Amel Ben Mrad

Nida Sheibat-Othman

Jorik Hill

Michael Bartke

Timothy F L Mckenna
email: timothy.mckenna@univ-lyon1.fr

Sanchez-Lacombe Eos

A Novel Approach for the Estimation of the Sanchez-Lacombe Interaction Parameters for the Solubility of Ternary Polyolefins Systems

Keywords: Sanchez-Lacombe, modeling, ternary system, Peng-Robinson, ICA, co-solubility

Experimental solubilities of multicomponent systems through gravimetric method

Introduction

Polyethylene (PE) is the most widely produced polymer on the planet because of its extensive range of properties and end-uses, low production cost, and relatively low environmental impact. Polyethylene can be made in solution, slurry and gas phase processes using ethylene, and very often an α-olefin comonomer. Gas phase processes are of particular interest as they are used for the production of approximately 75% of the linear low-density polyethylene (LLDPE), and 25% of high-density polyethylene (HDPE) worldwide. These processes will be the focus of the current work.

In industrial gas-phase reactors, polyethylene is produced using a catalyst particle that is exposed to a mixture of reactive species that include ethylene, hydrogen, and usually different amounts of an α-olefin comonomer such as 1-butene or 1-hexene [START_REF]Polyolefin Microstructural Modeling[END_REF] . As the ratio of ethylene to comonomer is used to control important physical properties such as the density, it is important to be able to predict accurately the concentration of the monomer at the active sites of the catalysts. In addition, inert compounds, typically alkanes as propane, isobutane, isopentane or n-hexane, often referred to as induced condensing agents (ICA), are used to regulate partial pressures and to improve heat transfer [START_REF] Mckenna | Condensed Mode Cooling of Ethylene Polymerization in Fluidized Bed Reactors[END_REF][START_REF] Iii | Method for fluidized bed polymerization[END_REF][START_REF] Iii | A method for controlling the temperature of a fluidized bed especially a process for producing polymers[END_REF][START_REF] Alizadeh | Condensed Mode Cooling in Ethylene Polymerisation: Droplet Evaporation[END_REF] . All of the species present in the gas phase, inert or not, will diffuse into the amorphous phase of the polymer, and will change the physical and possibly the chemical structure of the final polymer, which in turn can influence important parameters such as the individual species solubility and diffusivity [START_REF] Namkajorn | Condensed-Mode Cooling for Ethylene Polymerization: The Influence of Inert Condensing Agent on the Polymerization Rate[END_REF][START_REF] Alizadeh | Condensed Mode Cooling for Ethylene Polymerization: Part I. The Effect of Different Induced Condensing Agents on Polymerization Rate[END_REF][START_REF] Namkajorn | Condensed Mode Cooling for Ethylene Polymerization: Part III. The Impact of Induced Condensing Agents on Particle Morphology and Polymer Properties[END_REF] . It is therefore quite important to know the solubility of the individual species in gas mixtures in the polymer with precision.

As has been discussed elsewhere [START_REF] Mckenna | Condensed Mode Cooling of Ethylene Polymerization in Fluidized Bed Reactors[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][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 sorption of ethylene, comonomers and alkanes (used as induced condensing agents, ICA, for instance) is non-ideal. In other words, the solubility of a mixture of gases in the amorphous PE 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) [START_REF] Yao | Modeling the solubility of ternary mixtures of ethylene,isopentane,n-hexane in semicrystalline polyethylene[END_REF][START_REF] Mckenna | Solubility and crystallinity data for ethylene/polyethylene systems[END_REF][START_REF] Novak | Ethylene and 1-hexene sorption in LLDPE under typical gas-phase reactor conditions: Experiments[END_REF][START_REF] Chmelař | Equilibrium Sorption of Propane and 1-Hexene in Polyethylene: Experiments and Perturbed-Chain Statistical Associating Fluid Theory Simulations[END_REF][START_REF] Yao | Modeling solubility of gases in semicrystalline polyethylene[END_REF][START_REF] Moore | Solubility of ethylene, 1-butene and 1-hexene in polyethylenes[END_REF][START_REF] Savatsky | Parameterization of Models for Vapor Solubility in Semicrystalline Polyethylene[END_REF][START_REF] Kanellopoulos | Prediction of Solubility of α-Olefins in Polyolefins Using a Combined Equation of State Molecular Dynamics Approach[END_REF][START_REF] Yoon | Solubility of α-olefins in linear low density polyethylenes[END_REF][START_REF] Moebus | Modeling Vapor Solubility in Semicrystalline Polyethylene[END_REF] . 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 [START_REF] Bashir | Modelling of multicomponent olefins solubility in polyolefins using Sanchez-Lacombe equation of state[END_REF] . For instance, Yao et al. [START_REF] Yao | Modeling the solubility of ternary mixtures of ethylene,isopentane,n-hexane in semicrystalline polyethylene[END_REF] measured ethylene-alkane-PE systems by varying the partial pressure of n-hexane and iso-pentane as alkanes. 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 co-monomers in ternary systems [START_REF] Novak | Ethylene and 1-hexene sorption in LLDPE under typical gas-phase reactor conditions: Experiments[END_REF][START_REF] Nath | Simulation of Ternary Mixtures of Ethylene, 1-Hexene, and Polyethylene[END_REF][START_REF] Sun | Solubility measurement of hydrogen, ethylene, and 1-hexene in polyethylene films through an intelligent gravimetric analyzer[END_REF] . Chmelar et al. [START_REF] Chmelař | Equilibrium Sorption of Propane and 1-Hexene in Polyethylene: Experiments and Perturbed-Chain Statistical Associating Fluid Theory Simulations[END_REF] showed that the sorption of 1-hexene increases ethylene solubility in the amorphous PE. This phenomenon is often referred to as the co-solubility effect. These same authors also demonstrated that increasing ethylene pressure can lead to a decrease in the solubility of 1hexene at a constant 1-hexene pressure, considered as the anti-solvent effect. Furthermore, a recent series of articles from our research group has shown that adding ICA to a gas phase polymerization will increase the reaction rate due to the co-solubility effect [START_REF] Alizadeh | Condensed Mode Cooling for Ethylene Polymerization: Part I. The Effect of Different Induced Condensing Agents on Polymerization Rate[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] . Alizadeh et al. [START_REF] Alizadeh | Condensed Mode Cooling for Ethylene Polymerization: Part I. The Effect of Different Induced Condensing Agents on 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 seen for n-hexane [START_REF] Alizadeh | Modeling Condensed Mode Operation for Ethylene Polymerization: Part III. Mass and Heat Transfer[END_REF] , where they showed 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. These observed increases in the polymerization rate were attributed in large part to the co-solubility effect, but also to an increase in the diffusivity of ethylene through the amorphous fraction of the polymer [START_REF] Namkajorn | Condensed-Mode Cooling for Ethylene Polymerization: The Influence of Inert Condensing Agent on the Polymerization Rate[END_REF][START_REF] Alizadeh | Condensed Mode Cooling for Ethylene Polymerization: Part I. The Effect of Different Induced Condensing Agents on Polymerization Rate[END_REF][START_REF] Namkajorn | Condensed Mode Cooling for Ethylene Polymerization: Part III. The Impact of Induced Condensing Agents on Particle Morphology and Polymer Properties[END_REF] .

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. 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. It is clear that one cannot use binary sorption models to estimate the concentration of a given species in the polymer for multicomponent systems and simply add them to describe penetrants concentration in the system in order to predict their solubilities. It is necessary to account for the way that they interact with the polymer in a ternary system. This is of course possible using equations of state. The Sanchez-Lacombe equation of state (SL EoS) and the Perturbated Chain Statistical Associating Fluid Theory (PC SAFT) are two such models that are widely used in the field of polymer reaction engineering. In the PC-SAFT model, the molecules are assumed to be constituted of spherical segments of fixed diameter, freely joined. Detailed information about this equation of state can be found elsewhere [START_REF] Gross | Modeling Polymer Systems Using the Perturbed-Chain Statistical Associating Fluid Theory Equation of State[END_REF][START_REF] Krallis | Application of Sanchez-Lacombe and Perturbed-Chain Statistical Associating Fluid Theory Equation of State Models in Catalytic Olefins (Co)polymerization Industrial Applications[END_REF] . Sanchez-Lacombe is a model based on the lattice theory, meaning that pure components are assumed to be broken into parts and placed into a lattice structure [START_REF] Sanchez | An elementary molecular theory of classical fluids. Pure fluids[END_REF] . The main feature of this model is that it introduces vacant lattice sites or holes, and varying the fraction of holes varies the density and the compressibility. Both of these EoS have been used successfully [START_REF] Alizadeh | Modeling Condensed Mode Operation for Ethylene Polymerization: Part I. Thermodynamics of Sorption[END_REF] , but we have chosen to use the SL EoS because of its relative simplicity, which can be easily adapted for the use in single particle and reactor models [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][START_REF] Alizadeh | Study of sorption, heat and mass transfer during condensed mode operation of gas phase ethylene polymerization on supported catalyst[END_REF] , and especially can be extended to multicomponent systems (as we will discuss in a future paper). The main drawbacks to both of these approaches are: (1) they require empirical, adjustable parameters that are specific to a given system; and (2) 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 the penetrant solubility in the amorphous phase à priori.

The adjustable parameters in the SL EoS are the interaction parameters between the penetrant molecules and the polymer (if we assume negligible the interactions between the small penetrant molecules), that need to be estimated experimentally. 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, 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 and (very occasionally) ternary systems. In terms of alkenes, data is available for 1hexene solubility for different temperatures, pressures and PE crystallinity [START_REF] Novak | Ethylene and 1-hexene sorption in LLDPE under typical gas-phase reactor conditions: Experiments[END_REF][START_REF] Chmelař | Equilibrium Sorption of Propane and 1-Hexene in Polyethylene: Experiments and Perturbed-Chain Statistical Associating Fluid Theory Simulations[END_REF][START_REF] Moore | Solubility of ethylene, 1-butene and 1-hexene in polyethylenes[END_REF][START_REF] Savatsky | Parameterization of Models for Vapor Solubility in Semicrystalline Polyethylene[END_REF][START_REF] Kanellopoulos | Prediction of Solubility of α-Olefins in Polyolefins Using a Combined Equation of State Molecular Dynamics Approach[END_REF][START_REF] Yoon | Solubility of α-olefins in linear low density polyethylenes[END_REF][START_REF] Moebus | Modeling Vapor Solubility in Semicrystalline Polyethylene[END_REF] . A more limited set of data is available for 1-butene [START_REF] Moore | Solubility of ethylene, 1-butene and 1-hexene in polyethylenes[END_REF][START_REF] Savatsky | Parameterization of Models for Vapor Solubility in Semicrystalline Polyethylene[END_REF][START_REF] Kanellopoulos | Prediction of Solubility of α-Olefins in Polyolefins Using a Combined Equation of State Molecular Dynamics Approach[END_REF][START_REF] Yoon | Solubility of α-olefins in linear low density polyethylenes[END_REF][START_REF] Moebus | Modeling Vapor Solubility in Semicrystalline Polyethylene[END_REF] and 1-octene [START_REF] Kanellopoulos | Prediction of Solubility of α-Olefins in Polyolefins Using a Combined Equation of State Molecular Dynamics Approach[END_REF][START_REF] Yoon | Solubility of α-olefins in linear low density polyethylenes[END_REF] . In the case of ICAs, the solubilities of n-hexane, isopentane, and isobutane 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,isopentane,n-hexane in semicrystalline polyethylene[END_REF][START_REF] Savatsky | Parameterization of Models for Vapor Solubility in Semicrystalline Polyethylene[END_REF][START_REF] Moebus | Modeling Vapor Solubility in Semicrystalline Polyethylene[END_REF] , but very limited data can be found for the sorption of propane in PE despite it being an important diluent for a number of processes [START_REF] Chmelař | Equilibrium Sorption of Propane and 1-Hexene in Polyethylene: Experiments and Perturbed-Chain Statistical Associating Fluid Theory Simulations[END_REF] . Finally, 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 solubility 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 study, 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. These systems were measured at 70°C and total pressures up to 5 bars. Since individual solubilities are not available with the suspension balance, we used 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 measured solubilities and compressibility factors were then fitted to the the Sanchez-Lacombe equation of state in order to identify the SL interaction parameters. 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, 1-butene, isobutane, 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) powders having a crystallinity of 48.3% were graciously supplied by Borealis.

2.2.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 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 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 [START_REF] Chmelař | Equilibrium sorption of ethylene in polyethylene: Experimental study and PC-SAFT simulations[END_REF][START_REF] Cancelas | Solubility and Diffusivity of Propylene, Ethylene, and Propylene-Ethylene Mixtures in Polypropylene[END_REF][START_REF] Kröner | Sorption of Olefins in High Impact Polypropylene -Experimental Determination and Mass Transport Modeling[END_REF] , 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 (

). 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.

(

) 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:

Because 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.

Where is the crystallinity fraction. Kosek et al. [START_REF] Chmelař | Equilibrium sorption of ethylene in polyethylene: Experimental study and PC-SAFT simulations[END_REF] demonstrated that taking into account the impact of the temperature on the crystallinity 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 the differential scanning calorimetry (DSC 3+, Mettler Toledo).

Thermodynamic modeling

3.1. Model development

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 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 [START_REF] Sanchez | An elementary molecular theory of classical fluids. Pure fluids[END_REF] . The main equation of state is given by:

(

where , are the reduced temperature, pressure and density, respectively. is the number of sites (mers) a molecule occupies in the lattice. These reduced parameters, depending on the characteristic properties, are defined as follows:

(

where , , and are the scale factors known as the characteristic temperature, pressure, and density respectively, which are used to characterize each pure component in the mixture.

is the reduced volume. These characteristic parameters depend on three lattice parameters, describing the thermodynamic properties of a pure components: ε * , and r, where ε * is the mer-mer interaction energy (J mol -1 ) and the closed-packed molar volume of a mer (m 3 mol -1 ).

(

) 4
Where MW is the molecular weight of the component and R is the universal gas constant (J K - 1 mol -1 ).

The parameters of SL EoS for each pure component are given in Table 1. The characteristic closed-packed molar volume of a "mer" of the mixture, is defined as follow:

(5)

where N c denotes the number of components in the mixture (6) Where is the volume fraction of the i th and j th component in the mixture and corrects the possible deviation of from the arithmetic mean value of and of the pure components. In the present study, the value of the interaction parameter 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, is defined as: (7) where is the mass fraction of the component i in the mixture.

The mixing rule for the characteristic interaction energy for the mixture is defined as

(8)
The cross-energy parameter between mers of component (i) and component (j), is introduced in order to take into account the interactions between components in mixtures. (9) in which and are the interation energy between mers of component i and j, respectively.

is a binary interaction parameter between the i th and j th component in the 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, , is given by (10) where is the number of sites occupied by molecule j in the lattice.

For calculation of sorption equilibrium for polymer-penetrant system, the chemical potential of the i th component (i.e. monomer, ICA, comonomer) in a multicomponent system can be expressed as (11) Where ( 12) (13) We consider that the chemical potential of each component at equilibrium is equal to the chemical potential of the polymer as follows:

i=1,2… (14)
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

(15) (16) (17)
Finally, the solubility of each component at equilibrium, in gram of component per gram of polymer, S, can be calculated as follows

i,j = 1,2
where 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. [START_REF] Alizadeh | Study of sorption, heat and mass transfer during condensed mode operation of gas phase ethylene polymerization on supported catalyst[END_REF]

Peng-Robinson EoS

The Peng-Robinson (PR) equation of state [START_REF] Peng | A New Two-Constant Equation of State[END_REF] for gas mixture is defined as (18) Where is molar volume. The parameter is a measure of the attractive forces between molecules in the mixture and the parameter is the van der Waals co-volume occupied by these molecules in the mixture. These parameters are defined as follow:

(19) (20)
and are the respective PR parameters for pure substances. is the mole fraction of each component in the mixture and is the binary interaction parameter. This interaction parameter is identified using experimental data, but very few experimental data are available in the literature. The measure of the compressibility factor when using identified 36 compared to that assuming shows a difference of less than 0.4% for ethylene/propane mixture, at the same conditions than the gravimetric experiments. The detailed calculation can be found in the Supporting Information. Therefore, as for SL model, we will consider for this study that there is no interaction between small molecules, meaning that .

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

At temperatures other than the critical temperature, we have:

(23) (24)
A relationship between and the reduced temperature has be made and linearized by Peng and Robinson [START_REF] Peng | A New Two-Constant Equation of State[END_REF] as follow (25) Since the acentric factors of all studied species are , , which is a constant characteristic of each substance, is defined as follow:

Equation ( 16) can be rewritten as:

Where ( 28)

(29) ( 30
)
Where Z is the compressibility factor. Table 2 indicates the required properties of few components to implement the PR equation. a The properties of 1-butene were taken from reference [START_REF] Ambrose | Vapour-Liquid Critical Properties[END_REF] .

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.

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 k 12 , k 13 , k 23 which are the interactions between the monomer and the penetrant, the monomer and the polymer and the penetrant and the polymer, respectively. k 12 is assumed equal to zero due to low chemical affinity between like molecules. The interaction parameters are used to define the cross interaction energy ; when =0, the cross potential follows Lorentz-Berthelot combining rules [START_REF] Pedrosa | Phase Equilibria Calculations of Polyethylene Solutions from SAFT-Type Equations of State[END_REF] (i.e. )

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 the global solubility 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), the compressibility factor is given as follows:

(

We can then extend Equation ( 2) from SL EoS to calculate the compressibility factor [START_REF] Mirzaei | Fluidized Bed Polyethylene Reactor Modeling in Condensed Mode Operation[END_REF] as follows:

(32)

The identification of the two interaction parameters is now possible by fitting three "experimental" data (one 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 [START_REF] Peng | A New Two-Constant Equation of State[END_REF] . 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 k ij interaction parameters is presented in In this work, a novel approach consisting of combining the compressibility factor calculated with the Peng-Robinson EoS plus experimental global solubility results has been used in order 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 solubility of species in ternary systems from the new experimental measurements of the overall solubility using a magnetic suspension balance.

4.1.Model validation on literature data

Cancelas et. al. [START_REF] Cancelas | Solubility and Diffusivity of Propylene, Ethylene, and Propylene-Ethylene Mixtures in Polypropylene[END_REF] 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 k 23 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 2 shows the experimental overall solubilities measured by Cancelas et al. [START_REF] Cancelas | Solubility and Diffusivity of Propylene, Ethylene, and Propylene-Ethylene Mixtures in Polypropylene[END_REF] 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 2, Figure 3 and Figure 4, the symbols represent experimental values of the total solubility, and the lines represent the theoretical solubilities calculated with the SL-PR method. The points are desorption measurements from [START_REF] Cancelas | Solubility and Diffusivity of Propylene, Ethylene, and Propylene-Ethylene Mixtures in Polypropylene[END_REF] and the curves are SL-PR predictions.

A comparison of the ternary interaction parameters of ethylene/propylene mixture in iPP identified in this study and those identified by Cancelas et al. [START_REF] Cancelas | Solubility and Diffusivity of Propylene, Ethylene, and Propylene-Ethylene Mixtures in Polypropylene[END_REF] is given in Table 3. Note that the interaction parameters are temperature-dependent. Indeed, they identified binary k ij 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 k ij were then used for ternary system of ethylene/propylene/iPP, but k 23 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 with the novel approach SL-PR and the ones determined by Cancelas et al. [START_REF] Cancelas | Solubility and Diffusivity of Propylene, Ethylene, and Propylene-Ethylene Mixtures in Polypropylene[END_REF] . However, in the SL-PR method, no readjustment 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 3 and The second system used to validate this approach is ethylene/1-hexene/LLDPE. Novak et al. [START_REF] Novak | Ethylene and 1-hexene sorption in LLDPE under typical gas-phase reactor conditions: Experiments[END_REF] 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 4 and compared to the interaction parameters obtained by Bashir et al. [START_REF] Bashir | Modelling of multicomponent olefins solubility in polyolefins using Sanchez-Lacombe equation of state[END_REF] . These last authors fitted the experimental values of the total solubility found by Novak et al. [START_REF] Novak | Ethylene and 1-hexene sorption in LLDPE under typical gas-phase reactor conditions: Experiments[END_REF] with the SL EoS in a manner similar to how Cancelas et al. [START_REF] Chmelař | Equilibrium sorption of ethylene in polyethylene: Experimental study and PC-SAFT simulations[END_REF] did for the ethylene/propylene/iPP system, i.e. they started from the binary interaction parameters identified in binary systems by Novak et al. [START_REF] Novak | Ethylene and 1-hexene sorption in LLDPE under typical gas-phase reactor conditions: Experiments[END_REF] 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 6 and Figure 7. These figures show the solubility of ethylene and 1-hexene in ethylene/1-hexene/LLDPE mixture, respectively, at 70, 90 and 150°C. 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 al. [START_REF] Novak | Ethylene and 1-hexene sorption in LLDPE under typical gas-phase reactor conditions: Experiments[END_REF] data and then fitted to the binary SL EoS. The correlations of binary k ij for ethylene/LLDPE and 1-hexene/LLDPE, with respect to temperature are listed in Table 5.

Figure 8 and Figure 9 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. [START_REF] Novak | Ethylene and 1-hexene sorption in LLDPE under typical gas-phase reactor conditions: Experiments[END_REF] (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 8 and Figure 9 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 1-hexene 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. 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. k ij for ternary system were taken from Table 4 and k ij for binary systems were taken from Table 5.

Figure 9. 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. k ij for ternary system were taken from Table 4 and k ij 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 6 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 be then correlated using Sanchez-Lacombe combined to Peng-Robinson (i.e. SL-PR) to predict the individual solubility data and the interaction parameters. The results of this approach are shown in Figure 10 through It can be seen from Figure 10 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 k ij from SL EoS for the different mixtures, the values of which are shown in Table 7. 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 Once the k ij identified by this new approach, we can calculate the partial solubility of each component in the mixture. Figure 14, Figure 15 and Figure 16 show the global and partial solubility of each component of the mixture in the amorphous phase of LLDPE, 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). isobutane (iC 4 ) and 1-butene (1C 4 ) at 70°C. Gas phase composition is given in Table 6 and k ij are defined in Table 7.

Figure 18 compares 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. The same conditions (i.e. pressure, gas-phase composition, temperature) as the experimental ones were taken for these simulations (c.f. Table 6). 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.

Note that the molar fractions were not exactly the same for all the components (c.f. Table 6). However, if we consider 1-butene for instance, its molar fraction was lower than isobutane and propane (x 1C4 =0.383, x iC4 =0.515 and x C3 =0.491). Even with a lower fraction of 1-butene, the co-solubility effect was higher than in presence of isobutane or propane. 7.

Comparison to binary solubility of C 2 /LLDPE, where points represent the experimental solubility and the line represents the solubility fitted from SL EoS. k ij are defined in Table 8 Figure 19 shows the solubility of propane, isobutane and 1-butene in binary (diluent/PE) and ternary (ethylene/diluent/PE) systems. The same conditions as the experimental ones (c.f. Table 6) 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 antisolvent to ICA by decreasing its partial solubility in the amorphous LLDPE. Besides, we can

Conclusion

The overall equilibrium solubility of ethylene/propane/LLDPE, ethylene/isobutane/LLDPE and ethylene/1-butene/LLDPE 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 overall solubilities were then fitted to the Sanchez-Lacombe equation of state.

In this work, a novel approach consisting of implementing Peng-Robinson 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 the 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 is 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 will be important when extending SL EoS to quaternary systems, since we will have three interaction parameters that needs to be identified from the SL EoS.

Figure 1 .

 1 Figure 1. The identification of the two interaction parameters in the SL EoS for ternary

Figure 2 .

 2 Figure 2. Overall solubilities of an equimolar ethylene/propylene mixture in iPP at 50, 70 and 85°C.

Figure 4 .

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

Figure 3 .

 3 Figure 3. Ethylene partial solubility in an equimolar ethylene/propylene mixture in iPP at 50, 70 and 85°C from SL. k ij were taken from

Figure 4 .

 4 Figure 4. Propylene partial solubility in an equimolar ethylene/propylene mixture in iPP at 50, 70 and 85°C from SL. k ij were taken from

Figure 5 ,

 5 Figure 5, the overall experimental solubility of ethylene/1-hexene mixture in LLDPE is

Figure 5 .

 5 Figure 5. 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 13 and the curves

Figure 13 . 11 -Figure 13 .

 131113 Figure 13. It can be seen from Figure 10 that, as expected, the total solubility increases as the

Figure 10

 10 Figure10 being on the order of 1 -4%, indicating that we can have confidence in these

Figure 11 -

 11 Figure13show 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.

Figure 10 .

 10 Figure 10. Total solubility of ethylene/penetrant mixture in LLDPE in presence of different penetrants

Figure 11 .

 11 Figure 11. Compressibility factor of ethylene (C 2 ) and propane (C 3 ) in C 2 /C 3 /LLDPE mixture. Points represent compressibility values calculated with Peng-Robinson and the lines represent the predictions

Figure 12 .

 12 Figure 12. Compressibility factor of ethylene (C 2 ) and isobutane (iC 4 ) in C 2 /iC 4 /LLDPE mixture. Points represent compressibility values calculated with Peng-Robinson and the lines represent the

Figure 13 .

 13 Figure 13. Compressibility factor of ethylene (C 2 ) and 1-butene (1C 4 ) in C 2 /1C 4 /LLDPE mixture. Points represent compressibility values calculated with Peng-Robinson and the lines represent the

Figure 14 .

 14 Figure 14. Total solubility S tot (SL), partial solubility of ethylene S C2 (SL) and partial solubility of propane S C3 (SL) in LLDPE at 70°C from SL-PR method (ternary system C 2 /C 3 /LLDPE). k ij are defined in Table7.

Figure 15 .

 15 Figure 15. Total solubility S tot (SL), partial solubility of ethylene S C2 (SL) and partial solubility of isobutane S iC4 (SL) in LLDPE at 70°C from SL EoS (ternary system C 2 /iC 4 /LLDPE). k ij are defined inTable 7.

Figure 17 .

 17 Figure 17. Degree of LLDPE swelling caused by the sorption of different penetrants; propane (C 3 ),

Figure 18 .

 18 Figure 18. Partial ternary solubility of ethylene from SL EoS in presence of C 3 , iC 4 and 1C 4 with the molar fractions: x 1C4 =0.383, x iC4 =0.515 and x C3 =0.491; k ij for ternary systems are defined in Table7.

Figure 19 .

 19 Figure 19. Solubility of propane, isobutane and 1-butene in binary (diluent/LLDPE) and ternary

Table 1 .

 1 Pure component parameters used in SL EoS

	Component	(K)	(bar)	(kg/m 3 )	Reference
	Ethylene	283	3395	680	29
	Propane	371	3090	690	28
	Isobutane	398	2840	720	28
	1-butene	410	3350	770	34
	1-hexene	450	3252	814	24
	Propylene	345.4	3788	755	32
	LLDPE	667	4370	900	29
	Polypropylene	690.6	3007	885.6

32

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. , , ) based on the respective values of the pure component values.

Table 2 .

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

		Ethylene Propane	Isobutane	1-butene a	1-hexene Propylene	Ref
	w (-)	0.089	0.153	0.183	0.191	0.28	0.144	37
	T c (K)	282.36	369.83	407.82	419.95	504.03	364.76	38
	P c (bar)	50.318	42.471	36.4	40.4	31.4	46.126	38
	MW (g.mol -1 )	28.054	44.1	58.12	56.108	84.16	42.08

Table 3 .

 3 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.[START_REF] Cancelas | Solubility and Diffusivity of Propylene, Ethylene, and Propylene-Ethylene Mixtures in Polypropylene[END_REF]

		T = 50 °C	T = 70 °C	T = 85 °C
	k 13 from 32	-0.01	-0.015	-0.02
	k 13 SL-PR	-0.0095	-0.0157	-0.0192
	k 23 from 32	0.01	0.005	-0.005
	k 23 SL-PR	0.0137	-0.00026	-0.00043

Table 3 .

 3

Table 4 .

 4 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.[START_REF] Novak | Ethylene and 1-hexene sorption in LLDPE under typical gas-phase reactor conditions: Experiments[END_REF] with the interaction parameters found byBashir et al.

21 T = 70 °C T = 90 °C T = 150 °C k 13 from 21

		0.038	0.038	-0.05
	k 13 SL-PR	0.04188	0.0313	-0.01393
	k 23 from 21	0.027	0.016	-0.03
	k 23 SL-PR	0.04894	0.0403	-0.03258

Figure 6. Ethylene partial solubility in ethylene/1-hexene/LLDPE ternary mixture at 70, 90 and 150°C calculated from SL EoS. k ij were taken from

Table 4 .

 4 29Figure 7. 1-hexene partial solubility in ethylene/1-hexene/LLDPE ternary mixture at 70, 90 and 150°C calculated from SL EoS. k ij were taken from

Table 4 .

 4

Table 5 .

 5 Temperature dependence of k ij for ethylene/LLDPE and 1-hexene/LLDPE binary systems. T

	is in °C
	System	k ij

hexene/LLDPE Figure 8.

Table 6 .

 6 Gas-phase molar composition of the different studied ternary mixtures

	System	Composition
		0 < P tot < 5 bar
	Ethylene / Propane / LLDPE
		x C2 = 0.509 / x C3 = 0.491
		0 < P tot < 5 bar
	Ethylene / Isobutane / LLDPE
		x C2 = 0.485 / x iC4 = 0.515
		0 < P tot < 4 bar
	Ethylene / 1-Butene / LLDPE
		x C2 = 0.617 / x 1C4 = 0.383

Table 7 .

 7 Identified ternary interaction parameters by fitting experimental solubility data to SL EoS for

	different systems and temperatures
	Systems	k ij at 70°C
		k 13 = -0.03329
	Ethylene / Propane / LLDPE
		k 23 = 0.03905
		k 13 = -0.06856
	Ethylene / Isobutane / LLDPE
		k 23 = 0.04839
		k 13 = -0.09495
	Ethylene / 1-Butene / LLDPE
		k 23 = 0.04618

Table 8 .

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

	System	Binary k ij	Ref
	Ethylene/LLDPE	0.0546	Our work
	Propane/LLDPE	0.023

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 k ij 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.

1-Butene/LLDPE0.0418 Fitting to[START_REF] Chmelar | Softening of polyethylene powders at reactor conditions[END_REF]

Acknowledgements

ABM, NSO and TFLM gratefully acknowledge the support of the Agence Nationale de la Recherche (Projet ANR-16-CE93-0001 -Thermopoly). The authors would also like to acknowledge Ms. Rita Ferreira Alves for her helpful discussions on the model development.

 7.

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 7). Figure 17 illustrates the degree of LLDPE swelling in presence of propane, isobutane and 1-butene at 70°C at the same experimental conditions (c.f. Table 6). The evolution of polymer swelling as a function of the total pressure follows the same trends as that observed for the solubility. It can be seen that 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.