-PE systems by varying the partial pressure of n-hexane and isopentane. They showed that the solubility of ethylene in ternary systems is higher than its solubility in binary system. Other authors showed the same effect in the presence of comonomers in ternary systems 6,15,16 . Chmelar et al. 7 showed that the sorption of 1-hexene increases ethylene solubility in the amorphous PE. This phenomenon is often referred to as the cosolubility effect. These same authors also demonstrated that increasing ethylene pressures can lead to a decrease in the solubility of 1-hexene at constant 1-hexene pressure, considered as the anti-solvent effect.

All these studies show the importance of accounting for the overall composition of the gas phase, since the solubility of monomer and comonomer will be directly impacted by the relative amounts of the different species present in the mixture. It is therefore necessary to account for the way that the different species present in the gas phase mixture interact with the polymer in a ternary system. Obviously, this means that if one needs to develop models to quantify these effects, appropriate thermodynamic models that can account for these phenomena are needed. This is of course possible using equations of state. The Sanchez-Lacombe equation of state (SL EoS) is one of the most widely used thermodynamic model in the field of polymer reaction engineering. The only adjustable parameters in the SL EoS are the interaction parameters between the penetrant molecules and the polymer (if we assume negligible interactions between the small molecule penetrants), that need to be estimated experimentally. To do so, this equation 2A novel approach for the estimation of the Sanchez-Lacombe interaction parameters for ternary polyolefins systems of state has to be fitted to experimental solubility data, and such data is sorely lacking for multicomponent systems under conditions similar to those used in a polymerization process.

Currently, gravimetric experiments based on a magnetic suspension balance (MSB) are perhaps the most widely used technique for this type of solubility measurement because of their relative simplicity and high precision. A large number of studies have used this method to investigate the impact of partial pressures and temperatures on the solubility of (mostly) binary 6,7,9-13, and

(very occasionally) ternary systems 11,7,17,20 . However, it is not straightforward to measure individual concentrations (and thus individual solubilities) in this type of apparatus, so MSB experiments often give only the total solubilities for a given mixture. The advantage of these experiments is that they are rapid once the device is calibrated, and one can also have access to overall diffusivity data.

In the present Chapter, the gravimetric technique was used to measure the total solubility of ternary systems that are not available in the literature, such as ethylene/propane/LLDPE-B, ethylene/isobutane/LLDPE-B and ethylene/1-butene/LLDPE-B. These systems were measured at 70°C and total pressures up to 5 bars. Since individual solubilities cannot be measured experimentally with the suspension balance, we used the compressibility factor of the gas mixture calculated with the Peng-Robinson EoS (PR EoS) as second data source to estimate the two necessary interaction parameters for the SL EoS. The measured overall solubilities and the compressibility factors from PR EoS were then used to fit the interaction parameters of the SL EoS. The observed trends were theoretically interpreted because no available data can be found in the literature for these studied systems.

Experimental section 2.1.Materials

Ethylene and argon with minimum purities of 99.5%, and propylene, butene, butane, and propane with minimum of purities of 99% were obtained from Air Liquide Germany. Argon was used as blank gas. Linear low-density polyethylene (LLDPE-B) powders were graciously supplied by Borealis.

2.2.Polymer characterization

The crystallinity of the polymer samples was measured using differential scanning calorimetry (DSC 3+ by Mettler Toledo). Polyethylene samples were weighed (i.e. around 5-7 mg) and placed in an aluminium capsule with a volume of 40 μL. The sample was first cooled to -20°C 2 A novel approach for the estimation of the Sanchez-Lacombe interactions parameters in ternary polyolefins systems

and then heated from -20 to 180 °C at a heating rate of 10°C min -1 , held for 5 minutes at that temperature and then cooled from 180 to -20 °C at a rate of 10°C min -1 . This temperature was maintained for 5 minutes, and the sample was reheated to 180°C at a rate of 10°C min -1 . The melting behavior of the samples was studied in the second heating cycle in order to detect the melting point and the melting enthalpy of the samples. The ratio of the heat of fusion of the sample to that of the completely crystallized polyethylene was used to calculate the degree of crystallization. A heat of fusion of 293 J g -1 for completely crystallized polyethylene was used for the calculation. The STARe software was used for data acquisition and data analysis. The crystallinity of the LLDPE-B samples used during this study was measured at 48.3%.

2.3.Gravimetric method

The gravimetric apparatus used to measure the equilibrium solubility data is a magnetic suspension balance (Rubotherm GmbH), connected to a pressure cell that contains the polymer sample. The weighing mechanism of the balance is separated from the pressure cell, so it is not exposed to high temperatures and pressures. The detailed experimental apparatus is shown in The apparatus is equipped with temperature and pressure sensors, vacuum pump, heating gas bottles, and recording computer. The data acquired from the gravimetric experiment are pressure, temperature, time and sample mass. The magnetic coupling between the balance and 2 A novel approach for the estimation of the Sanchez-Lacombe interaction parameters for ternary polyolefins systems the sample in the pressure chamber allows an accuracy of 0.1 mg for weight measurements.

The setup is described in more details in references [17][18][19] , but we will briefly describe the measuring procedure and the corrections that have to be done.

1-2 grams of polymer particles (powder withdrawn directly from the reactor) are used for each sorption experiment (𝑚 𝑠𝑎𝑚𝑝𝑙𝑒 ). The polymer sample is first degassed under 40 mbar of vacuum for at least 45 minutes at the desired temperature. The process gas mixture is then injected into the pressure cell at the desired pressure and temperature. Pressure steps are then performed in the chamber, and as this pressure is increased, the gas density increases which results in higher buoyancy force acting on the container and the sample. The reading of the balance must be corrected for this buoyancy force. First, a blank run using nitrogen and without polymer sample determines the buoyancy of the sample container. Then, a second run with the polymer sample is performed to estimate the volume of the measured sample required for the buoyancy correction of the sample mass.

𝑚 corrected = 𝑚 balance + 𝑉 container+sample 𝜌 gas (1)
Where the volume of the container with the sample has been calculated during the second run.

Gas densities of propane, isobutane and 1-butene and their mixture at the temperature and pressure of interest were calculated for each set of experimental condition using the ideal gas law corrected for compressibility calculated with Peng-Robinson EoS.

The mass of the sample with the adsorbed gas is given by the following expression:

𝑚 = 𝑚 corrected -𝑚 container
Since PE is semicrystalline, it is necessary to evaluate the solubility by taking into account the crystalline phase where the solubility of the penetrants is zero. Thus, the solubility in the amorphous phase of the polymer is expressed as specific mass change of the sample due to the sorption.

𝑆 am = 𝑚 -𝑚 sample 𝑚 sample (1 -𝜒)
Where 𝜒 is the crystallinity fraction. Kosek et al. 17 demonstrated that taking into account the impact of temperature on the crystallinity of the polymer sample is an essential step for the solubility measurement. Therefore, the mass fraction of crystalline material in the polymer at the temperature of interest (i.e. T=70°C) has been estimated at 47.6% using differential scanning calorimetry (DSC 3+ by Mettler Toledo). sites to account for the compressibility and the density changes. Thus, the system volume or density can vary by changing the fraction of holes in the lattice structure 20 . The main equation of state is given by:

𝜌̅ 2 + 𝑃 ̅ + 𝑇 ̅ [ln(1 -𝜌̅ ) + (1 - 1 𝑟 ) 𝜌̅ ] = 0 (2)
where 𝑇 ̅ , 𝑃 ̅ , 𝜌̅ are the reduced temperature, pressure, volume and density, respectively. 𝑟 is the number of sites (mers) a molecule occupies in the lattice. These reduced parameters, depending on the characteristic properties, are defined as follows:

𝑇 ̅ = 𝑇 𝑇 * ⁄ 𝑃 ̅ = 𝑃 𝑃 * ⁄ 𝜌̅ = 𝜌 𝜌 * ⁄ = 1 𝑉 ̅ ⁄ (3)
where 𝑇 * , 𝑃 * , and 𝜌 * are the scale factors known as the characteristic temperature, pressure, and density respectively, which are used to characterize each pure component in the mixture.

The parameters of SL EoS for each pure component are given in Table 9. properties of a pure components: ε * , 𝜐 * and r, where ε * is the mer-mer interaction energy (J mol - 1 ) and 𝜐 * the closed-packed molar volume of a mer (m 3 mol -1 ).

𝑇 * = 𝜀 * 𝑅 ⁄ 𝑃 * = 𝜀 * 𝑣 * ⁄ 𝜌 * = 𝑀𝑊 (𝑟𝑣 * ) ⁄ 𝑉 * = 𝑁(𝑟𝑣 * ) (4)
Where MW is the molecular weight of the component and R is the universal gas constant (J K - 1 mol -1 ).

When the SL EoS is used for a mixture of components, it is necessary to define combining rules based on the Van der Waals mixing rule for the estimation of mixture properties (i.e. ε mix * , 𝜐 mix * , 𝑟 mix ) based on the respective values of the pure component values.

The characteristic closed-packed molar volume of a "mer" of the mixture, 𝜐 mix * is defined as follows:

𝑣 mix * = ∑ ∑ 𝜙 𝑖 𝜙 𝑗 𝑣 𝑖𝑗 * 𝑁 c 𝑗=1 𝑁 c 𝑖=1 (5) With 𝑣 𝑖𝑗 * = 𝑣 𝑖𝑖 * +𝑣 𝑗𝑗 * 2 (1 -𝑛 𝑖𝑗 ) (6)
Where 𝜙 is the volume fraction of the i th and j th component in the mixture, 𝑁 c is the number of component in the mixture and 𝑛 𝑖𝑗 corrects the possible deviation of 𝑣 𝑖𝑗 * from the arithmetic mean value of 𝑣 𝑖𝑖 * and 𝑣 𝑗𝑗 * of the pure component. In the present study, the value of the interaction parameter 𝑛 𝑖𝑗 is assumed to be equal to zero.

The closed-packed volume fraction of the i th component at the limit of zero temperature or incompressible state, 𝜙 𝑖 gas is defined as:

𝜙 𝑖 gas = 𝜔 𝑖 gas 𝜌 𝑖 * 𝑣 𝑖 * ∑ ( 𝜔 𝑗 gas 𝜌 𝑗 * 𝑣 𝑗 * ) 𝑁 c 𝑗=1 ⁄ (7)
where 𝜔 𝑖 gas is the mass fraction of the component i in the gas phase.

The mixing rule for the characteristic interaction energy for the mixture 𝜀 mix * is defined as:

𝜀 mix * = 1 𝑣 mix 2
A novel approach for the estimation of the Sanchez-Lacombe interactions parameters in ternary polyolefins systems in which 𝜀 𝑖𝑖 * and 𝜀 𝑗𝑗 * are the interation energy between mers of the i th and j th component respectively. 𝑘 𝑖𝑗 is a binary interaction parameter between the different components of the mixture, which is employed as adjustable parameter to fit the model to experimental solubility data.

The mixing rule for the number of sites (mers) occupied by a molecule of the mixture, 𝑟 mix , is given by: (10) where 𝑟 𝑗 is the number of sites occupied by molecule j in the lattice.

1 𝑟 mix , = ∑ 𝜙 𝑗 𝑟 𝑗 𝑁 c 𝑗=1
For the calculation of sorption equilibrium for polymer-penetrant system, the chemical potential of the i th component in a multicomponent system can be expressed as

𝜀 * mix gas = (𝜐 1 * -𝜐 2 * )𝜙 1 𝑔𝑎𝑠 +𝜐 2 * (13)
We consider that the chemical potential of each component at equilibrium is equal to the chemical potential of the polymer as follows:

𝜇 𝑖 pol = 𝜇 𝑖 gas i=1,2… (14)
In which 𝜇 𝑖 gas can be the chemical potential of the monomer, the comonomer, or the ICA.

By solving equations ( 2) and ( 11) using a nonlinear algebraic equation solver (e.g. in Matlab), for each component in the ternary system, the mass fraction of each sorbed component in the amorphous polymer (where 1 refers to the monomer, 2 to the penetrant molecule (i.e. comonomer or ICA) and 3 to the polymer, and the superscript pol means in the polymer phase) can be calculated as follows Finally, the solubility of each component at equilibrium, in gram of component per gram of polymer, S, can be calculated as follows

𝑆 𝑖 am = 𝜔 𝑖 am 𝜔 pol am (1 -𝜒) i,j = 1,2
where 𝜒 is the crystallinity of the polymer.

For more details on the development and the application of SL EoS to ternary systems, readers may refer to Alizadeh et al. 24 .

Peng-Robinson EoS

The Peng-Robinson (PR) equation of state 25 for gas mixture is defined as

𝑃 = 𝑅 𝑇 𝜐-𝑏 m - 𝑎 m (𝑇) 𝜐(𝜐+𝑏 m )+𝑏 m (𝜐-𝑏 m ) (18)
Where 𝜐 is molar volume. The parameter 𝑎 m is a measure of the attractive forces between molecules in the mixture and the parameter 𝑏 m is the van der Waals co-volume occupied by these molecules in the mixture. These parameters are defined as follow:

𝑎 m = ∑ ∑ 𝑥 𝑖 𝑥 𝑗 (1 -𝑙 𝑖𝑗 ) √ 𝑎 𝑖 𝑎 𝑗 2 𝑗=1 2 𝑖=1 (19) 𝑏 m = ∑ 𝑥 𝑖 𝑏 𝑖 2 𝑖=1 (20)
𝑎 and 𝑏 are the respective PR parameters for pure substances. 𝑥 is the mole fraction of each component in the mixture and 𝑙 𝑖𝑗 is the binary interaction parameter. This interaction parameter is identified using the experimental compressibility data, but very few experiments are available in the literature. The measure of the compressibility factor when using identified 𝑙 𝑖𝑗 compared to that assuming 𝑙 𝑖𝑗 = 0 shows a difference of less than 0.4% for ethylene/propane mixture, at the same conditions than the gravimetric experiments 31 . Therefore, as for SL model, we will consider for this study that there is no interaction between small molecules, meaning that 𝑙 𝑖𝑗 = 0.

Applying the first equation at the critical point, we have:

𝑎 𝑖 (𝑇 c,𝑖 ) = 0.45724 𝑅 2 𝑇 c,𝑖 2
A novel approach for the estimation of the Sanchez-Lacombe interactions parameters in ternary polyolefins systems

𝑏 𝑖 (𝑇 c,𝑖 ) = 0.07780 𝑅 𝑇 c,𝑖 𝑃 c,𝑖 (22)
At temperatures other than the critical temperature, we have:

𝑎 𝑖 (T) = 𝑎 𝑖𝑖 (𝑇 𝑐,𝑖 ) . 𝛼 𝑖 (𝑇 r,𝑖 , 𝑤 𝑖 ) (23)
𝑏 𝑖 (T) = 𝑏 𝑖 (𝑇 c,𝑖 )

A relationship between 𝛼 and the reduced temperature 𝑇 𝑟 can be made and linearized by Peng and Robinson 25 as follow

𝛼 𝑖 (𝑇 r,𝑖 , 𝑤 𝑖 ) = (1 + 𝑚 𝑖 (1 -𝑇 r,𝑖 1 2 ⁄ )) 2 (25)
Since the acentric factors of all studied species are 𝑤 ≤ 0.49, 𝑚 𝑖 , which is a constant characteristic of each substance, is defined as follow:

𝑚 𝑖 = 0.37464 + 1.54226 𝑤 𝑖 -0.26992 𝑤 𝑖 2 (26)
Equation ( 16) can be rewritten as:

𝑍 3 -(1 -𝐵)𝑍 2 + (𝐴 -3𝐵 2 -2𝐵)𝑍 -(𝐴𝐵 -𝐵 2 -𝐵 3 ) = 0 ( 27
)
Where Z is the compressibility factor. The roots function of Matlab is used to solve this polynomial equation. This equation yields one or three roots depending upon the number of phases in the system. In the two-phase region, the largest root is for the compressibility factor of the vapor while the smallest positive root corresponds to that of the liquid.

𝐴 = 𝑎 𝑃 𝑅 2 𝑇 2 𝐵 = 𝑏 𝑃 𝑅 𝑇 𝑍 = 𝑃 𝜐 𝑅 𝑇

3.2.Model solution procedure

We will define our ternary system as one containing ethylene (1), a penetrant (2) which can be a comonomer or an ICA, and a polymer (3). The corresponding interaction parameters in order to solve SL EoS are k12, k13, k23 which are the interactions between the monomer and the penetrant, the monomer and the polymer and the penetrant and the polymer, respectively. k12 is assumed equal to zero due to low chemical affinity between like molecules. The interaction parameters are used to define the cross interaction energy 𝜀 𝑖𝑗 * = (𝜀 𝑖 * 𝜀 𝑗 * ) 0.5 (1 -𝑘 𝑖𝑗 ); when 𝑘 𝑖𝑗 =0, the cross potential follows Lorentz-Berthelot combining rules 32 (i.e. 𝜀 𝑖𝑗 * = (𝜀 𝑖 * 𝜀 𝑗 * ) 0.5 ) while if the value is far from 1, then there is a deviation from these rules. Therefore, the heavier the penetrant, the more the interaction with the polymer deviates from Lorentz-Berthelot combining rules. This leaves us two interaction parameters that need to be estimated to solve the Sanchez-Lacombe EoS. In order to do that, one needs to fit this model to experimental solubility data that have been measured for this study using a magnetic suspension balance.

However, as we have only global solubilities from this apparatus, this measurement leaves us with only one experimental data to identify two interactions parameters. The calculation of the compressibility factor of the different gaseous species in the mixture represents therefore a crucial input data as it can be compared to the compressibility factor calculated through SL EoS.

From equations ( 3) and ( 4), we can calculate the compressibility factor as follows:

𝑍 = 𝑃 𝑣 𝑅 𝑇 = 𝑃 ̅ 𝑣 ̅ 𝑇 ̅ 𝑟 (14)
We can then extend Equation (2) from SL EoS to calculate the compressibility factor 29 as follows:

𝑍 = -𝑟 [ 1 𝜌 ̅ ln(1 -𝜌̅ ) + (1 - 1 𝑟 ) + 𝜌 ̅ 𝑇 ̅ ] (15)
2 A novel approach for the estimation of the Sanchez-Lacombe interactions parameters in ternary polyolefins systems

The identification of the two interaction parameters is now possible by fitting three "experimental" data (one overall solubility, two compressibility factors) to the Sanchez-Lacombe model. Since, no available experimental data can be found for the compressibility factor, it was calculated with another equation of state, for instance the Peng-Robinson EoS 25 .

This will be acceptable as input to the estimation procedure as the compressibility factor for the vapor phase does not have interactions with the polymer and we can therefore have confidence that the value is correct. The detailed procedure of the development of SL EoS associated with PR EoS for the identification of kij interaction parameters is presented in Figure 11. The identification of the two interaction parameters in the SL EoS for ternary systems is achieved by minimizing the average relative deviation, ARD, between the experimental overall solubility and the one calculated with SL EoS, as well as between the compressibility factors of each specie in the mixture calculated with PR EoS with the ones calculated with SL EoS, as follows:

𝐴𝑅𝐷 = ∑ |𝑆 tot SL -𝑆 tot exp | 𝑆 tot exp 𝑡 𝑖=1 + ∑ |𝑍 1 SL -𝑍 1 PR | 𝑍 1 PR 𝑡 𝑖=1 + ∑ |𝑍 2 SL -𝑍 2 PR | 𝑍 2 PR 𝑡 𝑖=1
Once the available experimental overall solubility as well as the calculated compressibility factors from PR EoS are fitted to SL EoS, we can therefore identify the kij interaction parameters, that will next lead to the calculation of the partial solubilities and the concentrations of the different species in the mixture, as well as the polymer swelling due to the dissolution of these species in the amorphous phase of the polymer.

Results and discussion

In this Chapter, a novel approach consisting of combining the compressibility factor calculated with Peng-Robinson plus experimental global solubility results has been used to predict the interactions parameters needed for SL EoS. We will first validate this new approach using previously published experimental data for ternary systems, and then use it to predict the partial 2 A novel approach for the estimation of the Sanchez-Lacombe interactions parameters in ternary polyolefins systems solubility of species in ternary systems from the new experimental measurements of the overall solubility using the magnetic suspension balance.

4.1.Model validation on literature data

Cancelas et al. 18 measured the overall solubility of an equimolar ethylene/propylene mixture in iPP at temperatures above 70°C and pressures up to 25 bar, using the same magnetic suspension balance. They used the interaction parameters from the two binary systems and readjusted k23 at low temperatures to fit the ternary data. In the current study, their experimental overall solubility data for ternary system of ethylene/propylene/iPP coupled with the Peng-Robinson

EoS were used in order to identify the ternary interaction parameters, without using binary interactions parameters. Figure 12 shows the experimental overall solubilities measured by Cancelas et al. 18 and the corresponding model predictions obtained from SL EoS employing the interaction parameters estimated from overall solubility plus the Peng-Robinson EoS (this approach will henceforth be referred to as SL-PR). In Figure 12, Figure 13 and Figure 14, the symbols represent experimental values of the total solubility, and the lines represent the theoretical solubilities calculated with SL-PR method. A comparison of the ternary interaction parameters of ethylene/propylene mixture in iPP identified in this study and those identified by Cancelas et al. 18 is given in Table 11. Note that the interaction parameters are temperature-dependent. Indeed, they identified binary kij interaction parameters for both binary systems of ethylene/iPP and propylene/iPP by fitting their experimental solubilities as well as solubility data from literature to binary SL EoS. These kij were then used for ternary system of ethylene/propylene/iPP, but k23 had to be readjusted for lower temperatures in order to fit better their experimental ternary global solubility. As can be seen, we find an excellent agreement between the ternary interactions parameters determined 2 A novel approach for the estimation of the Sanchez-Lacombe interaction parameters for ternary polyolefins systems with the novel approach SL-PR and the ones determined by Cancelas et al. 18 . However, in the SL-PR method, no re-adjustment of the parameters is required, which makes this novel methodology more robust. Partial solubilities of each component in the mixture can then be calculated using SL EoS, as can be seen in Figure 13 and The second system used to validate this approach is ethylene/1-hexene/LLDPE. Novak et al. 6 measured ethylene/1-hexene mixture overall solubility in LLDPE-1-hexene copolymer with a constant composition of 4.3 mol% 1-hexene and 95.7 mol% ethylene at 70, 90 and 150°C. In 12 compared to the interaction parameters obtained by Bashir et al. 14 . These last authors fitted the experimental values of the total solubility found by Novak et al. 13 with the SL EoS in a manner similar to how Cancelas et al. 31 did for the ethylene-propylene-iPP system, i.e. they started from the binary interaction parameters identified in binary systems by Novak et al. 13 and re-adjusted these parameters. Partial solubilities of each component in the mixture are then predicted using SL EoS, as can be seen in Figure 16 and Table 12. Comparison of ternary interaction parameters of ethylene/1-hexene mixture in LLDPE identified with the novel approach SL-PR by fitting experimental solubilities of Novak et al. 6 with the interaction parameters found by Bashir et al. We can use these results to study the co-and anti-solvent effects of 1-hexene and ethylene in this system. Binary data for ethylene/LLDPE and 1-hexene/LLDPE were taken from Novak et 2 A novel approach for the estimation of the Sanchez-Lacombe interaction parameters for ternary polyolefins systems al. 6 data and then fitted to the binary SL EoS. The correlations of binary kij for ethylene/LLDPE and 1-hexene/LLDPE, with respect to temperature are listed in Table 5. Figure 18 and Figure 19 show the solubility of ethylene and 1-hexene in binary (ethylene/PE and 1-hexene/PE) and ternary (ethylene/1-hexene/PE) systems under the same composition as Novak et al. 6 (4.3 mol% 1-hexene and 95.7 mol% ethylene) for the ternary simulations. The solubilities calculated assuming a binary and/or a ternary system are compared in Figure 18 and Figure 19 for 70 and 90°C, and the simulations indicate a co-solubility effect which is enhanced at higher temperature. It is shown that the partial solubility of ethylene in the ternary system is higher than its binary solubility, meaning that ethylene solubility is enhanced by the addition of 1hexene to the gas phase (co-solubility effect). On the other hand, it is shown that the ternary partial solubility of 1-hexene is lower than its binary solubility, meaning that ethylene acts as an anti-solvent to 1-hexene. It is also shown that the anti-solvent effect of ethylene is higher than the co-solvent effect of 1-hexene. Figure 18. Ethylene and 1-hexene partial solubility in ternary system at 70°C with 5 mol% 1-hexene and 95 mol% ethylene and their corresponding partial pressures for each pure component for binary simulations. kij for ternary system were taken from Table 12 and kij for binary systems were taken from Table 5.

127 2 A novel approach for the estimation of the Sanchez-Lacombe interactions parameters in ternary polyolefins systems Figure 19. Ethylene and 1-hexene partial solubility in ternary system at 90°C with 5 mol% 1-hexene and 95 mol% ethylene and their corresponding partial pressures for each pure component for binary simulations. kij for ternary system were taken from Table 12 and kij for binary systems were taken from Table 5.

The novel approach of SL combined with Peng-Robinson has been validated with the available experimental data for different conditions (i.e. temperature, composition, pressure, polymer type) and show reasonable results. This technique can then be used for the different studied overall solubilities measured with a magnetic suspension balance.

4.2.Model application to new experimental data

In general, the solubility of penetrants in the amorphous phase of the polymer depends on the pressure, the temperature, the type of penetrant and PE properties. For the following solubility studies, we kept the temperature, PE type and gas phase composition constant; while increasing the total pressure of the injected mixture.

Table 14 shows the total pressure and the composition of the gas phase mixtures studied here.

The overall solubility of these three systems will be measured for a pressure up to 5 bars and a constant temperature of 70°C. These solubility data will then be correlated using Sanchez-Lacombe combined to Peng-Robinson (i.e. SL-PR) to predict the individual solubility data and the interaction parameters. 20 that, as expected, the total solubility increases as the total pressure of the system increases. It is also important to note that the more condensable the penetrant is, the higher the total solubility of the mixture is at a given pressure. Indeed, for a fixed total pressure, the total solubility of the mixture in presence of 1-butene is higher than the solubility in presence of isobutane, which is higher than the solubility in presence of propane.

These data points are used to fit the kij from SL EoS for the different mixtures, the values of which are shown in Table 15. Indeed, when one has only global solubilities, identifying these interaction parameters is crucial in order to have the partial solubilities of each component in the mixture. These parameters are also useful to predict the solubility at different pressures. In order to do so, the approach discussed above was used to fit both global solubility from SL EoS to experiments and compressibility factor from SL EoS to Peng-Robinson EoS. Note that the model fits the data well, with the difference between the model and the experiments in Figure 20 being on the order of 1 -4%, indicating that we can have confidence in these values for the pressures and temperature used in the experiments. Figure 21-Figure 23 show that the calculated compressibility factors from SL EoS are in good agreement with the ones calculated with Peng-Robinson EoS, with a maximum difference of 0.8%. It should be pointed that this set of interaction parameters are only valid at temperature equal to 70°C. Once the kij identified by this new approach, we can calculate the partial solubility of each component in the mixture. Figure 24, Figure 25 and Figure 26 show the global and partial solubility of each component of the mixture in the amorphous phase of LLDPE-B, in presence of different penetrants; propane, isobutane and 1-butene, respectively. It is shown, as expected for the considered molar fraction, that the partial solubility of ethylene is lower than the partial solubility of the heavier components (i.e. propane, isobutane, 1-butene). The degree of polymer swelling caused by the sorption of the different ethylene/penetrants mixture in the amorphous phase of the polymer was calculated using SL EoS and is shown in isobutane and 1-butene at 70°C at the same experimental conditions (c.f. A novel approach for the estimation of the Sanchez-Lacombe interactions parameters in ternary polyolefins systems Table 14). The evolution of polymer swelling as a function of the total pressure follows the same trends as that observed for the solubility. Heavier alkanes cause higher LLDPE swelling compared to lighter ones. At a total pressure of 4 bars, the swelling in presence of propane is about 1.2%, 1.9% for isobutane and 2.55% for 1-butene. It can be concluded that the degree of polymer swelling is insignificant over the range of pressures used in these experiments. 14 and kij are defined in Table 15 Figure 28 shows the experimental solubility of ethylene in the binary system of ethylene/LLDPE and the ternary solubility of ethylene in presence of propane, isobutane and 1-butene in LLDPE. It shows that the ternary solubility of ethylene is higher than its corresponding binary solubility due to the co-solubility effect. Indeed, adding heavier components will increase the solubility of ethylene and vice-versa. Besides, it shows that the heavier the penetrant, the higher the partial solubility of ethylene in the amorphous phase of LLDPE-B. Note that the molar fractions were not exactly the same for all the components (c.f. Comparison to binary solubility of C2/LLDPE, where points represent the experimental solubility and the line represents the solubility fitted to SL EoS, with kij defined in Table 8 Figure 29 shows the solubility of propane, isobutane and 1-butene in binary (diluent/PE) and ternary (ethylene/diluent/PE) systems. The same compositions as in Table 14 were used for the ternary simulations and their corresponding partial pressures for each pure component were used for the binary simulation. This figure shows the anti-solvent behavior of ethylene. Indeed, adding ethylene to the gas phase composition decreases the solubility of ICAs, with respect to the binary solubility. Ethylene acts therefore as an anti-solvent to ICA by decreasing its partial solubility in the amorphous LLDPE-B. Besides, we can see that ethylene anti-solvent effect is higher (about 20%) for isobutane than for propane (note that they were added at almost the same fraction xiC4=0.515 and xC3=0.491).

2

A novel approach for the estimation of the Sanchez-Lacombe interaction parameters for ternary polyolefins systems

Conclusion

The overall equilibrium solubility of ethylene/propane/LLDPE-B, ethylene/isobutane/LLDPE-B and ethylene/1-butene/LLDPE-B were determined gravimetrically at 70°C in the pressure range of 0-5 bar. The measured solubilities were evaluated in gram of the mixture per gram of amorphous PE. The observed trends were theoretically interpreted because no available data can be found in the literature for these studied systems. The results showed, as expected, an anti-solvent behavior of ethylene and a co-solvent behavior of the penetrant. Adding heavy components to the binary ethylene/PE system increases the solubility of ethylene in the amorphous phase of the polymer, whereas ethylene decreases the solubility of heavier components in the amorphous polymer. Besides, the more condensable the penetrant, the higher will be these effects.

The measured solubilities were fitted to the Sanchez-Lacombe equation of state, in order to identify the interactions parameters, kij, between the different components and the polymer. In this Chapter, a novel approach consisting of implementing Peng-Robinson EoS to SL EoS (i.e. SL-PR) has been used in order to predict the interactions parameters needed for SL EoS.

Introducing the compressibility factor as input data is crucial when having only overall solubilities since we can therefore have an additional data for the fitting of SL EoS. This approach was first validated with the experimental available data and showed reasonable results, leading to its application to the experimental measured overall solubilities through the gravimetric method. Good agreement was observed between experimental overall solubility and SL EoS combined to Peng-Robinson, with a difference of 1-4% for overall solubilities and a maximum of 0.8% for the compressibility factor.

Capturing partial solubilities of the different penetrants in the polymer in multicomponent systems leads to a better understanding of the thermodynamic occurring during ethylene polymerization in gas phase. Furthermore, this will be a crucial tool when extending SL EoS to quaternary systems, since we will have three interaction parameters that needs to be identified from the SL EoS.

2 A novel approach for the estimation of the Sanchez-Lacombe interactions parameters in ternary polyolefins systems

Chapter 3

Experimental and Theoretical Solubility for Multicomponent Systems

Introduction

As was discussed in Chapter 2, there is a significant lack of solubility data for multicomponent gas/liquid/solid mixtures in polyolefin systems. Very few solubility studies are related to ternary systems (penetrant 1penetrant 2 -PE) [1][2][3][4][5][6][7][8][9][10] , and as far as we have been able to determine, never for quaternary systems (penetrant 1penetrant 2 -penetrant 3 -PE). The knowledge of the solubility in quaternary systems represents an important contribution to the modelling of olefin polymerizations since these systems are more realistic, approaching the conditions found in industry.

In the previous chapter, we began by looking at the solubility in ternary systems using gravimetry (i.e. a magnetic suspension balance) to measure the overall solubility. The total solubility data could be combined with an estimate of the compressibility factor of the gas mixture from an external source (i.e. Peng-Robinson EoS) to successfully calculate the Sanchez-Lacombe interaction parameters for the 2 penetrants. However, it is necessary to add a third data source if we wish to measure individual solubilities from SL EoS and estimate the kij for quaternary and higher order systems.

In the current chapter, we will discuss an experimental set-up based on the pressure decay method that uses rapid micro gas-chromatography measurements to calculate individual solubilities of multicomponent gas mixtures in different polyethylene samples at industrial conditions. The advantage of this method is that it gives the overall solubility, as gravimetric technique, but also the partial solubilities of each component in the gas mixture. Ternary and quaternary individual and overall solubilities will be measured using the pressure decay method, and the overall experimental solubility will be validated with the sorption experiments performed on the magnetic suspension balance (MSB). The ternary studied systems are ethylene/propane/PE, ethylene/isobutane/PE and ethylene/1-butene/PE at temperatures between 70-90°C. Then, quaternary experiments of ethylene/propane/1-butene/PE and ethylene/isobutane/1-butene/PE will be performed at 70 and 85°C. All this experimental data will be key for thermodynamic modeling, and will be fitted to SL EoS model for ternary and quaternary systems in order to identify the interactions parameters for these specific studied systems. To the best of our knowledge, this is the first time that the solubility is measured experimentally and theoretically in quaternary polyolefin systems.

Materials and methods

2.1.Materials

Ethylene (C2) with a minimum purity of 99.5%, and propane (C3a), propylene (C3e), 1-butene (1C4), isobutane (iC4) and n-butane (nC4), all with a minimum of purity of 99% were obtained from Air Liquide France. Argon with a minimum purity of 99.5% was also obtained from Air Liquide France and used as blank gas. Polyethylene and polypropylene powders were graciously supplied by Borealis, such as HDPE-B, LLDPE-B and by Ineos, such as HDPE-I, LLDPE-I and HiPP.

2.2.Polymer characterization

The crystallinity of the polymer samples was measured using differential scanning calorimetry (DSC 3+ by Mettler Toledo). Polyethylene samples were weighed (i.e. around 5-7 mg) and placed in an aluminium capsule with a volume of 40 μL. The sample was first cooled to -20°C

and then heated from -20 to 180 °C at a heating rate of 10°C min -1 , held for 5 minutes at that temperature and then cooled from 180 to -20 °C at a rate of 10°C min -1 . This temperature was maintained for 5 minutes, and the sample was reheated to 180°C at a rate of 10°C min -1 . The melting behavior of the samples was studied in the second heating cycle in order to detect the melting point and the melting enthalpy of the samples. The ratio of the heat of fusion of the sample to that of the completely crystallized polyethylene was used to calculate the degree of crystallization. A heat of fusion of 293 J g -1 for completely crystallized polyethylene was used for the calculation. The STARe software was used for data acquisition and data analysis.

Molar mass measurements of polyethylene (PE) were performed using a Viscotek High-Temperature Triple Detection GPC (HT-GPC) system (Malvern Instruments) that incorporates a differential refractive index, a viscometer, and a light scattering detector. The Omnisec software was used for data acquisition and data analysis. The molecular weight distributions (MWD) were calculated by means of a conventional calibration curve based on linear polyethylene standards from 300 to 130 000 g mol -1 (Polymer Standards Service).

Sample densities, ambient temperature crystallinities and polydispersities are summarized in A 2-liter stirred vessel is used to measure the gas phase solubility of pure components and their mixtures under standard gas phase polymerization conditions in order to generate data for thermodynamic models. The set-up is based on the pressure decay method 1,10 .

The experimental set-up describing the pressure decay method for sorption measurements is The reactor is then degassed and put under vacuum again. A known amount of process gas (or mixture of process gases) is then injected from the mixture ballast to the reactor. This process gas (or mixture) will occupy the volume of void space inside the reactor but will also absorb inside the polymer. The number of moles of the process gas in the mixture that occupies the void space of the reactor (i.e. 𝑉 void ) is calculated as follows:

𝑛 𝑖 gas = 𝑥 𝑖 𝑃 gas tot 𝑉 void 𝑍 𝑖 gas 𝑅 𝑇
Where 𝑃 gas tot , 𝑍 𝑖 gas and 𝑥 𝑖 are the total pressure, the compressibility factor of component i, and the gas phase composition of component i given by the microGC. T is the temperature of the reactor and R is the ideal gas constant.

3

Experimental and theoretical solubility for multicomponent systems

The pressure drop inside the mixture ballast indicates the amount of gas that is fed into the vessel, 𝑛 𝑖 feed . The difference between the number of moles of gas fed into the reactor and the one that occupies the void space of the reactor gives the solubility of these gases inside the amorphous phase of the polymer.

𝑛 𝑖 pol = 𝑛 𝑖 feed -𝑛 𝑖 gas
Where 𝑛 𝑖 pol is the number of moles of component i inside the polymer.

The partial solubility of component i inside the amorphous phase of the polymer, 𝑆 𝑖 am (g of solvent/ g of amorphous polymer), is given by:

𝑆 𝑖 am = 𝑛 𝑖 pol 𝑀𝑊 𝑖 𝑚 PE (1 -𝜒)
Where 𝑀𝑊 𝑖 is the molecular weight of component i, 𝑚 PE is the mass of polymer used for the experiment, and 𝜒 is the degree of crystallinity of the polymer.

The overall solubility of the mixture of gases inside the amorphous phase of the polymer, 𝑆 mix am , is therefore:

𝑆 mix am = ∑ 𝑆 𝑖 am 𝑁 𝑐 𝑖=1
Where 𝑁 𝑐 is the number of components in the gas mixture injected into the reactor.

Note that this treatment neglects the change in the polymer volume due to swelling upon sorption of the penetrants. Bashir et al. 11 showed that the degree of polymer swelling at 70°C is about 3% for ethylene/propane/LLDPE system and 8% for ethylene/isobutane/LLDPE system, at a total pressure of 6 bars. Consequently, the swelling of polyethylene by the different penetrants will be neglected in the evaluation of the experimental solubility, as it is below the accuracy of the pressure decay measurments. We will see from the validation steps below that this simplification has no measurement impact on the solubility measurements.

Validation in binary systems

Binary solubilities of ethylene and propylene in different polymer powders (i.e. HDPE-B and HiPP) were first measured in order to validate our approach, as data for the solubility of binary systems is available in the literature.

The first studied system is ethylene in HiPP. The first measurement point is taken at T=52.5°C

and then temperature steps were performed until 91.9°C. The experiment is then repeated at different ethylene pressures and at the same temperature steps. Each experiment was repeated twice at the same conditions and Figure 33 shows the average of both experiments. Sato et al. 12 measured the solubility of ethylene and propylene in HiPP of crystallinity of 58.1 wt.% with a magnetic suspension balance, and their data was compared to our experiments, since the crystallinity of the used HiPP in this study is about 60 wt.% (c.f. Table 17). Figure 33 shows that the experimental solubility of ethylene in HiPP agrees well with the data of Sato et al. 12 , with an average difference of 5.7%. Propylene solubility is first measured at 51.9°C and 4.40 bar and then temperature steps were performed until T=91.8°C. Figure 34 shows that the experimental solubility of propylene in HiPP follows nicely Sato et al. 12 data with an average difference of 6.7%. Ethylene solubility is first measured at 55°C and 9.5 bar and then temperature steps were performed until T=74°C. measured by gravimetry the solubility of ethylene in HDPE of crystallinity of 70%, which is close to that of HDPE-B (c.f. Table 17).

Figure 34. Propylene solubility in the amorphous phase of HiPP at different temperatures and propylene pressures. The lines represent the solubility from Sato et al. 12 and the points represent the experimental solubility measured with the pressure decay method. Once our experimental approach validated for the binary systems, the solubility of mixtures of gases in polyethylene powders can be performed with confidence. Ternary and quaternary systems will therefore be studied in Section 3 since those systems are lacking in the literature, and they are very important in order to have a more realistic description of ethylene polymerization in gas-phase.

2.4.Gravimetric method

Solubility measurements were achieved using a high-pressure magnetic suspension balance (Rubotherm GmbH), connected to a pressure cell that contains the polymer sample. The detailed experimental apparatus and measuring procedure are shown in Chapter 2 and in references 3,4,[13][14][15] .

The total solubility of ternary and quaternary systems will therefore be measured with the magnetic suspension balance to confirm the results of the pressure decay method. However, the experimental set-up with the magnetic suspension balance can only be used to measure the overall solubility of mixture of gases in the amorphous phase of the polymer.

Sanchez-Lacombe EoS for quaternary systems

The Sanchez-Lacombe equation of state has been used in order to determine the thermodynamic properties that are important in gas-phase ethylene polymerization, including the solubility of the different species in the amorphous phase of the polymer as well as the swelling of the polymer phase due to the sorption of these species, for a number of systems. However, this equation of state has not been extended to more than 2 penetrants and a polymer in the literature, mostly because experimental data does not exist for validation. Therefore, we will need to extend SL EoS to describe the thermodynamic properties of quaternary systems including a monomer, a comonomer, an ICA and a polymer.

3.1.Model development

The Sanchez-Lacombe EoS (SL EoS) is a lattice-fluid model in which pure components are assumed to be broken into parts and placed into a lattice structure. The SL EoS is basically an extension of the classic Flory-Huggins theory. The main advance is that this model introduces vacant lattice sites or holes to account for the compressibility and the density changes. Thus, the system volume or density can vary by changing the fraction of holes in the lattice structure.

The main equation of state is given by:

𝜌̅ 2 + 𝑃 ̅ + 𝑇 ̅ [ln(1 -𝜌̅ ) + (1 - 1 𝑟 ) 𝜌̅ ] = 0 (1)
where 𝑇 ̅ , 𝑃, ̅ 𝜌̅ are the reduced temperature, pressure, and density respectively, given by:

𝑇 ̅ = 𝑇 𝑇 * ⁄ 𝑃 ̅ = 𝑃 𝑃 * ⁄ 𝜌̅ = 𝜌 𝜌 * ⁄
where 𝑇 * , 𝑃 * , and 𝜌 * are the scale factors known as the characteristic temperature, pressure, and density respectively, which are used to characterize each pure component in the mixture.

Three lattice parameters describe the thermodynamic properties of a pure components: 𝜀 * , 𝑣 * and r which are the mer-mer interaction energy, the closed-packed molar volume of a mer and the number of mers a molecule occupies in the lattice, respectively. The scale factors are defined from these lattice parameters as follows:

𝑇 * = 𝜀 * 𝑅 ⁄ 𝑃 * = 𝜀 * 𝑣 * ⁄ 𝜌 * = 𝑀𝑊 (𝑟𝑣 * ) ⁄
Where R is the universal gas constant.

These parameters of SL EoS for each pure component are given in Table 2. where 𝜔 𝑖 gas is the mass fraction of the component i in the gas phase.

In quaternary system, the mass fraction of component 1 is defined as follows: The mixing rule for the characteristic interaction energy for the mixture 𝜀 mix * is defined as

𝜀 mix * = 1 𝑣 mix * ∑ ∑ 𝜙 𝑖 𝜙 𝑗 𝜀 𝑖𝑗 * 𝑣 𝑖𝑗 * 𝑁 c 𝑗=1 𝑁 c

𝑖=1

For quaternary systems, the characteristic interaction energy for the gas mixture is defined as follows:

𝜀 * mix gas = 1 𝜐 * mix gas (𝜀 1 * 𝑣 1 * 𝜙 1 gas 2 + 𝜀 2 * 𝑣 2
mixture, which is employed as adjustable parameter to fit the model to experimental solubility data.

Finally, the mixing rule for the number of sites (mers) occupied by a molecule of the mixture, 𝑟 mix , is given by:

1 𝑟 mix , = ∑ 𝜙 𝑗 𝑟 𝑗 𝑁 c 𝑗=1
where 𝑟 𝑗 is the number of sites occupied by molecule j in the lattice.

For quaternary systems, the number of sites occupied in the lattice by a molecule of gas phase mixture is defined as follows: Assuming that the number of site occupied by the gaseous species are negligible compared to that of the polymer, 𝑟 4 >>𝑟 1 , 𝑟 2 , 𝑟 3 , the number of sites occupied in the lattice by a molecule of polymer phase mixture is defined as follows:

1 𝑟 mix pol = 𝜙 1 pol 𝑟 1 + 𝜙 2 pol 𝑟 2 + 𝜙 3 pol 𝑟 3
For the calculation of sorption equilibrium for polymer-solvent system, the chemical potential of the i th component in each phase of the mixture of multicomponent species can be expressed as:

We consider that the chemical potential of each solute component at equilibrium is equal to the chemical potential of the polymer as follows:

𝜇 𝑖 pol = 𝜇 𝑖 gas i=1,2,3,4
The same expression of the chemical potential in the gas phase (2) can be rewritten for the polymer phase, with all the characteristic parameters calculated for the polymer. This equation will lead to the identification of the closed-packed volume fractions of solute components in the polymer phase, 𝜙 1 pol , 𝜙 2 pol and 𝜙 3 pol .

By solving simultaneously the three non-linear equations based on equation (2) for solute species 1, 2 and 3 and the equation of state (1) of SL for the polymer phase; the reduced polymer

𝜇 𝑖 gas = R 𝑇 [ln 𝜙 𝑖 gas + (1 - r i 𝑟 mix gas )] (2)
+𝑟 𝑖 {-𝜌̅ gas [

𝑆 𝑖 am = 𝜔 𝑖 am 𝜔 pol am (1 -𝜒) i = 1,2,3
where 𝜒 is the crystallinity of the polymer.

3.2.Modelling algorithm

For quaternary systems, the interaction parameters between the penetrating molecules and the polymer are required and cannot be taken from ternary systems a priori. We consider here a system containing a monomer (1), an ICA (2), a comonomer (3) and a polymer (4). The corresponding interaction parameters in order to solve the SL EoS are : k12, k13, k14, k23, k24, k34

which are the interaction parameters between the monomer and the ICA, the monomer and the comonomer, the monomer and the polymer, the ICA and the comonomer, the ICA and the polymer and the comonomer and the polymer, respectively. Only k14, k24, k34 parameters are estimated by fitting SL EoS to experimental data, while k12, k13 and k24 are equal to zero because we assume that the interaction between small olefin molecules does not deviate from Lorentz-Berthelot combining rules (used to calculate the potential) 20 .

The novel approach used in Chapter 2 where SL EoS is combined to Peng-Robinson EoS (i.e.

PR EoS) cannot be used for quaternary systems when only overall solubility data is available because three interaction parameters need to be identified from only two input data. This makes the pressure decay method, from which partial solubilities are measured, crucial to model the solubility of quaternary systems with SL EoS. The procedure allowing to identify SL interaction parameters through overall and partial solubility in the amorphous phase of the polymer is shown in Figure 36.

Partial solubilities of the different components in the mixture are calculated with SL EoS and are then fitted to the experimental data in order to identify the three interaction parameters, k14, k24, k34. The fitting procedure minimizes the percent average relative deviation, ARD, between the experimental and calculated partial and total solubilities, as follows:

Results and discussion

4.1.Ternary systems

Different ternary systems containing (1) ethylene, (2) penetrant and (3) polyethylene were experimentally studied using the pressure decay method. The same polymer was used in order to compare these ternary systems, and therefore to quantify the impact of adding different penetrants to ethylene/PE system. The overall solubility of the gas mixture as well as their partial solubility in the amorphous phase of the polymer were measured. Table 19 shows the range of pressure and temperature and the composition of the gas mixtures for the ternary systems considered here. ethylene/propane/LLDPE-I, ethylene/propylene/LLDPE-I and ethylene/1-butene/LLDPE-I, at a total pressure up to 8 bars and temperature up to 90°C. Figure 37 shows that, as expected, the total solubility increases as the total pressure of the system increases and as the temperature of the system decreases. It is important to note that the heavier the penetrant is, the higher is the total solubility of a mixture at a given total pressure. Indeed, at the same total pressure of the system and the same temperature, the overall solubility in presence of 1-butene is higher than in presence of propylene, which is higher than in presence of propane.

3 Experimental and theoretical solubility for multicomponent systems

Figure 37. Total solubility of ethylene/propane, ethylene/propylene and ethylene/1-butene in LLDPE-I at temperatures between 68-90°C. Gas compositions of the different systems is given in Table 19

Figure 38 through Figure 43 show the overall and partial solubilities of the different penetrants in the amorphous phase of the polymer at different temperatures, for each studied systems. The experimental solubility is then correlated to SL EoS in order to estimate the interaction parameters, kij, for each system at each temperature. The kij correlations thus obtained are shown in Table 20. The figures show that SL EoS is in good agreement with the experimental solubility data performed with the pressure decay method. The maximum difference between the experiments and SL EoS is 1% for ethylene/propane/LLDPE-I system, 5% for both ethylene/propylene/LLDPE-I and ethylene/1-butene/LLDPE-I systems. It can also be seen that in general, when the overall solubility increases with pressure, also the individual solubility of ethylene and the other penetrant increases, except for 1-butene (c.f. Figure 43 and Figure 44).

For this later system, at 68°C, increasing the total pressure leads to an increase of ethylene solubility and to a decrease of that of 1-butene while the inverse occurs at 84°C. Note however that the composition for ethylene/

Figure 44. Temperature dependent correlation of k13 for the different studied systems of ethylene/propane/LLDPE-I (C2/C3a/LLDPE-I), ethylene/propylene/LLDPE-I (C2/C3e/LLDPE-I) and ethylene/1-butene/LLDPE-I (C2/1C4/LLDPE-I).

It is known that the interaction parameters are function of the components composing the gaseous mixture, the polymer type and the temperature. Table 20 shows the temperature dependent correlations of the interaction parameter, kij, for the different studied systems. It can be seen that the interaction parameter between ethylene and the polymer, 𝑘 13 , is dependent on the nature of the second penetrant. For the temperatures of interest, heavier penetrants lead to lower 𝑘 13 , but higher in the absolute value, as can be seen in When 𝑘 𝑖𝑗 =0, the cross-potential follows Lorentz-Berthelot combining rules 19 (i.e. 𝜀 𝑖𝑗 * = (𝜀 𝑖 * 𝜀 𝑗 * ) 0.5 ) while if the value is far from 1, then there is a deviation from these rules. So, there is a higher deviation of the interaction between the polymer and 1C4, than C3e, than C3a. This was also observed for the ternary experiments performed on the magnetic suspension balance in Chapter 2. A similar study was also considered by Chmelar et al. 21 , and they found that the interaction parameter between 1-hexene and PE was lower than the one between propane and PE. They explained that the purpose of kij is to correct the energy interactions of unlike chain segments of the penetrant and the polymer. Therefore, the heavier the penetrant, the more the interaction with the polymer deviates from Lorentz-Berthelot combining rules. Furthermore, Bashir et al. 23 noted that unlike ethylene, propylene and hexene at 90°C, only 1-butene showed a negative kij in a binary system. Thus, while perhaps not immediately expected, this result is not attributable to experimental error. This negative value of the interaction parameter suggests that 1-butene has a higher interaction energy with polyethylene than do the other compounds studied here.

Validation with the SL-PR method

The advantage of the pressure decay method is that it leads to the measure of both the overall and partial solubility of each component in the gaseous mixture (so three data points) in order to estimate the interaction parameters, kij. As a further validation of the experimental set-up presented in this work, the overall solubilities measured using the pressure decay method are treated using the Sanchez-Lacombe -Peng-Robinson (SL-PR) approach used in Chapter 2. This method will therefore validate our experimental approach by comparing the kij interaction parameters identified when fitting SL EoS to the overall and partial solubilities of the ternary systems above with those estimated using the SL-PR. Overall and partial solubility of ethylene/propane/LLDPE-I ternary system at 80°C. The estimation of the partial solubility is achieved by using the SL-PR method. Molar composition of the system is given in Table 19 and kij used for this study is given in Table 20 Figure

Compressibility factor of ethylene (Z1) and propane (Z2) in ethylene/propane/LLDPE-I system at 80°C. The lines represent compressibility values calculated with Peng-Robinson and points

represent the predictions of the compressibility factor from the ternary SL EoS. Overall and partial solubility of ethylene/propylene/LLDPE-I ternary system at 68°C. The estimation of the partial solubility is achieved by using the SL-PR method. Molar composition of the system is given in Table 19 and kij used for this study is given in Table 20 21 compares the kij interaction parameters identified when fitting SL EoS to the overall and partial solubilities of the ternary systems and when only overall solubility of the ternary system is available and is therefore combined to the calculation of the compressibility factor of each component in the system through Peng-Robinson EoS (i.e. SL-PR). It is shown that the interaction parameters are quite similar, giving therefore confidence in the pressure decay experiments for the estimation of the individual solubilities.

. Comparison between binary and ternary systems

A comparison between binary and ternary systems will be achieved in this section in order to show the impact of ethylene as well as the different penetrants on the overall solubility of the gas mixture in the amorphous phase of the polymer. In order to do so, the binary solubilities of ethylene, propane, propylene and 1-butene in LLDPE will be compared to the ternary solubility of ethylene/penetrant/LLDPE-I at 80°C. The binary solubilities in LLDPE were estimated from Chmelar et al. 21 since they used an LLDPE with a density of 923 kg m -3 and a crystallinity of 50.3 wt%, that are close to the characteristic of the LLDPE-I sample used for the pressure decay experiments (c.f. Table 17). The different binary interaction parameters are given in Table 22 at 80°C. As mentioned above, the co-solvent effect refers to the observed increase in the concentration of a given species due to the presence of another species in the mixture. This can have important implications in terms of polymerization rate and polymer properties. Figure 52 and Figure 53 show ethylene solubility in the binary system and its individual solubility in the ternary mixtures with 5 and 30 mol% of propane, propylene and 1-butene at 80°C, respectively. It is clear that adding even 5 mol% of the different studied penetrants leads to an important increase of the partial solubility of ethylene in the ternary system. For instance, it can be seen in Figure 52 that at a total pressure of 4 bars, the partial solubility of ethylene is increased by a factor of 1.3 in presence of propane, 1.8 in presence of propylene and 3.3 in presence of 1-butene. It can be concluded that a co-solvent effect of the different penetrants on the solubility of ethylene in the amorphous phase of the polymer is preponderant in the ternary systems. Besides, it is clearly shown that the heavier the penetrant added in the ternary system is, the higher is the co-solvent effect of the different penetrants on ethylene concentration in the amorphous phase of the polymer. 21 (dotted lines) and from SL EoS (points) and the partial ternary solubility of ethylene in presence of 30 mol% of propane, propylene and 1-butene in LLDPE-I at 80°C. kij for the studied ternary systems at the temperature of interest are defined in Table 20 and for the ethylene/LLDPE binary system in Table 22.

Figure 52. Binary solubility of ethylene in LLDPE taken from 21 (dotted lines) and from SL EoS (points) and the partial ternary solubility of ethylene in presence of 5 mol% of propane, propylene and 1-butene in LLDPE-I at 80°C. kij for the studied ternary systems at the temperature of interest are defined in

Anti-solvent effect in ternary systems

The co-solvent effect of heavier components on ethylene is accompanied by an opposite antisolvent effect of ethylene on the other species. Figure 54 and Figure 55 show the solubility of pure propane, propylene and 1-butene and their individual solubilities in the ternary mixtures with 5 and 30 mol% of ethylene in LLDPE. It can be observed that adding ethylene to binary penetrant/LLDPE system leads to a decrease of the penetrant partial solubility in the ternary system, due to the anti-solvent effect. As an example, Figure 54 shows that adding 5 mol% of ethylene decrease the partial solubility of propane by a factor of 0.46, 0.58 for propylene, and 0.38 for 1-butene at a total pressure of 4 bar. It is clear that the heavier the penetrant is, the higher is the anti-solvent effect of ethylene on the penetrants concentration in the amorphous phase of the polymer. Furthermore, we can see that at an equivalent concentration, the cosolvent effect of the heavier molecules is more pronounced than the anti-solvent effect of ethylene.

Pressure decay experiments

There is no reason to believe that co-and anti-solvent effects would not be present in quaternary systems; rather one would expect the complexity of the interactions to increase. For instance, when the vapour phase contains only 2 components it is relatively straight forward to envisage how changing the fraction of one or the other would have on overall and partial solubilities. In quaternary systems this is less obvious. Furthermore, as mentioned above, it is known that the degree of crystallinity of a PE can influence the solubility of penetrants even if one considers only the amorphous fraction of the material. This means that if one wishes to have thermodynamic data for different mixtures and different polymer grades, it would be necessary to perform a large number of experiments.

To show the importance of performing these higher order experiments, and to better discern the impact of polymer properties on the equilibrium solubility, quaternary experiments were performed with both the pressure decay method and the magnetic suspension balance (gravimetric method). These quaternary systems contain (1) ethylene, (2) ICA, (3) comonomer and (4) polyethylene Some of these experiments were performed using polyethylenes with different crystallinities and molecular weight distributions in order to show the impact of changing the polymer type on the solubility modeling since kij are function of the polymer type and crystallinity. The characteristics of the different polymer samples are shown in Table 1.

The studied system is ethylene/propane/1-butene/PE at different temperatures and a total pressure up to 5 bars. The studied temperatures, pressures and the gas phase molar composition of the different quaternary studied system are shown in Table 23. 1.

Figure 56 shows the overall solubility of ethylene/propane/1-butene in HDPE-I and LLDPE-I, at different temperatures up to 84°C and different molar composition of the gas phase. It is shown that as for ternary systems, the solubility increases with increasing total pressure and with decreasing the temperature. Furthermore, it can be seen that the solubility is higher in the amorphous phase of LLDPE-I than in the amorphous HDPE-I, as expected. This observation has been attributed to the elastic effects due to the presence of more tie molecules in HDPE than in LLDPE 22 . It is proposed that these tie molecules stop the amorphous phase of the polymer from swelling, thereby reducing gas sorption in the amorphous phase of the polymer.

Figure 57 and Figure 58 show the overall and partial solubilities of ethylene/propane/1-butene mixture in LLDPE-I at 69 and 84°C and total pressure up to 5 bars. It is shown that increasing the temperature decreases the overall solubility of the quaternary system in the amorphous phase of LLDPE-I, as expected. Furthermore, the partial solubility of ethylene, propane and 1butene are also decreased when increasing the temperature. It is important to note that ethylene partial solubility decreases when increasing the total pressure of the system since its fraction in the gas phase in the second pressure step is lower compared to the first pressure step, which is not the case for propane and 1-butene. Total and partial solubility of ethylene/propane/1-butene/LLDPE-I system at 69°C. S_tot, S1_am, S2_am and S3_am are the total solubility of the mixture and the overall solubility of ethylene, propane and 1-butene in the amorphous phase of the polymer, respectively. Gas phase composition of the mixture is given in Table 23 Figure 58. Total and partial solubility of ethylene/propane/1-butene/LLDPE-I system at 84°C. S_tot, S1_am, S2_am and S3_am are the total solubility of the mixture and the overall solubility of ethylene, propane and 1-butene in the amorphous phase of the polymer, respectively. Gas phase composition of the mixture is given in As with LLDPE, increasing the total pressure increases the overall solubility. Note that a lower fraction of propane and a higher fraction of ethylene have been used for HDPE-I and HDPE-B than was the case with the LLDPE-I, as can be seen in Table 14.

This explains the fact that partial solubility of ethylene increases slightly with respect to the LLDPE experiments, and propane partial solubility is less increased. Furthermore, as explained above, the overall and partial solubility are lower in the amorphous phase of HDPE compared to LLDPE. It can be seen in Figure 56 that there is a significant difference in the solubility of the quaternary system in the amorphous phase of LLDPE-I compared to HDPE-I. Indeed, the difference in the overall solubility in HDPE-I is about 60% at a total pressure of 2.5 bars compared to LLDPE-I at almost the same temperature. This can also be explained by the difference that can be seen in the crystallinity of these PE in Total and partial solubility of ethylene/propane/1-butene/HDPE-B system at 75°C. S_tot, S1_am, S2_am and S3_am are the total solubility of the mixture and the overall solubility of ethylene, propane and 1-butene in the amorphous phase of the polymer, respectively. Gas phase composition of the mixture is given in Table 23 4.

Gravimetric experiments

Similar sorption experiments for quaternary systems were also performed using the gravimetric method to validate our approach using the pressure decay method combined to the µGC. Note that the gravimetric method gives only the overall solubility, whereas the pressure decay method gives both overall and partial solubility of the gaseous mixture in the amorphous phase of the polymer.

For each solubility experiment, we kept the temperature, PE type and gas phase composition constant; while increasing the total pressure of the injected mixture. Two quaternary systems were studied: ethylene/propane/1-butene/LLDPE-B and ethylene/isobutane/1-butene/LLDPE-B. Table 8 shows the total pressure and the composition of the gas phase for each studied quaternary system. The overall solubility of these two systems will be measured for a pressure up to 6.2 bars and a constant temperature of 70°C. Each experiment was repeated 2-3 times to verify its repeatability. However, the gas phase composition varies slightly between each repeated experiment.

Validation with SL EoS for quaternary systems

The experimental solubility data achieved using the pressure decay method will be fitted to the SL EoS for quaternary systems in order to identify the interaction parameters needed for SL modeling. Having four experimental data (i.e. overall solubility and partial solubility of the three penetrants) makes it possible to identify the three SL interaction parameters, k14, k24, k34.

Note that these kij are temperature dependent and function of the polymer type. It is therefore interesting to have different PE grades and different temperatures, to see their impact on the identified kij.

Table 25 shows the identified kij (representing the interactions between the different penetrants and the polymer) for the studied quaternary systems using pressure decay method. It is shown that these interaction parameters are temperature dependent but are also a function of the type of polymer. Figure 63 shows that the kij parameters for ethylene/propane/1-butene/LLDPE-I decreases with increasing the temperature. Furthermore, it is shown that the interaction parameter between the lighter component and the polymer is the lowest one, and the interaction parameter between the heavier component and the polymer is the highest one. The same trend was also observed for the ternary systems where the interaction between ethylene and the polymer, k13, was lower than the one between the penetrant and the polymer, k23, for all studied ternary systems. 9.

The Sanchez-Lacombe model was extended in order to describe quaternary systems (c.f.

Section 3) and the detailed procedure of the model calculation is shown in the Appendix A. information alone could not give the partial solubilities of each component in the gas mixture from SL EoS, even with the combined SL-PR method developed in Chapter 2.

Figure 64. Experimental and SL modeling of overall and partial solubility of each component in ethylene/propane/1-butene/LLDPE-I at 69°C. Gas phase composition of the mixture is given in

Figure 68 shows the overall solubility of ethylene/propane/1-butene/PE quaternary system normalized by the partial solubility of each component in HDPE-I and LLDPE-I at 73°C, from SL EoS, using the kij identified in Table 9. Note that both systems with HDPE-I and LLDPE-I have been simulated with the same gas phase composition of the mixture (𝑥 𝐶 2 =0.66, 𝑥 𝐶 3𝑎 =0. 16 and 𝑥 1𝐶 4 =0. 16). The purpose of this simulation is to compare the changes in the partial solubility of the different components in the gas phase mixture in different polyethylene samples (i.e.

HDPE-I and LLDPE-I).

If the solubility of a component changes in the same relative manner from HDPE-I to LLDPE-I, it is not unreasonable to suppose that the relative solubilities of the same 3 components for a similar gas mixture will be the same in another polymer. It can be seen from Figure 68 that the solubility of the different species in the mixture in both HDPE-I and LLDPE-I follows the same tendency. This means that the same changes in the partial solubility occur, with a difference of 2% for the partial solubility of propane up to 9% for the partial solubility of 1-butene when comparing HDPE-I to LLDPE-I. In order to overcome the issues of measuring only the overall solubility when using the gravimetric method, the SL-PR method will be used and combined to the measure of the overall solubility normalized by the partial solubility of the different penetrants in HDPE-B quaternary system. This will permit the calculation of the partial solubilities of the different species in ethylene/propane/1-butene/LLDPE-B system. Gas phase composition of the quaternary system in both HDPE-B and LLDPE-B is given in Table 23 and Table 24 and it can be seen that almost the same composition have been used for both PE grades. Therefore, we would expect to see the same tendency of the partial solubilities of the different components for both polymers, as can be seen for HDPE-I and LLDPE-I in Figure 68.

The application for the quaternary systems of the SL-PR method combined to the measure of the normalized solubility is shown in Figure 69 where it can be seen that the model fits the data well, with an average difference between the model and the experiments being on the order of 2%. Figure 70-Figure 72 show that the calculated compressibility factors from SL EoS are in good agreement with the ones calculated with Peng-Robinson EoS, with an average difference of 0.24% for ethylene, 0.42% for propane and 0.37% for 1-butene. Once the kij identified, we can therefore model these systems for different gas phase composition in order to better understand the effect of gas phase composition in quaternary systems.

Figure 69. Experimental and SL modeling of overall and partial solubility of each component in ethylene/propane/1-butene/LLDPE-B at 70°C. Gas phase composition of the mixture is given in

The effect of polymer properties

Let us look briefly at the impact of the polymer properties on the overall solubility of vapors in different types of PE. As discussed above, the more crystalline the polymer is, the lower the solubility of a given penetrant in the amorphous phase is expected to be. This is evident from the graphs in However, differences in the overall crystallinity of a sample, especially measured at room temperature are not enough to explain the differences observed in catalyst was used to make them. It is well known that one characteristic of polymers made with metallocenes is that this type of catalyst is a uniform incorporator of comonomerin other words all polymer chains have the same chemical composition. On the other hand, it is reasonable to propose that the "B" polymers, with polydispersities on the order of 8.8 for the LLDPE-B and 10.9 for the HDPE-B, are made with a Ziegler-type catalyst. Ziegler catalysts are known to have a non-uniform incorporation of comonomers, with the shorter chains containing more comonomer than the longer ones. Is it possible that the difference in terms of the homogeneity of chain structure can lead to a different organization of the amorphous phase, where higher degrees of inhomogeneity favor the swelling of the amorphous phase more than is the case in homogeneously organized polymers? Obviously, the observations shown here do not permit us to interpret the results to this degree, but a future study linking different facets of macromolecular architecture to the swelling of the amorphous phase might be interesting and useful.

Figure 73. Overall solubility of ethylene/propane/1-butene in HDPE on the left and LLDPE on the right at different temperatures, using gravimetry for HDPE-B, HDPE-I and LLDPE-I and pressure decay method for LLDPE-B. Gas phase composition is shown in

Figure 74. Molecular weight distribution of the different used PE samples performed with the HT-GPC

4.3.Importance of describing quaternary systems

The co-and anti-solubility effects are obviously going to be somewhat more complex in multicomponent mixtures, but once the kij have been identified (Table 25) can model the solubility to obtain some insight into the effect of gas phase composition in quaternary systems.

Effect of ICA on quaternary systems

A comparison between the ternary system of ethylene/1-butene/LLDPE-I and the quaternary system of ethylene/propane/1-butene/LLDPE-I at 80°C will be achieved in this section. The effect of propane as ICA will be highlighted, in order to see its effect on the overall quaternary solubility as well as on the partial solubility of the different components in the mixture. To do so, two simulations were performed: the first one was to keep ethylene molar fraction constant and the second one was to keep 1-butene molar fraction constant, by maintaining the same total pressure of the system, with respect to the ternary system. Therefore, the same molar fraction of propane, xC3a=0.16, was used for both simulations. In order to simulate the quaternary system at 80°C with SL EoS. A temperature dependent correlation of the kij interaction parameter for ethylene/propane/1-butene/LLDPE-I system was obtained from Table 9, and gives k14=-0.0992, k24=0.0147 and k34=0.0758. The kij for the ternary systems at 80°C were taken from the temperature correlations given in Table 20. 77 show the overall and partial solubilities of ethylene and of 1-butene, respectively, for the ternary ethylene/1-butene/LLDPE-I and quaternary ethylene/propane/1butene/LLDPE-I systems, at 80°C. Figure 75 shows that adding propane to the ternary ethylene/1-butene/LLDPE-I system decreases the overall solubility. Besides, it is shown that higher fraction of ethylene and therefore smaller fraction of 1-butene leads to a lower overall solubility of the quaternary system, about 28% (i.e. green line) compared to 12% (i.e. orange line). In order to better interpret these trends, it is better to look at the partial solubilities of ethylene and 1-butene in order to understand the different interactions that occurs between the species in the quaternary system. Figure 76 shows that adding propane to the ternary ethylene/1-butene/LLDPE-I system leads to a decrease of ethylene partial solubility. This is not an expected result as we have seen for ternary systems that heavier alkanes acts as co-solvents for the lighter ones (i.e. ethylene), and therefore increasing the solubility of the lighter component in the amorphous phase of the polymer. However, the trends in Figure 77 follows what we have seen in the ternary systems concerning anti-solvent effect. Figure 77 shows that adding propane to the ternary ethylene/1-butene/LLDPE-I system leads to a decrease in the partial solubility of 1-butene. This can be explained by the anti-solvent effect of propane (i.e. lighter component) on 1-butene (i.e. heavier component), leading to a decrease of the partial solubility of the heavier component in the system. To conclude, it has been shown that propane decreases both ethylene and 1-butene partial solubility when added in quaternary system, leading to a decrease of the overall solubility of the quaternary mixture in the amorphous phase of the polymer.

Figure 75-Figure

Figure 75. Overall solubility calculated from SL EoS for ethylene/1-butene/LLDPE-I ternary system and for ethylene/propane/1-butene/LLDPE-I quaternary system at 80°C. kij for the ternary system was

taken from Table 20 and for the quaternary system from Table 25 Figure 76. Ethylene partial solubility calculated from SL EoS for ethylene/1-butene/LLDPE-I ternary system and for ethylene/propane/1-butene/LLDPE-I quaternary system at 80°C. kij for the ternary system was taken from Table 20 and for the quaternary system from 20 and for the quaternary system from Table 25 4.

Effect of comonomer on quaternary systems

In this section, the comparison of the ternary system of ethylene/propane /LLDPE-I and the quaternary system of ethylene/propane/1-butene/LLDPE-I at 80°C will be examined. The effect of 1-butene as comonomer will be underlined. As in the previous section, the same simulations will be performed: keeping ethylene molar fraction constant first, and then keeping 1-butene molar fraction constant, by maintaining the same total pressure of the system, with respect to the ternary systems. The same mole fraction of 1-butene, xC3a=0.16, is used for both simulations.

The kij interaction parameter for ethylene/propane/1-butene/LLDPE-I system at 80°C are given in the previous section. The kij for the ternary systems at 80°C were taken from the temperature correlations given in Table 20.

Figure 78-Figure 80 show the overall and partial solubilities of ethylene and of propane, respectively, for the ternary ethylene/propane /LLDPE-I and quaternary ethylene/propane/1butene/LLDPE-I systems, at 80°C. Figure 78 shows that adding 1-butene to the ternary ethylene/propane /LLDPE-I system leads to an increase in the overall solubility of the quaternary system to about 18% for the simulated system having higher fraction of ethylene (i.e. green line) and 30% for a higher fraction of propane (i.e. orange line). Contrarily to propane behavior shown in the previous section, adding 1-butene to the ternary system leads a significant increase of the overall solubility. This increase can be explained by looking at the partial solubility of the different component in the gaseous mixture. Figure 79 demonstrates the effect of adding 1-butene in the ternary system on the partial solubility of ethylene in the amorphous phase of the polymer. It is shown that adding 1-butene leads to an increase of the partial solubility of ethylene, demonstrating the co-solvent effect of 1-butene on ethylene.

Figure 80 shows the effect of adding 1-butene in the ternary system on the partial solubility of propane. It is shown that, when keeping ethylene molar fraction constant, adding 1-butene to the ternary system decreases propane partial solubility in the amorphous phase of the polymer to about 54%. This is not what we expected to see, as we would suppose that 1-butene which is the heavier component acts as a co-solvent towards propane, whereas the results shows that 1butene acts as an anti-solvent to propane, decreasing its partial solubility in the amorphous phase of the polymer. However, adding 1-butene to the ternary system and decreasing ethylene molar fraction in the quaternary system leads to an increase of the partial solubility of propane.

It is clear that as we increase the number of components in the system (i.e. quaternary and higher order systems), the competing co-and anti-solvent effects make it difficult to estimate the impact of each component on the solubility à priori. In fact, this demonstrates the need for the type of thermodynamic model proposed here, and also suggests that more extensive measurements need to be made.

Figure 78.

Overall solubility calculated from SL EoS for ethylene/propane /LLDPE-I ternary system and for ethylene/propane/1-butene/LLDPE-I quaternary system at 80°C. kij for the ternary system was taken from Table 20 and for the quaternary system from

Conclusion

The solubility of ternary and quaternary polyolefin systems has been studied experimentally and theoretically. The solubility of ternary systems of ethylene/propane/LLDPE, ethylene/propylene/LLDPE and ethylene/1-butene/LLDPE have been measured experimentally using the pressure decay method at temperatures between 68 -90°C and a total pressure up to 8 bars. The quaternary system of ethylene/propane/1-butene in different PE grades and at temperatures between 69 -84°C and a total pressure up to 7 bars have been studied using pressure decay and gravimetric methods. The Sanchez-Lacombe EoS have been used in order to interpret the experimental results. An important data base of the solubilities of the ternary and more importantly the quaternary systems, as well as the kij interaction parameters from SL EoS, have been provided in this Chapter. It is important to note that these interaction parameters are valid for the range of pressure and temperature used during this study, and some changes might occur if very low temperatures and/or high pressures are used to describe the studied systems. Besides, the quaternary studied sorption represents an important contribution to the modelling of olefin polymerizations since it is the first time in the open literature that the solubilities of quaternary polyolefin systems are studied under conditions approaching those found in industry.

For ternary systems, it was demonstrated that, as expected, the heavier component acts as a cosolvent to the lighter one, and that the lighter component acts as an anti-solvent to the heavier one. Furthermore, it has been shown that the heavier the component is, the stronger is its cosolvent effect. It has also been revealed that the co-solvent effect has a more significant impact than the anti-solvent effect on the overall solubility. Finally, it has been shown that the heavier the component is, the stronger is its co-solvent effect, leading therefore to an important increase in the overall solubility of the system in the amorphous phase of the polymer.

Concerning quaternary systems, it has been shown that the overall solubility of the quaternary system of ethylene/propane/1-butene/LLDPE is higher than the one in the ternary system of ethylene/propane/LLDPE and lower than the one in the ternary system of ethylene/1butene/LLDPE at the same temperature and total pressure. The obtained results from the quaternary studied systems demonstrate that the complexity of the changes in the gas phase composition needs to be interpreted by an accurate thermodynamic model. To do so, more experimental data is needed in order to be able to identify the SL interaction parameters. It would therefore be interesting to look at the effect of other quaternary systems involving for example other ICAs (i.e. isobutane, n-hexane) and other comonomers (i.e. 1-hexene) in order 3 Experimental and theoretical solubility for multicomponent systems to compare the effect of the different ICAs and co-monomers, and their combination, on the overall solubility in quaternary systems. Besides, it would be helpful to study the ternary system of propane/1-butene/LLDPE in order to better understand the effect of ethylene on the studied quaternary system.

As expected, the crystallinity of the polymer used in the solubility study has at least as much impact on the solubility of penetrants in the amorphous phase as the composition of the vapour phase mixture does. However, we unexpectedly found that for polymers with the same overall crystallinity, one can observe significant differences in the overall solubility of a gas mixture. In the current study, it was observed that polymers having a significantly broader MWD had higher solubilities than did polymers of similar crystallinity but with narrow MWD. It was postulated that this difference in solubility might be attributable to differences in the chain architecture, attributable to the catalyst used to make the polymers. However, it should be made clear that there is not yet any strong evidence to support this proposition, so it should be viewed more as a testable hypothesis for future studies than a real explanation.

Diffusivity of multicomponent gas mixtures in polyethylene

Chapter 4

Diffusivity of Multicomponent Gas Mixtures in Polyethylene Diffusivity of multicomponent gas mixtures in polyethylene

Introduction

As was discussed in the previous chapters, there is a significant lack of data concerning both solubility and diffusivity of multicomponent gas phase mixtures. This information is crucial for a better understanding of mass transfer and concentrations of reactive species in the growing polymer particles during gas phase ethylene polymerization. It has been shown in the literature 1,2 that accounting for the co-solubility and co-diffusivity effects can help to explain the increase in the productivity. The sorption of multiple species can change the way that they interact with the amorphous phase of the polymer. Furthermore, the solubility of the different species will have an impact on the free volume of the amorphous phase of the polymer, which in turn can influence the diffusivities of the penetrants in the system. Indeed, it has also been shown that taking into account the interactions between the different species when calculating the diffusion coefficients improves the capacity of single particle models to explain high initial rates of polymerization in the presence of ICAs 1 . It is therefore crucial to have more experimental and theoretical studies about the diffusion of mixtures of penetrants in the polymer amorphous phase since it has a great importance in ethylene polymerization reaction in gas phase.

For this reason, the diffusivity of binary, ternary and quaternary systems will be studied experimentally and theoretically in the current Chapter. The overall diffusivity of ternary systems such as ethylene/propane/LLDPE-B, ethylene/isobutane/LLDPE-B and ethylene/1butene/LLDPE-B at 70°C will be studied and compared to the binary ethylene/LLDPE-B system. Then, the quaternary overall diffusivity of ethylene/propane/1-butene/LLDPE-B and ethylene/isobutane/1-butene/LLDPE-B will be investigated at 70°C and compared to ternary diffusivity. These studies will therefore help to understand the interactions between the different penetrants in order to have a more realistic description of the diffusivity when modeling ethylene polymerization in gas phase. To the best of our knowledge, this is the first time that the diffusivity is investigated for such ternary and quaternary systems.

Fickian diffusion in spherical particles

One way of interpreting the transient portion of the sorption curve is to use Crank's solution to the dynamic form of Fick's law 12 . Indeed, it has been shown that this model predicts well the shape of the experimentally measured degassing curves for compact polyolefin particles (in other words, homogeneous spheres). Crank's solution to Fick's law in spherical coordinates is written as:

𝑀 𝑡 𝑀 ∞ = 1 - 6 𝜋 2 ∑ 1 𝑛 2 ∞ 𝑛=1 𝑒𝑥𝑝 (-𝐷𝑛 2 𝜋 2 𝑡 𝑟 2 )
Where 𝑀 𝑡 is the mass sorbed at a given time t, 𝑀 ∞ is the total mass sorbed at equilibrium, D is the diffusivity of the process gases, r is the average radius of the polymer powders. From an 4 Diffusivity of multicomponent gas mixtures in polyethylene experimental point of view, this means that if we know D, we can use the sorption experiments to calculate the characteristic length scale for diffusion, r. And of course, the reverse method is also true. Bobak et al. 13 and Zubov et al. 14 noticed that applying this solution did not yield to satisfactory results for porous particles. Indeed, they found that Fick's law with a single value of D and of r was not capable of predicting the shape of the degassing curves. This approach could fit either the rapid rise in the mass sorbed, or the slower transition toward steady state, but not both. This was assumed to be due to the fact that the polymer particle is supposed to be an agglomerate of smaller granules having different sizes (so different surface areas). As an approximation of these properties, they proposed a particle model including two sizes of compact polymer granules, one with a small radius and one with a larger radius in order to better fit the experimental sorbed weight. This particle model with two sizes of granules will be used in this study. As we will see below, this is a satisfactory way to account for both the initial quick increase of the sorbed weight caused by the presence of a certain fraction of small granules of radius 𝑟 S with high exchange surface area, as well as the following slower increase of the sorbed weight caused by the big granules of radius 𝑟 B . This is of course not strictly true, as the size of particles is much more distributed and it is likely that the structure of particles in the powder is more complex (and can vary from particle to particle). However, as we will see, the aforementioned authors were correct to assume that a minimum of two "families" of particle substructures is required to be representative of mass transfer in these powders. We will assume for this study that mass transport resistance in the particle pores is negligible (so the pores are assumed to be at the bulk concentration instantaneously). Therefore, the experimental measured sorbed weight relates only to the sorbed mass of penetrants in the polymer phase.

In this case, assuming that small and big spheres have the same diffusivity, Crank's equation for the mass uptake in a heterogeneous particle is defined as follows:

𝑀 𝑡 𝑡𝑜𝑡 𝑀 ∞ 𝑡𝑜𝑡 = 𝑤 1 [1 - 6 𝜋 2 ∑ 1 𝑛 2 ∞ 𝑛=1 𝑒𝑥𝑝 (-𝐷𝑛 2 𝜋 2 𝑡 𝑟 S 2 )] + (1 -𝑤 1 ) [1 - 6 𝜋 2 ∑ 1 𝑛 2 ∞ 𝑛=1 𝑒𝑥𝑝 (-𝐷𝑛 2 𝜋 2 𝑡 𝑟 B 2 )]
Where 𝑟 S and 𝑟 B are the radii of small and big granules, respectively and 𝑤 1 and 𝑤 2 are the weight fractions of small and big granules, respectively. We will discuss below how the representative length scales and weight fractions of the substructures are identified.

Results and discussion

Binary systems

The binary system of ethylene in LLDPE-B of crystallinity of 48.3% was first studied because a good estimate of the pure components diffusivity can be obtained from the literature. The importance of this will become apparent below, but in short, this gives us a reference for the analysis of the sorption experiments. Alves et al. 11 studied the diffusivity of ethylene in LLDPE at 70°C and ethylene pressure varying from 3-15 bar. These experiments could be compared to the present work since ethylene pressures as well as the crystallinity of the polymer powders are close, with a value of 48.6% for Alves et al. 11 . The diffusivity data taken from Alves et al. From this fitting procedure, we can therefore identify the radii of the small and big granules, 𝑟 S and 𝑟 B , as well as the weight fraction of these granules, 𝑤 1 and 𝑤 2 . In order to identify these values, we studied the sorption in four ethylene/LLDPE-B systems with varying pressures, using the data of Alves et al. 11 for the diffusivity. The identified parameters are given in Table 26 and the sorption results are shown in Figure 85. Since the same powder is used for all the sorption experiments in the current paper, the identified parameters will then be used for the ternary and quaternary systems, and the solution to Crank's equation will be used to determine the total diffusivity for each gas mixture. Figure 82 to Figure 85 show that the Crank's diffusion model for polymer particles with two sizes of granules represents the experimental sorption curves correctly for the ethylene/LLDPE-B binary system. Crank's diffusion equation for small particle radius is in good agreement with the first moments of the experimental sorption curves and for big particle radius with the entire time of the reaction. This model has therefore been validated with our experimental data for the ethylene/LLDPE-B binary system at different pressures. Besides, from these simulations, the radii of the small and big granules, 𝑟 S and 𝑟 B , as well as the fractions of these granules, 𝑤 1 and 𝑤 2 were found to be identical for each pressure step of ethylene in LLDPE-B (c.f. Table 26).

Based on this analysis, we will henceforth use 𝑟 S =100 µm and 𝑟 B =210 µm, and 𝑤 1 =0.48 and 𝑤 2 =0.52 in the rest of our analysis, since the same LLDPE-B powder is used for all the following ternary and quaternary systems.

Ternary systems

The ternary systems of ethylene in LLDPE-B with different penetrants such as propane, isobutane and 1-butene were studied in this section. The overall diffusivity of these three systems will be measured for a pressure up to 5 bars and a constant temperature of 70°C. The gas phase composition of the different ternary systems is presented in Table 27. Several observations can be made from these figures. First of all, the usefulness of the particle model with two sizes of granules can clearly be seen from Figure 89-Figure 93; as the pressure increases, the rate of sorption at the beginning of the experiments increases quite quickly with respect to lower pressures. The use of large and small structures allows us to capture this quite precisely. Furthermore, it is important to point out that the same values of 𝑟 S and 𝑟 B are used in all of these fits, giving us confidence that this representation is reasonable. Diffusivity of multicomponent gas mixtures in polyethylene First, ethylene/isobutane/LLDPE-B system is studied at 70°C and total pressures of 2.86 and 5.07 bar. Gas phase composition of ethylene and 1-butene is presented in Table 27. In order to confirm the previous results, we compared the one level and two levels Crank's diffusion model for ethylene/isobutane/LLDPE-B ternary system. Figure 88 compares the sorption curves, as well as the prediction of the mass uptake through Crank's equation using an optimized value for a single value of r, as well as the prediction for two substructure sizes. The value of the used diffusivity in Figure 88, 8.52x10 -11 m 2 s -1 , was identified through the fitting of the particle model with two sizes of granules with the experimental sorption curve, and was used for the fitting of the one size of granules particle model in order to identify the radius r.

The calculated radius for the one level model is r = 150 μm. It can be seen that the particle model with two sizes of granules allows us to better predict the experimental sorption curves. The second studied system is ethylene/1-butene/LLDPE-B at 70°C and total pressures of 2.6 and 3.93 bar. The gas phase composition of ethylene and 1-butene is presented in Table 27. The last studied system is ethylene/propane/LLDPE-B at 70°C and total pressures of 2.56, 3.08 and 5.06 bar. Gas phase composition of ethylene and 1-butene is presented in Table 27. It can be seen that the overall diffusivity of all systems increases with increasing the total pressure of the mixture, as expected. Besides, it has been shown that the overall diffusivity of all ternary systems are higher than the binary system of ethylene/LLDPE.

This can be explain by the co-diffusion effect leading to an increase of ethylene diffusivity in presence of different penetrants, leading therefore to an increase of the overall diffusivity of the system. The overall diffusivity of ethylene/1-butene/LLDPE-B is higher than ethylene/isobutane /LLDPE-B, as expected. The overall diffusivity of ethylene/propane /LLDPE-B appears to be higher than with isobutane and 1-butene. In order to have better explanations about the obtained results, a diffusivity model, capable of taking into account the interactions between the different species in a mixture, is needed.

Quaternary systems

The overall diffusivity of quaternary systems is studied in this section in order to show the effect of the gas phase composition in the system of ethylene/1-butene/ICA/LLDPE with different ICAs. To the best of our knowledge, this is the first time that quaternary overall diffusivity is experimentally measured and theoretically analyzed, especially at polyolefins industrial conditions.

Quaternary systems of ethylene as monomer, 1-butene as comonomer and both isobutane and propane as ICAs in LLDPE-B were studied in this section. The overall diffusivity of these systems will be measured for a pressure up to 6.2 bars at a constant temperature of 70°C, using Crank's diffusion equation for particles with two sizes of granules. Gas phase composition of the studied quaternary systems is given in Table 28. The first studied quaternary system is ethylene/propane/1-butene/LLDPE-B at 70°C and a total pressure up to 6.2 bars. The composition of the gas phase of the studied system is given in Table 28. Two cases will therefore be studied, with a slightly different composition of the gas phase:

- As for ternary systems, the radii of small and big granules as well as their respective fractions that have been identified for ethylene/LLDPE-B binary system (c.f. Table 26), and validated with different ternary systems, will be used for this study as the same PE powder is used. The gas phase composition of both systems are slightly different, with a very small increase of the fractions of ethylene and propane for the Composition 2, with respect to the Composition 1. However, we do not expect that this small differences will have an appreciable impact on the diffusivity estimates. The total diffusivity of these systems is shown in Figure 101.

First, it can be seen for both systems that increasing the total pressure of the mixture leads to faster diffusion through the amorphous phase of the polymer as the time to reach equilibrium decreases. Furthermore, it appears that at high pressures, sorption is almost instantaneous, as can be seen in Figure 96 and Figure 97. In the same manner, it is shown that the diffusivity of big particles is also quick, leading to more difficult fitting of the model to the experimental data, especially at higher pressures, as can be seen in Figure 97 and Figure 100. This could be an indication that the particle model with two sizes of granules is not correct. However, there is no apparent reason that it should work for the ternary systems and not here (as the powders are from the same batch). Furthermore, we noticed that the initial flow rate into the MSB could have an impact on the recorded sorption rate, especially if it was rapid. It is more likely that this is the cause for the poor fit of this portion of the curves.

The fitting of the experimental overall diffusivity of ethylene/propane/1-butene/LLDPE-B quaternary system with Crank's diffusion model for particles with two sizes of granules for the Figure 101 presents the overall diffusivity of Composition 1 and Composition 2 with very little variation of the gas phase composition. A difference of 14% can be seen for both systems, which can be explained by the fact that Composition 2 contains slightly more ethylene and propane compared to Composition 1, which might increase the overall diffusivity of the system. Note also that the experimental errors are higher when measuring diffusivity compared to the solubility since the diffusivity corresponds to the few moments after the injection whereas the solubility corresponds to the equilibrium. As we mentioned above, the way of injecting the gas (i.e. quick or slow) to the reactor might influence these diffusivity results. Nevertheless, these results lie between the values for the ternary systems ethylene/propane/LLPDE-B and ethylene/1-butene/LLDPE-B seen in Figure 94, suggesting that the estimates are reasonable despite the poor fit during the initial 5 or 10 seconds of the experiments. The second studied quaternary system is ethylene/isobutane/1-butene/LLDPE-B at 70°C and a total pressure up to 7 bars. The composition of the gas phase of the studied system is given in Table 28. Two cases will therefore be studied with a slightly different composition of the gas phase:

- The fit of the initial portions of the sorption curves is more satisfactory with this system. This is possibly due to paying particular attention to the feed rate to the MSB and to the fact that we expect that the diffusion process in these experiments to be slower than in the previously shown quaternary experiments. Indeed, at a total pressure of 2.86 bar, the diffusion of small particles lasts during the first 25 seconds compared to the previous quaternary system where the diffusion of small particles was almost instantaneous. Diffusivity of multicomponent gas mixtures in polyethylene

The fitting of the experimental overall diffusivity ethylene/isobutane/1-butene/LLDPE-B quaternary system with Crank's diffusion model for particles with two sizes of granules for and different total pressures. Gas phase composition are given in Tables 3 and4. the diffusivity increases with increasing the total pressure of all systems. Furthermore, it can be seen that adding an ICA or comonomer increases the total diffusivity of the ternary systems with respect to the diffusivity of ethylene alone. The overall diffusivity of ethylene/propane/LLDPE-B appears to be higher than the diffusivity with isobutane and 1butene. The overall diffusivities of both quaternary systems of ethylene/ICA/1-butene/LLDPE-B are lower than their respective ternary diffusivities of ethylene/ICA/LLDPE-B. Indeed, for lower total pressure up to 3 bar, the quaternary and ternary diffusivities are almost the same.

Increasing the total pressure leads to higher slope of increase of diffusivity of ternary systems (i.e. ethylene/ICA/LLDPE-B) compared to quaternary systems (ethylene/ICA/1butene/LLDPE-B). Besides, it might be interesting to see that the quaternary diffusivity of ethylene/isobutane/1-butene/LLDPE-B is very low, and is equal to the binary diffusivity of ethylene in LLDPE-B at a total pressure around 6 bar. This might explain that the diffusivity of small particles when fitting to Crank's diffusion model was very slow for ethylene/isobutane/1butene/LLDPE-B. It has first been shown that the simple Crank's model for porous particle was not capable of predicting correctly the shape of the sorption curves. A particle model based on two sizes of compact polymer granules was used and showed good agreement with the experimental data.

This model could estimate the size as well as the fractions of the small and large compact granules in the polymer particle, giving important information about the polymer particle morphology. The fast initial increase in the sorption curves was explained by the presence of a significant amount of small granules, whereas the following slower increase of the sorbed weight is caused by the presence of a certain amount of big granules.

Besides, it has been presented that the overall diffusivity of the ternary systems (i.e.

ethylene/penetrant/LLDPE) were higher than the binary system of ethylene/LLDPE. This was explain by the co-diffusion effect leading to an increase of ethylene diffusivity in presence of different penetrants, leading therefore to an increase of the overall diffusivity of the system.

Moreover, it has been seen that the overall diffusivity of the ternary systems in presence of propane is higher than in presence of 1-butene which is higher than in presence of isobutane.

This might be explained by the fact that the lighter the penetrant, the easier its diffusion in the amorphous phase of the polymer. More experimental diffusivity studies with other penetrants would be useful in order to better understand the diffusion phenomena that occurs inside the polymer particle.

Finally, the overall diffusivity of the quaternary systems of ethylene/ICA/1-butene/LLDPE with propane and isobutane as ICA were also investigated in this chapter. It has been shown that the overall diffusivity of both quaternary systems is close to their respective ternary systems (i.e.

ethylene/ICA/LLDPE) at the studied temperature and pressure. Besides, it is interesting to see that the quaternary overall diffusivity of ethylene/isobutane/1-butene/LLDPE is very low compared to the overall diffusivity of ethylene/propane/1-butene/LLDPE at 70°C. This is an important information for modeling quaternary systems in gas-phase polyolefin systems. More experimental studies are needed if one needs to better understand the diffusivity of quaternary 4 Diffusivity of multicomponent gas mixtures in polyethylene systems in polyethylene, such as different compositions of the gaseous mixture and different ICAs and comonomers.

The particle model including two sizes of compact polymer granules might be improved by taking into account the changes in the radius of both the big and small granules in function of the reaction time as well as their fraction since distributions of sizes and shapes remain in real porous particles. It is well known that the particle morphology changes rapidly during the course of the polymerization reaction, especially at the beginning of the reaction, where we would expect to see an important change in the distributions of sizes and fractions of big and small granules.

Introduction

A growing polymer particle can be thought of as a dynamically evolving microreactor, which exchanges heat and mass with the continuous phase of the main reactor, and where the polymer is formed. Upon injection into a reactor, the catalyst particle will change rapidly, first by undergoing fragmentation, and then growing by expansion 1,2,3,4 . The rate at which this expansion occurs is of course linked to the rate of production of polymer, and thus to both the intrinsic activity of the used catalyst, as well as to the concentration of reactive species and the local temperature at the active sites; the same is true for the molecular weight 5 . The monomer concentration at an active site will be a trade-off between the rate at which it is consumed at a given point in the particle, and the rate at which it diffuses through the particle. During ethylene polymerization, monomer diffuses from the continuous phase of the reactor, and into the pores of the supported particle, where it sorbs in, then diffuses through the amorphous phase of the polymer surrounding the active sites. Heat is produced when the monomer reacts at the active site, and this heat must be transported out of the particle to avoid overheating. Thus, to describe the polymerization reaction, it is imperative to be able to predict the dynamics of mass and heat transfer in the growing particle.

Most modelling studies now employ a version of the polymer flow model (PFM) [6][7][8] to describe particle growth and estimate the concentration and temperature gradients. These models assume that transport is by Fickian diffusion. The choice of which model to use has been extensively discussed in the literature 2,5,[9][10][11] , so interested readers may refer to these references for further information. In the current work we choose to use the Random Pore Polymer Flow Model (RPPFM) to describe the single particle behavior since it combines a certain amount of geometric simplicity with a mechanistic description of the diffusion process 12 . This model takes into account the transport phenomena through the pore phase and the mass transfer through the semicrystalline polymer phase.

In that sense, Alizadeh et al. 13 extended the work of Kanellopoulos et al. 12 to include an accurate thermodynamic description through the Sanchez-Lacombe EoS as sorption model and Vrentas and Duda 14,15 as diffusion model. They suggested that combined cosolubility and codiffusion effects had a noticeable impact on the mass and heat transfer, as well as on the polymerization rate. They used the SL EoS model to account for the co-solubility effects with n-hexane, but did not try to link the reaction rate to the molecular weight data.

In this chapter, a comprehensive mathematical particle growth model, accounting for mass and heat transfer limitations appearing during the early growth of a Ziegler-Natta catalyst in gas phase olefin polymerization is developed. The RPPFM model will be adapted in order to estimate the overall particle polymerization rate as well as mass balances of the different species (i.e. monomer, ICA, "live" and "dead" polymer chains) and energy balance, in order to predict the temporal-spatial evolution of concentration and temperature profiles in the growing polymer particle and the evolution of molecular weight distribution. However, the accurate determination of both the penetrants concentration at the catalyst active site and the effective penetrant diffusion coefficient is crucial in any particle growth modeling study. This study will therefore show the importance of having an accurate thermodynamic model when developing a particle growth model. The Sanchez-Lacombe EoS for multicomponent systems (i.e. monomer, penetrants), as thermodynamic model, is incorporated in this particle model, in order to calculate the equilibrium concentration of penetrants in the amorphous phase of the polymer.

Vrentas and Duda diffusivity model is applied to calculate the diffusivity of penetrant molecules in the semicrystalline polymer. Finally, a kinetic model for homopolymerization is included to predict the changes in molecular weights in the growing particle as a function of the gas phase composition.

Model Development

2.1.Single Particle Model (SPM) -RPPFM

In the present study, the Random-Pore Polymer Flow Model (RPPFM) 12,16 has been adapted to predict the spatial-temporal evolution of monomer and ICA concentrations, temperature, polymerization rate and polymer molecular weight of gas-phase ethylene polymerization. For the purposes of modelling, the following assumptions have been made:

-The particle is assumed to be spherical; -The particle is considered to be a pseudo-homogeneous medium;

-The active sites are uniformly dispersed throughout the particle;

-The fragmentation process is completed instantaneously; -The crystallinity as well as the porosity of the produced polymer are assumed constant during the reaction;

-The hydrogen concentration in the polymer is very low compared to the other penetrants (i.e. ethylene, n-pentane) and is therefore assumed constant. These assumptions simplify the numerical calculations, but as discussed in the literature 2,5,9-11 , a good number of them are not entirely true. We know that the morphology of the particles evolves rapidly during the first minutes of the polymerization, and that the porosity changes with particle size and time. During the initial times, the crystallinity, critical length scale for diffusion, and perhaps even the number and nature of the active sites also change. Furthermore, it is possible that the relatively high ICA concentrations with respect to polymer during the initial phases of the reaction can retard crystallization, which in turn could have an impact on the diffusion of monomer to the active sites. Moreover, swelling of the amorphous phase by a penetrant might not be uniform in the whole particle, as it can be affected by the tie molecules.

As the main purpose of this paper is to quantify the impact of the gas phase composition (i.e. monomer, ICA, hydrogen) on the reaction rate and molecular weight distributions, we will not attempt to model the start-up of the polymerization. Rather, we choose to fit the kinetic parameters (see below) to the experimental data after 5 minutes or so of the polymerization. This will allow us to calculate the mass and heat transfer in the growing polymer particle for a long time, and let us approximate the impact of the ICA on the rate and molecular weight data reasonably well.

The RPPFM consists of a system of partial differential equations (PDEs) combining material and energy balances over space and time. The full set of material and energy balances with their corresponding initial and boundary conditions is presented in Table 29. In all of the balance equations, t and r represent the time and radial position inside the particle, respectively. 1 represents ethylene, 2 is the ICA. R is the radius of the pseudohomogeneous particle at each moment during its growth.

Table 29. Material and energy balance with initial and boundary conditions of the single particle model

In the material balance, 𝐷 𝑖,ov is the overall diffusion coefficient of penetrant molecules in the polymeric particle, [𝑀 𝑖 ] is the overall concentration of penetrant molecules and [𝑀 𝑖 ] eq is the overall equilibrium concentration of penetrant molecules in the particle calculated from Sanchez-Lacombe EoS.

In the energy balance, T, ΔHp, kc,p, ρov, and Cp,p represent the temperature, the enthalpy of ethylene polymerization, the thermal conductivity of the polymer phase, the overall particle density calculated from 𝜌 ov = 𝜌 p (1 -𝜀) where 𝜌 p is the density of the polymer phase, and the heat capacity of the polymer phase, respectively. In the boundary condition, h, and Tb are the heat transfer coefficient and the bulk temperature of the reactor, respectively. The use of the Ranz-Marshall correlation for the calculation of the external film heat transfer coefficient has been critically examined, as it can result in an overestimation of the particle temperature 12 . Nevertheless, we will use it here as the polymerization rates of the experiments serving as a basis for the estimation of the model parameters are moderate so no large temperature excursions are expected. In addition, it is worth underlining once again that the purpose of this chapter is to provide a quantitative analysis of the impact of the ICA on gas phase polymerization, not to do an "exact fit" model of a specific system.

As has been shown in Table 29, the present particle model consists of a system of non-linear partial differential equations as well as a number of algebraic equations like the polymerization rate, the external mass and heat transfer and the diffusivity. The partial differential equations, which are the energy balance equation, the material balance equations and the particle growth equation, and their boundary and initial conditions must be solved simultaneously. This will lead to a prediction of the concentration and the temperature profiles within the growing particle during ethylene polymerization in gas phase. However, in order to do that, one must have an accurate description of both the equilibrium concentration of the different penetrants as well as the diffusion of these penetrants in the amorphous phase of the polymer.

2.2.Thermodynamics 2.2.1. Sanchez-Lacombe equation of state

A number of papers have shown that Henry's law is not applicable to polymer-penetrant systems involving heavy penetrants vapors and high pressures 17,18 . It is therefore essential to use a more sophisticated thermodynamic model in order to be able to have a realistic thermodynamic description of the system, by taking into account the interactions between the different penetrants and the polymer. In the present study, the Sanchez-Lacombe equation of state (SL EoS) is employed to calculate the equilibrium concentration of the different species (i.e. monomer, penetrants) in the amorphous phase of the polymer. SL EoS is a lattice-fluid model in which pure components are assumed to be broken into parts and placed into a lattice structure.

The SL EoS introduces the concept of vacant lattice sites or holes to account for the compressibility and the density changes. Thus, the system volume or density can vary by changing the fraction of holes in the lattice structure 19 . The main equation of state is given by The development of SL EoS has been described in more details in Chapter 2 and in other papers, so will not be described here. For more details, the reader is referred to references 21,22 .

𝜌̅ 2 + 𝑃 ̅ + 𝑇 ̅ [𝑙𝑛(1 -𝜌̅ ) + (1 - 1
The only adjustable parameters from SL EoS are the interaction parameters between the different species in the mixture, kij. We believe that the interaction parameters between small molecules are equal to zero due to low chemical affinity between like molecules, which leaves us the interaction parameters between vapor species and the polymer to be identified. (1) represents ethylene, (2) the ICA and (3) the polymer, meaning that k13 and k23 are the only adjustable parameters for ternary systems. These parameters are identified by fitting SL EoS to experimental solubility data from Yao et al. 23 . They measured experimentally the solubility of ethylene/n-pentane in semicrystalline PE of crystallinity of 48.6% at temperatures between 70-90°C and total pressure of 20 bar with n-pentane partial pressure of 0.8-1.8 bar. These identified kij can therefore be used in the present particle model since the produced PE has a crystallinity of 45% and n-pentane partial pressure are between 1-2 bar which are close to the experimental conditions of Yao et al. 23 . kij used in the different simulations are given in Table 31. The equilibrium concentration of the different species in the polymer calculated from SL EoS for ternary systems have been incorporated in the RPPFM model as boundary conditions of the material balances. The implementation of an accurate sorption model will allow a better description of the thermodynamics during ethylene polymerization in gas-phase, especially since a large number of papers have shown that the gas phase composition has a great impact on the reaction rate, as well as on the concentration and temperature gradients inside a growing polymer particle 13,25,26 .

Since the solubility of hydrogen in the amorphous phase of the polymer is quite low compared to that of ethylene or n-pentane, the presence of hydrogen is assumed to have no effect on the solubility or diffusivity of ethylene in the particle model (and vice versa). The concentration of hydrogen, calculated with Henry's law, is therefore assumed to be constant in the whole particle over time, and not influencing the concentration/diffusion of ethylene 27 .

[𝐻 2 ] = 𝑘 He,H 2 𝑃 H 2 𝜌 p 𝑀𝑊 H 2
(1) with 𝑘 He,H 2 = 8.2533 × 10 -5 g Pa -1 g -1 amorphous polymer.

Vrentas and Duda diffusion model

The knowledge of the overall diffusion coefficient is also important in calculating the material balances of the RPPFM. However, the Sanchez-Lacombe EoS only gives equilibrium concentrations, while the system is not at equilibrium due to the high reaction rate. The diffusion of species is considered to take place from the bulk phase to the catalyst active sites through the pores and the amorphous polymer that make up the pseudo-homogeneous medium supposed by the RPPFM. This phenomena has been described in the random-pore model proposed by Wakao and Smith 28 and has then been applied by Kanellopoulos et al. [START_REF]Polyolefin Reaction Engineering[END_REF][START_REF] Mckenna | Condensed Mode Cooling of Ethylene Polymerization in Fluidized Bed Reactors[END_REF] and Yiagolopoulos et al. 27 in order to show its applicability for olefin polymerization. The effective (or overall) diffusion coefficient depends not only on the temperature and concentration of the penetrants in the amorphous phase of the polymer, but also on the particle porosity. Based on these works, the overall diffusion coefficient of species (i) can be expressed as follows:

𝐷 𝑖,ov = 𝜀 𝜏 f 2 𝐷 𝑖,g + (1 -𝜀)(1 + 3𝜀)𝐷 𝑖,pol
in which τf denotes the tortuosity factor and is introduced to take into account the tortuous nature of the pore phase present in the particle. The value of τf is assumed to be equal to 10, as this value has been approximated elsewhere 13,16 . The first term on the right-hand side of this equation represents the transport of the penetrants through the pores while the second term accounts for mass transfer through the semicrystalline polymer phase. The diffusion in the porous polymer particle is calculated from Chapman-Enskog correlation, and details on calculating this can be found elsewhere 29 . 𝐷 𝑖,pol = 𝐷 𝑖,am ter /(𝜏𝛽 𝑖 ) , with 𝛽 𝑖 the chain immobilization factor accounting for the reduction in amorphous chain segmental mobility due to the proximity of crystallites and 𝜏 denotes the tortuosity factor corresponding to the magnitude of extension in the diffusion path associated with bypassing impermeable crystallites by the penetrant 13 .

The diffusion of small molecules in the semicrystalline polymer has been studied extensively and has shown that the free volume theory refined by Vrentas and Duda 14,30,31 correlates accurately the diffusion coefficient at various temperatures and over a large concentration range. According to the free-volume theory 15 , the diffusion coefficient of ethylene (1) and an ICA (2) in the amorphous PE (3) in the ternary system is given by: 𝐷 in which D0i is the pre-exponential term for component (i), 𝜔 𝑖 is the weight fraction of component (i) in the amorphous polymer phase calculated from SL EoS, 𝑉 ̂𝑖 * is the specific hole free volume of component (i) required for a diffusion jump, K1i and K2i are the free-volume parameters for component (i), Tgi is the glass transition temperature of species (i) and γ is the overlap factor introduced because the same free volume is available for more than one molecule.

Table 32 shows the free-volume theory parameters for the current ethylene/n-pentane/LLDPE studied system. The 𝜉 parameter is the ratio of polymer and solvent molar jumping units 32 , defined as follows:

𝜉 𝑖3 = 𝑉 ̅ 𝑖 0 (0) 𝑉 ̅ 3𝑗 * = MW 𝑖 * 𝑉 𝑖 * V ̅ 3𝑗 *
Where 𝑉 ̅ 𝑖 0 (0)is the penetrant molar volume at 0 K (m 3 mol -1 ), 𝑉 ̅ 3𝑗 * is the critical molar volume of the polymer jumping unit (j) (m 3 mol -1 ) and 𝑀𝑊 𝑖 is the molecular weight of penetrant (i).

The detailed calculation of the parameter 𝜉 is proposed in references 20,30 . It is well accepted that Ziegler-Natta (ZN) catalysts are multi-site catalysts; in other words, each catalyst particle will contain a certain number of "families" of active sites. Each family will propagate at different rates, insert comonomer at different rates and eventually respond to hydrogen differently. One can use MWD deconvolution to identify the minimum number of (families) of active sites needed to describe the behavior of a catalyst system. We used the method described by Soares and McKenna [START_REF]Polyolefin Reaction Engineering[END_REF] to deconvolute the molecular weight distributions of a series of ethylene polymerizations in the presence of n-pentane and hydrogen over a commercial ZN catalyst. The used experimental conditions are given in Table 33 and the detailed experimental set-up is given in Appendix B. Where 𝐶 * is the active site concentration, 𝑀𝑊 Ti is the molecular weight of titanium, 𝜌 cat is the density of the catalyst and 𝑓 cat is the efficiency factor of the catalyst. 𝐶 max * , the maximum concentration of active site for the TiCl4 supported on MgCl2 Ziegler-Natta catalyst, was calculated using the amount of Titanium in the catalyst, 𝑤 Ti . Since the same catalyst was used by Andrade 35 , 𝑤 Ti was taken equal to 2.8 wt%.

Kinetic Parameters

Following from the deconvolution analysis, a four-site kinetic model is employed in the present study to describe ethylene polymerization in multicomponent system (i.e. ethylene, n-pentane) over a Ziegler-Natta catalyst. The kinetic mechanism shown in Table 35 comprises a classic series of elementary reactions including initiation and propagation reactions, transfer to monomer, molecular weight control reactions (i.e. initiation of 𝑃 𝐻 𝑘 , transfer to 𝐻 2 ), and a first order spontaneous deactivation reaction. We assume that site activation is instantaneous.

As a result, the particle model combined with the kinetic model can be used to predict not only the polymerization rate, but also the evolution of the temperature and concentration gradients, and the molecular weight distribution as a function of the gas phase composition during ethylene polymerization in gas phase. Based on the proposed kinetic scheme in Table 35 and the definition of the moments of "live" (𝜇 𝑖 𝑘 ) and "dead" (𝜂 𝑖 𝑘 ) total number chain length distributions, we obtain the moment equations summarized in Erreur ! Source du renvoi introuvable.. Where 𝑁 S is the number of catalyst sites and 𝑘 is the catalyst site type.

The volumetric rate of ethylene polymerization, 𝑅 v , is calculated by:

𝑅 v = 𝑅 p ( 1 -𝜖 𝜑 3 )
And φ is the overall growth factor defined as:

𝜑 = 𝑅 pol 𝑟 cat
where 𝑅 pol is the equivalent radius particle only, excluding the volume of pore phase, at each time step during the reaction and 𝑟 cat is the radius of initial catalyst particle.

All the rate constants are described by the Arrhenius law as follow:

𝑘 = 𝑘 0 exp (- 𝐸 𝑅 𝑇 )
in which 𝑇, 𝐸 and 𝑅 are the temperature, the activation energy and the universal gas constant, respectively.

The cumulative number average molecular weight of the polymer:

𝑀 ̅ n (𝑟, 𝑡) = ∑ (𝜇 1 𝑘 + 𝜂 1 𝑘 ) 𝑁 S 𝑘=1 ∑ (𝜇 0 𝑘 + 𝜂 0 𝑘 ) 𝑁 S 𝑘=1 MW 1
The cumulative weight average molecular weight of the polymer:

𝑀 ̅ w (𝑟, 𝑡) = ∑ (𝜇 2 𝑘 + 𝜂 2 𝑘 ) 𝑁 S 𝑘=1 ∑ (𝜇 1 𝑘 + 𝜂 1 𝑘 ) 𝑁 S 𝑘=1 MW 1
This method of moments gives the cumulative weight and number average molecular weight as well as the individual ones for each active site.

The MWD was reconstructed from these average values assuming that each catalyst site type produces a polymer where the chain lengths follow a lognormal distribution: Some adjustment of 𝑘 p0 and 𝑓 cat were needed when fitting to experiments with hydrogen.

These two parameters have the same impact on the reaction rate, and so they might not be identifiable based on the measurement of 𝑅 p alone. However, they do not have the same impact on the molecular weight (as 𝑓 cat does not influence the polymer molecular weight), therefore using both measurements may improve their identification. The adjusted value of 𝑓 cat was taken equal to 0.9.

Results and discussion

The development of a single particle model associated with a kinetic model was employed to analyze the change in the gas phase composition (i.e. n-pentane, hydrogen) on the evolution of the concentration profiles, the reaction rate, the growth and overheating of the polymer particle, as well as the molecular weight distribution. This model was validated with the previous experimental data on reaction rates and molecular weight distributions (c.f. Figure 111, Figure 113 and Figure 113). The temperature and ethylene partial pressure were kept constant (i.e. 7 bar of ethylene and 80°C, respectively) and only the partial pressures of n-pentane and hydrogen are changing between each experiment 𝑅 p𝑖 .

3.1.Effect of thermodynamic and diffusion models

To the best of our knowledge, with the exception of a recent paper from Alizadeh et al. 13 , virtually all single particle modelling studies appear to rely on binary solubility and diffusion models to describe ternary systems. However, it has been shown that the interactions between the two penetrants in ternary systems are crucial and lead to important changes when describing ethylene polymerization in gas phase. Adding ICA to the gas phase composition improves the heat capacity of the gas phase but also increases ethylene diffusion through the polymer phase which will lead to higher concentration of ethylene in the amorphous phase of the polymer. The instantaneous rate of ethylene polymerization is therefore higher in presence of ICA, as can be seen in Figure 114. This was partly attributed to the co-solubility effect and to co-diffusion effects. Adding ICA to the gas phase composition leads to a co-solubility effect of ICA on ethylene and an anti-solvent effect of ethylene on ICA and to an enhanced diffusivity of ethylene. Accurate sorption and diffusion models are therefore important tools when modeling a single particle model in order to better analyze the influence of the gas phase composition on the polymerization rate.

Figure 114 compares the reaction rate as well as ethylene concentration gradients at 7 bar of ethylene, 1 bar of hydrogen and 1 (i.e. Rp8) and 2 (i.e. Rp9) bar of n-pentane. The binary model corresponds to the use of binary diffusivity and concentration of ethylene and n-pentane using SL EoS for binary systems, without taking into account the interactions between these two penetrants. The ternary corresponds to the use of ternary diffusivity and concentration of the gas phase mixture in the polymer, estimated by ternary Sanchez Lacombe EoS It can be seen that using the binary thermodynamic model leads to an underestimation of the concentration of ethylene, leading to an underestimation of the polymerization rate. In the binary model, changing the concentration of ICA may only affect the temperature gradients inside the particle.

However, the thermal effect was found to be negligible in this case, so no effect of the ICA is observed on the reaction rate when using the binary model. Note that the equilibrium concentration (i.e. the concentration of ethylene at the surface of the particle), is higher when ICA pressure is higher due to the co-solubility effect estimated by SL EoS. It is shown that the concentration of ethylene at the surface of the particle is about 121.92 mol m -3 with 1 bar of C5 (i.e. Rp8) and 126.96 mol m -3 with 2 bar of C5 (i.e. Rp9); so about 3.97% increase. It can be seen that higher amount of ICA leads to higher concentration of ethylene at the center of the particle, with an increase of 17% when adding 1 bar of n-pentane, and of 31% when 2 bars of n-pentane are added. This difference does not explain alone the observed effect of ICA on Rp which increases from about 3.23 to 3.65 kg g -1 h -1 from 1 to 2 bar of ICA, at 10 minutes of the reaction; so about 13% increase. Therefore, the co-diffusion effects are playing an important role too.

3.2.Effect of particle porosity

The overall diffusion coefficient of a given system depends on the temperature and concentration of the different gases in the amorphous phase of the polymer, but also on the particle porosity. However, the porosity of the produced polymer has not been measured Rp8 -Ternary Rp8 -Binary Rp9 -Ternary Rp9 -Binary 5 A single particle model to predict the impact of induced condensing agents on polymerizing particles experimentally. The effect of particle porosity on ethylene diffusivity in ethylene/npentane/LLDPE ternary system calculated with the parameters given in The diffusion coefficient of ethylene increases with n-pentane partial pressure for a particle having a low degree of porosity (up to 𝜀=0.5). It is shown that for a highly compact particle, increasing n-pentane partial pressure increases ethylene diffusion in the amorphous phase of the polymer, due an increase in the free volume available to of the amorphous phase of the polymer and therefore higher diffusion coefficient of ethylene.

The observed effect of the particle porosity on ethylene diffusivity, especially at higher npentane partial pressure, will have an impact on ethylene concentration gradients inside the growing polymer particle. Figure 116 shows ethylene concentration as a function of the normalized particle radius for different polymer porosities and ICA pressures, 1 (i.e. Rp8) and 2 bar (i.e. Rp9) of n-pentane. It is shown that increasing the polymer porosity has a noticeable effect on ethylene concentration gradient inside the particle. Furthermore, increasing n-pentane partial pressure from 1 bar to 2 bar increases ethylene concentration gradient for all particle porosities because of the co-diffusion effect. However, the co-diffusion effect is stronger for a particle having a lower degree of porosity (as co-diffusion concerns diffusion into the amorphous polymer, not in the pores). As one would expect, faster diffusion corresponds to less significant gradients.

Figure 116. Effect of polymer porosity on ethylene concentration gradient at 40 minutes of the reaction as a function of the normalized particle radius for 7 bar of ethylene, 1 bar of hydrogen and 1 (Rp8) and 2 bar (Rp9) of n-pentane

The porosity of the polymer was chosen 𝜀=0.05 as we do not have this information about the used polymer, and all the next simulations will use this value of porosity.

3.3.Effect of the gas phase composition on ethylene polymerization 3.3.1. Effect of the gas phase composition on the polymerization rate

As have been showed in the previous sections, adding ICA will lead to co-solubility and codiffusion effects. Figure 117 shows the simulated polymerization rate with respect to the polymerization time, for different amounts of hydrogen and n-pentane; 0-2 bar for n-pentane and 0-3 bar for hydrogen. We can see that the difference between the polymerization rate in the absence of n-pentane (binary) and with 1 bar of n-pentane is higher than the difference between 1 and 2 bar of n-pentane, for all hydrogen pressures, due to the co-solubility and co-diffusion effects in ternary systems. function of the amount of ICA injected into the reactor during ethylene polymerization in gas phase. The overall equilibrium concentration of ethylene at the surface increases as we increase ICA partial pressure. This is due to co-solubility effect; heavier alkanes increase the solubility of lighter ones. Also, increasing ICA partial pressure leads to less concentration gradients (between the particle surface and its center) which is due to co-diffusion effects; heavier alkanes increase the overall diffusivity of lighter ones through the growing particle.

The effect of hydrogen partial pressure can also be seen in Figure 118. It is shown that increasing the hydrogen partial pressure decreases the gradient of concentration of ethylene inside the growing polymer particle, with an increase of almost 66% of ethylene concentration at the center of the particle from 0 to 3 bar of hydrogen with 1 bar of n-pentane. This can be explained by the lower reaction rate when adding more hydrogen.

Effect of the gas phase composition on the temperature gradient

The difference between the temperature of the particle surface Ts and the bulk temperature Tb represents the overheating of the particle. Figure 119 shows the overheating of the polymer particle during reaction time at the same conditions as previous simulations. We can see that the particle temperature rises significantly at the beginning of the reaction and then declines at 20 minutes to a steady-state value. This is expected as the reaction rate is higher at the beginning, and the surface area for heat exchange is the lowest. These temperature gradients were calculated using the Ranz-Marshall correlation, so it is possible that they are larger than might be the case in a fluidized bed reactor. Nevertheless, the relative changes between the different cases will be similar regardless of the correlation chosen. Regarding the effect of ICA, we can see that adding ICA leads to a slight increase of the boundary layer temperature gradient at the beginning of the polymerization, which is due to the higher monomer concentration and reaction rate. With the kinetic parameters chosen here, the increased heat capacity of the gas phase is outweighed by the increase in rate due to the addition of the ICA. On the other hand, as shown in Figure 111, Figure 112 and Figure 118, increasing the hydrogen partial pressure decreases the reaction rate which leads to a lower reaction rate.

5 A single particle model to predict the impact of induced condensing agents on polymerizing particles

Effect of the gas phase composition on the molecular weight distribution

Figure 120 and Figure 121 show the effect of hydrogen and n-pentane partial pressure, respectively, on the molecular weight distribution. Hydrogen is the most important factor for controlling the MWD, because chain transfer to hydrogen is the main reaction producing dead polymer, even though the temperature as well as ethylene concentration may impact importantly the MWD. Increasing the hydrogen concentration leads to lower MWD. However, increasing ICA partial pressure will lead to higher MWD since it will increase the concentration of ethylene in the amorphous phase of the polymer, producing longer chains. The effect of ICA partial pressure is nevertheless less significant than the effect of hydrogen partial pressure on the MWD, as can be seen in Figure 121.

3.4.Effect of ICA type

As has been shown in previous papers [38][START_REF] Fernandes | Fluidized bed reactor for polyethylene production. The influence of polyethylene prepolymerization[END_REF][START_REF] Mirzaei | Fluidized Bed Polyethylene Reactor Modeling in Condensed Mode Operation[END_REF] , different ICA can influence the polymerization rate in different ways due to different co-solubility effects. One could therefore suspect that changing the ICA will lead to changes in the internal concentration gradients and MWD as well.

To test this idea, simulations were run with the same kinetic parameters defined above, but with different parameters for the thermodynamic model (i.e. solubility and diffusivity). The SL parameters are given in Table 30 and Table 31 and the Vrentas and Duda parameters for the diffusivity are given in Table 32 for both n-pentane and n-hexane at 80°C. The interaction parameters for the ethylene/n-hexane/HDPE system were taken from Alizadeh et al. 13 . Note that the crystallinity of the polymer made in the presence of n-pentane in this work is 45%, whereas that of Alizadeh et al. was 60%. For this reason the simulation results are shown as normalized polymerization rates, normalized ethylene concentration gradient and normalized diffusivity to be able to quantify the change from the dry mode (pure ethylene) to the condensed mode in presence of n-hexane and n-pentane as ICAs. 1.4 for n-pentane compared to 1.62 for n-hexane. This trend was validated with Alizadeh et al. 38 who compared the polymerization rate in presence of 1 bar of n-pentane with 0.8 bar of nhexane. They found out that the increase in the polymerization rate in presence of 1 bar of npentane was about 1.5 and in presence of n-hexane about 1.85. It is important to note that they found a higher increase of the polymerization rate in presence of n-hexane because their crystallinity is lower than the one used in the present study. It is also interesting to note that the slopes of the curves are different, showing that there are also diffusion effects involved, and that the higher diffusion coefficients for n-pentane and for n-hexane lead to faster rates at the beginning of the polymerization. This is due to the co-solubility and co-diffusion effects that are presented in Figure 123 and shows the diffusivity of ethylene in the ternary system of ethylene/ICA/PE normalized by the diffusivity of ethylene in the binary system as a function of ICA partial pressure for 1 bar of n-pentane and 1 bar of n-hexane as ICA at 80°C. It is shown that adding 1 bar of n-hexane leads to a higher increase of ethylene diffusivity with respect to 1 bar of npentane. This is explained by the co-diffusion effect, where adding ICA increases ethylene diffusivity and the heavier the ICA is, the higher the diffusivity will be. calculated from SL EoS is slightly higher in presence of n-hexane than in presence of n-pentane.

The difference is expected to be higher, but the crystallinity of LLDPE in presence of n-hexane is about 60% compared to 45% in presence of n-pentane. However, it demonstrates that the concentration of ethylene at the surface is function of ethylene pressure but also on the type and amount of ICA used during the polymerization reaction. Besides, it is shown that adding ICA leads to higher concentration gradient of ethylene inside polymer particle due to co-solubility effect of ICA increasing ethylene concentration in the amorphous phase of the polymer.

Furthermore, heavier ICA lead to higher concentration gradient of ethylene due to the codiffusion effect of ICA increasing the diffusivity of ethylene through the growing polymer particle. The effect of the gas phase composition has been evaluated through the polymerization rate, ethylene concentration gradients inside the polymer particle, the growth and overheating of the polymer particle, and finally the molecular weight distribution. In all the results, the higher the partial pressure of hydrogen is, the lower is the productivity and the molecular weight of the polymer, whereas increasing n-pentane partial pressure increases the productivity and the polymer molecular weight.

Introduction

As have been discussed in the previous chapters, the sorption of pure gases in polyolefins has been widely studied. However, it turns out that very few studies have been performed

concerning the sorption of liquid diluents in polyethylene. This is crucial thermodynamic data for slurry polymerization, where the growing polymer particles are dispersed in a liquid diluent in which the monomer is dissolved. Obviously, the swelling of the polymer by the diluent, and the eventual dissolution of the polymer in the continuous phase are both going to be important points when developing an in-depth understanding of a slurry process. The temperature sensitivity of these two phenomena will also be important, and it is expected that slurries will exhibit a different temperature response than gas phase systems.

As with gas phase reactions, increasing temperature can lead to higher catalyst productivity, but unlike in the gas phase, this also enhances the risk of polymer swelling since the latter tends to become more soluble in the continuous phase at higher temperatures 1 . If the temperature reaches a critical point, polymer particles might swell significantly and this in turn can lead to the production of a more viscous or adhesive polymer slurry that can slow down recirculation in the reactor or lead to deposition on the reactor wall 2,3 . As an example, Zhou et al. 4 calculated the swelling of dodecane-PE using mass transfer rate measurements and developed a swellingbased fluid dynamic model in order to quantify the aggregation behavior of the swollen particles, and found out that solid accumulation appears at the bottom of the slurry loop reactors.

Reactor fouling by polymer adhesion is often linked to the swelling of the amorphous phase of the polymer due to the sorption of diluents like isobutane, or n-hexane.

Knowledge of this thermodynamic data is crucial if one needs to control the molecular weight of the produced polymer or to optimize the degassing process. Besides, this is all the more important since these liquid diluents will change the physical properties of the produced polymer as they cause its swelling and softening due to their high solubility in the amorphous phase of the polymer. Up to now, only a limited number of studies of phase equilibria for PE/liquid diluent systems can be found in the literature [5][6][7][8][9][10][11] . Recently, Krajakova et al. 10 studied the sorption of liquid diluents (i.e. n-octane, n-hexane, n-pentane) in different PE samples of varying density from 902 to 967 kg m -3 . They showed that longer hydrocarbons diluents have higher solubility in all PE samples. They also showed a partial dissolution of PE samples in the liquid diluents using GPC analysis, but this dissolution was not quantified.

This chapter is built upon these previous studies in order to quantify the solubility of diluents in different PE grades, the swelling of the PE in presence of different diluents, and the dissolution of PE in these diluents. The sorption and swelling of the amorphous phase of the polymer are investigated in presence of different liquid diluents (i.e. n-pentane to n-decane) at temperature between 25-100°C and various PE grades with densities from 925-960 kg.m -3 .

Materials and Methods

Part of the experiments described in this Chapter were carried out by an intern in the C2PM, M. Léo Mermet.

2.1.Materials

Normal alkanes with carbon numbers 4 through 10 with a minimum purity of 99% were used as received from Air Liquide France. All seven alkanes were used for the swelling experiments, but only the n-pentane and heavier alkanes were used in the solubility experiments.

HDPE granules and LLDPE powders were graciously supplied by Ineos (HDPE-I, LLDPE-I), and HDPE and MDPE powders were graciously supplied by Borealis (HDPE-B, MDPE-B).

The crystallinity of the polymer samples was measured using differential scanning calorimetry (DSC 3+ by Mettler Toledo). Polyethylene samples were weighed (i.e. around 5-7 mg) and placed in an aluminium capsule with a volume of 40 μL. The sample was first cooled to -20°C

and then heated from -20 to 180 °C at a heating rate of 10°C min -1 , held for 5 minutes at that temperature and then cooled from 180 to -20 °C at a rate of 10°C min -1 . This temperature was maintained for 5 minutes, and the sample was reheated to 180°C at a rate of 10°C min -1 . The melting behavior of the samples was studied in the second heating cycle in order to detect the melting point and the melting enthalpy of the samples. The ratio of the heat of fusion of the sample to that of the completely crystallized polyethylene was used to calculate the degree of crystallization. A heat of fusion of 293 J g -1 for completely crystallized polyethylene was used for the calculation. The STARe software was used for data acquisition and data analysis.

Polymer densities and crystallinities are summarized in Table 38.

LLDPE-I, HDPE-B and MDPE-B were hot-pressed at 180°C for about 2 hours, and then cooled, in order to produce a thin film. The final cylindrical film has a diameter of 20 mm and a thickness of 1 mm. The PE solubility in alkanes is measured as following: 5g of polymer is added to 15 mL of diluent in a test tube. This is the same ratio of polymer to solvent as was used in the reactor for the swelling experiments. A magnetic stirrer bar is inserted in the tube. The tube is then closed and kept at the desired temperature for 48 hours. After 48 hours, we would expect a partial dissolution of the polymer in the solvent, and therefore a supernatant solution of solvent and polymer. This solution is then withdrawn with a syringe and weighted. The solution is left to dry under the hood for 24 hours and then in an oven at 100°C for an additional 24 hours and weighted again, as can be seen in Figure 125. Each experiment was repeated 2-3 times.

The amount of dissolved polymer (g of polymer dissolved / g of solvent) is calculated as follows: All the experiments were carried out three times at the same conditions (identical temperature, PE granules) and were found to be quite repeatable. The solubility S is calculated as for solubility experiments at room temperature:

𝑆 PE diss = 𝑚 4 -𝑚 1 𝑚 2 -𝑚 4
𝑆 = 𝑚(𝑡 = 0) -𝑚 PE end 𝑚 PE end
Where 𝑚(𝑡 = 0) is the mass of the swollen polymer measured at zero time before the desorption and 𝑚 PE end is the mass of the dry polymer after the desorption experiment. At high temperatures, the dry polymer mass after the experiments is very different from the mass of the polymer before the desorption experiment since the dissolution of the polymer in the solvents is greater with temperature. It is therefore crucial to take into account PE dissolution in the solvents when measuring the solubility at high temperatures. The experimental temperature dependent swelling is then calculated in % of volume change, as follows:

𝑆𝑊 exp (%) = ℎ(𝑇) -h(𝑇 𝑖 ) h(𝑇 𝑖 ) 100
Where ℎ(𝑇) is the height of polymer powder inside the reactor at different temperature steps taken from the sapphire window and h(𝑇 𝑖 ) is the height of polymer powders inside the reactor at the beginning of the experiment at room temperature taken from the sapphire window.

However, the swelling of the polymer has to be corrected for the partial dissolution of the amorphous phase of the polymer in the liquid solvents, especially for light solvents (i.e. n-

100

It is important to note that the maximum temperature in each experiment varies as a function of the solvents. For heavier solvents, temperature steps were performed up to 100°C for the swelling experiments. However, for some of the lighter solvents (i.e. n-butane, n-pentane, nhexane) which can strongly plasticize the polymer at lower temperatures than heavier solvents, and can cause the particles to lose their integrity, the maximum temperature achieved was around 80°C. Figure 129 shows an example of plasticization of MDPE powders in presence of n-hexane at temperature of 90°C. When this occurred, the swelling experiments were stopped. These experimental results have shown that the partial dissolution of the amorphous phase of the polymer is not negligible, and can reach almost 20% of dissolution of the polymer, with respect to the mass of solvent used during swelling experiments. This data is therefore crucial in order to correct the experimental sorption of liquid diluents in polyethylene and the swelling of the latter when exposed to these diluents.

3.2.Alkane sorption in polymers -Room Temperature Experiments

The solubility of n-pentane, n-hexane, n-heptane, n-octane, n-nonane and n-decane in PE granules (i.e. HDPE-I) and films (i.e. HDPE-B, MDPE-B, LLDPE-I) was first measured at room temperature. The desorption time was constant for each granule and film in order to have reproducible results. Each desorption experiment was repeated at least 3 times under the same conditions in order to show the repeatability of these experiments. Figure 133 and Figure 134 show good repeatability with a difference at time zero of the desorption curve of 1.2% between each sorption experiment in presence of n-hexane in HDPE-I granules, and a maximum difference of 5% between each sorption experiment in presence of n-pentane in HDPE-I granules, respectively. The solubility of each alkane presented in Table 6 was taken as the average of all the experiments repeated at the same conditions. Furthermore, since n-butane is in a gaseous state at room temperature, it is hard to evaluate the solubility of this diluent. In order to correct for the solubility of polymer in n-butane during the swelling experiments, we approximated the solubility by extrapolating the results in Figure 138 to a chain length of 4. While inexact, we can most likely use these values as an approximation. Figure 139 shows the solubility of n-pentane, n-hexane, n-heptane and n-octane in different polyethylene samples (i.e. HDPE, MDPE and LLDPE). We can see that the solubility of all solvents in polyethylene at room temperature decreases with increasing the polymer density and crystallinity. This trend is expected since these solvents can only penetrate into the amorphous phase of the semicrystalline polymer, and increasing polymer density leads to a decrease of the amorphous phase fraction. Besides, it is shown that the solubility of the different solvents at room temperature can be considered constant for polyethylene densities up to 935 kg m -3 . This trend was confirmed by Krajakova et al. 11 who studied the solubility of n-hexane in different polyethylene samples of densities from 900 to 970 kg m -3 . They explained this trend by the new morphological model of free and constrained amorphous phase in polyethylene introduced by Chmelar et al. 12 . They assumed that polyethylene consists of the crystalline phase (rigid), constrained amorphous phase inside the lamellar stacks (semirigid) and the free amorphous phase outside the stacks (mobile). The free amorphous phase, which is not present for PE having crystallinities above 62 wt.%, corresponding to a density of 940 kg m -3 (i.e. HDPE), can sorb more penetrants than the constrained amorphous phase which is affected by the surrounding crystalline lamellar stacks. Knowing that the liquid diluents can only sorb in the amorphous phase of the polymer, we can relate the solubility to the mass of the polymer amorphous phase as follows:

𝑆 am = 𝑆 (1 -𝜔 c )
Where 𝜔 c is the crystallinity of the different polymers given in Table 1.

The solubility of the different solvents with respect to the amorphous phase of the polymer is therefore presented in solvents (i.e. n-pentane, n-hexane, n-heptane) is bigger at higher temperature than with heavier solvents. However, this significant increase of the solubility of light solvents at high temperature can cause the polymer to swell considerably, leading to reactor fouling by polymer adhesion and plasticization.

In the same manner as Figure 138, the solubility of liquid n-butane in function of temperature up to 60°C can be extrapolated from Figure 20 since the solubilites for n-pentane, n-hexane and n-heptane follow the same trend. Indeed, the solubility of liquid n-butane extrapolated with the experiments at room temperatures and the ones with TGA method are in good agreements, with a difference of 10% between both methods. Once again these values have to be taken with caution. Swelling of HDPE-B in presence of liquid solvents such as n-decane, n-nonane, n-octane, nheptane, n-hexane, n-pentane and n-butane expressed in volume percent units at different temperatures will be presented in this section. Figure 145 shows that the swelling of HDPE-B in all solvents is almost the same at temperature below 70°C, except for nC10. Then, the swelling increases considerably between 70 and 100°C, which corresponds to the breakthrough curve. Indeed, the difference of the degree of swelling of the polymer in presence of the different solvents can be seen for temperatures above 70°C.

The breakthrough curve corresponds to a temperature where solvents start increasing the swelling drastically. The swelling of the amorphous phase of the polymer can therefore increase in 90 minutes from 15 %vol to 57 %vol for a temperature increase of 10°C for the lighter solvent (i.e. n-pentane). This significant increase of swelling is very important to control in an industrial reactor, and this data, especially for butane and hexane, is therefore crucial for reactor model development.

The swelling of LLDPE-I in presence of liquid solvents such as n-decane, n-octane, n-heptane, n-hexane and n-pentane expressed in volume percent units at different temperatures will be presented in this section. Figure 146 shows the same tendency for LLDPE-I as can be seen in Figure 145 for HDPE-B.

Increasing the temperature increases the swelling of the polymer and the swelling is higher for lighter solvents. Note that the swelling of the polymer has been corrected for n-pentane and nhexane. Besides, the swelling is almost constant for temperature lower than 60°C, expect for nheptane which is higher than all other solvents.

The swelling of MDPE-B in presence of liquid solvents such as n-decane, n-octane, n-heptane, n-hexane and n-pentane expressed in volume percent units at different temperatures will be presented in this section.

Conclusion

The sorption of liquid diluents in PE samples as well as the swelling and the dissolution of PE samples was studied in this chapter. Indeed, a profound lack of data concerning phase equilibria for PE/liquid diluent systems can be found in the open literature. The thermodynamic of liquid diluents in PE samples was therefore studied at relevant industrial conditions; temperatures up to 100°C and isobutane to n-decane as liquid diluents. To the best of our knowledge, this is the first time in the literature that the swelling of different PE samples with varying density from 925-960 kg.m -3 are studied at such conditions.

Nevertheless, we were able to show that the smaller the hydrocarbon is, the higher is its solubility in the amorphous phase of the polymer. This implies that the swelling of PE samples in presence of light hydrocarbons is also higher than with heavier hydrocarbons. It was also shown that increasing the temperature increases the solubility of the different liquid diluents in the amorphous phase of the polymer. However, this significant increase of the solubility at high temperature can cause the polymer to swell considerably, leading to reactor fouling by polymer adhesion and plasticization. It was shown that the swelling of PE samples is almost constant for temperatures below 70°C for all types of polymer, and then increases significantly;

corresponding to the breakthrough curves where polymer starts to plasticize. This drastic increase of swelling is a very important property that needs to be controlled in industrial reactors, and is therefore crucial for reactor model development. Moreover, the effect of density and crystallinity of PE samples was also studied in this chapter. Indeed, it was shown that the solubility decreases with increasing polymer density and crystallinity for all PE samples, as expected since the crystalline phase of the polymer is impenetrable to liquid diluents. However, the solubility is assumed constant for PE densities up to 935 kg.m -3 , which was explained by the new morphological model comprising free and constrained amorphous phase in polyethylene. Finally, the dissolution of PE samples was also studied in order to correct swelling experiments.

It would be interesting to push the limits on these experiments to test the swelling of the polymer in supercritical propane (also used in slurry processes) and liquid propylene. Ideally, it would be useful to adapt the Sanchez-Lacombe or PC-SAFT equations of state to predict swelling (which is the ultimate goal here) in diluents used in commercial slurry processes. This data will then be used in reactor models to predict the impact of process conditions, not just on polymer properties, but also to explore operating limits on polymer loading, slurry viscosity and heat transfer. Furthermore, it would also be useful to combine swelling and solubility experiments.

General conclusion

The focus of the current thesis was to show the importance of an accurate description of penetrant-polymer thermodynamics in order to obtain a realistic model for the solubility, the diffusion and the swelling of the polymer particles in multicomponent systems. We have shown that a good thermodynamic model allows us to predict the impact of process conditions on the rate of polymerization and on the properties of the polymer produced from the gas phase polymerization of ethylene.

The primary objective of this thesis was to overcome the lack of experimental data available in the literature concerning the sorption and diffusion in multicomponent systems, especially in ternary and quaternary systems. Gravimetric and pressure decay methods were used to measure the overall solubility and diffusivity (gravimetry) and the overall and partial solubility (pressure decay) of penetrants in multicomponent systems at conditions relevant to those found in industrial processes.

The ternary systems were first studied for ethylene/propane/PE, ethylene/isobutane/PE and ethylene/1-butene/PE in order to show the impact of the different penetrants on ethylene polymerization in gas phase. The overall solubility of these systems was measured using the gravimetric method. The measured solubilities were then fitted to the Sanchez-Lacombe EoS and combined with the Peng-Robinson EoS (SL-PR). This novel approach used the experimental overall solubility as well as the compressibility factor of the gas mixture calculated with the Peng-Robinson EoS as second data source to estimate the two necessary interaction parameters for the SL EoS. The advantage of this method is that it leads to the identification of the interaction parameters with a measurement of only the overall solubility of the system. This approach was first validated with experimental available data and showed reasonable results, leading to its application to the experimental overall solubility measured with the magnetic suspension balance (MSB). Capturing partial solubilities of the different penetrants in ternary system leads to a better understanding of the thermodynamic phenomena that occurs during ethylene polymerization in gas phase. The co-solubility effect was observed when adding ICAs/comonomer to the gas mixture containing ethylene. This means that adding heavier components increases the solubility of ethylene in the amorphous phase of the polymer.

An anti-solvent effect was also highlighted, where lighter components like ethylene decrease the solubility of heavier ones. These thermodynamic properties are very important in the sense that it has been proved that using binary thermodynamic data for ternary system is not correct, as the interactions between the different components in ternary system have to be taken into account.

The SL EoS was extended to quaternary systems containing three penetrants and a polymer.

These quaternary solubility studies represent an important contribution to the modelling of olefin polymerizations since it is the first time in the open literature that solubilities involving polyolefin quaternary systems are studied under conditions approaching those found in industry. Ethylene/propane/1-butene/PE and ethylene/isobutane/1-butene/PE were studied experimentally using the gravimetric and pressure decay methods in order to validate both approaches. It has been shown that the overall solubility of the quaternary system of ethylene/propane/1-butene/LLDPE is higher than that of the ternary system of ethylene/propane/LLDPE and lower than that of ethylene/1-butene/LLDPE at the same temperature and total pressure. However, the results obtained from the quaternary systems demonstrate that the complexity of the changes in the gas phase composition needs to be interpreted by an accurate thermodynamic model, and therefore more experimental data is needed.

As was the case for the solubility, diffusion in multicomponent polymeric systems is so far not completely understood, especially in ternary and more importantly in quaternary systems.

Experimental overall diffusivity in ternary and quaternary systems was measured using the gravimetric method. Crank's solution to the Fickian diffusion model in spheres based on two sizes of compact polymer granules was used in order to theoretically interpret the experimental data, and showed good agreement with the latter. This model was capable of estimating the size as well as the fractions of the small and large compact granules in the polymer particle, giving important information about the polymer particle morphology. The rapid initial increase in the sorption curves was explained by the presence of a significant amount of small granules, whereas the following slower increase of the sorbed weight is caused by the presence of a certain amount of big granules. It has been shown that the overall diffusivity of the different studied ternary systems were higher than the binary system of ethylene/LLDPE. This was explained by the co-diffusion effect of heavy penetrants increasing ethylene partial diffusivity in the amorphous phase of the polymer. Furthermore, it has been shown that the quaternary overall diffusivities were found to be close to their respective ternary overall diffusivity at a given temperature and total pressure. However, this observation needs a more detailed thermodynamic analysis based on the changes in the partial molar volumes for example, in order to better understand the diffusion effects occurring in multicomponent systems.

Once all this thermodynamic data acquired, the main goal was to investigate the effect of accurate thermodynamic description on ethylene polymerization in gas phase reactors. This study explored the impact of changes in the gas phase composition (i.e. ICA, hydrogen) through the polymerization rate, ethylene concentration gradients inside the polymer particle, the growth and overheating of the polymer particle, and finally the molecular weight distribution.

In all the results, the higher the partial pressure of hydrogen is, the lower the productivity and the molecular weight of the polymer are, whereas increasing n-pentane partial pressure increases the productivity and the polymer molecular weight. In order to do so, the Sanchez-Lacombe EoS was used to describe the solubility of the different penetrants in the amorphous phase of the polymer and the diffusivity was modelled using this data and the free volume model of Vrentas and Duda.

As in gas phase systems, a profound lack of data concerning phase equilibria for PE/liquid diluent systems related to slurry polymerization was found in the open literature. These liquid diluents are expected to change the physical properties of the produced polymer as they cause its swelling and softening due to their high solubility in the amorphous phase of the polymer.

The thermodynamic of liquid diluents in PE samples was therefore studied at relevant industrial conditions; temperatures up to 100°C and isobutane to n-decane as liquid diluents. The sorption and swelling of the amorphous phase of the polymer in presence of these liquid diluents were experimentally studied with different grades of polyethylene (i.e. density varying from 925-960 kg.m -3 ). To the best of our knowledge, this is the first time in the literature that the swelling of different PE samples is studied at such conditions. It was found that the solubility of alkanes in the amorphous phase of the polymer was greater for lighter hydrocarbons than for heavier ones.

This implies that the swelling of PE samples in presence of light hydrocarbons is also higher than with heavier hydrocarbons. It was also shown that the solubility could be considered constant for polyethylene densities up to 940 kg m -3 because the free amorphous phase is not present for these PE grades (i.e. HDPE) and it can sorb more penetrants. Finally, increasing the temperature increases the solubility of liquids in the polymer, and therefore the swelling of the amorphous phase of the polymer. However, a significant increase of the swelling at high temperatures, especially with light hydrocarbons (i.e. n-pentane, n-hexane) can cause the polymer to plasticize, leading therefore to reactor fouling, which is an important parameter to take into account when modeling slurry reactors.

Pertinent thermodynamic data have been presented in this thesis, which represents an important tool for understanding the evolution of the growing polymer particle during ethylene polymerization, in gas and slurry phase. This thermodynamic data is crucial if one needs to obtain more accurate models describing ethylene polymerization in industrial reactors.

Perspectives and future work

The current thesis focused on important thermodynamic and kinetic effects related to industrial polyolefin issues. However, some questions are still not resolve, or might lead to further studies.

This could lead to new and very interesting research topics, concerning the impact of the thermodynamics in gas and slurry phase reactors producing polyethylene.

The RPPFM developed in the present thesis highlighted the importance of using an accurate thermodynamic model able to describe the solubility and the diffusivity of multicomponent systems. However, due to the level of complexity of the physical and chemical phenomena that occurs simultaneously during ethylene polymerization in gas phase, a more precise description of the evolution of the morphology of the polymer particle would be needed, as the catalyst/polymer particle evolves rapidly in terms of structure and morphology (i.e. porosity, crystallinity), especially during the initial steps of the polymerization reaction. The measure of the porosity of the polymer particle during the first time of the reaction would be an important parameter to take into account. The crystallinity of the polymer as a function of time would also be a crucial parameter to take into account since the amorphous phase of the polymer changes due to the solubility of the different species in the latter. Besides, it has been shown that the use of the Ranz-Marshall correlation to calculate the external film heat transfer coefficient leads to an overestimation of the particle temperature. Another correlation should be used in order to better predict the change in the overheating of the polymer particle, as this will also have an important impact on the productivity. In the present RPPFM model, ethylene polymerization in presence of n-pentane and hydrogen has been shown. However, the use of only ternary system of ethylene/n-pentane/PE have been used in this study. The concentration of hydrogen have been calculated with Henry's law. It would be more useful to measure the solubility of ethylene/n-pentane/hydrogen/PE in order to take into account the possible interactions between ethylene and n-pentane with hydrogen, in order to have a more accurate kinetic model. Finally, this model can be extended to copolymerization reaction by integrating quaternary solubility and diffusivity data. The measured quaternary solubility and diffusivity data from gravimetric and pressure decay method could be used in order to describe the polymerization of the mixture of ethylene/isobutane/1-butene/PE or ethylene/propane/1-butene/PE.

One of the main concern in the field of polyolefin reaction engineering is the lack of experimental thermodynamic data. In order to overcome these issues, two approaches have been used in order to measure the overall and partial solubility of the different components in a mixture, as well as the overall diffusivity; gravimetric and pressure decay methods. The gravimetric method measures the overall solubility and diffusivity data whereas the pressure decay method measures the overall and partial solubility of multicomponent systems.

Significantly more data for gas mixtures and polymers of different densities and molecular weight distributions needs to be collected in order to refine our thermodynamic models and to eventually help explain the relationship between solubility/swelling and polymer structure.

The same is true for diffusion. Concerning the diffusivity experiments, we have shown that the two level Crank's model is capable of predicting the shape of the degassing experiments.

However, we have seen that some systems (i.e. ethylene/propane/1-butene/LLDPE) lead to more difficult fitting of this model to the experimental data. An improvement of this model might be to take into account the changes in the radius of both the big and small granules as a function of time, as well as their fractions in the polymer particle. In that sense, we would expect to have different fractions of big and small granules during ethylene polymerization. Indeed, the particle morphology changes rapidly, especially at the beginning of the reaction by undergoing a fragmentation step where smaller particles are produced followed by the expansion of the polymer particles through the diffusion and sorption of the different penetrants.

The effect of the evolution of the radius of the big and small granules on the productivity of the polymerization could therefore be analyzed. Besides, this two-level model could be integrated inside the RPPFM in order to have a better description of the diffusivity of the different penetrants inside the amorphous phase of the polymer.

The quaternary experiments performed (i.e. ethylene/propane/1-butene/PE and ethylene/isobutane/1-butene/PE) have shown the impact of different ICAs on the solubility of the mixture. It would be interesting to look at the effect of other quaternary systems involving for example other ICAs (i.e. isobutane, n-pentane, n-hexane) and other comonomers (i.e. 1hexene) in order to compare the effect of different ICAs and co-monomers, and their combination, on the overall solubility in quaternary systems. Furthermore, the study of structurally close ICA and comonomer could be interesting in order to show the impact of the interactions between them and with ethylene on the overall solubility in quaternary sytems. As an example, the systems of ethylene/isobutane/1-butene/PE and ethylene/n-hexane/1butene/PE could be compared. Finally, the study of the solubility in different polymer grades, using different catalyst (i.e. metallocene, Ziegler-Natta) could be achieved in order to show the impact of the interaction parameters kij on the different polymer samples. The interaction parameters, kij, being temperature dependent, broader temperature could be performed in order to have more reliable correlations of these interaction parameters, that are crucial when modeling thermodynamic behaviors with SL EoS. Finally, quintary system could also be realized using the pressure decay method in order to show the impact of hydrogen on the solubility of the quaternary systems. The SL EoS could be extended to quintary systems in order to model systems that are closer to the industrial polymerization conditions.

Concerning the experiments in slurry phase, the effect of crystallinity changes during the experiments would be a useful characteristic to analyze. It would be expected to see a change in the crystallinity of the polymer due to a high degree of solubility and swelling of the amorphous phase of the polymer in presence of liquid diluents. It would be interesting as well to push the limits on these experiments to test the swelling of the polymer in supercritical propane (also used in Borstar slurry processes) and liquid propylene. Ideally, it would be useful to adapt the Sanchez-Lacombe or PC-SAFT equations of state to predict swelling (which is the ultimate goal here) in diluents used in commercial slurry processes. This data could be used in reactor models to predict the impact of process conditions (i.e. Hostalen and Innovene processes), not just on polymer properties, but also to explore operating limits on polymer loading, slurry viscosity and heat transfer. Furthermore, it would also be useful to combine solubility experiments in slurry and gas phase. This study would explore the influence of liquid diluent on the solubility of gaseous components in the polymer. temperature, T. All of the other properties will therefore be calculated from the extent of the solubility of the different component in the polymer.

First, it is necessary to calculate the chemical potential of component 1, 2 and 3 in the gas phase.

These values will later be used in the equilibrium condition, in which the chemical potential of Knowing the partial pressure of component 1, 2 and 3 in the gas phase at the specific temperature, the reduced density of each component in the gas phase is calculated from the SL EoS, (1) by solving equation ( 13), ( 14) and ( 15), respectively, as follow

𝜌̅ 1 𝑔𝑎𝑠 =
The density of component 1, 2 and 3 in the gas phase will be equal to

Figure 1 .

 1 Figure 1. Schéma représentant les différents paramètres à prendre en compte lors de la modélisation d'une particule de polymère en croissance

4 .

 4 CONCLUSION ................................................................................................................................... 5. REFERENCES ................................................................................................................................... CHAPTER 5 : A SINGLE PARTICLE MODEL TO PREDICT THE IMPACT OF INDUCED CONDENSING AGENTS ON POLYMERIZING PARTICLES 229 1. INTRODUCTION .............................................................................................................................. 2. MODEL DEVELOPMENT ...............................................................................................................

4 . 2 .

 42 CONCLUSION ................................................................................................................................... 1. INTRODUCTION .............................................................................................................................. MATERIALS AND METHODS ........................................................................................................ 2.1. MATERIALS ...................................................................................................................................

4 .A

 4 CONCLUSION ................................................................................................................................... 5. REFERENCES ................................................................................................................................... surface area of contact between the exchanger wall (m 2 ) a attractive forces between molecules b co-volume occupied molecules 𝐶 * concentration of active sites at the surface of the catalyst (mol.site.m -3 cata) 𝐶 d concentration of deactivated catalyst site (mol.m -3 ) 𝐶 ICA ICA concentration (mol.m -3 ) 𝐶 M monomer concentration (mol.m -3 )

  d 𝑘 deactivation constant for k catalyst site type (s -1 ) 𝑘 𝑖H 𝑘 initiation of PH rate constant for k catalyst site type (m 3. mol -1 .s -1 ) 𝑘 I 𝑘 initiation rate constant for k catalyst site type (m 3. mol -1 .s -1 ) 𝑘 p 𝑘 propagation rate constant for k catalyst site type (m 3. mol -1 .s -1 ) 𝑘 tH 𝑘 transfer to hydrogen constant for k catalyst site type (m 3. mol -1 .s -1 ) 𝑘 tM 𝑘 transfer to monomer constant for k catalyst site type (m 3. mol -1 .s -1 ) XXVII 𝑘 tsp 𝑘 spontaneous transfer constant for k catalyst site type (s -1 ) 𝐾 𝑖1 free volume parameters for the solvent 𝐾 𝑖3 free volume parameters for the polymer 𝑚̇g ,in inlet mass flow rates of the process gases (kg.s -1 ) P pressure (bar) 𝑃 H concentration of polymer chains containing hydrogen (mol.m -3 ) Pr concentration of "live" polymer chains of length r (mol.m -3 )

1. 1 .

 1 Characteristics of polyolefinsIn 2020, polyolefins represent approximately 45% of the production of plastic materials, which makes them the most significant family of polymers and the most widely used polymer in the world. The worldwide market for polyethylene is expected to rise substantially to about 2.6% over the next five years: from 108.3 million mt in 2016 to 150 million mt by 2027 2 .

Figure 2 .

 2 Figure 2. Evolution of polyethylene worldwide market from 2017 to 2025 3

Figure 4 .

 4 Figure 4. Scheme of a gas phase process operating in a fluidized bed reactor

Figure 5 .

 5 Figure 5. Scheme of a fluidized bed reactor for olefin polymerization. Adapted from 5

Figure 6 .

 6 Figure 6. Impact of ICA during condensed mode operation in gas-phase FBR.

Figure 7 .

 7 Figure 7. Schematic representation of temperature and concentration gradient through the growing polymer particle. Adapted from 47

4. 4 .

 4 Particle modeling studies for the polymerization of olefins A great number of mathematical models can be found in the open literature to describe the growth of polymer particles during olefins polymerization, with different degrees of complexity.

Figure 9 .

 9 Figure 9. Schematic representation of PC-SAFT EoS and Sanchez-Lacombe EoS describing sorption equilibrium of multicomponent species during gas-phase ethylene polymerization. Adapted from 61

Figure 10 .

 10 Figure 10. P&ID of the sorption reactor based on gravimetric method

2 A

 2 novel approach for the estimation of the Sanchez-Lacombe interactions parameters in ternary polyolefins systems 3. Thermodynamic modeling 3.1. Model development 3.1.1. Sanchez-Lacombe EoS The Sanchez-Lacombe EoS is a lattice-fluid model in which pure components are assumed to be broken into parts and placed into a lattice structure. The SL EoS is basically an extension of the classic Flory-Huggins theory. The main advance is that this model introduces vacant lattice

2 AFigure 11 .

 211 Figure 11. Flowchart for the modeling of ternary systems with SL EoS

Figure 14 .

 14 These figures show the solubility of ethylene and propylene in ethylene/propylene/iPP mixture, respectively, at 50, 70 and 85°C.

Figure 12 .

 12 Figure 12. Overall solubilities of an equimolar ethylene/propylene mixture in iPP at 50, 70 and 85°C.The points are desorption measurements from18 and the curves are SL-PR predictions.

Figure 13 .Figure 14 .

 1314 Figure 13. Ethylene partial solubility in an equimolar ethylene/propylene mixture in iPP at 50, 70 and 85°C from SL. kij were taken from Table 11

Figure 15 ,

 15 Figure 15, the overall experimental solubility of ethylene/1-hexene mixture in LLDPE is compared with the corresponding model predictions obtained from SL EoS using the SL-PR method at the different temperatures. In Figure 15 the symbols represent experimental values of the total solubility, and the lines represent the theoretical solubilities calculated with SL-PR method. Ternary interaction parameters were then obtained by fitting SL EoS to the experiments by Novak et al. 6 at 70, 90 and 150°C inTable 12 compared to the interaction parameters obtained

Figure 17 .

 17 These figures show the solubility of ethylene and 1-hexene in ethylene/1-hexene/LLDPE mixture, respectively, at 70, 90 and 150°C.

2 AFigure 15 .

 215 Figure 15. Overall solubilities of an ethylene/1-hexene mixture in LLDPE with 4.3 mol% 1-hexene and 95.7 mol% ethylene at 70, 90 and 150°C. The points are sorption measurements from 6 and the curves are SL-PR predictions.

14 TFigure 16 .

 1416 Figure 16. Ethylene partial solubility in ethylene/1-hexene/LLDPE ternary mixture at 70, 90 and 150°C calculated from SL EoS. kij were taken from Table 12

Figure 17 . 1 -

 171 Figure 17. 1-hexene partial solubility in ethylene/1-hexene/LLDPE ternary mixture at 70, 90 and 150°C calculated from SL EoS. kij were taken from Table 12

2 ATable 14 .

 214 novel approach for the estimation of the Sanchez-Lacombe interaction parameters for ternary polyolefins systems Gas-phase molar composition of the different studied ternary mixtures System Composition Ethylene / Propane / LLDPE-B 0 < Ptot < 5 bar xC2 = 0.509 / xC3 = 0.491 Ethylene / Isobutane / LLDPE-B 0 < Ptot < 4 bar xC2 = 0.485 / xiC4 = 0.515 Ethylene / 1-Butene / LLDPE-B 0 < Ptot < 4 bar xC2 = 0.617 / x1C4 = 0.383 The results of this approach are shown in Figure 20 through Figure 23. The symbols represent experimental values of the total solubility in Figure 20, and the different compressibility factors for the vapor mixtures calculated with Peng-Robinson EoS in Figure 21-Figure 23. It can be seen from Figure

2 AFigure 20 .

 220 Figure 20. Total solubility of ethylene/penetrant mixture in LLDPE-B in presence of different penetrants at 70°C. Points represent experimental values (average of 2 or 3 runs) and the lines represent the predictions of the SL EoS with the fitted parameters. kij are defined in Table 15

Figure 21 .Figure 22 .Figure 23 .

 212223 Figure 21. Compressibility factor of ethylene (C2) and propane (C3) in C2/C3/LLDPE-B mixture. Points represent compressibility values calculated with Peng-Robinson and the lines represent the predictions of the compressibility factor from the ternary SL EoS

Figure 24 .

 24 Figure 24. Total solubility Stot (SL), partial solubility of ethylene SC2 (SL) and partial solubility of propane SC3 (SL) in LLDPE-B at 70°C from SL-PR method (ternary system C2/C3/LLDPE-B). kij are defined in Table15

2 AFigure 25 .

 225 Figure 25. Total solubility Stot (SL), partial solubility of ethylene SC2 (SL) and partial solubility of isobutane SiC4 (SL) in LLDPE-B at 70°C from SL EoS (ternary system C2/iC4/LLDPE-B). kij are defined in Table 15

Figure 26 .

 26 Figure 26. Total solubility Stot (SL), partial solubility of ethylene SC2 (SL) and partial solubility of 1butene S1C4 (SL) in LLDPE-B at 70°C from SL EoS (ternary system C2/1C4/LLDPE-B). kij are definedin Table15

Figure 27 .

 27 Figure 27. This figure illustrates the degree of LLDPE swelling in presence of propane,

Figure 27 .

 27 Figure 27.Degree of LLDPE swelling caused by the sorption of different penetrants; propane (C3), isobutane (iC4) and 1-butene (1C4) at 70°C. Gas phase composition is given inTable 14 and kij are defined in Table 15

shown in Figure 31 ,Figure 31 .Figure 32 .

 313132 Figure 31. Experimental set-up of the pressure decay method for sorption measurements

Figure 33 .

 33 Figure 33. Ethylene solubility in the amorphous phase of HiPP at different temperatures and ethylenepressures. The lines represent the solubility from Sato et al.12 and the points represent the experimental solubility measured with the pressure decay method.

Figure 35 3

 35 Figure35 shows that the experimental points are in good agreement with the literature data

Figure 35 .

 35 Figure 35. Ethylene solubility in the amorphous phase of HDPE-B at different temperatures and ethylene pressures. The lines represent the solubility from Moore and Wanke 6 and the points represent the experimental solubility measured with the pressure decay method.

3Figure 36 .

 36 Figure 36. Flowchart for the modeling of quaternary systems with Sanchez-Lacombe Equation of State

Figure 37

 37 Figure37 compares the overall solubility of the three studied ternary systems:

Figure 44 .

 44 Remember that the /LLDPE-I interaction parameters are used to define the cross interaction energy 𝜀 𝑖𝑗 * = (𝜀 𝑖 * 𝜀 𝑗 * ) 0.5 (1 -𝑘 𝑖𝑗 );

Figure 45 and

 45 Figure47 show the SL-PR fitting of the overall solubility of ethylene/propane/LLDPE-I at 80°C and ethylene/propylene/LLDPE-I at 68°C, respectively.

Figure 46 and

 46 Figure46 and Figure48show the compressibility factor calculated with Peng-Robinson EoS to that calculated from SL EoS, for ethylene/propane/LLDPE-I mixture at 80°C and ethylene/propylene/LLDPE-I mixture at 68°C, respectively. The calculated compressibility factors from SL EoS are in good agreement with the ones calculated with Peng-Robinson EoS, with a maximum difference of less than 1%. It is also shown that the SL-PR model fits the overall solubility data well, with a difference between the model and the experiments of 2% for ethylene/propane/LLDPE-I and 12% for ethylene/propylene/LLDPE-I.

Figure 45 .

 45 Figure45. Overall and partial solubility of ethylene/propane/LLDPE-I ternary system at 80°C. The estimation of the partial solubility is achieved by using the SL-PR method. Molar composition of the system is given in Table19and kij used for this study is given in Table20

Figure 47 .

 47 Figure47. Overall and partial solubility of ethylene/propylene/LLDPE-I ternary system at 68°C. The estimation of the partial solubility is achieved by using the SL-PR method. Molar composition of the system is given in Table19and kij used for this study is given in Table20

Figure 48 . 3

 483 Figure 48. Compressibility factor of ethylene (Z1) and propylene (Z2) in ethylene/propylene/LLDPE-I system at 68°C. The lines represent compressibility values calculated with Peng-Robinson and pointsrepresent the predictions of the compressibility factor from the ternary SL EoS.

3

  Figure57. Total and partial solubility of ethylene/propane/1-butene/LLDPE-I system at 69°C. S_tot, S1_am, S2_am and S3_am are the total solubility of the mixture and the overall solubility of ethylene, propane and 1-butene in the amorphous phase of the polymer, respectively. Gas phase composition of the mixture is given in Table23

Figure 59 and

 59 Figure59 and Figure60show the solubility of ethylene/propane/1-butene mixture in different HDPE grades, HDPE-I and HDPE-B, respectively. As with LLDPE, increasing the total

Figure 59 . 3

 593 Figure59. Total and partial solubility of ethylene/propane/1-butene/HDPE-I system at 73°C. S_tot, S1_am, S2_am and S3_am are the total solubility of the mixture and the overall solubility of ethylene, propane and 1-butene in the amorphous phase of the polymer, respectively. Gas phase composition of the mixture is given in Table23

3Figure 63 .

 63 Figure 63. Temperature dependent kij interaction parameters between ethylene (k14), propane (k24), 1butene (k34) and the polymer for ethylene/propane/1-butene/LLDPE-I quaternary system.

Figure 64 -

 64 Figure 64-Figure67compares the overall and partial solubility measured from the pressure decay experimental method and SL model for quaternary systems using kij shown in Table9.

3

  Experimental and theoretical solubility for multicomponent systems

Figure 68 .

 68 Figure 68. Overall solubility of ethylene/propane/1-butene/PE quaternary system normalized by the partial solubility of each component in HDPE-I (lines) and LLDPE-I (dotted lines) at 73°C. Gas phasecomposition for both system is 𝑥 𝐶 2 =0.66, 𝑥 𝐶 3𝑎 =0.16 and 𝑥 1𝐶 4 =0.16. kij are given in Table25

3 Figure 70 .

 370 Figure 70. Compressibility factor of ethylene in C2/C3/1C4/LLDPE-B quaternary system at 70°C. The red points represent the compressibility values calculated with Peng-Robinson (Z1 -PR EoS) and the black points represent the predictions of the compressibility factor from the ternary SL EoS (Z1 -SL EoS). The dashed lines are simply trend lines to guide the eye.

Figure 71 .Figure 72 .

 7172 Figure 71. Compressibility factor of propane in C2/C3/1C4/LLDPE-B quaternary system at 70°C. The red points represent the compressibility values calculated with Peng-Robinson (Z1 -PR EoS) and the black points represent the predictions of the compressibility factor from the ternary SL EoS (Z1 -SL EoS). The dashed lines are simply trend lines to guide the eye.

Figure 73 . 3

 733 While the temperatures are not identical for all of the experiments in this figure, the difference between the solubility of ethylene/propane/1-butene mixture in LLDPE-B and in HDPE-I cannot be explained by 3°C difference in the temperatures of the two experiments. Experimental and theoretical solubility for multicomponent systems

Figure 73 .

 73 For instance, both of the LLDPE have almost the same global crystallinity, as do the two HDPE powders. Note that the thermograms of the LLDPE-B and LLDPE-I in Figure30have different shapes, indicating that there are clearly going to be differences in the macromolecular architectures of the different polymers. However, if one considers the HDPE products, the differences in the thermograms are not particularly striking, yet there is at least as much difference in the total solubility of the gaseous mixture in the two HDPE products as in the two LLDPE products. It is therefore unlikely that the degree of crystallinity alone can be responsible for the observed differences in the solubility. Where one can see a difference between the two LLDPE and the two HDPE is in the molecular weight distributions, shown in Figure74. HDPE-B and LLDPE-B have broader molecular weight distribution compared to HDPE-I and LLDPE-I. It is possible that the broader molecular weight distribution has a different molecular structure (less homogeneous) that favors the sorption of penetrants. Note that we are not proposing that a difference in chain length is directly responsible for the solubility differences, but rather something linked to the organization of the chains in the matrix that is correlated with the solubility. The polydispersity of the "I" polymers is very low, going from just over 2 for the LLDPE-I to 3 for the HDPE-I. While no information is available on the type of catalyst used to make these products, one would suspect from these polydispersity values that a metallocene

3

 3 Experimental and theoretical solubility for multicomponent systems

4 2 .

 2 Diffusivity measurements 2.1.Gravimetric method A gravimetric method is used to measure the overall diffusivity of the different studied systems. The experimental set-up is briefly described in Chapters 2 and 3. From the pressure, temperature, time and sample mass that are acquired during the sorption experiments, we can obtain the sorption curve as a function of time, as can be shown in Figure 81. The equilibrium solubility of the penetrant(s) can be obtained from the sorbed mass at steady state. The rate of sorption during the transient initial phase can be used to estimate the total diffusivity of the same system.

Figure 81 .

 81 Figure 81.Sorbed weight measured with a magnetic suspension balance

  11 at 70°C at different ethylene pressures are given in Table26. The experimental sorbed weight were therefore fitted to Crank's diffusion equation for small particles for the first moments of the experiment and for big particles for the entire time of the sorption experiment. The fitting procedure minimizes the percent average relative deviation, ARD, between the experiments and Crank's diffusion model: Crank is the normalized sorbed weight calculated with Crank's diffusion model and 𝑀 norm exp is the experimental normalized sorbed weight. 𝑡 is the time of the reaction in seconds.

Figure 82 .Figure 83 .

 8283 Figure 82.Sorption rates of ethylene/LLDPE-B binary system at a total pressure of 4.305 bar at 70°Cobtained through gravimetric method and Crank's diffusion equation for particles with two sizes of granules

Figure 84 .Figure 85 .

 8485 Figure84. Sorption rates of ethylene/LLDPE-B binary system for a total pressure of 10.298 bar at 70°C obtained through gravimetric method and Crank's diffusion equation for particles with two sizes of granules

Table 27 . 86 -Figure 87 (Figure 90 ,

 27868790 Figure 87 (at two different pressures), for ethylene/1-butene/LLDPE-B system in Figure 89-Figure 90, and for the ethylene/propane/LLDPE-B system in Figure 91-Figure 93. The fitting of the Crank's model with the experimental sorption rates will therefore yield the overall diffusivity of the two gases: ethylene and the added penetrant (i.e. propane, isobutane, 1butene). The total diffusivities for these systems are shown in Figure Figure 94.

Figure 86 .

 86 Figure 86. Sorption rates of ethylene/isobutane/LLDPE-B ternary system at total pressure of 2.857 bars and 70°C obtained through gravimetric method and Crank's diffusion equation for particles with two sizes of granules. Molar composition is: xC2 = 0.485 and xiC4 = 0.515.

Figure 87 .

 87 Figure 87. Sorption rates of ethylene/isobutane/LLDPE-B ternary system at total pressure of 2.857 bars and 70°C obtained through gravimetric method and Crank's diffusion equation for particles with two sizes of granules. Molar composition is: xC2 = 0.485 and xiC4 = 0.515.

Figure 88 . 52 4

 8852 Figure 88. Experimental sorption curves (blue lines) and Crank's solution for one level (green lines)and two levels (red line) particle morphology at 70°C, a total pressure of 2.86 bar and a total diffusivity of 8.52x10 -11 m 2 s -1 for ethylene/isobutane/LLDPE-B system. 𝑟=150 µm for one size of granules model and 𝑟 𝑆 =100 µm, 𝑤 1 =0.48, 𝑟 𝐵 =210 µm and 𝑤 2 =0.52 for two sizes of granules model

Figure 89 .

 89 Figure 89. Sorption rates of ethylene/1-butene/LLDPE-B ternary system at a total pressure of 2.6 bars and 70°C obtained through gravimetric method and Crank's diffusion equation for particles with two sizes of granules. Molar composition is: xC2 = 0.617 and x1C4 = 0.383.

Figure 90 .

 90 Figure 90. Sorption rates of ethylene/1-butene/LLDPE-B ternary system at a total pressure of 3.927 bars and 70°C obtained through gravimetric method and Crank's diffusion equation for particles with two sizes of granules. Molar composition is: xC2 = 0.617 and x1C4 = 0.383.

Figure 91 .

 91 Figure 91. Sorption rates of ethylene/propane/LLDPE-B ternary system at total pressure of 2.593 bars and 70°C obtained through gravimetric method and Crank's diffusion equation for particles with two sizes of granules. Molar composition is: xC2 = 0.509 and xiC4 = 0.491.

Figure 92 . 4 Figure 93 .

 92493 Figure 92. Sorption rates of ethylene/propane/LLDPE-B ternary system at total pressure of 3.082 bars and 70°C obtained through gravimetric method and Crank's diffusion equation for particles with two sizes of granules. Molar composition is: xC2 = 0.509 and xiC4 = 0.491.

Figure 94 .

 94 Figure 94. Overall diffusivity of ethylene in LLDPE-B in binary system and in presence of propane, isobutane and 1-butene at 70°C with the molar fractions: x1C4=0.383, xiC4=0.515 and xC3=0.491.

Figure 94

 94 Figure 94 compares the overall diffusivity of the binary system of ethylene/LLDPE-B and the different ternary systems (i.e. ethylene/propane/LLDPE-B, ethylene/isobutane/LLDPE-B, ethylene/1-butene/LLDPE-B) in order to show the effect of the gas phase composition on the

Figure 95 -

 95 Figure 95-Figure 97 show the overall quaternary diffusivity at 70°C for the mixture ethylene/propane/1-butene/LLDPE-B (Composition 1) at total pressures of 2.63, 4.83 and 6.21 bar, respectively.Figure 98-Figure 100 show the mixture of ethylene/propane/1butene/LLDPE-B (Composition 2) at total pressures of 2.61, 4.14 and 5.56 bar respectively.

2 3. 3 . 2 .

 232 Ethylene/Isobutane/1-butene/LLDPE-B

Composition 3 : 40 - 40 Figure 102 -Composition 3 .

 340401023 Figure 102-Figure 104 show the overall quaternary diffusivity at 70°C of ethylene/isobutane/1butene/LLDPE-B Composition 3 at total pressures of 2.86, 4.6 and 6.08 bar and Figure 105-Figure 107 of ethylene/isobutane/1-butene/LLDPE-B Composition 3 at total pressures of 2.5, 4.19 and 5.68 bar. The gas phase composition of both systems are slightly different, with an increase of isobutane and a decrease of 1-butene for Composition 4, with respect to Composition 3. Figure 108 shows the overall diffusivity of both systems.

Figure 107 .Figure 108 .

 107108 Figure 107. Sorption rates of ethylene/isobutane/1-butene/LLDPE-B quaternary system at total pressure of 5.679 bars and 70°C obtained through gravimetric method and Crank's diffusion equation for particles with two sizes of granules for Composition 4. Molar composition is: xiC4=0.43, x1C4=0.17 and xC2=0.40

Figure 108 compares the overall diffusivity of Composition 3 and4Figure 109 .

 3109 Figure108compares the overall diffusivity of Composition 3 and Composition 4, having a slight variation in the gas phase composition of the quaternary system. Indeed, the slight variation corresponds to an increase of isobutane fraction and a decrease of 1-butene fraction, which leads to very close overall diffusivity values, with a difference of 3%. This very small variation of isobutane and 1-butene composition in the gas phase might be considered equivalent. More experimental data with bigger variation of the gas phase composition might help explain these results.

  e. ethylene/propane/LLDPE-B, ethylene/isobutane/LLDPE-B and ethylene/1-butene/LLDPE-B) and quaternary (i.e. ethylene/propane/1-butene/LLDPE-B and ethylene/isobutane/1-butene/LLDPE-B) overall diffusivities have been compared in Figure 109. It can be seen from this figure that, as expected,

  𝟐,𝐨𝐯 𝒓 𝟐 𝝏[𝑴 𝟐 ] 𝝏𝒓 ) [𝑀 2 ] (𝑟,0) = 0 [𝑀 2 ] (𝑅,𝑡) = [𝑀 2 ] eq 𝜕[𝑀 2 ] (0,𝑡) 𝝏𝑻 𝝏𝒓 ) + (-∆𝑯 𝐩 )𝑹 𝐯 𝑇 (𝑟,0) = 𝑇 b -𝑘 c,p 𝜕𝑇 (𝑅,𝑡) 𝜕𝑟 = ℎ(𝑇 -𝑇 b )

5 A

 5 single particle model to predict the impact of induced condensing agents on polymerizing particles

  𝑟) 𝜌̅ ] = 0 where 𝑇 ̅ , 𝑃 ̅ and 𝜌̅ are the reduced temperature, pressure, and density respectively. Three lattice parameters describe the thermodynamic properties of a pure component: the mer-mer interaction energy, the closed-packed molar volume of a mer and the number of mers a molecule occupies in the lattice. Three reduced parameters are defined from these three lattice parameters as follows𝑇 ̅ = 𝑇 𝑇 * ⁄ 𝑃 ̅ = 𝑃 𝑃 * ⁄ 𝜌̅ = 𝜌 𝜌 * ⁄where 𝑇 * , 𝑃 * , and 𝜌 * are the scale factors known as the characteristic temperature, pressure, and density respectively, which are used to characterize each pure component in the mixture.These parameters of SL EoS for each pure component used in the present study are given in the

Figure 111 ,

 111 Figure 111, Figure 113 and Figure 113 compare the experimental rates of polymerization without hydrogen, with 3 bar of hydrogen and the molecular weight distributions with the model predictions, respectively. These results shows that the predicted model and the experimental

Figure 112 .Figure 113 .

 112113 Figure 112. Polymerization rate calculated with the PFM and the experimental data of 7 bar of ethylene, with 3 bar of hydrogen and different amounts of n-pentane, at 80°C

Figure 114 .

 114 Figure 114. Effect of thermodynamic model (i.e. binary, ternary) on the polymerization rate during the reaction and ethylene concentration gradients as a function of the normalized particle radius. The gas phase composition of Rp8 (1 bar C5, 1 bar H2) and Rp9 (2 bar C5, 1 bar H2) is described in Table33. The concentration are taken at 40 minutes of the reaction. The porosity is ε=0.05.

Figure 120 . 5 AFigure 121 .

 1205121 Figure 120. Effect of hydrogen on the molecular weight distribution calculated with the method of moments at 7 bar of ethylene and 0 and 1 bar of n-pentane, at 80°C

Figure 122

 122 Figure122compares the polymerization rate in presence of 7 bar of ethylene with either 1 bar of n-pentane or 1 bar of n-hexane normalized by the polymerization rate in dry mode (i.e. pure ethylene). This figure shows that the polymerization rate in presence of n-hexane is higher than in presence of an equivalent amount of n-pentane, with respect to the one with pure ethylene.The increase in the polymerization rate in condensed mode (i.e. in presence of ICA) is about

Figure 122 .

 122 Figure 122.Polymerization rate during the reaction time for 7 bar of ethylene with 1 bar of n-pentane and n-hexane normalized by the polymerization rate without ICA at 80°C

Figure 124.

  Figure 124. shows the diffusivity of ethylene in the ternary system of ethylene/ICA/PE normalized by the diffusivity of ethylene in the binary system as a function of ICA partial

  Figure 124 shows ethylene concentration gradient at 40 minutes of the reaction with either 1 bar of n-pentane or 1 bar of n-hexane normalized by the concentration gradient of ethylene in dry mode as a function of the normalized particle radius. This figure shows that the equilibrium concentration

Figure 123 .Figure 124 .

 123124 Figure123.Ethylene diffusivity in the ternary system of ethylene/ICA/PE normalized by ethylene diffusivity in the binary system of ethylene/PE as a function of ICA partial pressure at 80°C.

Figure 125 .

 125 Figure 125. Experimental scheme of the dissolution of polyethylene powders in liquid diluents at different temperatures

Figure 126 )

 126 Figure126): a PE film or granule is weighted and placed into different liquid solvents at room temperature for 48 hours, during which the sorption equilibrium was established (it was found that increasing the sorption time to 72 hours does not have any measurable difference). After reaching equilibrium, the polymer sample is removed from the liquid, any droplets present on the surface of the film/granule are rapidly removed by sponging with a filtration paper, and the polymer sample is placed on an analytical balance. The rate of decrease of the mass of the polymer at room temperature was recorded manually until a stable value was observed.

Figure 126 .

 126 Figure 126. Scheme of the desorption experiments of PE films and granules at room temperature.

2. 4 .

 4 Swelling Experiments HDPE-B, MDPE-B and LLDPE-I powders were used for the swelling experiments in order to show the effect of the crystallinity of the PE sample on its degree of swelling, as well as the temperature of the reactor. The reactor used for the swelling experiments is shown in Figure 128.

Figure 128 .

 128 Figure 128. Experimental set-up for the swelling experiments in slurry

Figure 129 .

 129 Figure 129. Example of plasticization of MDPE-B powders in presence of n-hexane at 90°C

Figure 131 .

 131 Figure 131. Percent mass dissolved of LLDPE-I powders in the presence of different solvents and at different temperatures.

6

  Sorption and swelling of liquid diluents in polyethylene polymer dissolved in liquid diluents because only the amorphous phase of the polymer partially dissolves in the solvent.

Figure 133 .6Figure 134 .

 133134 Figure 133. Solubility of n-hexane in HDPE-I granules at room temperature (the same experiment atthe same conditions was repeated three times)

Figure 135 .Figure 136 .Figure 137 .

 135136137 Figure 135. Solubility of n-heptane, n-nonane and n-decane in HDPE-B films at room temperature(the same experiment at the same conditions was repeated 2 times)

Figure 138 .

 138 Figure 138. Solubility (g of solvent / g of polyethylene) at room temperatures in the different PE samples with respect to the solvent chain length.

Figure 139 .

 139 Figure 139. Solubility (g of solvent / g of polyethylene) at room temperature of the different solvents in three different PE samples having different densities.

Figure 140 .6Figure 141 ,

 140141 Figure 140. Solubility (g of solvent / g of amorphous polyethylene) at room temperature of the different solvents in three different PE samples having different densities

Figure 141 .Figure 142 .Figure 143 .6

 141142143 Figure 141. Solubility of different solvents (nC5, nC6, nC7, nC8, nC9, nC10) in HDPE-I granules at 25°C (Each experiment was repeated at least two times under the same conditions)

Figure 144 .

 144 Figure 144. Solubility of different solvents (nC5, nC6, nC7, nC8, nC9, nC10) in HDPE-I granules as a function of temperature

Figure 145 .

 145 Figure 145. Swelling of HDPE-B (in % vol) powders in presence of different solvents at different temperatures. (Each experiment was repeated at least twice under the same conditions)

6

  Sorption and swelling of liquid diluents in polyethylene

Figure 146 .

 146 Figure 146. Swelling of LLDPE-I (in % vol) powders in presence of different solvents at different temperatures. (Each experiment was repeated at least twice under the same conditions)

Figure 147 .

 147 Figure 147. Swelling of MDPE-B (in % vol) powders in presence of different solvents at different temperatures. (Each experiment was repeated at least twice under the same conditions).

Figure 147 5 .6

 1475 Figure147shows the same tendency as for LLDPE-I and HDPE-B. Increasing the temperature increases the swelling of the polymer and higher degree of swelling can be seen for lighter solvents. Note that the swelling of the polymer has been corrected for n-pentane and n-hexane, since lighter solvents leads to higher degree of polymer dissolution. Besides, swelling experiments for lighter solvents (i.e. n-pentane, n-heptane) were stopped at lower temperatures; around 60°C for n-pentane and 80°C for n-hexane because MDPE-B powders starts plasticizing, as shown in Figure5.

Figure 149 .

 149 Figure 149. Swelling measured experimentally and corrected with MDPE-B dissolution in the solventgiven in Table42

  each of the components are equal in all phases. In other words,

  At given volume of the gas phase mixture, the mass fraction of component 1, 2 and 3 in the gas phase is given by

gas

  Assuming that there is no polymer molecules in the gas phase, The reduced gas phase density, 𝜌̅ 𝑔𝑎𝑠 , is calculated base on (1) as follow 𝜌̅ 𝑔𝑎𝑠 = 1 -exp (-𝜌 ̅𝑔𝑎𝑠 2

  .1. GRAVIMETRIC METHOD ...................................................................................................................... 2.2. FICKIAN DIFFUSION IN SPHERICAL PARTICLES ....................................................................................

1. INTRODUCTION TO POLYOLEFINS ..............................................................................................

1.1. CHARACTERISTICS OF POLYOLEFINS..................................................................................................... 1.2. CATALYSTS FOR ETHYLENE POLYMERIZATION ..................................................................................... 1.3. PROCESSES FOR THE PRODUCTION OF POLYOLEFINS ............................................................................. 1. INTRODUCTION .............................................................................................................................. 2. DIFFUSIVITY MEASUREMENTS .................................................................................................. 23. RESULTS AND DISCUSSION .......................................................................................................... 3.1. BINARY SYSTEMS ............................................................................................................................... 3.2. TERNARY SYSTEMS ............................................................................................................................ 3.3. QUATERNARY SYSTEMS ..................................................................................................................... 3.3.1. Ethylene/Propane/1-butene/LLDPE-B ...................................................................................... 3.3.2. Ethylene/Isobutane/1-butene/LLDPE-B .................................................................................... 3.4. COMPARISON OF MULTICOMPONENT SYSTEMS ...................................................................................

  EFFECT OF THERMODYNAMIC AND DIFFUSION MODELS ...................................................................... 3.2. EFFECT OF PARTICLE POROSITY .......................................................................................................... 3.3. EFFECT OF THE GAS PHASE COMPOSITION ON ETHYLENE POLYMERIZATION ....................................... Effect of the gas phase composition on the polymerization rate ............................................... 3.3.2. Effect of the gas phase composition on ethylene concentration gradient .................................

	3.1.
	3.3.1.

2.1. SINGLE PARTICLE MODEL (SPM) -RPPFM ....................................................................................... 2.2. THERMODYNAMICS ............................................................................................................................ 2.2.1. Sanchez-Lacombe equation of state .......................................................................................... 2.2.2. Vrentas and Duda diffusion model ............................................................................................ 2.3. KINETIC MODEL ................................................................................................................................. 2.3.1. Polymer Molecular Weight Data and Deconvolution ............................................................... 2.3.2. Kinetic Parameters ................................................................................................................... 3. RESULTS AND DISCUSSION .......................................................................................................... 3.3.3. Effect of the gas phase composition on the temperature gradient ............................................ 3.3.4. Effect of the gas phase composition on the molecular weight distribution ............................... 3.4. EFFECT OF ICA TYPE ..........................................................................................................................

  ..... 2.2. DISSOLUTION OF POLYETHYLENE IN LIQUID SOLVENTS ...................................................................... 2.3. SOLUBILITY OF ALKANES IN PE ......................................................................................................... 2.3.1. Room temperature sorption ...................................................................................................... 2.3.2. High temperature sorption ........................................................................................................ 2.4. SWELLING EXPERIMENTS ...................................................................................................................

3. RESULTS AND DISCUSSION ..........................................................................................................

3.1. POLYMER SOLUBILITY IN ALKANES .................................................................................................... 3.2. ALKANE SORPTION IN POLYMERS -ROOM TEMPERATURE EXPERIMENTS .......................................... 3.3. ALKANE SORPTION IN POLYMERS -TGA METHOD ............................................................................ 3.4. SWELLING EXPERIMENTS ....................................................................................................................

Table 1 .

 1 Density and crystallinity ranges for different commercial polyethylene resins

	Literature Review	1

Table 2

 2 summarizes the main characteristics of coordination catalyst for olefin polymerization.

	Literature Review	1

Table 2 .

 2 Main characteristics of coordination catalyst for olefin polymerization adapted from Asua10 .

	Type	Physical state	Polymer type
	Ziegler-Natta	Heterogeneous	Non-uniform
		Homogeneous	Uniform
	Phillips	Heterogeneous	Non-uniform
	Metallocene	Heterogeneous	Uniform
		Homogeneous	Uniform
	Ziegler-Natta and Phillips catalysts are considered to have multiple-site types, which explains
	why they make polymer with non-uniform properties: each site type produces polymer

populations with different average microstructural properties. Ziegler-Natta catalysts can be homogeneous, meaning soluble in the reaction medium, or heterogeneous. When these catalysts are homogeneous, they have only one site type and synthesize polyolefins with uniform properties. Phillips catalysts, on the other hand, are always heterogeneous, meaning that short chains have higher comonomer content.

1.3.Processes for the production of polyolefins

Different industrial processes are used to produce polyolefins; high-pressure free radical and low-pressure catalyst polymerization.

Table 3 .

 3 Typical process condition for slurry polyethylene reactors

	Process	Reactor type	Diluent	Comonomer
	Mitsui	2 stirred autoclaves	Hexane	1-Hexene
	Basell (Hostalen)	2-3 stirred autoclaves	Hexane	Butene
	Equistar -Maruzen -Nissan	1-2 stirred autoclaves	Hexane	1-Hexene / 1-Butene
	Chevron Phillips	Single loop	Isobutane	1-Hexene
	Borealis -Borstar	2 loops / FBR	Supercritical Propane	1-Butene
	Innovene S	1-2 loops	Isobutane	1-Hexene / 1-Butene
	Total	1-2 loops	Isobutane	1-Hexene

Table 4 .

 4 Heat capacity of gaseous components used in the polymerization of ethylene in gas phase processes

	Gaseous component	Heat Capacity (25°C) (cal.K -1 .mol -1 )	Reference
	Hydrogen	6.9	52
	Nitrogen	7.0	52
	Ethylene	10.4	53
	Propylene	15.3	53
	Propane	17.4	53
	Iso-butane	23.1	53
	Iso-pentane	28.4	53
	n-hexane	34.0	53

  Where 𝑅 v is the rate of the reaction per unit of volume of catalyst. The initial and boundary conditions used for the energy balance are

								Literature Review	1
				𝜌 ov 𝐶 𝑝,ov	𝜕𝑇 𝜕𝑡	=	𝑘 c,p 𝑟 2	𝜕 𝜕𝑟	(𝑟 2 𝜕𝑇 𝜕𝑟	) + (-∆𝐻 p )𝑅 v
	𝑇 (𝑟,0) = 𝑇 b
	-𝑘 c,p	𝜕𝑇 (𝑅,𝑡) 𝜕𝑟	= ℎ(𝑇 -𝑇 b )
	𝜕𝑇 (0,𝑡) 𝜕𝑟	= 0

Table 5 .

 5 Kinetic scheme of ethylene homopolymerization over a Ziegler-Natta multiple site catalyst

	Description	Chemical reactions	Rate constants
	Initiation	𝐶 𝑘 * + M	𝑃 1 𝑘	𝑘 𝑖 𝑘
	Propagation	𝑃 𝑟 𝑘 + M	𝑃 𝑟+1 𝑘	𝑘 𝑝 𝑘
	Transfer to H2	𝑃 𝑟 𝑘 + 𝐻 2	𝑃 H 𝑘 + 𝐷 𝑟 𝑘	𝑘 𝑡𝐻 𝑘
	Initiation of PH	𝑃 H 𝑘 + M	𝑃 1 𝑘	𝑘 𝑖𝐻 𝑘
	Transfer to monomer	𝑃 𝑟 𝑘 + M	𝑃 1 𝑘 + 𝐷 𝑟 𝑘	𝑘 𝑡𝑀 𝑘
	Spontaneous chain transfer	𝑃 𝑟 𝑘	𝐶 𝑘 * + 𝐷 𝑟 𝑘	𝑘 𝑡𝑠𝑝 𝑘
	Deactivation	𝑃 𝑟 𝑘	𝐶 d 𝑘 + 𝐷 𝑟 𝑘	𝑘 𝑑 𝑘

Table 6 .

 6 Literature data for modeling methods of binary and ternary systems in polyolefins

	1 Literature Review
	Binary / Ternary systems	Modeling method	References
	Methane / PE	SL EoS	65
	Ethylene / PE	SL EoS PC SAFT	57,61,65,68,129,140,141 61,128-130,145-147
	Ethylene / PP	SL EoS PC SAFT	65,68,129,140 129
	Propane / PE	SL EoS PC SAFT	129 128-130,145,145,146,148
	Propylene / PE	PC SAFT	130,146
	Propylene / PP	SL EoS PC SAFT	65,68,129,140 129,145
	1-butene / PE	SL EoS PC SAFT	68,129,140 128-130,145-147
	n-butane / LDPE	SL EoS PC-SAFT	127,149,150 127,128,146,147
	Isobutane / LDPE	SL EoS PC-SAFT	149,150 128
	n-pentane / PE	SL EoS PC-SAFT	65,127 127,128,148
	1-hexene / PE	SL EoS PC SAFT	57,68,140,149 128,130
	n-hexane / PE	SL EoS PC SAFT	61,65,150 61,128
	Ethylene / propylene / PE	SL EoS	65,140
	Ethylene / 1-butene / PE	SL EoS	68,140
	Ethylene / 1-hexene / PE	SL EoS PC SAFT	57,68,129,140 129
	Ethylene / n-hexane / PE	SL EoS PC SAFT	61,65 61

Table 7 .

 7 Literature data of sorption of binary systems in polyolefins. P is the pressure in bar and T is the temperature in °C.

	Literature Review	1

166 n-butane / PE

Permeation T=

[30-60]

; P=[0

-10]

155 Gravimetry T=[0-45] ; P=[0-9] 159 Isobutane / PE Pseudo-gravimetry T=[150-260] ; P=[0-2.3] 157 Gravimetry T=[50-85] ; P=[0-14] 131,132 1-butene / PE Gravimetry T=[30-100] ; P=[0-12] 131,141,162-164 n-pentane / PE Gravimetry T=[0-45] ; P=[0-7] 159 Isopentane / PE Gravimetry T=[50-85] ; P=[0-5] 131 n-hexane / PE Gravimetry T=[0-85] ; P=[0-2] 166 Pressure decay T=[75-110] ; P=[0-80] 137 1 Literature Review 5.2.2. Sorption studies for ternary systems As we stated above, ternary data for PE systems is less common in the open literature, undoubtedly because it is more challenging to measure accurately. Also, usually only overall solubility is available.

Table 8 .

 8 Literature data of sorption of ternary systems in polyolefins. P is the pressure in bar and T is the temperature in °C.

	Systems	Experimental techniques	Experimental conditions	Reference
	Ethylene / Propylene / PE	Gravimetry	T=[30-100] ; P=[1-3]
		Gravimetry	T=69 ; P=[0-35]	56,164
		IGA	T=[50-90] ; P=[0-10]

162 Ethylene / Butene / PE Pressure decay T=[70-94] ; P=[1-21] 140 Ethylene / Isopentane / PE Pressure decay T=[70-90] ; P=[0-20] 55 Gravimetry T=80 ; P=[0-13] 131,132 Ethylene / n-hexane / PE Pressure decay T=[70-90] ; P=[0-10] 55 Gravimetry T=80 ; P=[0-13] 131,132 Ethylene / 1-hexene / PE 57 Ethylene / Propylene / PP Gravimetry T=[50-85] ; P=[0-25] 140 Isobutane / Isopentane / PE Gravimetry T=80 ; P=[0-12] 131,132

1

  Literature Review (between 0.5 and 1) which is introduced because the same free volume is available to more than one molecule183 . 𝐾 11 and 𝐾 21 are the free volume parameters for the solvent, while 𝐾 13 and 𝐾 23 are for the polymer. 𝜙 1 is the volume fraction of component 1 and 𝜒 is the solvent-polymer interaction parameter.

Table 9 .

 9 Pure component parameters used in SL EoS

	Component	𝑻

* (K) 𝑷 * (bar) 𝝆 * (

kg/m 3 ) Ref

	Ethylene	283	3395	680	21
	Propane	371	3090	690	20
	Isobutane	398	2840	720	20
	1-butene	410	3350	770	22
	1-hexene	450	3252	814	23
	Propylene	345.4	3788	755	18
	LLDPE	667	4370	900	21
	Polypropylene	690.6	3007	885.6

18

𝑃 ̅ , 𝑇 ̅ , 𝜌̅ and 𝑉 ̅ are the reduced pressure, temperature, density and volume, respectively. These characteristic parameters depend on three lattice parameters, describing the thermodynamic 2

A novel approach for the estimation of the Sanchez-Lacombe interaction parameters for ternary polyolefins systems

Table 10

 10 indicates the required properties to implement the PR equation of the components studied in this chapter.

Table 10 .

 10 Critical temperature and pressure, acentric factors and molecular weight for selected fluids.

Ethylene Propane Isobutane 1-butene a 1-hexene Propylene Ref

	2	A novel approach for the estimation of the Sanchez-Lacombe interaction parameters for ternary polyolefins systems
		w (-)	0.089	0.153	0.183	0.191	0.28	0.144	26
		Tc (K)	282.36	369.83	407.82	419.95	504.03	364.76	27
		Pc (bar)	50.318	42.471	36.4	40.4	31.4	46.126	27
		MW (g.mol -1 )	28.054	44.1	58.12	56.108	84.16	42.08

Table 11 .

 11 Comparison of ternary interaction parameters of ethylene/propylene mixture in iPP identified with the novel approach SL-PR by fitting experimental solubilities of Cancelas et al.18

		T = 50 °C	T = 70 °C	T = 85 °C
	k13 from 18	-0.01	-0.015	-0.02
	k13 SL-PR	-0.0095	-0.0157	-0.0192
	k23 from 18	0.01	0.005	-0.005
	k23 SL-PR	0.0137	-0.00026	-0.00043

Table 13 .

 13 Temperature

dependence of kij for ethylene/LLDPE and 1-hexene/LLDPE binary systems. T is in °C System kij Ethylene/LLDPE -2.93 × 10 -4 𝑇 + 5.87 × 10 -2 1-hexene/LLDPE -8.23 × 10 -4 𝑇 + 8.89 × 10 -2

Table 15 .

 15 Identified ternary interaction parameters by fitting experimental solubility data to SL EoS for different systems at 70°C

	Systems	kij at 70°C
	Ethylene / Propane / LLDPE-B	k13= -0.03329, k23= 0.03905
	Ethylene / Isobutane / LLDPE-B	k13= -0.06856, k23= 0.04839
	Ethylene / 1-Butene / LLDPE-B	k13= -0.09495, k23= 0.04618

Table 14 )

 14 . However, if we consider 1-butene for instance, its molar fraction was lower than isobutane and propane (x1C4=0.383, xiC4=0.515 and xC3=0.491). Even with a lower fraction of 1-butene, the co-solubility effect was higher than in presence of isobutane or propane.

2

A novel approach for the estimation of the Sanchez-Lacombe interaction parameters for ternary polyolefins systems Figure 28. Partial ternary solubility of ethylene from SL EoS in presence of C3, iC4 and 1C4 with the molar fractions: x1C4=0.383, xiC4=0.515, xC3=0.491; kij for ternary systems are defined in Table 15.

Table 15 .

 15 kij for binary systems are defined in

2 A novel approach for the estimation of the Sanchez-Lacombe interactions parameters in ternary polyolefins systems Figure 29. Solubility of propane, isobutane and 1-butene in binary (diluent/LLDPE) and ternary (ethylene/diluent/LLDPE) systems using SL EoS.kij for ternary systems are defined in

Table 8 .

 8 The lines represent diluent partial solubility in ternary systems and the dotted lines represent diluent solubility in binary systems.

Table 16 .

 16 Binary interaction parameters at T=70°C used in this work

	System	Binary kij	Ref
	Ethylene/LLDPE	0.0546	Our work
	Propane/LLDPE	0.023	3
	Isobutane/LLDPE	0.0265	3
	1-Butene/LLDPE	0.0418	Fitting to 30

Table 17 ,

 17 and the thermograms of these same polymers are presented in Figure30. It can be seen from this last figure that while the crystallinity of the 2 HDPE powders remains virtually constant at temperatures up to 90°, that of the LLDPE samples is temperature dependent over the range used in the solubility experiments (up to 90°C). For that reason, the fraction of amorphous polymer in the LLDPE samples was calculated at the temperature of the experiment.

Table 17 .

 17 Characteristics of the polymer powders used for sorption measurements

		Density (kg.m -3 )	Crystallinity (%)	PI
	HDPE-B	945	65.7	10.9
	LLDPE-B	923	48.3	8.8
	HiPP	880	60	-
	HDPE-I	954	69.5	2.97
	LLDPE-I	926

𝜒 = -0.378 × 𝑇 + 75.85 2.03 Figure 30. DSC thermograms of the 4 PE shown in Table 17. (a) LLDPE and (b) HDPE.

Table 18 .

 18 Pure component parameters used in SL EoS

	Component	𝑻

* (K) 𝑷 * (bar) 𝝆 * (

kg m -3 ) Ref

  For a mixture of components, it is necessary to define combining rules based on the Van der Waals mixing rule for the estimation of 𝜀 mix * , 𝜐 mix * , and 𝑟 mix , to be able to use the equation of state to calculate the properties of the mixture.Where 𝜙 is the volume fraction of the i th and j th component in the mixture, 𝑁 c is the number of component in the mixture and 𝑛 𝑖𝑗 corrects the possible deviation of 𝑣 𝑖𝑗

	3 Experimental and theoretical solubility for multicomponent systems
	The characteristic closed-packed molar volume of a "mer" of the mixture, 𝜐 mix *	is defined as
			𝑁 c	𝑁 c
	𝑣 mix *	= ∑ ∑ 𝜙 𝑖 𝜙 𝑗 𝑣 𝑖𝑗 *
			𝑖=1	𝑗=1
	With
	𝑣 𝑖𝑗 * =	𝑣 𝑖𝑖 * + 𝑣 𝑗𝑗 * 2	(1 -𝑛 𝑖𝑗 )
	𝜐 * mix gas = 𝑣 11 * 𝜙 1 gas + 𝑣 22 * 𝜙 2 gas + 𝑣 33 * 𝜙 3 gas
	𝜐 * mix pol = 𝑣 11 * 𝜙 1 pol + 𝑣 22 * 𝜙 2 pol + 𝑣 33 * 𝜙 3 pol + 𝑣 44 * 𝜙 4 pol
	The closed-packed volume fraction of the i th component at the limit of zero temperature or
	Ethylene incompressible state, 𝜙 𝑖 gas is defined as 283	3395	680	16
	𝜙 𝑖 gas	Propane Isobutane = 𝜔 𝑖 gas 𝜌 𝑖 * 𝑣 𝑖 * ∑ ( 𝑁 c 𝑗=1 ⁄	gas 𝜔 𝑗 𝜌 𝑗 * 𝑣 𝑗 * )	371 398	3090 2840	690 720	17 17
			1-butene		410	3350	770	18
			Propylene		345.4	3788	755	14
			HDPE			650	4250	905	11
			LLDPE		667	4370	900
	152

* from the arithmetic mean value of 𝑣 𝑖𝑖 * and 𝑣 𝑗𝑗 * of the pure component. In the present study, the value of the interaction parameter 𝑛 𝑖𝑗 is assumed to be equal to zero.

For quaternary systems, the characteristic closed-packed molar volume of the gas and polymer phase, respectively, are defined as follows:

  𝜌̅ pol , as well as the closed-packed volume fractions of solute components in the

							Experimental and theoretical solubility for multicomponent systems	3
	phase density, polymer phase 𝜙 1 pol , 𝜙 2 pol , 𝜙 3 pol , can be identified.
	All the properties of interest can be deduced from these parameters. The volume fraction of
	each sorbed component in the amorphous polymer can therefore be calculated as
	𝜔 𝑖 am =	𝜌 1 * 𝜐 1 * 𝜙 1 pol	𝜌 𝑖 * 𝜐 𝑖 * 𝜙 𝑖 pol * 𝜙 2 * 𝜐 2 + 𝜌 2 pol + 𝜌 3 * 𝜐 3 * 𝜙 3 pol	+ 𝜌 4 * 𝜐 4 * 𝜙 4 pol
	𝜔 𝑖 am =	𝜌 𝑖 * 𝜐 𝑖 * 𝜌 1 * 𝜐 1 *	𝜙 𝑖 pol 𝜙 1 pol 𝜔 1 am	i = 2,3
	𝜔 4 am = 1 -𝜔 1 am -𝜔 2 am -𝜔 3
						2 𝜐 * mix gas (∑ 𝜙 𝑗 gas 𝑁c 𝑗=1	υ ij * ε ij *	-𝜀 * mix gas ∑ 𝜙 𝑗 gas 𝑁c 𝑗=1	υ ij *	) + 𝜀 * mix gas	]
	+	R 𝑇 𝜌̅ gas	[(1 -𝜌̅ gas )ln(1 -𝜌̅ gas ) +	𝜌̅ gas 𝑟 𝑖	𝑙𝑛𝜌̅ gas ]
	+	𝑃 𝜌 ̅ gas	(2 ∑ 𝜙 𝑗 gas 𝑁c 𝑗=1	υ ij * -𝜐 * mix gas	)}

am

Finally, the solubility of each component at equilibrium, in gram of component per gram of amorphous polymer can be calculated as follows

Table 19 .

 19 Gas phase molar composition of the different ternary studied system using pressure decay method

	System	Temperature (°C)	Composition
			Ptot = [5, 7.9] bar
	Ethylene/Propane/LLDPE-I	T=[80, 90]	𝑥 𝐶 2 = 0.497
			𝑥 𝐶 3𝑎 = 0.503
			Ptot = [2.2, 5.2] bar
	Ethylene/Propylene/LLDPE-I	T=[68, 78]	𝑥 𝐶 2 = [0.475, 0.395]
			𝑥 𝐶 3𝑒 = [0.525, 0.605]
			Ptot = [3.2, 4] bar
	Ethylene/1-Butene/LLDPE-I	T=[68, 84]	𝑥 𝐶 2 = [0.834, 0.496]
			𝑥 1𝐶 4 = [0.166, 0.504]

Table 20 Figure 41

 2041

	1-butene/LLDPE-I at the two studied temperatures are
	different (i.e. 𝑥 C 2 =0.834 and 𝑥 1C 4 =0.166 at 68°C compared to 𝑥 C 2 =0.496 and 𝑥 1C 4 =0.504 at
	84°C, c.f. Table 19), so we should expect a different behavior of the gas phase between both
	temperatures.

Figure 38. Total (S_tot) and partial solubility of ethylene (S1_am) and propane (S2_am) in ethylene/propane/LLDPE-I system at 80°C. The lines represent the experimental solubility and the red points represent the solubility calculated with SL EoS. Gas phase composition is xC2=0.497 and xC3a=0.503 at 80°C. kij correlations are given in

Table 20 Figure 39. Total (S_tot) and partial solubility of ethylene (S1_am) and propane (S2_am) in ethylene/propane/LLDPE-I system at 90°C. The lines represent the experimental solubility and the red points represent the solubility calculated with SL EoS. Gas phase composition is xC2=0.497 and xC3a=0.503 at 90°C. kij correlations are given in Table 20 . Total (S_tot) and partial solubility of ethylene (S1_am) and propylene (S2_am) in ethylene/propylene/LLDPE-I system at 68°C. The lines represent the experimental solubility and the red points represent the solubility calculated with SL EoS. Gas phase composition is xC2=0.395 and xC3e=0.605 at 68°C. kij correlations are given in Table 20

Table 20 .

 20 kij correlation with respect to temperature (°C) for ethylene/propane/LLDPE-I, ethylene/propylene/LLDPE-I and ethylene/1-butene/LLDPE-I systems

	Correlation

Table 21 .

 21 Comparison of the predicted kij for ethylene/propane/LLDPE-I and ethylene/propylene/LLDPE-I with the SL EoS and with SL EoS combined to PR EoS (SL-PR)

		SL EoS with overall and	SL-PR with only
		partial solubilities	overall solubility
	Ethylene/Propane/LLDPE-I	𝑘 13 = -0.062 𝑘 23 = 0.011	𝑘 13 = -0.0628 𝑘 23 = 0.0121
	Ethylene/Propylene/LLDPE-I	𝑘 13 = -0.089 𝑘 23 = 0.0281	𝑘 13 = -0.081 𝑘 23 = 0.0281
	4.1.2

Table 22 .

 22 Binary interaction parameters at 80°C for ethylene, propane, propylene and 1-butene in LLDPE. Figure51show the predicted (i.e. SL EoS) binary solubilities of ethylene and the different penetrants, estimated from Chmelar et al.

	System	Binary kij	Ref
	Ethylene/LLDPE	0.00387	Fitting to 21
	Propane/LLDPE	-0.0068	Fitting to 21
	Propylene/LLDPE	-0.00812	Fitting to 21
	1-Butene/LLDPE	0.0155	Fitting to 21
	Figure 49-

21

, and the predicted total solubilities of mixtures of ethylene and the same penetrants for different fractions each in ternary systems at 80°C. kij interaction parameters at 80°C were taken from Table

22

for all binary systems and from Table

20

for the ternary studied systems. It is shown that the overall solubility of ternary systems of ethylene/penetrant in LLDPE is between the binary solubility of ethylene and that of the penetrant, at each gas phase composition.

Figure 49. Solubility of binary ethylene/LLDPE (C2) and propane/LLDPE (C3a) systems and overall solubility of their mixture in ternary system of ethylene/propane/LLDPE at 80°C, calculated with SL

EoS. Different composition of the gaseous mixture in the ternary system is shown. kij for binary systems are defined in Table

22

and for ternary systems in Table

20

with SL EoS. Different composition of the gaseous mixture in the ternary system is shown. kij for binary systems are defined in Table

22

and for ternary systems in Table

20

Figure

51

. Solubility of binary ethylene/LLDPE (C2) and 1-butene/LLDPE (1C4) systems and overall solubility of their mixture in ternary system of ethylene/1-butene/LLDPE at 80°C, calculated with SL EoS. Different composition of the gaseous mixture in the ternary system is shown. kij for binary systems are defined in Table

22

and for ternary systems in Table

20

Table 20

 20 Binary solubility of ethylene in LLDPE taken from

		0,02	C2=95mol% + 1C4=5mol%
	(g/g.am.LLDPE)	0,015	C2=95mol% + C3e=5mol% C2=95mol% + C3a=5mol% C2/LLDPE C2/LLDPE -SL EoS
	solubility	0,01
	Ethylene	0,005
		0
		0	2	4 Total pressure (bar) 6	8	10

and for the ethylene/LLDPE binary system in Table

22

.

Table 23 .

 23 Gas-phase molar composition of the different studied quaternary systems using pressure decay method

	System	Polymer Type	Temperature (°C)	Composition
		LLDPE-I	T=[69 ; 84]	Ptot = [2.5, 5] bar 𝑥 𝐶 3 = 0.54, 𝑥 1𝐶 4 = 0.18
	Ethylene / Propane / 1-Butene	HDPE-I	T=73	Ptot = [2.5, 5] bar 𝑥 𝐶 3 = 0.43, 𝑥 1𝐶 4 = 0.17
		HDPE-B	T=75	Ptot = [2.7, 5] bar 𝑥 𝐶 3 = 0.46, 𝑥 1𝐶 4 = 0.19

Figure 56.

Overall solubility of ethylene/propane/1-butene in different PE types (HDPE-I and LLDPE-I) and at different temperatures, using pressure decay method. Gas phase composition is shown in Table

23

. PE characteristics are shown in Table

Table 23

 23

Table 17 ;

 17 56.2% for LLDPE-I compared to 69.5% for HDPE-I.

Table 24 .

 24 Gas-phase molar composition of the different studied quaternary systems with the magnetic suspension balance, at 70°C Overall solubility of quaternary system of ethylene/isobutane/1-butene/LLDPE-B at 70°C with different composition of the gas-phase. Gas-phase composition for each experiment is 𝑥 𝑖𝐶 4 =0.43 and 𝑥 1𝐶 4 =0.17 for(1), 𝑥 𝑖𝐶 4 =0.4 and 𝑥 1𝐶 4 =0.23 for (2) and 𝑥 𝑖𝐶 4 =0.44 and 𝑥 1𝐶 4 =0.21 for (3)

	System	Composition
		Ptot = [2.5 -6.2] bar
	Ethylene / Propane / 1-butene / LLDPE-B	0.39 < 𝑥 𝐶 3 < 0.42
		0.17 < 𝑥 1𝐶 4 < 0.23
		Ptot = [2.5 -6.5] bar
	Ethylene / Isobutane / 1-butene / LLDPE-B	0.4 < 𝑥 𝑖𝐶 4 < 0.43
		0.17 < 𝑥 1𝐶 4 < 0.23

Figure

61

and Figure

62

show the overall solubility of ethylene/propane/1-butene/LLDPE-B and ethylene/isobutane/1-butene/LLDPE-B, respectively, measured with the magnetic suspension balance at 70°C. It is first shown that, as expected, increasing the total pressure increases the overall solubility of both quaternary systems. Furthermore, it is interesting to see that the overall solubility of the quaternary mixture with propane as ICA is higher than the one with isobutane for the same composition of the gas-phase. These results cannot be explained yet since only overall solubilities are available. The partial solubilities of each component in these quaternary systems are needed in order to better understand the different interactions that occur between the gaseous species in the mixture. Figure 61. Overall solubility of quaternary system of ethylene/propane/

1-butene/LLDPE-B at 70°C with different composition of the gas-phase. Gas-phase composition for each experiment is 𝑥 𝐶 3 =0.42 and 𝑥 1𝐶 4 =0.17 for (1), 𝑥 𝐶 3 =0.39 and 𝑥 1𝐶 4 =0.23 for (2) and 𝑥 𝐶 3 =0.34 and 𝑥 1𝐶 4 =0.25 for (3)

Table 23 .

 23 kij are given in Table25Experimental and SL modeling of overall and partial solubility of each component in ethylene/propane/1-butene/HDPE-I at 73°C. Gas phase composition of the mixture is given in

	Figure 65. Experimental and SL modeling of overall and partial solubility of each component in
	ethylene/propane/1-butene/LLDPE-I at 84°C. Gas phase composition of the mixture is given Table 23.
	kij are given in Table 25

Figure 66.

Table 23 .

 23 kij are given in

Table 25 Figure 67 .

 2567 Experimental and SL modeling of overall and partial solubility of each component in ethylene/propane/1-butene/HDPE-B at 75°C. Gas phase composition of the mixture is given in

Table 23 .

 23 kij are given in Table25

	0,02	S_tot -Exp
		S_tot -SL EoS
	0,012 0,016 (g/g.am.HDPE-I)	S1_am -Exp S1_am -SL EoS S2_am -Exp S2_am -SL EoS S3_am -Exp
	0,004 0,008 Solubility
	0
	2	2,5	3 Total pressure (bar) 3,5 4	4,5	5

Concerning the quaternary system of ethylene/propane/1-butene/LLDPE-B performed with the magnetic suspension balance, the overall quaternary solubility is the only measured data. This

Table 24

 24

. kij are given in Table

25

Table 23 and

 23 Table 24. PE characteristics are shown in

Table 1

 1

Table 25

 25

	(g/g.am.LLDPE-I)	0,03 0,04 0,05	Stot | 0.66 C2+0.34 1C4 Stot | 0.66 C2 + 0.16 C3a + 0.16 1C4 Stot | 0.5 C2 + 0.16 C3a + 0.34 1C4
	solubility	0,02
	Overall	0,01
		0
		0	3	6	9	12
				Total pressure (bar)

Figure 77. 1-Butene partial solubility calculated from SL EoS for ethylene/1-butene/LLDPE-I ternary

system and for ethylene/propane/1-butene/LLDPE-I quaternary system at 80°C. kij for the ternary system was taken from Table

Table 25

 25

	Overall solubility (g/g.am.LLDPE-I)	0,01 0,02 0,03 0,04	Stot | 0.66 C2+0.34 C3a Stot | 0.66 C2 + 0.16 C3a + 0.16 1C4 Stot | 0.5 C2 + 0.34 C3a + 0.16 1C4
		0
		0	3	6	9	12
				Total pressure (bar)

Figure 79. Ethylene partial solubility calculated from SL EoS for ethylene/propane /LLDPE-I ternary

system and for ethylene/propane/1-butene/LLDPE-I quaternary system at 80°C. kij for the ternary system was taken from Table

20

and for the quaternary system from Table

25

Table 26 .

 26 Identified radii of small and big granules, and their respective fractions in the polymer particle. Diffusivity data were taken from Alves et al.11 at 70°C.Figure 82 to Figure 85 show the sorbed weight of ethylene in LLDPE at 70°C at different pressures up to 14 bars. The diffusivity data was taken from Alves et al. 11 at 70°C for different ethylene pressures, as can be seen in

	4

Table 26 .

 26 The parameter n in Crank's equation was assumed equal to 50 as large value of n satisfies the analytical solution convergence criterion.

Table 28 .

 28 Gas-phase molar composition of the different studied quaternary systems

	System	Composition
		0 < Ptot < 6.2 bar
	Ethylene / Propane / 1-butene / LLDPE-B	0.39 < xC3 < 0.42
		0.17 < x1C4 < 0.23
		0 < Ptot < 6 bar
	Ethylene / Isobutane / 1-butene / LLDPE-B	0.4 < xiC4 < 0.43
		0.17<x1C4 < 0.23

  Molar composition is: xC3=0.39, x1C4=0.23 and xC2=0.38. Comparison of the diffusivity of ethylene/propane/1-butene/LLDPE-B quaternary system obtained from the fitting of Crank's diffusion equation for particles with two sizes of granules from Composition 1 and Composition 2.

	4				Diffusivity of multicomponent gas mixtures in polyethylene Diffusivity of multicomponent gas mixtures in polyethylene	4
		1,2 1,2
		1 1
	1 1,2 0,4 0,6 0,8 M(t)/M∞ 0,6 0,8 M(t)/M∞ 0,4			Exp. P = 6.207 bar Crank's model Exp. P = 4.14 bar Crank's model
	M(t)/M∞	0,4 0,6 0,8 0 0,2 0 0,2	0 0	50 50	Exp. P = 2.626 bar Crank's model 150 Time (s) 100 Crank's model BIG 100 150 Time (s) Crank's model BIG Crank's model SMALL Crank's model SMALL 200 200
	Composition 1. Molar composition is: xC3=0.39, x1C4=0.23 and xC2=0.38. particles with two sizes of granules for Composition 2. Molar composition is: xC3=0.42, x1C4=0.17 and 0 0,2 Crank's model BIG Crank's model SMALL Figure 97. Sorption rates of ethylene/propane/1-butene/LLDPE-B quaternary system at total pressure of 6.207 bars and 70°C obtained through gravimetric method and Crank's diffusion equation for Figure 99. Sorption rates of ethylene/propane/1-butene/LLDPE-B quaternary system at total pressure of 4.14 bars and 70°C obtained through gravimetric method and Crank's diffusion equation for Figure 101.
			0	50	100 xC2=0.41	150	200
					Time (s)
		1,2
		1
	0,4 0,6 0,8 1 1,2 0,2 M(t)/M∞ 0,4 0,6 0,8 1 1,2 0 0,2 0,4 0,6 0,8 M(t)/M∞ M(t)/M∞	0	50	Crank's model Crank's model BIG 100 150 Time (s) Exp. P = 4.832 bar Crank's model Exp. P = 5.557 bar Crank's model Crank's model BIG Exp. P = 2.613 bar Crank's model SMALL	200
	0 0,2 0 Figure 100. Sorption rates of ethylene/propane/1-butene/LLDPE-B quaternary system at total Crank's model BIG Crank's model SMALL 0 50 100 150 Crank's model SMALL 0 50 100 150 200 pressure of 5.557 bars and 70°C obtained through gravimetric method and Crank's diffusion equation Time (s) for particles with two sizes of granules for Composition 2. Molar composition is: xC3=0.42, x1C4=0.17 200 Figure 98. Sorption rates of ethylene/propane/1-butene/LLDPE-B quaternary system at total pressure Time (s) and xC2=0.41
	of 2.613 bars and 70°C obtained through gravimetric method and Crank's diffusion equation for
	Figure 96. Sorption rates of ethylene/propane/1-butene/LLDPE-B quaternary system at total pressure particles with two sizes of granules for Composition 2. Molar composition is: xC3=0.42, x1C4=0.17 and
	of 4.832 bars and 70°C obtained through gravimetric method and Crank's diffusion equation for xC2=0.41
	particles with two sizes of granules for Composition 1. Molar composition is: xC3=0.39, x1C4=0.23 and
					xC2=0.38.

Composition 1 is given in

Figure 95-Figure 97.

Figure 95. Sorption rates of ethylene/propane/1-butene/LLDPE-B quaternary system at total pressure of 2.626 bars and 70°C obtained through gravimetric method and Crank's diffusion equation for particles with two sizes of granules for Composition 1. The fitting of the overall diffusivity of ethylene/propane/1-butene/LLDPE-B quaternary system with Crank's diffusion model for particles with two sizes of granules for the Composition 2 is given in Figure 98-Figure 100.

  total pressure of 2.864 bars and 70°C obtained through gravimetric method and Crank's diffusion equation for particles with two sizes of granules for Composition 3. Molar composition is: xiC4=0.37, x1C4=0.23 and xC2=0.40. Sorption rates of ethylene/isobutane/1-butene/LLDPE-B quaternary system at total pressure of 6.078bars and 70°C obtained through gravimetric method and Crank's diffusion equation for particles with two sizes of granules for Composition 3. Molar composition is: xiC4=0.37, x1C4=0.23 and xC2=0.40. Sorption rates of ethylene/isobutane/1-butene/LLDPE-B quaternary system at total pressure of 4.185 bars and 70°C obtained through gravimetric method and Crank's diffusion equation for particles with two sizes of granules for Composition 4. Molar composition is: xiC4=0.43, x1C4=0.17 and xC2=0.40

				Diffusivity of multicomponent gas mixtures in polyethylene
	1,2
	1
	0,2 0,4 0,6 0,8 1 1,2 0 0,2 0,6 0,8 M(t)/M∞ 0,4 M(t)/M∞	0	50	Exp. P = 2.864 bar Crank's model BIG Time (s) Crank's model Exp. P = 6.078 bar 100 150 Crank's model Crank's model BIG Crank's model SMALL	200
	Figure 103. Sorption rates of ethylene/isobutane/1-butene/LLDPE-B quaternary system at total and xC2=0.40 x1C4=0.23 and xC2=0.40. for particles with two sizes of granules for Composition 4. Molar composition is: xiC4=0.43, x1C4=0.17 for particles with two sizes of granules equation for Composition 3. Molar composition is: xiC4=0.37, pressure of 2.504 bars and 70°C obtained through gravimetric method and Crank's diffusion equation pressure of 4.604 bars and 70°C obtained through gravimetric method and Crank's diffusion equation 0 0 50 100 150 200 Time (s) Crank's model SMALL 0 0,2 0,4 0,6 0,8 1 1,2 0 50 100 150 200 M(t)/M∞ Time (s) Exp. P = 4.604 bar Crank's model Crank's model BIG Crank's model SMALL 0 0,2 0,4 0,6 0,8 1 1,2 0 50 100 150 200 M(t)/M∞ Time (s) Exp. P = 2.504 bar Crank's model Crank's model BIG Crank's model SMALL Figure 104. Figure 105. Sorption rates of ethylene/isobutane/1-butene/LLDPE-B quaternary system at total Figure 106.

Composition 3 is given in

Figure 102-Figure 104.

Figure 102. Sorption rates of ethylene/isobutane/1-butene/LLDPE-B quaternary system at

The fitting of the experimental overall diffusivity ethylene/isobutane/1-butene/LLDPE-B quaternary system with Crank's diffusion model for particles with two sizes of granules for Composition 4 is given in Figure 105-Figure 107.

Table 30 .Table 30 .

 3030 Pure component parameters used in SL EoS

	Component	𝑻 * (K)	𝑷 * (bar)	𝝆 * (kg m -3 ) Reference
	Ethylene	283	3395	680	20
	n-pentane	445	3060	755	20
	n-hexane	476	2979.1	775	20
	LLDPE	653	4360	903	20

Table 31 .

 31 Binary and ternary interaction parameters of ethylene/alkane /LLDPE system from Sanchez-Lacombe EoS at 80°C.

	System	kij	Ref
	Ethylene/LLDPE	k13 = -0.022	24
	n-pentane/LLDPE	k13 = 0.024	24
	n-hexane/LLDPE	k13 = 0.010	13
	Ethylene/n-pentane/LLDPE k13 = -0.027, k23 = 0.0501	23
	Ethylene/n-hexane/LLDPE	k13 = -0.022, k23 = 0.029	13

Table 32 .

 32 Parameters of the free-volume theory

		ethylene	n-pentane	n-hexane	PE	Units
	D0i	2.96 × 10 -7	3.11 × 10 -8	3.50×10 -8	-	m 2 s -1
	V ̂i*	1.341	1.158	1.133	1.006	cm 3 g -1
	K1i / γ	1.97 × 10 -3	2.41 × 10 -3	1.96×10 -3	1.02 × 10 -3	cm 3 g -1 K -1
	K2i ─ Tgi	42.38	-38.89	-41.08	-228.70	K
	MWi	28.05	72.15	86.18	-	g mol -1
	ξi3	0.4548	0.95664	0.9184	-	(─)

Table 33 .

 33 Ethylene, n-pentane and hydrogen partial pressure for each polymerization experiment. All pressures are in bar.Figure110shows the experimental MWD and the results of the deconvolution. It appears from this analysis that the MWD can be adequately described with four distributions corresponding to the four site types, for experiments Rp4 and Rp5. This fit gives the fraction of each site; then the average of the fraction of each site for both experimental MWD was calculated and is presented in

		𝑹 𝐩𝟏	𝑹 𝐩𝟐	𝑹 𝐩𝟑	𝑹 𝐩𝟒	𝑹 𝐩𝟓
	𝑷 𝐂 𝟐			7
	𝑷 𝐂 𝟓	0	1	2	0	2
	𝑷 𝐇 𝟐		0		3

Table 34 .

 34 This average weight fraction of each site type was then used to identify the polymerization parameters by fitting the measured reaction rate and MWD to the experiments in the following sections. The deconvolution of the experimental molecular weight distributions (red dots, wnorm) for 𝑹 𝒑 𝟒 and 𝑹 𝒑 𝟓 suggests that it is reasonable to use a minimum of 4 site types to describe this catalyst, designated by 1, 2, 3 and 4, respectively. wFlory represents the calculated MWD.

	𝑹 𝐩𝟒

Table 34 .

 34 Fraction of each catalyst active site obtained from the deconvolution

		Site 1	Site 2	Site 3	Site 4
	𝑹 𝐩 𝟒	0.1	0.32	0.40	0.18
	𝑹 𝐩 𝟓	0.09	0.34	0.40	0.17
	𝒘 𝒎 𝒊	0.095	0.33	0.4	0.175
	The initial concentration of active sites, that is used as an initial condition for the calculation of
	the active site concentration, is considered as follow:
		𝐶 0,𝑘 * = 𝑤 𝑚 𝑖 𝑘 𝐶 * = 𝑤 𝑚 𝑖 𝑘 𝑓 cat 𝐶 max *	= 𝑤 𝑚 𝑖 𝑘 𝑓 cat	𝑤 Ti 𝑀𝑊 Ti	𝜌 cat

Table 35 .

 35 Kinetic mechanism of ethylene homopolymerization over a Ziegler-Natta 4 sites catalyst (k =1 -4)

	Description	Chemical equations	Rate constants
	Initiation	𝐶 𝑘 * + M	𝑃 𝑟 𝑘	𝑘 i 𝑘
	Propagation	𝑃 𝑟 𝑘 + M	𝑃 𝑟+1 𝑘	𝑘 p 𝑘
	Transfer to H2	𝑃 𝑟 𝑘 + 𝐻 2	𝑃 H 𝑘 + 𝐷 𝑟 𝑘	𝑘 tH 𝑘
	Initiation of PH	𝑃 𝐻 𝑘 + M	𝑃 1 𝑘	𝑘 iH 𝑘
	Transfer to monomer	𝑃 𝑟 𝑘 + M	𝑃 1 𝑘 + 𝐷 𝑟 𝑘	𝑘 tM 𝑘
	Spontaneous chain transfer	𝑃 𝑟 𝑘	𝐶 𝑘 * + 𝐷 𝑟 𝑘	𝑘 tsp 𝑘
	Deactivation	𝑃 𝑟 𝑘	𝐶 d 𝑘 + 𝐷 𝑟 𝑘	𝑘 d 𝑘
	In

Table 35 ,

 35 𝐶 𝑘 * is the concentration of vacant catalyst active sites, M is the monomer concentration, 𝑃 𝑟 𝑘 is the concentration of "live" polymer chains, 𝑃 𝐻 𝑘 is the concentration of the activated vacant catalyst sites occupied by a hydrogen, 𝐷 𝑟 𝑘 is the concentration of dead polymer chains, 𝐻 2 is the hydrogen concentration in the bed, 𝐶 𝑑 𝑘 is the concentration of deactivated catalyst site. The symbol k denotes the four catalyst active sites: 1, 2, 3 and 4. A single particle model to predict the impact of induced condensing agents on polymerizing particles

Table 36 .

 36 Mass balances of active sites and moments of chain lengths, with the initial conditions:The rate of ethylene polymerization at the active site, Rp, for multiple catalyst active sites, is

	𝝁 𝒊 𝒌 (𝒓, 𝟎) = 𝜼 𝒊 𝒌 (𝒓, 𝟎) = 𝟎, 𝑷 𝑯 𝒌 (𝒓, 𝟎) = 𝟎, 𝑪 𝒌 * (𝒓, 𝟎) = 𝑪 𝟎,𝒌 * , 𝒊 = 𝟎, 𝟏, 𝟐, 𝒌 = 𝟏, 𝟐 , 𝟑, 𝟒.
	Description				Equations	Initial
					conditions
	0 th moment of living chains	𝑑𝜇 0 𝑘 𝑑𝑡	= (𝑘 i 𝑘 𝐶 𝑘 * + 𝑘 iH 𝑘 𝑃 H 𝑘 )[𝑀] -(𝑘 tH 𝑘 [𝐻 2 ] + 𝑘 d 𝑘 + 𝑘 tsp 𝑘 )𝜇 0 𝑘	𝜇 0 𝑘 (0) = 0
	1 st moment of living chains	𝑑𝜇 1 𝑘 𝑑𝑡	= (𝑘 i 𝑘 𝐶 𝑘 * + 𝑘 iH 𝑘 𝑃 H 𝑘 + (𝑘 p 𝑘 + 𝑘 tM 𝑘 )𝜇 0 𝑘 )[𝑀] -(𝑘 tH 𝑘 [𝐻 2 ] + 𝑘 d 𝑘 + 𝑘 tM 𝑘 [𝑀] + 𝑘 tsp 𝑘 )𝜇 1 𝑘	𝜇 1 𝑘 (0) = 0
	2 nd moment of living chains	𝑑𝜇 2 𝑘 𝑑𝑡	= (𝑘 i 𝑘 𝐶 𝑘 * + 𝑘 p 𝑘 (2𝜇 1 𝑘 + 𝜇 0 𝑘 ) + 𝑘 iH 𝑘 𝑃 H 𝑘 + 𝑘 𝑡𝑀 𝑘 𝜇 0 𝑘 )[𝑀] -(𝑘 tH 𝑘 [𝐻 2 ] + 𝑘 d 𝑘 + 𝑘 tM 𝑘 [𝑀] + 𝑘 tsp 𝑘 )𝜇 2 𝑘	𝜇 2 𝑘 (0) = 0
	0 th moment of dead chains :			𝑑𝜂 0 𝑘 𝑑𝑡	= (𝑘 tH 𝑘 [𝐻 2 ] + 𝑘 d 𝑘 + 𝑘 tM 𝑘 [𝑀] + 𝑘 tsp 𝑘 )𝜇 0 𝑘	𝜂 0 𝑘 (0) = 0
	1 st moment of dead chains			𝑑𝜂 1 𝑘 𝑑𝑡	= (𝑘 tH 𝑘 [𝐻 2 ] + 𝑘 d 𝑘 + 𝑘 tM 𝑘 [𝑀] + 𝑘 tsp 𝑘 )𝜇 1 𝑘	𝜂 1 𝑘 (0) = 0
	2 nd moment of dead chains			𝑑𝜂 2 𝑘 𝑑𝑡	= (𝑘 tH 𝑘 [𝐻 2 ] + 𝑘 d 𝑘 + 𝑘 tM 𝑘 [𝑀] + 𝑘 tsp 𝑘 )𝜇 2 𝑘	𝜂 2 𝑘 (0) = 0
	Active site concentration				𝑑𝐶 𝑘 * 𝑑𝑡	= 𝑘 tsp 𝑘 𝜇 0 𝑘 -(𝑘 i 𝑘 [𝑀] + 𝑘 d 𝑘 )𝐶 𝑘 *	𝐶 0,𝑘 *
	Catalyst sites bearing hydrogen, 𝑷 𝑯			𝑑𝑃 H 𝑘 𝑑𝑡	= 𝑘 tH 𝑘 𝜇 0 𝑘 [𝐻 2 ] -𝑘 iH 𝑘 𝑃 H 𝑘 [𝑀] -𝑘 d 𝑘 𝑃 H 𝑘	𝑃 H 𝑘 (0) = 0
	defined as follow:

Table 37 .

 37 Values of the kinetic parameters used for the model of ethylene polymerization in gas-phase. The same parameters are used for all sites, except transfer to hydrogen (a the given value is for catalyst site type n° 4).This leaves us with 5 parameters that need to be identified by fitting the experimental data: 𝑘 i0 , 𝑘 p0 , 𝑘 d0 , 𝑘 tH0 4 and 𝑓 cat . In order to do so, the experiments without hydrogen were first fitted to the model in order to identify 𝑘 i0 , 𝑘 p0 , 𝑘 d0 and 𝑓 cat . The parameters 𝑘 p0 and 𝑓 cat were used to adjust the level of 𝑅 p 𝑖 model during the entire period of the reaction, while 𝑘 i0 was used to tune the initial amplitude of 𝑅 p 𝑖 model and 𝑘 d0 was used to adapt the slope of 𝑅 p 𝑖 model . Then, the experiments with hydrogen were fitted to the model by keeping the previously identified parameters constant, and by changing only 𝑘 tH04 which serves to adjust the MWD with 𝑅 p 𝑖 model .

		Pre-exponential factor (m 3 mol -1 s -1 )	Activation energy (J mol -1 )	Reference
	Initiation	94	37656	This work
	Reinitiation of sites bearing hydrogen	94	37656	This work
	Propagation	1.410 6	37656	This work
	Transfer to H2	20.6 a	33472	This work
	Transfer to monomer	0.186	33472	36
	Spontaneous chain transfer	8.84	33472	36,37
	Deactivation	33	33472	This work

Table 32 is shown in Figure 115. Figure 115.

  Ethylene diffusion coefficient in ethylene/n-pentane/LLDPE at 80°C as a function of npentane partial pressure for different polymer porosity.

Table 38 .

 38 Density and crystallinity characteristics of the studied polymer samplesIt is known that amorphous PE and low molecular weight fractions of polymer can dissolve in liquid alkanes, and that the amount of polymer that dissolves will be a function of the temperature and alkane chain length.9 Furthermore, if we wish to study the swelling of PE by liquid alkanes, it is necessary to know what fraction of that PE actually dissolves in the liquid diluents.

		Density (kg.m -3 )	Crystallinity (%)
	HDPE-I	960	75
	HDPE-B	945	65.7
	MDPE-B	935	59
	LLDPE-I	926	52.6
	2.2.Dissolution of polyethylene in liquid solvents

Table 40 ,Table 41 and

 4041 Table 42. The same method is performed for HDPE-B, MDPE-B and LLDPE-I samples.The data in Table39appears to be roughly linear over the studied range of temperature, as can be seen from the plots in Figure130. The correlations used to estimate the solubility of HDPE-B in the different alkanes as a function of temperature are given in

Table 40 .

 40 The results of similar experiments for LLDPE-I are shown in Figure131and Table41, and MDPE in Figure132 and

Table 42 .Table 39 .

 4239 Percent mass dissolved of HDPE-B powders in the presence of different solvents and at different temperatures.

	T (°C)	80	60	40
	nC10	2.48	1.67	0.99
	nC9	2.52	1.72	1.24
	nC8	2.65	1.72	1.26
	nC7	2.59	1.83	1.26
	nC6	2.94	2.09	1.4
	nC5	4.16	3.48	1.56

Table 40 .

 40 Temperature dependence correlation for the mass of dissolution of HDPE-B powders in presence of different solvents

	Solvents	Correlation for HDPE-B dissolution
	nC10	% 𝑚𝑎𝑠𝑠 𝑑𝑖𝑠𝑠 = 0.0374 × 𝑇 -0.528
	nC9	% 𝑚𝑎𝑠𝑠 𝑑𝑖𝑠𝑠 = 0.0317 × 𝑇 -0.079
	nC8	% 𝑚𝑎𝑠𝑠 𝑑𝑖𝑠𝑠 = 0.0347 × 𝑇 -0.206
	nC7	% 𝑚𝑎𝑠𝑠 𝑑𝑖𝑠𝑠 = 0.0333 × 𝑇 -0.107
	nC6	% 𝑚𝑎𝑠𝑠 𝑑𝑖𝑠𝑠 = 0.0386 × 𝑇 -0.171

Table 41 .

 41 Temperature dependence correlation for the mass of dissolution of LLDPE-I powders in presence of different solvents

	20	nC5
	18
	14 16 of LLDPE-I	nC6 nC7
	8 10 12 dissolution	nC8 nC9
	% mass of	4 6	nC10
		2
		0
		20	30	40	50 Temperature (°C) 60	70	80	90
	Solvents			Correlation for LLDPE-I dissolution
	nC10				% 𝑚𝑎𝑠𝑠 𝑑𝑖𝑠𝑠 = 0.229 × 𝑇 -4.247
	nC9				% 𝑚𝑎𝑠𝑠 𝑑𝑖𝑠𝑠 = 0.239 × 𝑇 -4.169
	nC8				% 𝑚𝑎𝑠𝑠 𝑑𝑖𝑠𝑠 = 0.241 × 𝑇 -3.916
	nC7				% 𝑚𝑎𝑠𝑠 𝑑𝑖𝑠𝑠 = 0.241 × 𝑇 -3
	nC6				% 𝑚𝑎𝑠𝑠 𝑑𝑖𝑠𝑠 = 0.235 × 𝑇 -1.807
	nC5				% 𝑚𝑎𝑠𝑠 𝑑𝑖𝑠𝑠 = 0.269 × 𝑇 -1.787

Table 42 .

 42 Temperature dependence correlation for the mass of dissolution of MDPE-B powders in presence of different solvents

	Solvents	Correlation for MDPE-B dissolution
	nC10	% 𝑚𝑎𝑠𝑠 𝑑𝑖𝑠𝑠 = 0.0893 × 𝑇 + 0.132
	nC9	% 𝑚𝑎𝑠𝑠 𝑑𝑖𝑠𝑠 = 0.1012 × 𝑇 + 0.163
	nC8	% 𝑚𝑎𝑠𝑠 𝑑𝑖𝑠𝑠 = 0.0919 × 𝑇 + 1.016
	nC7	% 𝑚𝑎𝑠𝑠 𝑑𝑖𝑠𝑠 = 0.0909 × 𝑇 + 1.444
	nC6	% 𝑚𝑎𝑠𝑠 𝑑𝑖𝑠𝑠 = 0.0875 × 𝑇 + 1.802
	nC5	% 𝑚𝑎𝑠𝑠 𝑑𝑖𝑠𝑠 = 0.14 × 𝑇 + 0.989
	We can see from these experimental results that the polymer dissolution increases with
	increasing the temperature from 40 to 80°C, and with lighter solvents. nC5 dissolves more
	polymer per gram than does nC10 for all types of polyethylene. Besides, the polymer dissolution

in the different solvents depends on the polymer type. Indeed, LLDPE dissolves more than MDPE which in turn dissolves more than HDPE with the same solvent and under the same conditions. Increasing the crystallinity of the polymer (i.e. HDPE) decreases the amount of

Table 43 .

 43 Comparison of the solubility of different liquid solvents in all PE samplesFigure138shows the solubility per gram of different alkanes in the different PE samples with respect to the solvent chain length (N.B. this is total polymer mass, not with respect to the

	1)
	nC5 (2)
	nC6 (1)
	nC6 (2)
	nC8 (1)
	nC8 (2)
	nC10 (1)
	nC10 (2)

amorphous phase). It is shown that increasing the solvent chain length decreases its solubility in the polymer. The difference of solubility between the heavier (i.e. n-decane) and the lighter (i.e. n-pentane) solvent in HDPE-B and HDPE-I is about 55%. Smaller solvents have higher solubility in the semicrystalline polymer as they are small enough to diffuse inside the amorphous phase of the polymer.

  Application of the Sanchez-Lacombe equation of state for quaternary systemsOnce the gas phase system and its characteristics defined, it is necessary to define the characteristic parameters for the polymer phase as the mixture of solute (1, 2, 3) and polymer (4) components. By considering that 𝜙 1Finally the number of sites (mers) occupied in the lattice by a molecule of polymer phase mixture will be given assuming 𝑟 4 >>𝑟 1 , 𝑟 2 , 𝑟 3 , as follow

	+ 𝑟 2	{	𝑅 𝑔 𝑇 𝜌̅ gas	-𝜌̅ gas [ 𝜐 * mix 2 gas ( (𝜙 1 gas -𝜀 * mix 𝜐 12 * 𝜀 12 * + 𝜙 2 gas gas (𝜙 1 gas 𝜐 12 * + 𝜙 2 𝜐 2 * 𝜀 2 * + 𝜙 3 gas gas 𝜐 2 * + 𝜙 3 𝜐 23 * 𝜀 23 * ) gas 𝜐 23 ) + 𝜀 * mix gas * ) [(1 -𝜌̅ gas ) ln(1 -𝜌̅ gas ) + 𝜌̅ gas 𝑟 2 𝑙𝑛𝜌̅ gas ] + 𝑃 𝜌̅ gas [2(𝜙 1 gas 𝜐 12 * + 𝜙 2 gas 𝜐 2 * + 𝜙 3 ] + gas	𝜐 23 * ) -𝜐 * mix gas	] }
	𝜇 3 gas	= 𝑅 𝑔 𝑇 [𝑙𝑛𝜙 3 gas	+ 1 -	𝑟 3 𝑟 mix gas ]	(29)
	+ 𝑟 3	{	𝑅 𝑔 𝑇 𝜌̅ gas	-𝜌̅ gas [ 𝜐 * mix 2 gas ( (𝜙 1 gas -𝜀 * mix 𝜐 13 * 𝜀 13 * + 𝜙 2 gas gas (𝜙 1 gas 𝜐 13 * + 𝜙 2 𝜐 23 * 𝜀 23 * + 𝜙 3 gas gas 𝜐 23 * + 𝜙 3 gas 𝜐 3 * 𝜀 3 * ) 𝜐 3 * ) [(1 -𝜌̅ gas ) ln(1 -𝜌̅ gas ) + 𝜌̅ gas 𝑟 3 𝑙𝑛𝜌̅ gas ] + 𝑃 𝜌̅ gas [2(𝜙 1 gas 𝜐 13 * + 𝜙 2 ) + 𝜀 * mix gas gas 𝜐 23 * + 𝜙 3 ] + gas	𝜐 3 * ) -𝜐 * mix gas	] }
							(31.9)
	𝐶 1 = 𝜀 4 * 𝜐 4 *			pol	+ 𝜙 2 pol	+ 𝜙 3 pol	+ 𝜙 4	(31.10)
	𝑇 𝑚𝑖𝑥 𝑔𝑎𝑠 -pol + 𝜎 2 𝜙 2 pol = 𝜎 1 𝜙 1 𝜐 * mix pol + 𝜎 3 𝜙 3 𝑃 𝑚𝑖𝑥 𝑔𝑎𝑠 𝑇 𝑚𝑖𝑥 𝑔𝑎𝑠 -(1 -pol + 𝜎 4	1 𝑟 mix gas ) 𝜌̅ 𝑔𝑎𝑠 )	(24) (30)
	in which 𝑇 𝑚𝑖𝑥 𝑔𝑎𝑠 = 𝜎 2 = 𝜐 2 * -𝜐 4 𝜀 * mix gas * 𝑅 𝑔 𝑇 in which 𝜎 1 = 𝜐 1 * -𝜐 4 * 1 𝑟 mix pol = 𝜙 1 pol 𝑟 1 +	𝜙 2 pol 𝑟 2	+	pol 𝜙 3 𝑟 3	(30.2) (24.1 ) (30.1) (32)
	𝑃 𝑚𝑖𝑥 𝑔𝑎𝑠 = 𝜎 3 = 𝜐 3 * -𝜐 4 𝑃 𝜐 * mix gas gas * The Sanchez-Lacombe EoS for polymer phase is obtained by substituting equations (30), (31), (24.2 ) (30.3) 𝜀 * mix 1 𝑟 mix gas = 𝜙 1 𝑔𝑎𝑠 𝑟 1 + 𝜙 2 𝑔𝑎𝑠 𝑟 2 + 𝑔𝑎𝑠 * (30.4) (32) into the equation (1) as follow 𝜎 4 = 𝜐 4 𝜙 3 𝑟 3 (24.3 ) Similarly, it is possible to obtain the characteristic mer-mer interaction energy of the polymer 𝜌̅ pol 2
	in which 𝜐 * mix gas and 𝜀 * mix gas are calculated based on (5) and (8), respectively, as follow 𝜐 * mix gas = 𝛼𝜙 1 𝑔𝑎𝑠 + 𝛽𝜙 2 𝑔𝑎𝑠 + ϒ 𝜀 * mix gas = 𝐴𝜙 1 𝑔𝑎𝑠 2 + 𝐵𝜙 2 𝑔𝑎𝑠 2 + 𝐶𝜙 1 𝑔𝑎𝑠 𝜙 2 𝑔𝑎𝑠 + 𝐷𝜙 1 𝑔𝑎𝑠 + 𝐸𝜙 2 𝑔𝑎𝑠 + 𝐹 𝛼𝜙 1 𝑔𝑎𝑠 + 𝛽𝜙 2 + 𝐵 2 𝜙 2 pol + 𝐵 3 𝜙 3 pol + 𝐶 1 ) 𝑔𝑎𝑠 + ϒ phase mixture, by expanding equation (8), as follow 𝜀 * mix pol = 1 𝜎 1 𝜙 1 pol + 𝜎 2 𝜙 2 pol + 𝜎 3 𝜙 3 + 𝐴 12 𝜙 1 pol 𝜙 2 pol + 𝐴 23 𝜙 2 pol 𝜙 3 pol + 𝐴 13 𝜙 1 pol 𝜙 3 pol + 𝐵 1 𝜙 1 pol pol + 𝜎 4 (𝐴 1 𝜙 1 pol 2 + 𝐴 2 𝜙 2 pol 2 + 𝐴 3 𝜙 3 pol 2 + 𝑃 (𝜎 1 𝜙 1 pol + 𝜎 2 𝜙 2 pol + 𝜎 3 𝜙 3 pol 2 + 𝜎 4 ) 𝐴 1 𝜙 1 pol 2 + 𝐴 2 𝜙 2 pol 2 + 𝐴 3 𝜙 3 pol 2 + 𝐴 12 𝜙 1 pol 𝜙 2 pol + 𝐴 23 𝜙 2 pol 𝜙 3 pol + 𝐴 13 𝜙 1 pol +𝐵 1 𝜙 1 pol + 𝐵 2 𝜙 2 pol + 𝐵 3 𝜙 3 + 𝐶 1 pol 𝜙 3 pol	(25) (26) (31) (33)

pol = 1, the characteristic closedpacked molar volume of "mer" of polymer phase mixture can be written, based on (5) as follow

Literature Review

𝑃 c,𝑖 𝑏 𝑖 (𝑇 c,𝑖 ) = 0.07780 𝑅 𝑇 c,𝑖 𝑃 c,𝑖 At temperatures other than the critical temperature, we have: 𝑎 𝑖 (T) = 𝑎 𝑖𝑖 (𝑇 𝑐,𝑖 ) . 𝛼 𝑖 (𝑇 r,𝑖 , 𝑤 𝑖 ) 𝑏 𝑖 (T) = 𝑏 𝑖 (𝑇 c,𝑖 )

Literature Review

A novel approach for the estimation of the Sanchez-Lacombe interaction parameters for ternary polyolefins systems

𝑃 c,𝑖(21)

a The properties of 1-butene were taken from reference28

Experimental and theoretical solubility for multicomponent systems

Diffusivity of multicomponent gas mixtures in polyethylene

A single particle model to predict the impact of induced condensing agents on polymerizing particles

Sorption and swelling of liquid diluents in polyethylene

Remerciements

Mme Isabelle Cermelli d'avoir

Where 𝑀 w is the polymer molecular weight, the variance is 𝜎 ln (𝑘) = √ln 𝑀 ̅ w,ov (𝑘) 𝑀 ̅ n,ov (𝑘) , 𝑤(𝑘, 𝑀 w ) is a density distribution as a function of ln 𝑀 w , and 𝑀 ̅ n,ov (𝑘) is the average value of 𝑀 ̅ n (𝑟, 𝑡)

over the whole particle. The total MWD is then the sum of the MWD produced by the different catalyst sites, which is then normalized and transformed to a molar mass density to permit comparison with the experimental MWD given from the HT-GPC: Where 𝑚 ̅ w,m and 𝑚 ̅ w,exp are the modes of the final model MWD, 𝑤 tot (𝑀 w ), and experimental 𝑤 logM w exp (𝑀 w ), respectively. Some parameters were taken from literature, such as the rate constants of transfer to monomer, 𝑘 tm0 , spontaneous termination, 𝑘 tsp0 , as well as the activation energy of the different constants 36 . The parameter 𝑘 iH0 was considered to be equal to 𝑘 i0 , as was assumed by McAuley et al. for instance 37 . For the sake of simplicity, the kinetic parameters were taken the same for all catalyst site types, except for transfer to hydrogen, 𝑘 tH0 . In order to be able to predict the experimental width of the MWD, it was found that the transfer to hydrogen parameters of the different active sites can be related as follows: 𝑘 tH0 1 = 8𝑘 tH0 4 , 𝑘 tH0 2 = 6𝑘 tH0 4 and 𝑘 tH0 3 = 4𝑘 tH0 4 .

Once again, the goal of this modelling study is to show realistic values of the impact of the ICA on the polymerization process, and not to accurately model the reaction precisely for this catalyst. These parameters give us however a reasonable approximation of ZN kinetics so that the thermodynamic effects can be highlighted independently.

reaction rates and MWDs are in good agreements at different compositions of the gas phase (i.e. n-pentane, hydrogen). The model can capture well the effect of n-pentane (as ICA) and hydrogen on the reaction rate and MWD. To the best of our knowledge, this is the first time that a detailed SPM, containing accurate thermodynamic models, is associated to a kinetic model in order to show the impact of the gas phase composition on the dynamics of polymerization of ethylene in gas-phase reactors. A great difference can be seen without hydrogen and with 1-3 bar of hydrogen at the beginning of the reaction, from 5.3 kg/g.cat/h without hydrogen to 2.7 kg/g.cat/h with 3 bar of hydrogen, with 1 bar of npentane. This difference is mainly due to the fact that chain transfer to hydrogen stops the reaction on the active site until PH is reinitiated, and this reinitiation is slower than the initiation of vacant sites. Note that the solubility of hydrogen in the amorphous phase of the polymer is quite low compared to that of ethylene or n-pentane, the concentration of hydrogen is therefore assumed to be constant in the whole particle over time, and not influence the concentration/diffusion of ethylene in the amorphous phase of the polymer. 5 A single particle model to predict the impact of induced condensing agents on polymerizing particles

Effect of the gas phase composition on ethylene concentration gradient

Figure 118 shows ethylene concentration gradients inside the polymer particle, from the surface to the center of the particle. Different amounts of ICA as well as hydrogen are used; 0-2 bar for n-pentane and 0-3 bar for hydrogen. The concentration of ethylene at the surface is the equilibrium concentration calculated from SL EoS for ternary system of ethylene/npentane/LLDPE (c.f. Table 30 and Table 31).

Figure 118. Ethylene concentration gradients as a function of the normalized particle radius at 40 minutes of the reaction for 7 bar of ethylene, 0-2 bar of n-pentane and 1-3 bar of hydrogen at 80°C

Figure 118 shows that ethylene concentration at the surface is not the same for the different amounts of ICA used. The concentration of ethylene at the surface is not only function of ethylene pressure, as one would expect from a basic particle model. This concentration is also Besides, the overheating of the polymer particle during ethylene polymerization in gas phase depends on hydrogen partial pressure. As can be seen in Figure 117 and Figure 118, increasing the hydrogen partial pressure decreases the reaction rate which leads to a reduction in the ethylene concentration inside the amorphous phase of the polymer. Producing less polymer leads to less overheating of the produced polymer particle, especially in presence of ICA that acts as heat dispersing agents. Nevertheless, the use of the Ranz-Marshall correlation to calculate the external film heat transfer coefficient may lead to an overestimation of the particle temperature. Thus the exact values of the temperature gradient reported here need to be taken with caution.

Sorption and Swelling of liquid diluents in polyethylene

The solubility of the liquid diluent in the polymer is calculated as follows:

Where 𝑚(𝑡 = 0) is the mass of the swollen polymer measured at zero time before the desorption and 𝑚 PE end is the mass of the dry polymer after the desorption experiment.

Generally, the dry polymer mass after the experiment is found to be lower than the mass of the dry polymer before the experiment. The reason is that a fraction of the polymer partially dissolves into the liquid diluent during the sorption experiment, as have been seen in the previous section. The dissolution of the amorphous phase of the polymer in the different diluents is therefore taken into account when calculating the solubility of the different liquid diluents in the polymer. The values obtained in the desorption analysis were treated using STARe Software.

Figure 127. Schema of the desorption experiments of PE granules at high temperature

General conclusion and perspectives

General Conclusion and Perspectives

Application of the Sanchez-Lacombe Equation of State for quaternary systems

Model Development Application of the Sanchez-Lacombe equation of state for quaternary systems

The details of the mathematical equations derived from the Sanchez-Lacombe Equation of State in order to describe quaternary systems is presented in this Appendix.

Description of Sanchez-Lacombe EoS model

The Sanchez-Lacombe Equation of State (SL EoS) is defined by

where 𝑇 ̅ , 𝑃 ̅ and 𝜌̅ are the reduced temperature, pressure, and density respectively which are defined as follows

where ε * is the mer-mer interaction energy, 𝑣 * , is the closed packed molar volume of a mer, MW is molecular weight, N is number of molecules, r is the number of sites (mers) a molecule occupies in the lattice, 𝑇 ̅ is the reduced volume and Rg is the universal gas constant. The parameters ε * ,𝜐 * and r are used to define T * , P * , and ρ * which are the characteristic temperature, pressure, and close-packed mass density.

For a mixture of components, defining combining rules for the estimation of 𝜀 mix * , 𝜐 mix * , and 𝑟 mix is necessary to be able to use the equation of state to calculate the properties of mixture.

The Van der Waals mixing rule is chosen and applied in our study.

For characteristic closed-packed molar volume of a "mer" of the mixture, the "Van der Waals" mixing rule, 𝜐 𝑚𝑖𝑥 * , is defined as

with

Where 𝜙 is the volume fraction of the i th and j th component in the mixture, 𝑁 c is the number of component in the mixture and 𝑛 𝑖𝑗 corrects the possible deviation of 𝑣 𝑖𝑗 * from the arithmetic mean value of 𝑣 𝑖𝑖 * and 𝑣 𝑗𝑗 * of the pure component.

The closed-packed volume fraction of the i th component at the limit of zero temperature or incompressible state, 𝜙 𝑖 gas is defined as

where 𝜔 𝑖 gas is the mass fraction of the component i in the gas phase, defined as follows:

The mixing rule for the characteristic interaction energy for the mixture, 𝜀 mix * , is defined as

Where the cross-energy parameter between mers of component i and component j, 𝜀 𝑖𝑗 * , is introduced in order to take into account the interactions between the different components in the mixture, as follows

where 𝜀 𝑖𝑖 * and 𝜀 𝑗𝑗 * are the characteristic mer-mer interaction energies for components i and j, and 𝑘 𝑖𝑗 is an adjustable parameter that accounts for the interactions between components i and j. Application of the Sanchez-Lacombe equation of state for quaternary systems Finally, the mixing rule for the number of sites (mers) occupied by a molecule of the mixture, 𝑟 mix , is given by

where 𝑟 𝑗 is the number of sites occupied by molecule j in the lattice.

For the calculation of sorption equilibrium for polymer-solvent system, the chemical potential of the i th component in each phase of the mixture of multicomponent species can be expressed as:

In order to solve SL EoS, we consider that the chemical potential of each penetrant at equilibrium is equal to the chemical potential of the polymer as follows:

As an example, in binary system composed of a gaseous specie with a polymer, we will have two phase in equilibrium: the gas phase composed of the gaseous specie only and the polymer phase, which is constituted of both polymer and the gaseous specie.

Sanchez-Lacombe EoS for quaternary systems

We consider here a quaternary system containing a monomer (1), an ICA (2), a comonomer (3) and a polymer (4). The corresponding interaction parameters in order to solve SL EoS are : k12, k13, k14, k23, k24, k34 which are the interaction parameters between the monomer and the ICA, the monomer and the comonomer, the monomer and the polymer, the ICA and the comonomer, the ICA and the polymer and the comonomer and the polymer, respectively. Only k14, k24, k34 parameters are estimated by fitting SL EoS to experimental data, while k12, k13 and k24 are equal to zero because we assume that there is no interaction between small olefin molecules.

The first objective here is to calculate the solubility of the different component in the mixture in the polymer phase, knowing their partial pressure in the mixture (i.e. P1, P2, P3) at a specific Application of the Sanchez-Lacombe equation of state for quaternary systems

As a result, according to the equation ( 7), the closed packed volume fraction of components in the gas phase can be found by ) and the reduced gas phase density (𝜌̅ gas ), it is possible to calculate chemical potential of components 1, 2 and 3 in the gas phase following equation (11) as follow The reduced polymer phase density, 𝜌̅ 𝑝𝑜𝑙 , as well as the closed-packed volume fractions of solute components in the polymer phase, 𝜙 1 pol , 𝜙 2 pol and 𝜙 3 pol are calculated by solving simultaneously four non-linear equations, ( 33), ( 34), ( 35) and (36).

Solubility calculation from SL EoS

In order to calculate the solubility of the different species in the mixture in the polymer, the mass fraction of each specie in the polymer has to be calculated. According to equation ( 7), the mass fraction of solute 1, 2 and 3 in the amorphous phase of the polymer can be calculated as follow

Where the mass fraction of the polymer phase is defined as

It is important to note that the sorption of solute species is assumed to take place only in the amorphous phase of the polymer, while the crystalline phase is assumed to be impenetrable to the solute species.

Where 𝑆 𝑖 am is the partial solubility of the i th component in the mixture in the amorphous phase of the polymer, 𝑆 123 am and 𝑆 123 tot are the overall solubility of all the species in the mixture in the amorphous phase of the polymer, and in the entire polymer phase (i.e. amorphous and crystalline), respectively.

Appendix B

Homopolymerization of ethylene and characterization of the produced polymer

Experimental set-up

Appendix B Homopolymerization of ethylene and characterization of the produced polymer

Homopolymerization of ethylene

The experiments described in this Appendix were carried out by a phD student in our group research, M. Niyi B. Ishola.

1.1.Materials

Ethylene and hydrogen, with a minimum purity of 99.5%, were obtained from Air Liquide France. Ethylene was passed over columns of zeolite and active carbon for purifying. n-pentane, with a minimum purity of 99%, was obtained from Sigma-Aldrich ICN -Germany. Argon, with a minimum purity of 99.5%, was obtained from Air Liquide France and was used to keep the reaction environment free of oxygen and other impurities. A commercial TiCl4 supported on MgCl2 Zeigler-Natta catalyst was used as the catalytic system with a Ti content of 2.8 wt% for all polymerizations, and triethylaluminium (TEA) was used as co-catalyst. Sodium chloride (NaCl), with a minimum purity of 99.8%, was carefully dried and used as a seedbed to disperse the catalyst particles.

1.2.Experimental set-up

The gas-phase polymerization experiments are performed in a spherical stirred-bed semi-batch reactor. The reactor operates at a pressure up to 25 bar and temperatures up to 90°C. The reactor is heated by a circulating water through an external jacket, and is equipped with temperature and pressure sensors, vacuum pump and recording computer. This stirred-bed reactor operates in semi-batch mode, meaning that all components involved in the polymerization reaction (i.e. ICA, hydrogen, catalyst, salt) are added to the reactor, and only ethylene is supplied continuously to the reactor during the entire time of the reaction in order to maintain a constant pressure. The detailed experimental set-up and measuring procedure are given elsewhere [1][2][3] .

The rate of polymerization was recorded by measuring the pressure drop of ethylene in the ballast, corresponding to the consumption of ethylene during the entire time of the polymerization. The pressure drop in the ballast was then converted into a molar flow rate using the Soave-Redlich-Kwong (SRK) cubic equation of state.

The experiments performed used 7 bar of ethylene, 0-2 bar of n-pentane and 0 and 3 bar of hydrogen, at 80°C. The experimental conditions of the different studied experiments are given Appendix B Homopolymerization of ethylene and characterization of the produced polymer in Table 1. The experimental results for the homopolymerization without hydrogen are given in Figure 1 and with 3 bar of hydrogen in Figure 2.

2.1.Crystallinity of the polymer

The crystallinity of the polymer samples was measured using differential scanning calorimetry (DSC 3+ by Mettler Toledo). Polyethylene samples were weighed (i.e. around 5-7 mg) and placed in an aluminium capsule with a volume of 40 μL. The sample was first cooled to -20°C and then heated from -20 to 180 °C at a heating rate of 10°C min -1 , held for 5 minutes at that temperature and then cooled from 180 to -20 °C at a rate of 10°C min -1 . This temperature was also maintained for 5 minutes, and the sample was reheated to 180°C at a rate of 10°C min -1 .

The melting behavior of the samples was studied in the second heating step in order to detect the melting point and the melting enthalpy of the sample. The ratio of the melting heat of the sample and the melting heat of the completely crystallized polyethylene was used to calculate the degree of crystallization. A melting heat of 293 J g -1 for completely crystallized polyethylene was used for the calculation. The STARe software was used for data acquisition and data analysis. The crystallinity of the produced PE from the homopolymerization of ethylene in presence of n-pentane was measured equal to 45%.