Estela Kamile Gelinski

N Sheibat-Othman

T F L Mckenna

Mass Transfer in Emulsion Polymerization: an experimental and modelling study

Keywords: Emulsion polymerization, mass transfer, agitation, vinylidene fluoride, modelling

A model is proposed to describe the impact of interphase monomer transfer on the emulsion polymerization of vinylidene fluoride (VDF). The model is validated with experimental data of the rate of polymerization and particle size distribution (PSD). During a typical emulsion polymerization process, the VDF monomer is in a gas or supercritical state, and thus much lighter than the aqueous phase in which the polymerization takes place. For this reason, the flux of monomer into the aqueous phase can depend on the type, number and distribution of agitators on the shaft, as well as the rotation rate. The new model incorporates the interphase mass transfer coefficient into a standard emulsion polymerization model to demonstrate the impact of the agitation (in the broadest sense) on the rate of polymerization and the PSD.

INTRODUCTION

As with many other chemical processes, the modelling of emulsion polymerization can be treated by levels of varying complexity. It is possible to use population balance methods to include any number of quality-related quantities including the particle size distribution (PSD), number of radicals in the particles, molecular weight and copolymer composition distributions. For instance, for the PSD, this approach allows us to account for different physical phenomena, including particle nucleation, growth, and coagulation. However, even if one wishes to use simpler approaches to calculate more basic information such as the rate of homopolymerization (𝑅 ) with particles of identical size using equation [START_REF] Ramirez | Kinetics of the styrene eçulsion polymerization above cmc. II. Agitation effect on molecular weight[END_REF], one thing remains common to all the models, namely the need to calculate the concentration of monomer in the polymerizing particles.

𝑅 𝑘 M (1)

Where 𝑘 is the rate constant for propagation, M the monomer concentration in the particles, 𝑛 the average number of radicals per particle, 𝑁 the number of particles per m 3 of latex, and 𝑁 is Avogadro's number.

In an emulsion polymerization most of the polymer is produced inside the polymer particles. Monomer, present in the form of droplets in the early stages or as a separate feed phase in semi batch processes, must therefore be transferred through the continuous phase to the polymer particles. While it is theoretically possible for this to happen by diffusion, such mass transfer generally requires a sufficient level of agitation to increase the surface area between the phases, and proceed at a commercially viable rate. Despite the potential importance of agitation on emulsion polymerizations, it appears that very little academic research has been invested in this aspect of a widely studied process. It is often assumed that just so long as there is a sufficient level of agitation to assure decent heat transfer and homogeneity in the reactor, there is no need to consider this aspect any further. However, in certain instances, it is necessary to integrate the impact of agitation to fully understand what is taking place in the reactor. Table 1 presents the results of several studies on the influence of agitation rate (N) in emulsion polymerization found in the literature. This is not an exhaustive treatment of the subject, and we have voluntarily excluded studies in round bottomed flasks with agitation by magnetic stirring, and only included polymers made by conventional free radical mechanism.

If we consider the different systems with only liquid monomers, there are 2 major trends that appear. It is important to maintain a minimum rate of agitation to avoid phase separation of the droplets present during the initial stages of the polymerization. Without this, the monomers that are lighter than water tend to pool on the surface [START_REF] Ramirez | Kinetics of the styrene eçulsion polymerization above cmc. II. Agitation effect on molecular weight[END_REF][START_REF] Kemmere | Colloidal Stability of High-Solids Polystyrene and Polyvinyl Acetate Latices[END_REF][START_REF] Roudsari | Experimental and CFD Investigation of the Mixing of MMA Emulsion Polymerization in a Stirred Tank Reactor[END_REF][START_REF] Arai | Agitation effect on the rate of soapless emulsion polymerization of methyl methacrylate in water[END_REF][START_REF] Zubitur | Agitation effects in the semicontinuous emulsion polymerization of styrene and butyl acrylate[END_REF][START_REF] Hong | Effects of Agitation in Emulsion Polymerization -Kinetic and Mechanistic Study of Coagulum[END_REF]. Beyond a certain point, increasing the intensity of agitation seems to provoke a drop in the polymerization rate, usually attributed to fewer particles. Fewer particles can be the result of high agitation rates causing monomer droplets to be more finely divided. This increases their surface area and thus uses surfactant for stabilization of the droplets that could otherwise be used to nucleate particles. [START_REF] Arai | Agitation effect on the rate of soapless emulsion polymerization of methyl methacrylate in water[END_REF][START_REF] Zubitur | Agitation effects in the semicontinuous emulsion polymerization of styrene and butyl acrylate[END_REF] Also higher agitation can provoke the orthokinetic coagulation of the particles. [START_REF] Ramirez | Kinetics of the styrene eçulsion polymerization above cmc. II. Agitation effect on molecular weight[END_REF][START_REF] Roudsari | Experimental and CFD Investigation of the Mixing of MMA Emulsion Polymerization in a Stirred Tank Reactor[END_REF][START_REF] Nomura | Effect of stirring on the emulsion polymerization of styrene[END_REF][START_REF] Roudsari | Using a Novel CFD Model to Assess the Effect of Mixing Parameters on Emulsion Polymerization[END_REF][START_REF] Roudsari | Mixing Effect on Emulsion Polymerization in a Batch Reactor[END_REF][START_REF] Oprea | Influence of agitation during emulsion polymerization of acrylic-styrene latexes on end product properties[END_REF][START_REF] Chern | Semi-batch Surfactant-Free Emulsion Polymerization of Butyl Acrylate[END_REF] Finally, in studies where highly insoluble species are present, increasing the agitation rate can have different kinetic effects. In the case of an emulsion polymerization in the presence of highly insoluble dodecyl mercaptan (DDM) as a chain transfer agent (CTA), increasing the agitation rate increases the rate of mass transfer of the DDM to the particles. The results show no impact on the observed Rp, but a reduction in the average molecular weight. [START_REF] Zubitur | Agitation effects in the semicontinuous emulsion polymerization of styrene and butyl acrylate[END_REF][START_REF] Mendoza | Kinetics of the styrene emulsion polymerization using n-dodecyl mercaptan as chain transfer agent[END_REF] In one of these studies, an effect was seen only when passing from 70 to 150 rpm, but no change in the average molecular weight was observed beyond 150 rpm. [START_REF] Zubitur | Agitation effects in the semicontinuous emulsion polymerization of styrene and butyl acrylate[END_REF] This underlines the fact that the design and geometry of the reactor and the agitation system can be quite important, and that it is necessary to be cautious when drawing conclusions on agitation from a single paper. Similar effects were seen in cases where very sparingly soluble monomer such as butyl methacrylate was used, [START_REF] Hong | Effects of Agitation in Emulsion Polymerization -Kinetic and Mechanistic Study of Coagulum[END_REF] or when studying the impact of inhibitors such as dissolved oxygen. [START_REF] Kiparissides | Continuous Emulsion Polymerization of Vinyl Acetate. Part I: Experimental Studies[END_REF][START_REF] Krishnan | Effects of agitation on oxygen inhibition, particle nucleation, reaction rates, and molecular weights in emulsion polymerization of n-butyl methacrylate[END_REF] The authors of reference [START_REF] Krishnan | Effects of agitation on oxygen inhibition, particle nucleation, reaction rates, and molecular weights in emulsion polymerization of n-butyl methacrylate[END_REF] proposed a process model based on the use of gas-liquid mass transfer coefficient (𝑘 𝑎, see below) to account for the increased inhibition by oxygen and induction period seen at higher agitation rates.  Some minimum level of dispersion is needed to avoid mass transfer limitations due to droplet coalescence.  This will depend on vessel size, monomer content and impeller type.  𝑅 increased with N from 250 to 360 rpm, and particle diameter (𝑑 ) decreased.  Beyond 360 rpm little change was noticed.  Scale-up with Rushton turbine is the best at constant tip speed, for pitched blade at constant energy dissipation.  𝑅 increases from 20 to 250 rpm, then coagulation of particles results in a decrease upon increase to 350 rpm.  Foaming observed at 350 rpm. [START_REF] Roudsari | Experimental and CFD Investigation of the Mixing of MMA Emulsion Polymerization in a Stirred Tank Reactor[END_REF][START_REF] Roudsari | Using a Novel CFD Model to Assess the Effect of Mixing Parameters on Emulsion Polymerization[END_REF][START_REF] Roudsari | Mixing Effect on Emulsion Polymerization in a Batch Reactor[END_REF]  Adding baffles causes drop in 𝑅 at 250 rpm, but a higher 𝑀 .  Addition of a second impeller leads to slightly higher 𝑅 at same N Methylmethacrylate (L)  Phase separation noted for N < 300 rpm.  Ran with low surfactant and low solids.  Induction periods seen to increase as N increases when oxygen present (no induction noted with no oxygen in reactor). Attribution to enhanced gas-liquid mass transfer at high N.  Increasing N in presence of oxygen led to an increase in 𝑁 from 7.010 16 at 300 rpm to 9.810 16 at 800 rpm  Propose a model based on gas-liquid mass transfer to explain the effect of oxygen.


[14]

Butyl Acrylate (L)  Surfactant-free semi-batch polymerization in a 1 L glass reactor with a 4-bladed agitator at 80 °C.  Total solids content up to 40 %.  Significant coagulation as N increased from 400 to 800 rpm. The authors estimate there is no coagulation at 400 rpm.  Lower yields at higher N, probably linked to lower 𝑁 . [START_REF] Chern | Semi-batch Surfactant-Free Emulsion Polymerization of Butyl Acrylate[END_REF] Vinyl Acetate (L)  Glass batch reactor with Teflon stirrer, and agitation rates of 75, 150 and 220 rpm.  Polymerization at 60°C.

 Phase separation at 75 rpm.  𝑅 increases with increasing N. [START_REF] Bataille | The Effect of Agitation on the Conversion of Vinyl Acetate Emulsion Polymerization[END_REF] Systems with at least one gaseous monomer

Ethylene (Vapor, V) + Vinyl Acetate (L)
 Semi-batch polymerization in 2 L reactor with ethylene introduction through a sparger, and a height to diameter ratio (H/D) of 2.6.  Pitched blade and Rushton turbines with N between 200 and 400 rpm.

 Importance of mixing because of the presence of the vapor phase ethylene.  A high H/D is important to enhance contact time between gas and colloids, and vortex formation is a useful feature with the gaseous monomer to enhance mass transfer.  Use of a porous sparger led to more foaming than introduction of ethylene through point source.  At 400 rpm could not sample because of foaming.  Increasing N enhances ethylene concentration and leads to lower rate of vinyl acetate consumption because it polymerizes more slowly.

[17]

 Combination of agitators and N allows one to reach point where ethylene mass transfer is no longer limited by the mixing.

Ethylene (V) + Vinyl Acetate (L)
 Polymerization in a 1.8 L semi-batch RC1 calorimeter with a stainless steel anchor agitator operated at 200 to 300 rpm.  Pressure up to 30 bar and reactor temperature between 65-80 °C.

 Increasing N increased ethylene dissolution rate and thus 𝑅 . This also has an impact on the copolymer composition.  Decrease in mass transfer rate as viscosity (solids content)

increases.

[18]

Ethylene (V) + Vinyl Acetate (L)
 Batch emulsion polymerization in a 7.5 L stainless steel reactor.

 Agitation rate from 200 to 500 rpm (agitator not clearly specified).

 Ethylene content in the copolymer increases as N is increased from 250 to 450 rpm due to greater ethylene solubilization (mass transfer).

[19]

Ethylene (V), vinyl acetate (L), N-methylol acrylamide (L)  18 L stainless steel semi-batch reactor with 2 baffles and a Rushton turbine.  Pressure not specified, T between 55 and 80 °C.

 Increasing N increases 𝑑 due to coagulation.

[20]

Tetrafluoroethy lene (V)  1 L reactor with anchor impeller and a baffle plate.

 N varies from 250-750 rpm.  P = 4 bars and T = 75 °C (P c = 38.9 bars, T c =33 °C [21], so monomer is vapor).

 Chose the anchor agitator because it gave a good vortex and a baffle inserted to enhance turbulence.  Solubility of TFE very low, measured at 0.002 mol/L at reaction conditions.  For N between 250 and 500 rpm, 𝑅 increases quickly then levels off, with the plateau being higher as N increases. At 750 rpm, get rapid activation, followed by rapid drop in 𝑅 (i.e. coagulation of particles).  Type of baffle and agitator are very to enhance contact between particles and gas.

[

Vinyl Chloride (V)  5 L stainless steel reactor with either a 3-blade propeller or an anchor agitator.  Reactor temperature 55 °C and pressure adjusted to be 97 % of saturation pressure.

 𝑅 and 𝑀 both increase noticeably from 500 to 1000 rpm. Mass transfer limitations at low N, but that no longer exist after 1000 rpm.  At low N, the anchor agitator seems to be more effective in incorporating the gaseous monomer, and mass transfer limitations disappear around 600 rpm.  Increasing N increases 𝑅 and 𝑀 , especially when using the hydrofoil.  No significant coagulation and final 𝑁 is independent of N.  Rate of polymerization was directly correlated with gas-liquid mass transfer coefficient k L a.

Vinylidene fluoride (V), Hexafluoroprop ylene (V)

 Semi-batch polymerization in a 10 L reactor with 2 variable width baffles and 1 three-blade turbines.  Reactor temperature 85 °C, total pressure less than 40 bars so both components in vapor phase.  Surfactant free polymerization.

 Reference N is 550 rpm. Although not shown in the publication, the authors state that the agitation conditions must be chosen to avoid mass transfer limitations that occur at N < 550 rpm.

[25]

Chlorotrifluoro ethylene (V), vinyl acetate (L), butyl acrylate (L), Veova 10 (L)  Batch emulsion polymerization at 60 °C in stirred autoclave. No other information given.

 Increasing N increases dissolution of the chlorotrifluoroethylene in the liquid phase, and thus increases monomer consumption and generates larger particles.

[26]

In systems with gaseous (V) (or supercritical, SC) monomers, the monomers are typically quite insoluble in water, and given the fact that they are significantly lighter than water, mechanical agitation and good reactor design are important in maintaining sufficient concentrations in the aqueous phase or polymer particles. For instance, reactor height to diameter ratios higher than normal (H/T > 1.5 [START_REF]Handbook of Industrial Mixing: Science and Practice[END_REF]) can help enhance the contact time between the bubbles of monomer and the continuous phase, thereby increasing monomer availability in the particles. [START_REF] Scott | Reactor Emulsion Design Considerations for Gas-Liquid Emulsion Polymerizations: the Ethylene-Vinyl Acetate Example[END_REF][START_REF] Ecoscia | Emulsion polymerization of vinylidene fluoride: Effects of mixing and reaction conditions on the initial rate of polymerization[END_REF][START_REF] Mendez Ecoscia | Reaction engineering of the emulsion homopolymerization of vinylidene fluoride: Progress and challenges[END_REF] Scott et al. [START_REF] Scott | Reactor Emulsion Design Considerations for Gas-Liquid Emulsion Polymerizations: the Ethylene-Vinyl Acetate Example[END_REF] suggested that using gas spargers in the bottom of the reactor can help as well, but these can lead to increased foaming if care is not taken. However, just like with the dissolved oxygen in reference [START_REF] Krishnan | Effects of agitation on oxygen inhibition, particle nucleation, reaction rates, and molecular weights in emulsion polymerization of n-butyl methacrylate[END_REF], studies with ethylene [START_REF] Scott | Reactor Emulsion Design Considerations for Gas-Liquid Emulsion Polymerizations: the Ethylene-Vinyl Acetate Example[END_REF][START_REF] Poljanšek | Emulsion copolymerization of vinyl acetateethylene in high pressure reactor-characterized by inline FTIR spectroscopy[END_REF][START_REF] Mobarakeh | Effect of different parameters in emulsion copolymerization of vinyl acetate-ethylene in 7.5 liter high pressure reactor[END_REF][START_REF] Choi | Effects of agitation in emulsion polymerization of vinyl acetate, ethylene, and N-methylol acrylamide[END_REF], vinyl chloride in the vapor phase [START_REF] Hjertberg | Polymerization of vinyl chloride at reduced monomer accessibility. IV. The effect of diffusion control on polymerization rate, molecular weight, and thermal stability[END_REF], and fluorinated monomers [START_REF] Kim | Emulsion polymerization of tetrafluoroethylene: effects of reaction conditions on the polymerization rate and polymer molecular weight[END_REF][START_REF] Ecoscia | Emulsion polymerization of vinylidene fluoride: Effects of mixing and reaction conditions on the initial rate of polymerization[END_REF][START_REF] Apostolo | Kinetics of the Emulsion Polymerization of Vinylidene Fluoride and Hexafluoropropylene[END_REF][START_REF] Liu | Kinetics of waterborne fluoropolymers prepared by one-step semi-continuous emulsion polymerization of chlorotrifluoroethylene, vinyl acetate, butyl acrylate and Veova 10[END_REF][START_REF] Mendez Ecoscia | Reaction engineering of the emulsion homopolymerization of vinylidene fluoride: Progress and challenges[END_REF] all showed that increasing the agitation rate (regardless of the shape of the reactor and type of agitator) led to a measurable increase in the rate of polymerization (and molecular weight [START_REF] Ecoscia | Emulsion polymerization of vinylidene fluoride: Effects of mixing and reaction conditions on the initial rate of polymerization[END_REF]). Once again, the magnitude of the changes will depend on the design of the reactor and agitator [START_REF] Kim | Emulsion polymerization of tetrafluoroethylene: effects of reaction conditions on the polymerization rate and polymer molecular weight[END_REF][START_REF] Ecoscia | Emulsion polymerization of vinylidene fluoride: Effects of mixing and reaction conditions on the initial rate of polymerization[END_REF].

If we consider a system with a supercritical or gaseous monomer, where the monomer is much lighter than the continuous phase, mechanical agitation of the reactor will be designed to entrain the gas in the head space of the reactor into the liquid where it will be dispersed in the form of bubbles. The flux of monomer from the bubbles into the continuous aqueous phase of the reactor will be governed by the following equation:

𝑉 𝑘 𝑎𝑉 𝐴 * 𝐴 𝑟𝑉 𝑄 𝐴 (2)
where 𝑉 is the volume of water in the reactor, 𝐴 the concentration of component A in the water, 𝑡 the time, 𝑘 the liquid mass transfer coefficient, 𝑎 the gas-liquid interfacial area per unit of volume of the dispersion, 𝐴 * the equilibrium concentration of A in the liquid, 𝑟 the reaction rate per unit volume and 𝑄 is the exit flow rate of liquid phase (equal to zero in batch and semi-batch operations). The equilibrium concentration 𝐴 * is a thermodynamic property (and can be approximated using for instance Henry's law) affected only by pressure and temperature, not by fluid dynamics or mixing. The concentration of A in the liquid (𝐴 ) is affected by the transport and reaction rate.

One of the challenges in solving this equation is to estimate the value of the product 𝑘 𝑎 (it is very difficult to estimate the values separately). While it would be desirable to be able to estimate 𝑘 𝑎 à priori, Krishnan et al. [START_REF] Krishnan | Effects of agitation on oxygen inhibition, particle nucleation, reaction rates, and molecular weights in emulsion polymerization of n-butyl methacrylate[END_REF] used this method to estimate the impact of changing levels of dissolved oxygen on the rate of polymerization, and in particular the induction period for the emulsion polymerization of n-butyl methacrylate (BMA).

While the method showed reasonable results, a good fit to the experimental data required that they use adjustable parameters in their model. As they pointed out, one of the problems is that they used correlations for 𝑘 𝑎, and many of the available correlations can show errors of ± 50 % in these situations as they are a function of many system dependent variables as noted above. As an example of what this means, Chaudhari et al. [START_REF] Chaudhari | Gas-liquid mass transfer in "dead-end" autoclave reactors[END_REF] compared 𝑘 𝑎 values for a dead-end reactor using different methods: dynamic physical absorption, catalytic hydrogenation of styrene and oxidation of sodium sulfite.

From the results, it was possible to state that the different methods led to a similar mass transfer coefficient and that the mode of introduction of gas in the system also influences the coefficient. However, when a dip tube was used to introduce the gas, the 𝑘 𝑎 values increased in 3 to 5 times in a manner not accounted for by any correlations. For this reason, experimental methods are often used to estimate the value of this product. One commonly used means of estimating this values is the dynamic pressure method [START_REF] Chaudhari | Gas-liquid mass transfer in "dead-end" autoclave reactors[END_REF][START_REF] Moucha | Gas hold-up, mixing time and gas-liquid volumetric mass transfer coefficient of various multiple-impeller configurations: Rushton turbine, pitched blade and techmix impeller and their combinations[END_REF][START_REF] Sardeing | Gas-Liquid Mass Transfer. A Comparison of Down-and Up-pumping Axial Flow Impellers with Radial Impellers[END_REF], which consists in monitoring the pressure drop caused by the absorption of gas A in the liquid over time. In the absence of reaction, the concentration of the gas A in the liquid is related to the pressure drop in the reactor as follows:

𝑉 ( 3
)
where 𝑉 is the volume of gas in the reactor, 𝑃 the pressure, 𝑍 is the compressibility factor, 𝑅 the gas constant and 𝑇 is the temperature. Mendez Ecosia et al. [START_REF] Ecoscia | Emulsion polymerization of vinylidene fluoride: Effects of mixing and reaction conditions on the initial rate of polymerization[END_REF] used this method to estimate 𝑘 𝑎 for the reactor used in our current study and showed that it was a very strong function of the agitation rate.

This review shows that automatically neglecting a priori mass transfer in emulsion polymerization can lead to a poor choice of operating conditions, where mixing can have an impact on polymer properties and the polymerization rate. This is especially true when dealing with highly insoluble reactive species, be they monomers or impurities. Some studies have shown that using a 𝑘 𝑎-based approach can account for the non-equilibrium partitioning of light gaseous or supercritical reactants, such as vinylidene fluoride (VDF) which is the object of the current paper. It appears necessary to use an experimental approach to estimate 𝑘 𝑎 rather than to use existing correlations as these do not account for the true nature of all types of agitators or reactor geometries, nor for changes in viscosity due to the accumulation of polymer in a semi-batch system. [START_REF] Scott | Reactor Emulsion Design Considerations for Gas-Liquid Emulsion Polymerizations: the Ethylene-Vinyl Acetate Example[END_REF] For these reasons, we propose a simplified model of the emulsion polymerization of VDF under supercritical conditions that will allow us to account for the impact of mass transfer in a manner similar to that of Krishnan et al. [START_REF] Krishnan | Effects of agitation on oxygen inhibition, particle nucleation, reaction rates, and molecular weights in emulsion polymerization of n-butyl methacrylate[END_REF]. The model is compared to experimental data, and it is shown that if one has a good estimate of 𝑘 𝑎 it is possible to accurately model the impact of stirring on the polymerization rate and particle size distribution under conditions where the number of particles in the latex is known (i.e. once particle nucleation is finished and coagulation is assumed negligible).

MATERIALS AND METHODS

Polymerization

The high pressure reactor used for polymerization in this study is a 3.8 L stainless steel 316 jacketed reactor with three 6-bladed 45° impellers of 5 cm diameter and one hydrofoil of 9.1 cm diameter, a dip tube and a thermocouple located 5 mm from the reactor wall. The reaction is performed with an initial charge of 1.6 liters of water and the following composition, The monomer is added by activating the pump manually when the reactor pressure is 0.5 bar below the set-point. The polymerization rate is calculated by calorimetry from data recorded each 10 seconds. The reactor geometry and internals disposition, as well as more details on the experimental mode of operation for semi-batch polymerization are presented elsewhere [START_REF] Ecoscia | Emulsion polymerization of vinylidene fluoride: Effects of mixing and reaction conditions on the initial rate of polymerization[END_REF]. The dimensions and arrangement of the impellers are shown in The difference between the two can be seen in Figure 2. The vinylidene fluoride (VDF) monomer and proprietary surfactant (TA) were kindly provided by Arkema (Pierre Bénite, France), and used without further purification. Potassium Persulfate (KPS) (99 %, Acros Organics) was used as initiator, Ethyl Acetate (99.5 %, Sigma Aldrich) was used as chain transfer agent (CTA) and Sodium Acetate Anhydrous (Salt) (99 %, Sigma Aldrich) was used as a buffer. Deionized water is used as initial charge, as rinse water and to prepare the initiator and salt solution and a commercial paraffin/wax is used as antifouling agent. The semi-batch emulsion polymerization is performed with a surfactant concentration below its critical micellar concentration, so the particles are created by homogeneous nucleation. The reaction is performed at 83 ºC and 88 bar (so under supercritical conditions of VDF, Pc = 44.3 bar and Tc = 30°C), using the setup 2 or setup 2D at 400 or 550 rpm as described in [START_REF] Ecoscia | Emulsion polymerization of vinylidene fluoride: Effects of mixing and reaction conditions on the initial rate of polymerization[END_REF].

The solids content (SC) is measured by gravimetry, and the particle diameter (𝑑 ) is measured using the Zetasizer®, so the number of particles (𝑁 ) can be calculated. The surface coverage (θ) is calculated based on the parking area of the surfactant, 25 Å 2 . [START_REF] Ecoscia | Emulsion polymerization of vinylidene fluoride: Effects of mixing and reaction conditions on the initial rate of polymerization[END_REF] The Size Exclusion Chromatography (SEC) technique is used to measure the molecular weight (𝑀 ) of the samples, using DMSO as solvent.

Semi-batch reactions were performed with the two different setups and agitation speeds for 120 minutes and the polymerization curves are shown in Figure 3. It is possible to observe that there is no significant difference in the reaction rate for the two setups when the speed is 550 rpm. There seems to be an optimal agitation configuration for which the mass transfer limitations are no longer acting, which is the case for 550 rpm with setup 2 and setup 2D.

As expected from the previous literature, the polymerization rate is slower for a lower agitation speed, as the mass transfer coefficients are in average 75 % lower than the values found for 550 rpm. It is interesting to observe that during the period 0 to 40 min, where the level of the latex and its viscosity are still low, so ensuring a good quality of mixing, the three curves are very similar (also leading to similar 𝑁 , see Table 2). The results of these reactions are presented in Table 2. The consequence of working with higher mass transfer coefficients (see below) is seen in the increase in the solids content (SC) and in the weight average polymer molecular weight (𝑀 ), especially when comparing the reactions performed with the same setup at different speeds. In the table, 𝑁 refers to the number of particles per unit volume, PdI is the polydispersity index of the particle size, PDI is the polydispersity index of the molecular weight, and θ is the surface coverage by the surfactant.

Estimation of 𝑘 𝑎

The experiments to determine 𝑘 𝑎 are based on the dynamic physical absorption method proposed by Chaudhari et al. [START_REF] Chaudhari | Gas-liquid mass transfer in "dead-end" autoclave reactors[END_REF] which consists in monitoring the pressure drop caused by the absorption of a gas A in a liquid over time, and were performed as indicated by Mendez Ecoscia et al. [START_REF] Ecoscia | Emulsion polymerization of vinylidene fluoride: Effects of mixing and reaction conditions on the initial rate of polymerization[END_REF]. The reactor was charged with a latex 10 wt % kindly provided by Arkema and the experiments were performed at the same temperature used to run the reactions. As the pressure drop in our system is reliable when performing the experiments below 40 bars, the experiments were performed at 30 bars, and we will assume that the pressure will not significantly impact the estimate of 𝑘 𝑎. Obviously changing the pressure will impact the density of the gaseous or supercritical fluid and thus the specific surface area a (more particularly for VDF, the fluid is supercritical at 88 bar and 83 °C, but not at 30 bars). However, the work of Teramoto et al. [START_REF] Teramoto | Effects of pressure on liquid-phase mass transfer coefficients[END_REF] on the estimation of several process gases suggests that the variation of 𝑘 𝑎 is negligible over variations of pressures from 10 to 100 atm. The experiments were performed for setups 2 and 2D and the results are presented in the Table 3. In most cases, setup 2D has a higher 𝑘 𝑎 value. Also, it is clear that the values at 400 rpm are lower than for 550 rpm, and thus not sufficient to maintain a full rate control of the polymerization. Note that in Figure 3 the initial latex volume was approximately 1.7 L (very close to the 2 nd impeller, thus ensuring a good mixing), so no difference on 𝑅 was observed between both setups at 550 rpm. The mass transfer coefficient can be estimated by the following equation, adapted from the work of Chaudhari et al. [START_REF] Chaudhari | Gas-liquid mass transfer in "dead-end" autoclave reactors[END_REF],

𝑘 𝑎 𝐴 𝑁 (4)
where 𝑘 𝑎 is the mass transfer coefficient in min -1 , 𝑉 is the volume of gas or headspace available in the reactor in m 3 , 𝑉 is the volume of liquid in m 3 and 𝑁 is the agitation speed in rpm. The constants for setup 2 (400 and 550 rpm) and setup 2D (550 rpm) are given in Table 4. Even knowing that the original equation from Chaudhari et al. [START_REF] Chaudhari | Gas-liquid mass transfer in "dead-end" autoclave reactors[END_REF] was obtained for a dead-end autoclave reactor with a single impeller, the estimated values we found with our correlations, are pretty close to the experimental ones as shown in Figure 4.

MODELLING EMULSION POLYMERIZATION

The rate of polymerization for the process carried out as an emulsion is given by equation [START_REF] Ramirez | Kinetics of the styrene eçulsion polymerization above cmc. II. Agitation effect on molecular weight[END_REF]. The emulsion polymerization model is based on the works and reactions constants of Apostolo et al. [START_REF] Apostolo | Kinetics of the Emulsion Polymerization of Vinylidene Fluoride and Hexafluoropropylene[END_REF] and Pladis et al. [START_REF] Pladis | Modelling of vinylidene fluoride emulsion polymerization[END_REF][START_REF] Pladis | Mathematical Modeling and Simulation of Vinylidene Fluoride Emulsion Polymerization[END_REF]. The current version of this model does not The partial differential equation representing the PSD was discretized to become a set of ordinary differential equations, using finite differences on diameters, and solved using Matlab®. The fitting parameters to the model can be: the efficiency of radical absorption (𝑓 , here taken equal to 1), the desorption coefficient (the model is taken from [START_REF] Pladis | Mathematical Modeling and Simulation of Vinylidene Fluoride Emulsion Polymerization[END_REF], but the parameters refitted) and the choice among different models for radical entry/exit (two exit models were compared). Note that the desorption coefficient is valid only for the considered operating conditions (i.e. concentration of CTA, temperature, particle size), and in order to make it valid for a wider range of operation more experiments and a more detailed model would be required.

Kinetics, mass balances and PSD

The free radical homopolymerization scheme taken into consideration as well as the material balance equations are presented below and the material balances are given in Table 5. We assume equi-partitioning of CTA between particles and water. Note that the PVDF produced by emulsion polymerization is semi-crystalline. Also, during the polymerization of VDF, the mobility of the hydrogen atoms of VDF is responsible for the chain transfer to monomer and to polymer (backbiting) reactions and bimolecular termination by disproportionation. While this influences the polymer molecular weight distribution, it has a minor effect on the reaction rate and therefore was not considered in the kinetic scheme where we are focusing only on predicting the rate of reaction. Finally, a better description of 𝐷 would be needed for a better description of the polymerization. Termination in water rate coefficient 𝑘 (m 3 mol -1 s -1 ) [START_REF] Ameduri | From Vinylidene Fluoride (VDF) to the Applications of VDF-Containing Polymers and Copolymers: Recent Developments and Future Trends[END_REF] .

Termination in particles rate coefficient 𝑘 (m 3 mol -1 s - 1 ) -analog to styrene 0.05 𝑘 Henry coefficient at 80 °C 𝐻 , (mol Pa -1 m -3 am polymer) [START_REF] Apostolo | Kinetics of the Emulsion Polymerization of Vinylidene Fluoride and Hexafluoropropylene[END_REF] 3.1 10

Equilibrium concentration of monomer in water 𝑀 (mol m -3 at 83 °C and 85 bar) [START_REF] Pladis | Mathematical Modeling and Simulation of Vinylidene Fluoride Emulsion Polymerization[END_REF] 10

Monomer diffusion coefficient in water 𝐷 (m² s -1 ) 1.5 10

Initiator efficiency 𝑓 (-) 0.6

Initial condition of the model

As the model does not take into consideration the nucleation stage, seeded reactions are performed to generate the experimental results to be modelled. In this system, repressurizing seeds was found to be delicate and not reproducible, as some coagulation was observed. As an alternative, first several replicate semi-batch runs were performed to ensure that the experiments were reproducible during the nucleation period, even when changing the agitator type or speed. We estimated that the main nucleation phase is over after 40 minutes of polymerization, when comparing the number of particles obtained in 40 minutes and after 120 minutes of reaction. The polymerization rate profiles for the starting point data are shown in Figure 5. The results of the reaction used as initial condition are presented in Table 7. The reaction stopped at the end of the nucleation period was used as initial point for the model to simulate longer duration experiments (starting from 40 min until the end of the reaction).

The reactions performed with different setups and agitation speeds presented in the previous section were simulated to evaluate the robustness of the model. Effect of mixing on the concentration of monomer in amorphous polymer Figure 6 presents the model prediction of the monomer concentration in the amorphous polymer particles and time zero correspond to the 40 minutes of the initial condition reaction. As already mentioned, the equilibrium concentration of monomer calculated by Henry's Law is only affected by pressure and temperature, and is not influenced by the agitation type or speed. If the model does not take into consideration the mass transfer limitations, it is assumed that the concentration of monomer in the polymer particles is at equilibrium. It is possible to observe that the concentration of monomer in the particles is below the equilibrium value, especially for the lowest agitation speed. An increase in the agitation speed increases this concentration, and as observed from the reaction results, the monomer concentration curves at 550 rpm are virtually the same with the different setups.

The results show that the use of a non-equilibrium model is essential when the agitation speed is low, but the results at 550 rpm are close to Henry's law. Influence of different models of the average number of radicals per particle

The impact of different models for the radical desorption rates are tested using Smith and Ewart [START_REF] Smith | Kinetics of emulsion polymerization[END_REF] and a model combining the works of Friis and Nyhagen [START_REF] Friis | A kinetic study of the emulsion polymerization of vinyl acetate[END_REF] and Ugelstad and Hansen [START_REF] Ugelstad | Kinetics and Mechanism of Emulsion Polymerization[END_REF]. The reaction performed at 550 rpm with the setup 2 is used to compare the models and the results are presented in Figure 7. The evolution of 𝑛 as a function of time is shown in Figure 8, and justifies the assumption of pseudo-bulk conditions as it is superior to 1 throughout the modelled period. It is possible to observe the general improvement in the particle size distribution and polymerization rate when using the Friis & Nyhagen (1973) [START_REF] Friis | A kinetic study of the emulsion polymerization of vinyl acetate[END_REF] Effect of mixing on the reaction rate Figure 9 shows that the model fits well the experimental data by calorimetry and that the polymerization rate using setup 2D is slightly higher than the one performed with setup 2. As the monomer concentration in the particles is lower for the agitation speed of 400 rpm, it was expected that the polymerization rate follows this behavior. Effect of mixing on the particle size distribution

Figure 10 shows the effect of mixing on the particle size distribution for the reactions performed with different setups and speeds. We can observe that the model fits the experimental data, and as only the particle growth is modelled, different polymerization rates will have an impact on the final particle size distribution. As the polymerization rate at 550 rpm was virtually independent of the setup used, the final particle distributions of both setups are also similar. At 400 rpm the polymerization rate is slower, not leading to same growth of particles, which is reflected by the slightly smaller particles obtained.

CONCLUSIONS

Experimental evidence of the importance of mass transfer in the emulsion polymerization of VDF has been presented, as well as the impact of changing the agitator.

A simplified model that does not take into account nucleation or coagulation allows us to account for the influence of the mass transfer resistances on the monomer concentration in the polymer particles. This model shows that at agitation rates of less than 550 rpm, the polymerization is limited by mass transfer from the entrained droplets. It is important to note that besides the type of impeller and agitation speed other parameters influence the 𝑘 𝑎, and so mass transfer, such as the viscosity of the latex (or solids content, particles size) and the level of latex in the reactor (e.g. vortex).

  steel reactors of 1.85 and 7.48 L, each equipped with geometrically similar six blade Rushton turbines (radial flow), or with a pitched (45°) six blade impeller (axial flow).  4 baffles inserted in the reactors.

 1 L 2 L

 12 glass reactor operated in batch mode with a paddle agitator.  Polymerization at 50 °C. No observable coagulation and both conversion and 𝑀 increase as N increases.  𝑑 does not change, suggesting some sort of mass transfer limitation due to agitation. batch reactor with different agitation setups: single and double pitched blade turbine, single and double Rushton turbine.  Polymerizations at 50-60 °C.

 4 L

 4 reactor with 4 impellers (either 4 pitched blade impellers with 4 blades at 45°, or 1 hydrofoil plus 3 pitched blades).  Temperature 88 °C, pressure 83 bars (constant in semi-batch, initial in batch).  N from 350 to 650 rpm in batch and semi-batch mode.

Figure 1 .

 1 Figure 1. The reactor has a height to diameter (H/T) of 5 and the height of the conical section (hc) is 3 cm. The position of the impellers in the agitation setup can be changed, and in this work two setups are used:  Setup 2: three 6-bladed 45° impellers plus hydrofoil A345 (pumping up).  Setup 2D: three 6-bladed 45° impellers plus hydrofoil A315 (pumping down).

Figure 1 :Figure 2 :

 12 Figure 1: a) Reactor geometry, impellers spacing for Setup 2/2D and volumes, b) Reactor internals showing the thermocouple and dip tube. a) b)

Figure 3 :

 3 Figure 3: Semi-batch polymerization rates of reactions performed with different setups and speeds.

Figure 4

 4 Figure 4 Estimated-Experimental kLa values (min -1 ).

  the nucleation and coagulation of particles, thus modelling only the growth of particles.

Figure 5 :

 5 Figure 5: Polymerization rate for the first 40 minutes of semi-batch reactions performed with different setups and speeds.

Figure 6 :

 6 Figure 6: Prediction of the monomer concentration in amorphous polymer.

Figure 7 :

 7 Figure 7: Influence of 𝑛 model a) Constant 𝑛 = 1.6, b) Using Smith & Ewart desorption model, and c) Using Friis & Nyhagen and Ugestald & Hansen radical desorption model (time 0 corresponds to the end of the nucleation period).

Figure 8 :

 8 Figure 8: Variation of 𝑛 justifying the assumption of pseudo-bulk behaviour.

Figure 9 :

 9 Figure 9: Polymerization rate a) from the model and b) compared to the experimental results from calorimetry (for setup 2D at 550 rpm)

Figure 10 :

 10 Figure 10: Initial (experimental) and final particle size distributions (experimental and by the model).Importance of a 𝑘 𝑎 ModelThe importance of a non-equilibrium model taking into account the mass transfer coefficient during the reaction is shown in Figure11and Figure12. It can be seen that when a model is considering that the monomer concentration in the polymer particles is at thermodynamic equilibrium, there is an overestimation of the polymerization rate. The overestimation of the polymerization rate will also lead to slightly overestimation the size of particles.

Figure 11 :

 11 Figure 11: Polymerization rate by non-equilibrium and Henry models.

Figure 12 :

 12 Figure 12: Initial and final particle size distributions for non-equilibrium and Henry models.

Table 1 .

 1 Impact of agitation on emulsion polymerizations (L = liquid, V = vapour, SC = supercritical)

	Monomer (Phase)	Reactor Operation

 No influence of 𝑁 on 𝑅 .  Increasing 𝑁 from 200 to 275 then 350 rpm decreases the molecular weight 𝑀 ) by improving mass transfer of highly insoluble CTA. [12] Styrene (L)  Cylindrical round-bottomed glass reactor with 4 baffles and anchor or half-moon agitators.  Batch polymerizations at 70 °C.  Polymerizations at 150, 200, 300 and 400 rpm.  With anchor, if N too low (150 rpm) we get phase separation and low reaction rate.  Highest 𝑅 with anchor observed at 200 rpm.  Above 200 rpm, 𝑁 drops as N increases, slowing the rate.

Table 2

 2 Semi-batch polymerizations performed with different setups and speeds.

		S e t u p 2	Setup 2	Setup 2D
		400 rpm	550 rpm	550 rpm
	Time (min)		120
	SC (%)	29.4	34.6	35.8
	𝒅 𝐩 (nm)	208	221	225
	PdI	0.012	0.021	0.007
	𝑵 𝐩 x10 -19 (m -3 )	4.3	4.3	4.3
	θ (%)	8.0	7.0	6.9
	𝑴 𝐰 (kDa) PDI	398 2.51	457 2.37	485 2.50

Table 3

 3 𝑘 𝑎 estimation results for Setups 2 and 2D for different volumes and speeds.

	Vw (L)	N (rpm)	𝒌 𝐋 𝒂 (min -1 )
			Setup 2	Setup 2D
	2.0	400	1.109	0.727
		550	2.681	2.876
	2.5	400	0.386	0.505
		550	1.320	1.705

Table 4

 4 Coefficients for kLa estimation.

		A	B	C
	Setup 2	2.57x10 -8 1.369	2.95
	Setup 2D	0.869	1.040 0.207

Table 5 .

 5 Model for emulsion polymerization.[START_REF] Gilbert | Emulsion Polymerization. A Mechanistic Approach[END_REF][START_REF] Asua | Polymeric Dispersions : Principles and Applications[END_REF][START_REF] Asua | Polymer Reaction Engineering[END_REF][START_REF] Coen | Modelling particle size distributions and secondary particle formation in emulsion polymerisation[END_REF]

	Mass balances
	Initiator		𝑘 I
	Oligomeric radicals in the aqueous phase (length 1 to 𝑗 )	•	2𝑓 𝑘 I 𝑘 𝑛 𝑣, 𝑡 𝑓 𝑣 d𝑣 •	𝑘 𝑣 𝑓 𝑣 d𝑣 𝑘 𝑅 •
	Reaction rate in particles (mole s -1 m -3 emulsion)	𝑅	𝑘	𝑘	,	𝑛 𝑣, 𝑡 𝑓 𝑣 d𝑣
	Volume of the swollen particle phase 𝑉	𝑉 𝑉	𝑉	,
	Volume of the unswollen polymer phase		, 𝑅 𝑉

Table 6 .

 6 Model Parameters.

	Parameter	Value
	Decomposition rate coefficient 𝑘 (s -1 ) [34]	4.56 10 exp
	Propagation rate coefficient 𝑘 (m 3 mol -1 s -1 ) [34]	2.2 10 exp
	Transfer to monomer rate coefficient 𝑘 (m 3 mol -1 s -1 ) [34]	1.2 10 exp
	Transfer to CTA rate coefficient in the particles 𝑘 (m 3 mol -1 s -1 )	1.1 10 exp

Table 7 :

 7 Results of semi-batch polymerization stopped at 40 minutes.

	Time (min)	40
	SC (%)	9.9
	𝒅 𝐩 (nm)	145
	PdI	0.01
	𝑵 𝐩 x10 -19 (1/m³)	3.7
	θ (%)	16.8
	𝑴 𝒘 (kDa) PDI	310 3.12

ACKNOWLEDGEMENTS

The authors thank Arkema for providing the monomer and surfactant used in this study and the ANR Clean-Poly project (ANR-18-CE06-0023) for financial support.


Kusuma Kulanjanpeng

Yesmine Bel

Hadj Hassine

Nida Sheibat-Othman

Wiwut Tanthapanichakoon

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

Kusuma Kulajanpeng

Yesmine Bel Hadj Hassine

Nida Sheibat -Othman

Multicomponent solubility in gas-phase polypropylene processes

Keywords: multicomponent solubility, co-solubility effect, anti-solubility effect, polypropylene, thermodynamic modelling

published or not. The documents may come   L'archive ouverte pluridisciplinaire

Introduction

Propylene can be polymerized on supported catalysts to make different kinds of polypropylene (PP). It can be homopolymerized to make isotactic PP (iPP), copolymerized with low levels of a co-monomer like ethylene to make random copolymers (RCP), or with higher levels of ethylene to make impact copolymers (ICP, a 2-phase material comprising a matrix phase of homopolymer or random copolymer and an amorphous ethylene-propylene rubber (EPR) phase, with EPR having an ethylene content up to 40 %) [START_REF] Soares | Introduction to Polyolefins[END_REF]. While both gas-and slurry-phase processes can be used to make iPP, it is necessary to use gas-phase processes to make RCP or ICP as the higher amorphous content of these polymers can pose a certain number of operational problems in slurries. In the gas-phase process, in addition to the monomers and hydrogen (used to control the molecular weight), propane is widely used as an inert gas to control partial pressures and aid with heat removal.

It is well-known that the presence of heavier components can enhance the solubility of the lighter component in the amorphous polymer. This is often referred to as the so-called "co-solvent (or co-solubility) effect", and the heavier the alkene or alkane, the greater the effect on lighter components. On the other hand, the lighter species in the gas composition decrease the solubility of the heavier ones via the "anti-solvent (or anti-solubility) effect". Therefore, precise values of the solubility of a sorbed monomer / co-monomer(s) in amorphous polymer must be measured as a function of the composition of the gas phase to predict accurate reactive species concentrations at the active sites [START_REF] Kulajanpeng | Multiscale modelling of multizone gas phase propylene (co)polymerization reactors -A comprehensive review[END_REF]. Different experimental methods [START_REF] Kanellopoulos | Prediction of solubility of α-olefins in polyolefins using a combined equation of state -Molecular dynamics approach[END_REF], [START_REF] Sadrzadeh | Fundamentals and Measurement Techniques for Gas Transport in Polymers[END_REF] have been used in the past to measure the solubility of gases in polymers, and these can be grouped into 3 main approaches:

(1) Gravimetric methods: A sensitive microbalance such as a magnetic suspension balance, or quartz microbalance is used to record changes in weight of a polymer sample during a sorption experiment. The mass gain of the polymer sample at equilibrium reveals the gas solubility in polymer films or powders [START_REF] Sadrzadeh | Fundamentals and Measurement Techniques for Gas Transport in Polymers[END_REF]. This method is adapted for measuring the total solubility;

(2) Pressure decay methods: The pressure decrease due to the gas sorption into a polymer is measured until an equilibrium is established. This method relies on the accuracy of the pressure sensors as well as the precise estimation of the system volume (i.e. volume of void space in the vessel containing the polymer sample) and temperature control. This method is commonly used for high-pressure sorption experiments [START_REF] Sadrzadeh | Fundamentals and Measurement Techniques for Gas Transport in Polymers[END_REF]. It can include GC measurements of the vapour space and can thus be used for multicomponent mixtures;

(3) Volumetric methods: The volume change in a constant-pressure vessel is used to determine gas sorption into a polymer sample. This technique is not as versatile compared to the other sorption methods due to the complexity to maintain constant pressure [START_REF] Sadrzadeh | Fundamentals and Measurement Techniques for Gas Transport in Polymers[END_REF].

More recently, spectroscopy was also used to measure the sorption of gases in amorphous polymers [START_REF] Ongkasin | Supercritical loading of gatifloxacin into hydrophobic foldable intraocular lenses -Process control and optimization by following in situ CO2 sorption and polymer swelling[END_REF], which appears to be a promising method.

Without installing gas chromatography (GC), one would obtain only the total pressure from the three above-mentioned methods. Of the three methods, only the pressure decay method can be easily adapted to accommodate a GC that could be used to directly measure the gas-phase composition once equilibrium has been achieved. With the other 2 approaches, sampling the gas phase from the experimental set-up would perturb the measurement process. In this work, we have chosen to employ the pressure decay method. These different methods have been used to measure solubilities of α-olefins in different types of PP over a wide range of temperatures, pressures, and polymer crystallinities (𝜒 ) shown in Table 1. Sato et al. [START_REF] Sato | Vaporliquid equilibrium ratios for hexane at infinite dilution in ethylene+impact polypropylene copolymer and propylene+impact polypropylene copolymer[END_REF] conducted sorption experiments for propylene, and for ethylene (separately) in iPP and in ICP) at pressures up to 30 bars and temperatures in the range of 50 -90 °C using the pressure decay method. They made it clear that the gas solubility measurement in the polymer at high pressure always requires a swelling correction. They found that polymer swelling caused an increase in gas solubility in polymer. They also reported that the solubility of propylene in ICP was about six to eight times higher than that of ethylene. However, in a different work, they [START_REF] Sato | Solubility of propylene in semicrystalline polypropylene[END_REF] reported solubilities of the binary system of propylene in PP having different crystallinities (i.e., different stereoregularities) and molecular weights. They found that the solubility of propylene in PP with low stereoregularity (19 % crystallinity and a weight average molecular weight, Mw, of 220 kDa) was about twice that in PP with higher stereoregularity (47 % crystallinity and Mw of 1,000 kDa). This is coherent with the measurements made by Krajakova et al. [START_REF] Krajakova | Sorption of Liquid Diluents in Polyethylene: Comprehensive Experimental Data for Slurry Polymerization[END_REF] who investigated the solubility of alkanes in polyethylenes (PE) of different densities and observed that the solubility of n-hexane in the amorphous phase of polyethylene (PE) with low density (less than 0.923 g/cm 3 ) was superior than the solubility in the amorphous phase with higher density PE. This is attributed to elastic constraints imposed by tie molecules linking neighboring crystallites that are more important at higher crystallinities.

Other groups also found a difference between solubilities in the amorphous phase of ICP and iPP. For instance, Tsuboi et al. [START_REF] Tsuboi | Sorption and partial molar volumes of C2 and C3 hydrocarbons in polypropylene copolymers[END_REF] measured solubility of propylene, and ethylene in iPP and ICPs. They found that the solubility coefficients of propylene in the amorphous ICPs were 1.8 times greater than those in the amorphous iPP. The group of Bartke [START_REF] Bartke | Sorption and diffusion of propylene and ethylene in heterophasic polypropylene copolymers[END_REF], [START_REF] Kröner | Sorption of olefins in high impact polypropylene -experimental determination and mass transport modeling[END_REF] measured the sorption equilibrium of propylene, and ethylene in iPP and ICP having different rubber contents.

They found that gas solubilities in ICP increased with the rubber content (i.e., more amorphous material).

Cancelas et al. [START_REF] Cancelas | Solubility and Diffusivity of Propylene, Ethylene, and Propylene-Ethylene Mixtures in Polypropylene[END_REF] carried out solubility measurements of propylene, ethylene, and an equimolar mixture of propylene and ethylene in amorphous iPP using a gravimetric method. Their ternary solubility experiments (i.e., propylene (1) / ethylene (2) / iPP (3)) also demonstrated the co-solubility effect of propylene on ethylene. The experimental solubilities were correlated with Sanchez-Lacombe equation of state (SL EoS) model.

Recently, Hill [START_REF] Hill | Kinetics and Catalyst Overheating in the Gas Phase Polymerization of Propylene[END_REF] measured solubilities of propylene in iPP over a wide range of temperatures. SL EoS and Perturbed-Chain-Statistical-Association-Fluid-Theory equation of state (PC-SAFT EoS) models were compared, and found that the SL EoS model was more suitable for this system.

Although several studies of the individual solubilities of propylene, and of ethylene in PP have been carried out, solubility data for propane in PP, and mixtures of propane and propylene under industrially relevant conditions are scarce in the open literature. For example, solubility data are required to study the influence of propane as an inert gas on the polymerization rate in gasphase propylene homo-and copolymerizations.

In this work, the SL EoS model, an extension of the well-known lattice theory of Flory and Huggins, was used to describe the sorption behavior of monomer / commoner(s) / inert gas in the amorphous phase of semicrystalline PP due to its relative simplicity and sufficient accuracy covering a wide range of operating conditions (i.e., pressure, temperature, degree of polymer crystallinity). The SL EoS (and other EoS), however, relies on binary interaction parameters (𝑘 ) which need to be identified based on experimental sorption data. The SL EoS model can be used as follows:

(1) SL EoS binary model, used to estimate the solubility of one component in the amorphous polymer phase (i.e., monomer (1) and polymer ( 2)). This model accounts for non-ideal mixtures, especially at high pressure; and

(2) SL EoS ternary model, used to estimate the solubility of two components in the amorphous polymer phase (i.e., monomer (1) / penetrate (2) / polymer (3)). The penetrant here can be a monomer, co-monomer, induced condensing agent, or an inert gas. This model not only accounts for non-ideal behavior but also for co-solubility and anti-solvent effects.

A full set of the SL EoS expression and their characteristic parameters can be found in the Supplementary Material.

In the current study, gas solubilities of the multiple penetrants (i.e., propylene, ethylene, propane, and their mixtures) in two different types of PP (i.e., iPP, and RCP) were measured at industrially relevant conditions (i.e., temperature of 75 °C and 85 °C, and pressure range of 0 -29 barG) by a pressure decay method that includes an in-line micro GC, meaning that both individual and total solubilities of multi-component mixtures can be measured. The obtained solubility results are used to fit the temperature-dependent binary interaction parameters (𝑘 ) of the SL EoS model. Gravimetric method

C (1) / iPP (2) 70 5-25 39 + - - - C (1) / iPP (2) 39 + C (1) / ICP (2) 10 𝜒 1 % - - 35 -64 C (1) / ICP (2)
Cancelas et al (2018) [START_REF] Cancelas | Solubility and Diffusivity of Propylene, Ethylene, and Propylene-Ethylene Mixtures in Polypropylene[END_REF] Gravimetric method ✭ determined with density method in vol % or wt % a C : ethylene, C : propylene, 1C : 1-butene, C : ethane, and C : propane b iPP: isotactic polypropylene, ICP: impact copolymer, EPR: ethylene propylene rubber, and RC: ethylene rubber content b % XS represents the percentage of soluble species in polypropylene homo-and co-polymers. This value is almost proportional to the amorphous content of the material.

C (1) / iPP (2) 50-85 3-25 35*/72 ✭ 300 - - C (1) / iPP (2) C (1) / C (2) / iPP (3)
Mass-fraction crystallinities of samples before and after the sorption were determined with a differential scanning calorimetry (DSC 3+ by Mettler Toledo) (Table 3). The heat of fusion of a perfect crystal of the PP was set as 209 J g -1 [START_REF] Karger-Kocsis | Polypropylene Handbook[END_REF]. The first heating cycle was used to determine the degree of crystallinity as it represents the nascent powder in the state that leaves the reactor, whereas the second heating cycle represents the inherent properties of the polymer [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]. Thus, the first value of crystallinity (bold text in table 3) was used for the solubility calculation, as it is not influenced by recrystallization during the analyses, and is assumed to be more representative of the polymer during the sorption experiment.

Pressure decay method

Setup

A simplified scheme of the high-pressure sorption apparatus based on the pressure decay method equipped with the micro-gas chromatography (micro-GC) is presented in Figure 1. The setup is equipped with temperature and pressure sensors connected with the data acquisition unit to record the measured data.

The temperature of the ballast and connecting lines is controlled by an electrical heater, and both the ballast and lines are completely insulated. The total volume of the ballast plus the connecting lines (V B+L ) is 3.82 liters. The total volume of the empty reactor plus any ends of lines between the reactor and nearest valves (i.e., pressure control valve) (𝑉 ) can be measured by a pressure method, which introduces a known number of moles of nitrogen from the ballast to the reactor (𝑛 . For this, it is necessary to know the volume of the ballast plus the connecting lines.

Therefore, the 𝑉 can be calculated as follows.

𝑉

(1)

𝑛 ∆𝑛 , , , , (2)
where P is the pressure (bar), T is the temperature (K), R is the ideal gas constant, Z is the compressibility factor of nitrogen (calculated using Peng-Robinson equation of state (PR model)).

Detailed equations are provided in the Supplementary Material). Superscripts are R: reactor, B:

Ballast, L: Line. Subscript N2: Nitrogen, 0: initial condition, 1: condition after gas injection.

Figure 1 A simplified high-pressure sorption apparatus based on the pressure decay method used in this work.

Estimation of the void volume of the reactor filled with polymer

The void volume (𝑉 ), corresponds to the unoccupied volume of the reactor plus the connecting lines (𝑉 ) minus the volume of polymer. It can be estimated by injecting a known mass of polymer (𝑚 with known density (𝜌 while including the swelling effect at the different studied temperatures and pressures, as follows [START_REF] Hill | Kinetics and Catalyst Overheating in the Gas Phase Polymerization of Propylene[END_REF]:

𝑉 𝑉 𝑉 (3)
where 𝜌 and 𝜌 are the densities of the crystalline and amorphous phases, respectively, and 𝜑 is the volume fraction of the crystalline phase.

The amorphous density (𝜌 at different temperatures and pressures was estimated using the SL EoS that includes the swelling effect due to the solubility of the penetrant in the polymer.

We used the temperature-dependent correlation of the binary interaction parameter from available literature [START_REF] Sato | Solubility of propylene in semicrystalline polypropylene[END_REF], [START_REF] Hill | Kinetics and Catalyst Overheating in the Gas Phase Polymerization of Propylene[END_REF], [START_REF] Yao | Solubility of Subcritical and Supercritical Propylene in the Semicrystalline Polyethylenes[END_REF] to provide an initial guess (𝑘 ) for determining the final binary interaction parameters (𝑘 ) based on the obtained solubility data. The crystalline region is assumed to remain constant under the studied pressure conditions. The crystalline density (𝜌 ) is estimated by Tait [START_REF] Sato | Solubility of propylene in semicrystalline polypropylene[END_REF] that includes the effect of thermal expansions and pressure using equation ( 4):

𝜌 , , (4)
where 𝑣 , is the specific volume of crystalline phase (𝑣 , 0.943 10 m 3 kg -1 ) [START_REF] Sato | Solubility of propylene in semicrystalline polypropylene[END_REF], 𝛼 , is the coefficient of thermal expansion (𝛼 , 3.774 10 K -1 ) [START_REF] Sato | Solubility of propylene in semicrystalline polypropylene[END_REF], 𝑃 0.1 MPa, c = 0.04467, 𝐵 4479.7 MPa, 𝐵 9.609 10 K -1 , and T is the temperature in K.

The crystalline volume fraction (𝜑 can be calculated via the relationship between crystalline mass percentage (𝜒 ) and density of the polymer:

𝜑 (5)
Note that 𝜒 is the degree of crystallinity of the polymer (% wt.) measured by the DSC, shown in Table 3; whereas, 𝜌 is the measured density of the powder (kg m -3 ).

Sorption measurements

The gaseous mixtures for the binary or ternary systems are prepared in the ballast at the target pressure and temperature. A measured quantity of polymer (400 -600 g) is placed in the reactor under vacuum for 1 hour to remove any dissolved gases. The reactor is then heated to the target sorption temperature by an external jacket. Subsequently, a known amount of the gas mixture of interest is injected from the ballast into the reactor. This gas will not only occupy the volume of void space inside the reactor but may also get sorbed inside the polymer. The system is left for 1-2 days to ensure equilibrium and full sorption (while ensuring constant temperature and no leakage in the reactor). The number of moles of each gas in the mixture that occupies the void of the reactor, is calculated from the measured pressure and temperature (in the reactor) as follows:

𝑛 ( 6
)
where 𝑃 is the final total pressure in the reactor and 𝑉 is obtained from Eq. ( 3), and 𝑥 is the gas-phase molar fraction of component i given by the calibrated micro-GC.

For the ternary solubility measurements, the molar fraction of component i in the mixture in the reactor headspace is measured by the calibrated micro-GC, initially before the sorption to represent the gas compositions of the mixture in the ballast (𝑥 , , in equation 7), and later at each equilibrium (𝑥 , ).

The pressure drop inside the ballast indicates the total amount of gas that is injected into the reactor (𝑛 . Combined with the micro-GC measurement, one may get the number of moles of each injected gas, as follows:

𝑛 ∆𝑛 , , , , , , (7)
The gas solubility inside the amorphous phase of polymer can be estimated by calculating the difference between the number of moles of gas fed into the reactor and that which occupies the void of the reactor at equilibrium, to deduce the number of moles of component i inside the amorphous polymer (𝑛 ), as follows:

𝑛 𝑛 𝑛 (8)
The partial solubility of component i inside the amorphous phase of the polymer, 𝑆 (g of penetrant per g of amorphous polymer), is defined by:

𝑆 ( 9
)
where 𝑀𝑤 is the molecular weight of component i.

The solubility of the penetrating gas in the polymer, 𝑆 (g of penetrant per g of polymer), is given by:

𝑆 1 𝑆 (10)
The overall solubility of the mixture of gases inside the amorphous polymer (𝑆 ) is given by:

𝑆 ∑ 𝑆 ( 11
)
where 𝑁 is the number of gas components injected into the reactor.

The interaction parameters (𝑘 ) were determined by fitting the SL EoS model with the measured partial solubility of the different components in the amorphous polymer to satisfy the minimum percentage mean square deviation (% MSD) between the measured experimental and calculated partial solubilities by the SL EoS model, as follows:

MSD % ∑ , , , 100 (12)
where the superscripts of "model" and "exp" represent the solubility predicted by the SL EoS model, and the experimental solubility value, respectively.

Iterative methodology for estimating binary interaction parameters

As the void volume (𝑉 , the swollen polymer volume, and the gas solubility (𝑆 are coupled, we proposed an iterative procedure to simultaneously solve for 𝑘 and 𝑆 (Figure 2).

The objective function to minimize was the % MSD between 𝑆 , and 𝑆 , , and "fslove" function in MATLAB® was used to determine the binary interaction parameters that include the polymer swelling effect.

Figure 2

Iterative procedure for the estimation of the binary interaction parameters and polymer swelling from the sorption data.

Experimental design

Table 4 shows the experimental conditions studied. The sorption experiments were performed at temperatures of 75 °C and 85 °C and pressures up to 29 barG.

Experimental results and SL EoS simulations

Polymer density and volume

As detailed above, the SL EoS model together with the identified binary parameters allows one to estimate the density of the swollen amorphous polymer (𝜌 . The predictions of the densities of iPP and ICP (by the combination of the SL EoS model for the amorphous density and the Tait equation for the crystalline density) were validated with the density data of polypropylene obtained from the pressure-volume-temperature (PVT) relationship of the molten iPP and ICP at different temperatures at 1 bar reported by Sato et al. [START_REF] Sato | Vaporliquid equilibrium ratios for hexane at infinite dilution in ethylene+impact polypropylene copolymer and propylene+impact polypropylene copolymer[END_REF], [START_REF] Sato | Precise measurement of the PVT of polypropylene and polycarbonate up to 330°C and 200 MPa[END_REF]. The simulation results for iPP (Figure 3a) and ICP (Figure 3b) provided good agreement with the reported experimental densities.

Therefore this approach can be used to estimate the volume of swollen polymer and void volume of the reactor. [START_REF] Cancelas | Solubility and Diffusivity of Propylene, Ethylene, and Propylene-Ethylene Mixtures in Polypropylene[END_REF]), (b) ICP densities (k12 = -0.0006T(K) + 0.2251 [START_REF] Sato | Vaporliquid equilibrium ratios for hexane at infinite dilution in ethylene+impact polypropylene copolymer and propylene+impact polypropylene copolymer[END_REF]).

The iterative procedure proposed above (Figure 2) also allows one to estimate the volume of swollen polymer. Figure 4 shows the model prediction of the volume of the RCP with the presence of the propylene or ethylene at various pressures at a temperature of 75 °C. It can be seen in Figure 4a that when the pressure increases, the volume of the RCP polymer (𝜒 = 32.34 % wt.) increases due to the swelling effect (propylene is only dissolved in the amorphous region).

The presence of ethylene in RCP (see Figure 4b) leads to a smaller increase in polymer volume when the pressure increases, compared to propylene. This is because the solubility of ethylene is lower than that of propylene in amorphous RCP.

This methodology for the estimation of polymer volume with the presence of penetrating gases can therefore be adopted for the calculation of void volume of the reactor containing polymer (i.e., iPP and RCP), and gas solubility in amorphous polymer.

Validation of binary solubility of propylene in iPP with literature

Experimental results for the solubility of propylene in amorphous iPP obtained from this work at various pressures were compared with the two literature at different temperatures (75 and 85 °C) (Figure 5). The measured propylene solubility in amorphous iPP is in agreement with the literature. Note that the solubility was originally calculated by Cancelas et al. [START_REF] Cancelas | Solubility and Diffusivity of Propylene, Ethylene, and Propylene-Ethylene Mixtures in Polypropylene[END_REF] based on the reported crystallinity of 72 wt % measured by the density method. However, they also reported the crystallinity of 35 wt % measured by DSC (1 st heating cycle), which we then used to calculate back their solubility and compare it with our measurements.

Binary solubility of propylene and of propane in amorphous iPP

Figure 6 shows the propylene solubility in amorphous iPP at various pressures and two different temperatures. The solubilities increased with the decrease in temperature, and increased nonlinearly with the increase in pressure, especially at the lower temperature. The solubility results

were used with the binary SL EoS model to identify the binary interaction parameters; k12 = 0.0460 at 75 °C, and k12 = 0.0554 at 85 °C. The obtained interaction parameters (𝑘 ) and their correlation as a function of temperature (K) are presented in the supplementary material (see Table S5).

Binary solubility of propylene and of ethylene in amorphous RCP

Figure 8 shows the propylene and ethylene solubility at various pressures in amorphous RCP at two temperatures. Figure 8a shows the ethylene solubility in amorphous RCP increases linearly with an increase in pressure, while that of propylene (Figure 8b) increases nonlinearly with the pressure. The solubility of propylene was about four to five times higher than that of ethylene, in the amorphous RCP.

Comparison of propylene solubility in amorphous iPP and RCP

Figure 9 shows that there is only a small difference between the solubility of propylene in the amorphous phase of RCP and the amorphous fraction of iPP (3.3 % higher in RCP at 75 °C, and 6.2% higher at 85 °C). It is difficult to interpret such small differences of solubility in terms of structural differences between the 2 polymers; it is just as possible that they are due to experimental variability and uncertainties associated with the exact measurement of the polymer crystallinity and the pressure drop, etc.

Ternary solubility of propylene and propane in amorphous iPP

The solubilities of mixtures of different compositions of propylene and propane in amorphous iPP were measured at 75 °C and 85 °C. The results (Figure 10) showed that the total and individual solubilities increased with the pressure as expected. The values of 𝑘 estimated using the ternary solubilities allow a very good fit of both the overall and individual solubilities.

If we compare the solubilities of propylene and propane in a mixed system with the corresponding values obtained for single penetrant (binary) systems under similar conditions, we can see an interesting result. Figure 11a shows the propylene solubility (at 75 °C) in the presence of 20 % mole of propane (ternary solubility prediction ─ solid line) is lower than the pure propylene solubility (binary solubility prediction ─ dashed line) due to the anti-solvent effect of propane on propylene. Similarly, (Figure 11b) propylene can act as an anti-solvent to propane, resulting in lowering propane solubility as well. Similar findings were observed at 85 °C (Figure 12).

Ternary solubility of propylene and ethylene in RCP

Ternary solubility of propylene and ethylene in amorphous RCP was measured at two temperatures (75 °C and 85 °C), and shown in Figure 13.

Figure 14b indicates that ethylene solubility in RCP in the presence of 92 % mole of propylene at 75 °C is significantly greater than pure ethylene solubility in amorphous RCP (estimated by SL EoS binary model). This confirms the co-solubility effect of propylene on ethylene.

On the other hand, Figure 14a shows that propylene solubility in amorphous RCP in the presence of 8 % mole of ethylene at 75 °C is slightly lower than its binary solubility (i.e., pure propylene solubility), indicating that ethylene acts as an anti-solvent to propylene. This antisolvent effect of ethylene on propylene shows greater impact at elevated pressure. Comparing the two effects, we clearly see that the co-solubility effect of propylene on ethylene is higher than the anti-solvent effect of ethylene on propylene. This is due to the higher propylene composition (92 % mole) compared with ethylene (8 % mole). Similar findings were observed at 85 °C (Figure 15). of propylene (ternary system), or without propylene (binary system).

Conclusions

Individual and total solubilities of the multiple components (i.e., propylene, ethylene, propane, and their mixtures) in iPP, and RCP were measured at industrially relevant conditions (i.e., temperature of 75 °C and 85 °C, and pressure range of 0 -29 barG) using a pressure decay method combined with a micro GC. An iterative procedure for the estimation of the binary interaction parameters was proposed to simultaneously predict the volume of the polymer including the swelling effect and the gas solubility in amorphous polymer. As expected, the results showed that the total and individual solubilities of gases in amorphous iPP and RCP increased with an increase in pressure and a decrease in temperature, as expected.

Considering the binary solubility results, it was found that solubilities of propylene and propane in amorphous iPP were quite similar due to their similarity in molecular weight and structure. However, the solubility of propylene in the amorphous RCP was about four to five times higher than that of ethylene, as reported in the literature. The solubility of propylene in amorphous RCP was slightly higher than in amorphous iPP, perhaps due to the differences in polymer crystallinity, polymer chains and architectures. Based on the ternary solubility results, an antisolvent effect of propylene on propane in the amorphous iPP was observed and vice versa (an antisolvent effect of propane on propylene). Also, we observed a co-solubility of propylene on ethylene in the amorphous RCP (as well as the anti-solvent effect of ethylene on propylene) at the observed operating conditions. The obtained solubility results were correlated with the SL EoS model to determine the values of its binary interaction parameters (𝑘 ) and the dependence of 𝑘 on temperature.

Acknowledgement

The authors are grateful to SCG Chemicals Public Company Limited for the financial support of this project. The authors would also like to thank Dr. Supawadee Shiohara of SCG Chemicals Public Company Limited for assistance with the polymer characterizations.

Notation

List of Abbreviations

  differential scanning calorimetry (DSC) + determined with X-ray diffraction (XRD)

Figure 3

 3 Figure 3Comparison of experimental densities of iPP (with 𝜒 = 66 % wt) and ICP (𝜒 = 58 % wt)[START_REF] Sato | Vaporliquid equilibrium ratios for hexane at infinite dilution in ethylene+impact polypropylene copolymer and propylene+impact polypropylene copolymer[END_REF],[START_REF] Sato | Precise measurement of the PVT of polypropylene and polycarbonate up to 330°C and 200 MPa[END_REF] and model predictions based on SL EoS model for the amorphous polymer and Tait equation for the crystalline phase, at different temperatures (1 bar); (a) iPP densities (k12 = -5.6757X10 -4 T(K) + 0.1988[START_REF] Cancelas | Solubility and Diffusivity of Propylene, Ethylene, and Propylene-Ethylene Mixtures in Polypropylene[END_REF]), (b) ICP densities (k12 = -0.0006T(K) + 0.2251[START_REF] Sato | Vaporliquid equilibrium ratios for hexane at infinite dilution in ethylene+impact polypropylene copolymer and propylene+impact polypropylene copolymer[END_REF]).

3 )Figure 4

 34 Figure 4 Predicted volume of polymer at 75 °C; (a) RCP with the presence of only propylene, (b) RCP with the presence of only ethylene. The solid line is the volume of both amorphous and crystalline phases, blue circles represent the amorphous volume estimated by SL EoS using the 𝑘 obtained from iterative procedure (Figure 2), and red circles represent the crystalline volume estimated by Tait equation.

Figure 5

 5 Figure 5 Comparison of propylene solubility in amorphous iPP obtained from this work with the literature data (Sato et al. [7], and Yao et al [14], Cancelas et al. [12] and Yao et al [14]); (a) 75 °C, (b) 85 °C.

Figure 6

 6 Figure 6 Propylene solubility in amorphous iPP at 75 °C and 85 °C.

Figure 7

 7 Figure 7 Propane solubility in amorphous iPP at 75 °C and 85 °C.

Figure 8

 8 Figure 8 Solubility in amorphous RCP at 75 °C and 85 °C; (a) Propylene, (b) Ethylene.

Figure 9

 9 Figure 9 Comparison of propylene solubility in amorphous parts of iPP and RCP at 75 °C and 85 °C.

Figure 10 Figure 11

 1011 Figure 10 (a) Ternary solubility of 80 % mole of propylene and 20 % mole of propane in amorphous iPP at 75°C, (b) Ternary solubility of 77 % mole of propylene and 23 % mol of propane in amorphous iPP at 85 °C.

Figure 12

 12 Figure 12 Comparison of ternary solubility of 77 % mole of propylene and 23 % mole of propane in amorphous iPP and binary solubility of propylene and propane in amorphous iPP at 85 °C; (a) propylene solubility as function of propylene partial pressure, (b) propane solubility as function of propane partial pressure.

Figure 13 Figure 14 Figure 15

 131415 Figure 13 (a) Ternary solubility of 92 % mole of propylene and 8 % mole of ethylene in amorphous RCP at 75 °C, (b) Ternary solubility of 91 % mole of propylene and 9 % mole of ethylene in RCP at 85 °C

  -Robinson Equation of State RCP Polypropylene random copolymers SL EoS Sanchez-Lacombe Equation of State Model Nomenclature 𝑀 , Molecular weight of gas i (kg mol -1 ) 𝑛 Total number of mole that is fed into the reactor vessel (mole) 𝑛 Total number of moles of each gas in the mixture that occupies the void space of the reactor (mole) 𝑃 Pressure (bar) 𝑃 Partial pressure of species i (bar) 𝑅 Ideal gas constant (J K -1 mol -1 ) 𝑇 Temperature (°C) or (K) 𝑉 Reactor volume (m 3 ) 𝑉 Void volumes of the reactor (m 3 ) 𝑍 Compressibility factor of component i (-)

Table 1

 1 Overview of experimental sorption studies of binary and ternary systems in amorphous PP at different temperaures, pressures,

	and degrees of crystallinity a,b
	Authors/ year	Method	System	Temperature	Pressure	Crystallinity
				(°C)	(barG)	(𝜒 , %

wt) Mw (kg mol -1 ) Co- monomer content (wt %) EPR or RC (wt %)

	Authors/ year	Method	System	Temperature	Pressure	Crystallinity	Mw	Co-	EPR or
				(°C)	(barG)	(𝜒 , % wt)	(kg mol -1 )	monomer	RC
								content	(wt %)
								(wt %)
	Yao et al. (2011)	Pressure decay	C (1) / iPP (2)	75-110	1-80	50.2*	229	-	-
	[14]	method
	Kroner and
	Bartke (2013)
	[11]
	Sato et al.	Pressure decay	C (1) / iPP (2)	50-90	3-30	66.9 ✭	190	-	-
	(2000) [6]	method	C (1) / ICP (2)			58.1 ✭	240	4.9 wt. %	-
			C (1) / ICP (2)			58.1 ✭	240		-
	Sato et al.	Pressure decay	C (1) / iPP (2)	50-75	3-30	47*	220	-	-
	(2000) [7]	method	C (1) / ICP (2)			19*	1000		-
	Tsuboi et al.	Gravimetric	C (1) / iPP (2)	14-92	0-10	50 ✭	-	-	-
	(2001) [9]	method	C (1) / ICP (2)			58 ✭	-	-	16
			C (1) / ICP (2)						16
			C (1) / ICP (2)						16
			C (1) / ICP (2)			58 ✭ ; 56 ✭			16; 31
	Bartke et al.	Gravimetric	C (1) / iPP (2)	70	10	55*; 56*	-	-	-
	(2007) [10]	method	C (1) / iPP (2)						-
			C (1) / ICP (2)			32* -40*		-	24.5 -
									42.7
			C (1) / ICP (2)						24.5 -
									42.7

2. High-pressure sorption experiments 2.1 Polymer characterizations

  Propylene C , ethylene C with a minimum purity of 99.5 %, and propane C with a minimum purity of 99.95 % were obtained from Air Liquide France. Nitrogen and argon, used as the inert gases, were also obtained from Air Liquide France with a minimum purity of 99.99 %, and 99.5 %, respectively. Isotactic polypropylene (iPP) was provided by SCG Chemicals Public Company Limited, while PP random copolymer (RCP) was supplied by another industrial partner.

Polypropylene powders (iPP, and RCP) used in this work were characterized by SCG Chemicals Public Company Limited, as given in Table

2

.

Table 2

 2 Characteristics of polymer samples (iPP and RCP) used in this work a, b

	Polymer Characterization	Testing method	iPP	RCP
	Density (kg m -3 )	ASTM	906.7	900.9
	Melt Index (g/10 min)	ASTM	2.91	49.24
	𝑀 (kg mol -1 )	GPC	57.7	30.5
	𝑀 (kg mol -1 )		394.9	218.9
	𝑀 (kg mol -1 )		1,347.2	742.9
	Polydispersity Index (PI)		6.84	7.17
	Pentad fraction (mmmm) (% mol) NMR: Tacticity	95.35	-
	Pentad fraction (rrrr) (% mol)	Evaluation	0.36	-
	% XS (xylene Soluble) (% wt)	Xylene Extraction	1.36	6.38
	RC2 (% mol)	FTIR	-	3.37

a Pentad fraction indicates the tacticity in a polypropylene molecule such as [mm] or [mmmm] for isotactic polypropylene (iPP) and [rr] or [rrrr] for syndiotactic polypropylene

[START_REF] Karger-Kocsis | Polypropylene Handbook[END_REF]

.

Table 3

 3 Degree of crystallinity of the PP samples used in this study

	PP type	Sorption temperature (°C)	Degree of crystallinity (𝝌 𝐩 , % wt)
	iPP	75	41.08* / 51.80 ◊
		85	41.59* / 51.69 ◊
	RCP	75	32.44* / 40.88 ◊
		85	31.01* / 40.52 ◊

*1 st heating cycle, and ◊ 2 nd heating cycle.

Table 4

 4 Sorption experimental conditions used in this study

	System	Temperature	Pressure	Gas compositions
		(°C)	(barG)	(% mol)
	Propylene (1) / iPP (2)	[75.8; 85.6]	0 -19.0	100
	Propane (1) / iPP (2)	[75.0; 85.1]	0 -16.1	100
	Propylene (1) / propane (2) / iPP (3)	[75.1; 85.6]	0 -13.1	80:20, 77:23
	Propylene (2) / RCP (2)	[75.5; 85.0]	0 -15.7	100
	Ethylene (1) / RCP (2)	[74.8; 85.7]	0 -29.1	100
	Propylene (1) / ethylene (2) / RCP (3)	[75.6; 84.8]	0 -13.1	92:8, 91:9

Supplementary material

The supporting information contains a description of the Sanchez-Lacombe equation of state, the Peng-Robinson equation of state, and the SL EoS binary interaction parameters.