Felipe Morais Bolner

Yashmin Rafante Blazzio

Rezende Bárbara

Fabricio Lara

Timothy F L Machado

Mckenna

Bárbara Rezende-Lara

Fabricio Machado

Timothy F L Mckenna

Impact of silica pore structure on the performance of metallocene catalysts in ethylene gas-phase polymerization

Introduction

Metallocene catalysts supported on a suitable solid carrier are used industrially for several applications in low pressure slurry and gas phase ethylene polymerisation processes. [START_REF] Stürzel | From Multisite Polymerization Catalysis to Sustainable Materials and All-Polyolefin Composites[END_REF][START_REF] Krallis | Comprehensive Study of Reactants Depletion in Catalytic Olefin Polymerization Industrial Loop Reactors[END_REF][START_REF] Qiao | Recent advances in polyolefin technology[END_REF] Among the various supports available, amorphous silica is by far the most commonly used material for this purpose because it is inexpensive, spherical, and has a balance of material properties needed for this type of application, including a high surface area and pore structure that can be tailored over a wide size range, proper mechanical properties to favor the catalyst fragmentation process, a controllable number of silanol functions on its surface that can be used to anchor catalyst and cocatalyst, " p " . [START_REF] Duchateau | Incompletely Condensed Silsesquioxanes: Versatile Tools in Developing Silica-Supported Olefin Polymerization Catalysts[END_REF][START_REF] Severn | Silica and a Complex: The "Holy Trinity" of Supported Single-Site Catalyst[END_REF] Commercially available amorphous silica gel supports for metallocenes can be either a product of spray-drying or emulsion technology. Typically, the silica used for this purpose have pore volumes in the range of 1-3 mLg -1 , average pore diameters on the order of 10-30 nm, with particle sizes in the range of 1-10 2 µm. [START_REF] Severn | Silica and a Complex: The "Holy Trinity" of Supported Single-Site Catalyst[END_REF] It is widely accepted that the physical propertiesparticle size and porosityof silica supports are as crucial as the chemical properties of their surface in determining catalyst performance. It is possible that they impact the distribution of the catalyst and co-catalyst throughout the support particles during catalyst synthesis, [START_REF] Nooijen | Natta catalysts in particle form ethylene polymerization: The effect of polymerization start-up on catalyst activity and morphology of the produced polymer[END_REF][START_REF] Nooijen | On the importance of diffusion of cocatalyst molecules through heterogeneous ziegler/natta catalysts[END_REF] (co)monomer(s) diffusion during the polymerisation, molar mass of the polyolefin [START_REF] Mcdaniel | Influence of Catalyst Porosity on Ethylene Polymerization[END_REF] and, last but not the least, crystallization process of nascent polyolefin chains within the porous support. [START_REF] Tioni | Influence of Reaction Conditions on Catalyst Behavior during the Early Stages of Gas Phase Ethylene Homo-and Copolymerization[END_REF] In addition, it is well known that how the surface is calcined can change the performance of a given catalyst in terms of the observed polymerisation rate. [START_REF] Bashir | Silica/MAO/(n-BuCp) 2 ZrCl 2 catalyst: effect of support dehydroxylation temperature on the grafting of MAO and ethylene polymerization[END_REF] Earlier papers from our group discussed the impact of particle size on the rate of polymerisation and molar mass distribution (MMD). [START_REF] Tisse | Influence of Silica Support Size on the Polymerisation of Ethylene Using a Supported Metallocene Catalyst[END_REF][START_REF] Tisse | Role of Silica Properties in the Polymerisation of Ethylene Using Supported Metallocene Catalysts[END_REF][START_REF] Bashir | Experimental proof of the existence of mass-transfer resistance during early stages of ethylene polymerization with silica supported metallocene/MAO catalysts[END_REF] It was shown that, for a given support and catalytic precursor, the rate of polymerisation decreased as the particle size increased, and that in certain cases, the average molar mass also decreased as the particles size increased. It was also briefly demonstrated that methylaluminoxane (MAO) impregnation can also be influenced by particle size, with larger particles requiring longer impregnation times that smaller ones. [START_REF] Tisse | Influence of Silica Support Size on the Polymerisation of Ethylene Using a Supported Metallocene Catalyst[END_REF][START_REF] Tisse | Kinetics and morphology of metallocene catalysts used in ethylene polymerisation[END_REF] However, the porosity of the catalyst supportsalso called texture by some authorsmust also be important. Despite the interest and quality of many of these works, it is often the case that they present an analysis that focuses on the pore volume distribution and ignores the fact that they are using many kinds of materials for supports, and do not control for the particle size distribution.

For instance, in two different studies Silveira et al. [START_REF] Silveira | Supported metallocene on mesoporous materials[END_REF][START_REF] Silveira | The Role of the Support in the Performance of Grafted Metallocene Catalysts[END_REF][START_REF] Silveira | Effect of the silica texture on the structure of supported metallocene catalysts[END_REF] studied ethylene homopolymerisation using metallocenes supported on Grace silicas, bare alumina, MCM-41, SBA-15, MCM-22, and non-conventional supports. Activities were found to be higher for the catalysts with larger final pore diameters, which were within 50 to 80 Å. The authors attributed this effect to the facilitated fixation of metallocenes inside of larger pores, along with easy access of MAO and monomer to the supported metallocene. The authors attributed low activities of the catalysts with low pore diameters to the possibility of the formation of inactive bidentate species. However, one needs to point out that the particle sizes of the supports used were also significantly different, and as discussed elsewhere the particle size also has a strong impact on observed activities. [START_REF] Tisse | Role of Silica Properties in the Polymerisation of Ethylene Using Supported Metallocene Catalysts[END_REF][START_REF] Bashir | Experimental proof of the existence of mass-transfer resistance during early stages of ethylene polymerization with silica supported metallocene/MAO catalysts[END_REF] In a similar study Jongsomjit et al. [START_REF] Wongwaiwattanakul | Copolymerization of ethylene/1-octene via different pore sized silica-based-supported zirconocene/dMMAO catalysts[END_REF] studied the impact of support geometries using a variety of supports, and also saw higher activities with larger pore sizes. Nevertheless, they too ignored the influence of particle size. Also, as with the previous studies, it is very difficult to compare supports of a different chemical nature based only on the pore structure, since we have just seen that for similar calcination conditions, different supports with the same sites have different kinetics.

McDaniel [START_REF] Mcdaniel | Influence of Catalyst Porosity on Ethylene Polymerization[END_REF] discussed the influence of pore structure on the fragmentation phenomenon of the catalyst support, stating that pore volume defines friability of the support, while pore diameter defines the mechanism of polymer growth from the pores. While these remarks were for chromium catalysts, there is no reason that this should not extend to other catalyst systems. This was also studied by Tisse et al, [START_REF] Tisse | Role of Silica Properties in the Polymerisation of Ethylene Using Supported Metallocene Catalysts[END_REF] who observed that silicas with small pore diameters (3.7 nm)

had no activity at all. They concluded that it was possible that MAO (especially if it forms dimers or trimers) might block the pores of the support making them unavailable for monomer diffusion.

If we return to the concept of MAO impregnation of the catalyst supports, Smit et al. [START_REF] Smit | Effects of methylaluminoxane immobilization on silica on the performance of zirconocene catalysts in propylene polymerization[END_REF] compared two supports, Grace 948 and PQ MS3040. They observed that the calcination conditions, as well as differences in porosity, led to the formation of core-shell Al distributions in the catalyst particles. Bashir [START_REF] Bashir | Reaction Engineering of Polyolefins -The role of catalyst supports in ethylene polymerization on metallocene catalysts In Polymer Reaction Engineering of Dispersed Systems[END_REF] also studied the same supports along with silica PQ MS1732 but also taking into account that PQ silicas present much more homogenous pore structures, while Grace presents interstitial voids that allow a better impregnation of MAO and metallocene.

Impregnation temperature might also be varied to obtain different graftings, as well as the washing method after grafting. Zheng [START_REF] Zheng | Fragmentation Behavior of Silica-Supported Metallocene/MAO Catalyst in the Early Stages of Olefin Polymerization[END_REF] affirmed that at room temperature, uniform impregnation is hard to achieve, whereas Severn [START_REF] Severn | Bound but Not Gagged" -Immobilizing Single-Site r-Olefin Polymerization Catalysts[END_REF] observed evenly distributed aluminum atoms in the silicas activated with MAO at 130 °C for 3 hours. Tisse [START_REF] Tisse | Kinetics and morphology of metallocene catalysts used in ethylene polymerisation[END_REF] observed higher activities for metallocene catalysts with MAO impregnation for 4 hours than the ones impregnated for 1 hour, both synthetized at 85 °C. Thus, impregnation time and temperature are variable in relation to the pore size distribution of the support used and to the final Al and metallocene distribution on the catalyst.

Therefore, in the present study we aim to investigate the impact of silica porosity on i) the distribution of metallocene/MAO catalyst inside the support particles, and ii) ethylene homopolymerisation in gas phase process in short times. As many studies focused on the impact of the porosity on the performance of silica supported metallocenes have been clear on whether the particle size was kept constant while changing silica porosity. Furthermore, the effect of the pore structure on catalyst and/or co-catalyst distribution inside the support particles does not appear to have been extensively analyzed in such studies. This point is of great importance when using a cocatalyst like MAO, which is bulky and has the tendency to agglomerate depending upon MAO/metallocene ratio [START_REF] Babushkin | Activation of Dimethyl Zirconocene by Methylaluminoxane (MAO) Size Estimate for Me-MAO -Anions by Pulsed Field-Gradient NMR[END_REF] as well as on its age. [START_REF] Zijlstra | Methylalumoxane -History, Production, Properties, and Applications[END_REF] 2. Experimental section

Materials

Several commercial silicas were in this study, the physical characteristics of which are shown in

Catalyst Synthesis Procedure

The incipient wetness method [START_REF] Bingel | Supported catalyst system, method for the production and use thereof in olefin polymerization[END_REF][START_REF] Schottek | Metallocene compound and polymerization catalyst comprising heterocyclic group[END_REF] was used to prepare all the catalysts used in this work unless otherwise mentioned. All silicas were dehydroxylated at 600 °C overnight under dynamic vacuum of 10 -3 to 10 -5 mbar prior to their impregnation with the metallocene/MAO mixture.

Dehydroxylations at 600 °C start with around 4 g of each silica left under 200 °C at a heating rate of 5 °C/min for 2 hours in a quartz Schlenk flask, followed by calcination at 600 °C under vacuum for over 12 hours. It should be noted that the dehydroxylation temperature of 600 °C was selected based on a detailed study done using the Grace 948 support as support for (n-BuCp) 2 ZrCl 2 . [START_REF] Bashir | Silica/MAO/(n-BuCp) 2 ZrCl 2 catalyst: effect of support dehydroxylation temperature on the grafting of MAO and ethylene polymerization[END_REF] There is no guarantee that this will be the optimal calcination procedure for the other supports, but for the sake of consistency the same procedure was used for all of the silica used in this study.

Two different incipient wetness catalyst synthesis methods were used for this study: a two-step and a three-step method. The first method was used for all (n-BuCp) 2 ZrCl 2 catalysts, while most of the THI catalysts were synthetized using the three-step method.

For the first method, the first step starts by the mixing of a weighed amount of (n-BuCp) 2 ZrCl 2 , in dry toluene and MAO (30 wt% toluene solution), which are left for 1 hour under stirring at room temperature inside the glove box. The amounts of metallocene and MAO were selected by aiming Al/Zr molar ratio of 150 in the final catalysts, whereas the volume of toluene used was in 150 % excess of the pore volume of the used silica support. More precisely, the volume of the MAO 30 wt% solution used per amount of silica was 1.77 mL/g for each catalyst. In the second step, the mixture containing (n-BuCp) 2 ZrCl 2 , toluene and MAO was added dropwise to the weighed amount of dehydroxylated silica followed by heating at 50 °C for 1 h without any stirring. The supported catalyst was dried under vacuum at 75 °C, without applying any wash, for 2 to 3 h and then stored inside the glovebox.

The three-step incipient wetness method, also known as the SMAO method, consisted of synthetizing an activated silica prior to its metallocene grafting, and was used only for the THIactivated catalysts. For such, also 1.75 mL of the MAO solution per gram of dehydroxylated silica were mixed with 150 % of the silica pore volume in toluene, and then the resulting solution was added dropwise to the dehydroxylated silica in a Schlenk flask inside of a glovebox. The resulting slurry was heated at 80 °C for 4 hours under argon atmosphere without stirring, and afterwards the supernatant solution over the slurry was removed with a syringe. Dry heptane in the same volume of the removed toluene was inserted into the slurry to wash any unreacted MAO, then gently stirred by hand, and removed with a syringe. Lastly, the slurry was dried under vacuum at 80 °C for 2 hours. The resulting free-flowing white powder, henceforth called SMAO, was stored in a glovebox prior to its second step, the metallocene impregnation. Following an Al/Zr molar ratio of 150, a solution of the metallocene compound is prepared in pure dry toluene prior to its dropwise addition onto the SMAO, followed by heating at 50 °C for 1 hour, heptane wash and vacuum drying at 50 °C for at least 2 hours.

Catalysts activated by (n-BuCp) 2 ZrCl 2 were free-flowing orange powders, while the ones activated by THI were free-flowing yellow powders.

Stopped-Flow Polymerisation Protocol

The procedure for the nascent polymerisation study in gas-phase reactions in the membrane-like fixed-bed reactor (Annular Stopped Flow Reactors, ASFR) and a description of the device have been presented in detail by Blazzio et al. [START_REF] Blazzio | Polymerization of ethylene in the gas phase-A new combined hardware and software tool[END_REF] Briefly, the ASFR is composed of concentric cylinders, with the internal cylinder being a fritted metal membrane filter, blocked at both ends by glass wool. A mixture of around 40 mg of catalyst and 1,500 mg of sodium chloride (NaCl) salt (purified by recrystallisation in acid citric solution, dried under vacuum at 200 °C for 5 h) is put in the annular space, the feed gas is heated to a desired reaction temperature and fed into the central cylinder. The gas flows through the membrane, reacting with the catalyst in the annular space and then out to the exit lines. The reactors were filled in a glovebox and sealed with rubber septa before being installed in the ASFR set-up. The configuration of this device allows us to run 3 reactions at a time.

Ethylene homopolymerisations were carried out in reaction times of 15, 30, 45 and 90 seconds at ethylene mass flow of 986 g/h at 11 bar pressure (corresponding to a velocity of 20 cm/s). Inlet and outlet temperatures, reactor pressure, as well as flowrate of nitrogenused as purge gas -, carbon dioxideused as catalyst poison for reaction quenching -, and ethylene were collected using inline thermocouples, pressure transmitters and mass flowmeters, respectively, and acquired data were sent to a computer every 0.5 second. All the injections were set in the software interface and done automatically.

After each reaction, the reactors were weighed again to calculate reaction yield, and the mixture of sodium chloride and polyethylene was placed in a beaker and water was used to dissolve the salt bed, followed by filtration of the mixture using a polyvinylidene difluoride membrane and dried under vacuum at 70 °C for 3 hours before being kept in a sealed container for later analyzes. Activity curves were calculated according to the model described by Blazzio et al. [START_REF] Blazzio | Polymerization of ethylene in the gas phase-A new combined hardware and software tool[END_REF]

Silica and Catalyst Characterization

Al and Zr content of the final catalysts were obtained by Inductively Coupled Plasma-Atomic Emission Spectroscopy (ICP-AES) at Mikroanalytisches Labor Pascher, Germany.

Nitrogen adsorption/desorption isotherms of silica and silica-supported final catalysts were obtained in a Micromeritics ASAP 2020 V3.04 H unit at -196 °C. All the silica samples were degassed for 4 h with no pre-heating. Specific surface areas were determined by the Brunauer-Emmett-Teller (BET) equation. [START_REF] Brunauer | Adsorption of Gases in Multimolecular Layers[END_REF] Desorption branch of Barrett-Joyner-Halenda (BJH) method [START_REF] Barrett | The Determination of Pore Volume and Area Distributions in Porous Substances. I. Computations from Nitrogen Isotherms[END_REF] which employed Halsey standards was used to estimate the mesopore size and pore volume distribution of all the samples.

Scanning Electron Microscopy (SEM) and coupled Energy-Dispersive X-ray (EDX) with SEM were done for some supported catalysts samples. First, each sample was put on a doubled-sided conductive carbon adhesive tape priorly attached to a sampler. Then, the samples were put inside of the vacuum chamber, which rotated them while they were metallized under vacuum with a 10 nm copper layer in a Bal-Tec MED-020 High Vacuum Coating System using 3 to 10 keV. In a ZEISS Merlin VP Compact microscope, the metallized samples were placed under vacuum, photographed, and had their particles diameters calculated using the SmartSEM software.

For the SEM-EDX analyzes, before being put under vacuum for their images acquirement, the samples were initially put in a low viscosity EpoFix resin, which was chemically hardened and then metallized and cut using a Diatome Ultra 45 diamond knife in a Reichert Ultracut S ultramicrotome.

Polymer Characterization

Thermal characterisations were performed via Differential Scanning Calorimetry (DSC) on a

Mettler Toledo DSC 1 equipped with an auto-sampler and a 120-thermocouple sensor. The temperature and the heat flow of the equipment were calibrated with an indium standard. All samples were accurately weighed (between 5 to 10 mg) and sealed in 40 μ aluminum pans.

They were heated from -20 °C to 180 °C at 10 °C min -1 with an empty 40 μL aluminum pan as reference. Two successive heating and cooling ramps with 5-minute intervals between the maximum and minimum temperatures were performed. Dry nitrogen gas with a flow rate set at 50 mL min -1 was used as the purge gas. Melting temperature (T m ) was measured at the top of the endothermic peak of the second heating ramp. The STAR e thermal analysis software is used to

u g p u (χ) f p : χ = H m / H m 0 where
H m (J g -1 ) is the second melting heat of the sample and H m 0 (293 J g -1 ) the melting heat of a 100% crystalline polyethylene.

Finally, SEM and SEM-EDX were also performed for the polymer samples following the same aforementioned methods.

Results and Discussion

Supports and catalysts porosity studies

The properties obtained via nitrogen porosimetry and elemental analysis are shown in TABLE 1.

The silica with lowest pore volume and diameter is S2, and its catalysts were the ones with the lowest pore volumes as well. Supports S2, S3 and S5 are very similar in their properties. Silica S4 had the biggest particle, and along with S1, presents the largest pore volumes and pore diameters of the studied supports. Finally, all catalysts had similar Al/Zr ratios independently on the support used and around the predicted values.

As mentioned before, it is important to keep the particle size of the silica support (and the resultant catalyst) constant while studying the impact of catalyst porosity on reaction kinetics and polyethylene properties. Since no stirring was applied during the catalyst synthesis, we can confidently assume that the final supported catalysts possess particle size distribution (PSD) like those of their parent silica fractions. In addition, the provider of the support showed that its silicas had PSD narrow enough not to require sieving. Another way to observe the original silica morphology integrity on the catalysts is the nitrogen sorption plots, as shown in FIGURE 1 andFIGURE 2.

FIGURE 1

Sorption isotherms for the dehydroxylated silicas and their corresponding supported metallocene catalysts, as described in TABLE 1.

Particle integrity can be shown by the maintenance of the shape of the sorption isotherms of the original supports in their resulting catalysts. This was also expected from the chosen synthesis methods since no stirring was used for both methods. In addition, one can also see the partially occupied pores in the catalysts by the lower amounts of adsorbed nitrogen compared to their original dehydroxylated silicas. However, it is also visible that silicas S4 and S1 had around the same values of maximum nitrogen gas sorption, with a difference in the shape of their curves.

Silicas S2 and S3 had also very similar sorption curves with similar adsorbed nitrogen amounts.

The presence of hysteresis in all sorption curves is an indication of their mesoporous nature.

Nevertheless, a very clear difference in these curves is seen when comparing S4 silica and its catalysts to the other supports. S4, S4-a, and S4-b present a desorption plateau instead of the smoother transition from adsorption and desorption seen in the other silicas. [START_REF] Thommes | Physisorption of gases, with special reference to the evaluation of surface area and pore size distribution (IUPAC Technical Report)[END_REF] The reason for this plateau is its narrow pore volume distribution, as shown in FIGURE 2.

FIGURE 2

Pore volume distributions for the dehydroxylated silicas and their corresponding supported metallocene catalysts.

From FIGURE 2 one can see that, though silicas S1 and S4 present almost the same average pore volume, their pore volume distributions are very different, with S1 presenting a much broader distribution. Only S4 from the four silicas studied presented a narrow pore volume distribution, with no significant porosity after 40 nm, while the rest of the silicas showed wider pore volume distributions that go up to 90 nm. This indicates that, different from the other three silicas, S4 is more homogeneous, with similar pore diameters distributed from the surface to the core of its particles.

Silicas S2 and S3, as expected from their nitrogen adsorption-desorption curves, show resembling pore distribution profiles. On the other hand, this time we can also observe a slight bimodal pore distribution for both, in which a smaller pore diameter population that peaks around 14 nm merges with the main population around 30 nm. In addition, though S1 also had a wide pore volume distribution, it has larger pores than S2 or S3, as seen from its larger peak in FIGURE 2 and higher amounts of adsorbed nitrogen in FIGURE 1.

Finally, for all silicas and their correspondent catalysts, it is seen smaller pore diameters than their original treated supports, indicating the partial filling of the pores with the grafting of cocatalyst and metallocene. It must be added also that a shift in the pore volume distribution to smaller pore diameters was observed for all catalysts in relation to their original supports, which indicates that bigger pores were the ones most filled with MAO and zirconocene.

Effect of Catalyst Porosity on the Nascent Morphology and Kinetics of Gas Phase Ethylene Homopolymerisation

The kinetics calculated via the inlet and outlet temperatures of the gases in the ASFR system are shown in FIGURE 3: but the porosity of the support the variable that produced the highest effect in the catalytic activities in the early stages of polymerisation. In general, the catalysts with narrow distributions, S4-a, S4-b, had higher activities than the ones with broad distributions. This occurred since the S4-supported catalysts were supported from shell to core since the silica has a very homogenous pore volume distribution. On the other hand, it is also visible that the THI catalystsrepresented by the dashed lineshad higher activities than its (n-BuCp) 2 ZrCl 2 counterpartsrepresented by the solid lineswith the exception for S2-supported catalysts. The activity profiles of the other reaction timespresented in FIGURES S1, S2 and S3 in Supporting Informationshow pore effect much more pronounced and visible in the kinetics after 45 seconds of reaction.

FIGURE 4

shows yields over time in kg PE /mol Zr , displaying a growing yield with time in practically all the reactions, with exception for a small deviation presented by S2-a catalyst (THI/MAO/S2) from 45 to 90 s. Catalysts S4-a and S4-b appear to highly increase activity from 45 to 90 seconds of reaction, while the other catalysts seem to present a gradual increase of yield over time. In general, yields were very similar up to 45 s, and after 90 s the highest yields were obtained by the S4-supported catalysts. The second melting temperatures obtained are within the normal range for HDPE samples, [START_REF] Peacock | Handbook of Polyethylene: Structures, Properties and Applications[END_REF] but are due to the evolution of the polymerisation. Catalyst fragmentation is also seen in FIGURE 5 and FIGURES S4-S7, where the presence of bi-and tri-modality in the crystallization curves are due to the pore confinement mechanism in the first seconds of reaction, as seen in other works using Stopped-Flow reactors [START_REF] Machado | Evaluation of the Initial Stages of Gas-Phase Ethylene Polymerizations with a SiO 2 -Supported Ziegler-Natta Catalyst[END_REF][START_REF] Tioni | Optimization of a tool to study the start-up of the gas phase olefin polymerization[END_REF][START_REF] Tioni | Morphological Interpretation of the Evolution of the Thermal Properties of Polyethylene during the Fragmentation of Silica Supported Metallocene Catalysts[END_REF] . Di Martino et al. [START_REF] Martino | Contributions to the Experimental Investigation of the Nascent Polymerisation of Ethylene on Supported Catalysts, 1. A Quenched-Flow Apparatus for the Study of Particle Morphology and Nascent Polymer Properties[END_REF][START_REF] Martino | Contributions to the Experimental Investigation of the Nascent Polymerisation of Ethylene on Supported Catalysts, 2. Influence of Reaction Conditions[END_REF] attribute this phenomenon in short-time slurry-phase ethylene homopolymerisation using Ziegler-Natta catalysts supported on magnesium chloride to the initial high rates of polymerisation producing disordered and entangled chains.

Furthermore, TABLE 2 shows a growing crystallinity over reaction time. S4-a and S4-b have consistently higher crystallinities than the other catalysts. This implies once more that they have less polymer formed in confined spaces at a given time and that fragmentation was faster. Once again, when looking at the two metallocenes separately, S1-and S4-supported catalysts presented the highest crystallinities for the THI metallocene, while for metallocene (n-BuCp) 2 ZrCl 2 this difference becomes only visible for S4. This can be explained by in-situ hydrogen production of (n-BuCp) 2 ZrCl 2 during ethylene polymerization, a property that the THI metallocene does not present. [START_REF] Bashir | Reaction Engineering of Polyolefins -The role of catalyst supports in ethylene polymerization on metallocene catalysts In Polymer Reaction Engineering of Dispersed Systems[END_REF] FIGURE 5 Crystallization profiles in the ASFR system for catalyst (A) S2-a, (B) S2-b, (C) S4-a, and (D) S4-b.

The evolution on the crystallization and melting profiles was seen for all studied catalysts.

Though reaction times were not short enough for the visualization of a first crystallization peak at 75 °C as suggested in the literature, a second and the third ones in the regions around 105-115 °C can be seen in all studied samples. In addition, the second peak gives place to the third one in all cases, where the second peak loses its initial predominance at 15 seconds and, by 90 seconds, it is mostly consumed, indicating the chains are no more confined after this reaction time. In some cases, usually at 45 seconds, one can observe from two to three simultaneous and/or overlapping peaksaround 111, 114 and 118 °Csuch as seen in FIGURES S4 to S7 in Supporting Information.

Catalysts supported in silicas S2 and S3 presented a slower evolution of these peaks, while silicas S1 and S4 presented a faster crystallinity evolution towards a more resolved peak, specially the ones supported on S4. This might suggest that the presence of larger pores lowered the pore confinement effect like seen in the small-pored silicas, where a bimodal crystallization could be seen up to 45 seconds. To further our studies, we selected two THI-activated catalysts for SEM-EDX, S4-a and S2-a, as depicted in FIGURE 10. respectivelyone can observe that S4-a particles present a more homogenous appearance, with mostly the same grey color from their cores to shells; meanwhile, S2-a has very different particles with different densities, many of them presenting a brighter grey color on their shells than their cores. Their Si atom distributions -Figures 6B and 6Fcorroborate with this observation: silica S4 presents a more even distribution of pores from the outside to the inside of all particles, and silica S2 presents a packed shell and more porous core, with major Si density discrepancies between particles.

Furthermore, the very different Al distributions seen in FIGURE 6C and Figure 6G are, however, related to their differences in MAO impregnation. Due to their very large size, the MAO molecules will have their diffusion restricted inside small pores, requiring longer impregnation times during the catalyst synthesis to reach inner pores of the supportas also shown in Figure S8 of the Supporting Information. Thus, considering the same synthesis method and MAO impregnation time, certain particles of small-pored catalysts might present no active sites whatsoever in its innermost pores, for it is required the presence of both MAO and zirconocene for their formation; meanwhile, catalysts with evenly distributed mesopores will present an even Al distribution from its surface to its core. This also explain why S1-a and S4-a, though supported in silicas with very similar average pore geometries, obtained different productivities and kinetic profilesas also explained by their pore volume distributions in FIGURE 2. The carbon distributions -FIGURES 6D and 6H in redshow nothing in particular, since no polymerisation reaction had been done for these particular samples but serve as references for the SEM-EDX images showed in FIGURES 7 and 8. It should be noted that one would ideally like to know how the pore size influenced the zirconium distribution.

Unfortunately, the Zr signal was not strong enough to detect because of the very low levels of the active metal on the site. Even multiplying its concentration by an order of magnitude was not enough to allow for reliable detection of the metal. Furthermore, the differences in fragmentation can be attributed to the evenly distributed Al in the S4-a catalyst, which allowed this catalyst to homogenously anchor the zirconocene and, therefore, had active sites from its shell to its core, thus obtaining higher activities during the first seconds of reaction. On the other hand, the core-shell Al distribution of the S2-a catalyst produced particles with an uneven distribution of active sites, producing particles with high amounts of unfilled pores and, therefore, a catalyst with low activities. Last, catalyst S5-a, such as S2-a, also displays different lack of homogeneity on the Al distribution. FIGURE 10 shows three types of catalyst particles observed for the same batch of supported catalyst: a particle with evenly distributed active sites; a particle with a core-shell distribution of sites; and a catalyst with very low concentration of sites. In addition, the SEM image of the polymer produced with S5-a shows the formation of hollow polymer particles due to the presence of core-shell Al distribution in the S5-a catalyst.

Conclusion

Silica pore morphology and dimensions can have a significant impact on the distribution of metallocene/MAO mixture throughout the particles. Ethylene polymerisation kinetics are also dependent upon silica support porosity. Silica-supported (n-BuCp) 2 ZrCl 2 /MAO and THI/MAO catalysts of equal particle sizes but different porosities (i.e., pore volumes, pore diameters and surface areas) evaluated in short time gas-phase ethylene homopolymerisation reveal that pore diameter and pore volume seem to significantly impact the instantaneous activity of the catalysts in both types and phases of polymerisations in a similar fashion. Higher activation rates were observed for the supported catalysts when the internal and external morphology of the catalyst particles was kept similar along with particle size. Since the level of mass transfer resistance can be assumed constant due to same particle sizes, this observation can be attributed to the slow fragmentation rate of catalysts with broad pore volume distributions observed by microscopy and porosimetry, as compared to that with homogenous and narrow pore volume distribution. The same effect of difference in pore volumes was noticed when two catalysts having different metallocenes were evaluated. With respect to surface area of the catalysts, no clear trend was observed.

FIGURE 3

 3 FIGURE 3 Instantaneous activity profiles over reaction time during the 90-second reactions in the ASFR system. The activity profiles seen in FIGURE 3 display three different kinetic behaviours: a very fast one for catalysts S4-a and S4-b; a rapid for S1-a, S1-b, and S3-b; and a slow and gradual S3-a, S2-b, and S2-a. One can see that, comparing these profiles with FIGURE 2, it was not the metallocene,

FIGURE 4

 4 FIGURE 4 Yields over reaction time for all ASFR reactions.

FIGURE 6 .

 6 FIGURE 6. SEM-EDX images for cross section of catalysts S2-a (upper row) and S4-a (lower row). Original image (a and e); and silicon (b and f, in green), aluminum (c and g, in orange) and carbon (d and h, in red) distributions.In these images, the intensity of the color indicates the density or concentration of the atom observed via X-rays. When analyzing the first images for S2-a and S4-a -Figures6A and 6E

FIGURE 7

 7 FIGURE 7 SEM-EDX images for cross section of S2-a after reactions with the ASFR for (upper row) 15 seconds and (lower row) 90 seconds. Original image (a and e); and silicon (b and f, in green), aluminum (c and g, in orange) and carbon (d and h, in red) distributions.

FIGURE 8 FIGURES 7

 87 FIGURE 8 SEM-EDX images for cross section of S4-a after reactions at the ASFR for (upper row) 15 seconds and (lower row) 90 seconds. Original image (a and e); and silicon (b and f, in green), aluminum (c and g, in orange) and carbon (d and h, in red) distributions.FIGURES 7 and 8 show the development of the polymerisation reactions over time. Again, the intensity of the color is proportional to the concentration of the related compound in the sample.The S2-supported catalyst presented slower particle growth than the S4-supported one. Observing Figure7Ato 7D and 8A to 8D, 15-second reactions, the formation of a fragmented layer of catalyst is clear in the images of catalyst S4-a but not observed for catalyst S2-a particles. After

FIGURE 9

 9 FIGURE 9 SEM photographs of 90-second reactions of (A) S4-a, (B) S4-b, (C) S2-a, and (D) S2-b catalysts.

FIGURE 3 to FIGURE 9 ,

 39 FIGURE3 to FIGURE 9, this can be attributed once more to the pore confinement effect. The larger pores caused lower mass transfer resistance and fast fragmentation on the initial seconds of reaction, followed by a decrease in the fragmentation rate caused by the filling of the pores with polymer. In the case of catalyst S4-a, one can see that, due to its very pronounced activity from the initial stages of polymerisation, the polymer particle did not reproduce the original morphology of the support. This suggests that the hydraulic pressure caused by the growth of polymer chains inside or the pores of the support was higher than the adhesive tension of the polymer on the fragmented parts of the support.

FIGURE 10 .

 10 FIGURE 10. SEM-EDX of catalyst S5-a particles (left, in green and orange) and SEM of polymer particles it produced in gas-phase ethylene homopolymerisation (right, in grey).

TABLE 1 .

 1 To

	obtain silica supports with well-defined particle sizes, the commercial silicas
	supplied by the Asahi Glass Company (AGC Si-Tech) were narrow enough in terms of size
	distribution that sieving was not necessary. Silica Grace Sylopol 948 was provided by Grace.
	The metallocene precursors used to make supported catalysts were (n-BuCp) 2 ZrCl 2 , and rac-
	ethylene bis(4,5,6,7-tetrahydro-1-indenyl)zirconium dichloride -	"THI" -, and
	both metallocenes were provided by Strem Chemicals Inc (Bischheim, France). Tri-
	isobutylaluminium (TIBA) was used as received from Witco Corporation. The MAO solution
	29.4 wt% in toluene was supplied by Grace with the following characteristics: 13.3 wt% Al, 4.94

TABLE 1

 1 Nitrogen porosimetry and ICP-AES results of as-received commercial supports and their respective metallocene catalysts.

	Silica	Support code	Surface Area (m 2 /g)	Pore Volume (cm 3 /g)	Pore Diameter (nm)	d 50 (µm)	Metallocene type	Catalyst Code	Surface Area (m 2 /g)	Pore Volume (cm 3 /g)	Pore Diameter (nm)	Al (%wt)	Zr (%wt)	Al/ Zr
				Support properties				Catalyst properties
							THI	S1-a	327	1.11	17.6	14.2	0.24
	DM-L-303	S1	296	2.2	30	32
							(n-BuCp) 2 ZrCl 2	S1-b	265	0.77	18.0	14.3	0.23
							THI	S2-a	294	0.47	10.4	15.1	0.26
	DM-H-302	S2	618	1.7	11	35
							(n-BuCp) 2 ZrCl 2	S2-b	188	0.33	12.0	13.9	0.22
							THI	S3-a	371	0.72	11.4	16.9	0.24
	DM-M-302	S3	580	1.8	13	35
							(n-BuCp) 2 ZrCl 2	S3-b	315	0.75	14.2	14.4	0.24
							THI	S4-a	326	0.82	12.1	16.9	0.26
	DM-L-403	S4	363	2.1	23	43
							(n-BuCp) 2 ZrCl 2	S4-b	217	0.61	12.2	14.6	0.24
	M-302-F	S5	500	1.8	15	30	(n-BuCp) 2 ZrCl 2	S5-a	330	0.76	13.9	14.4	0.35

TABLE 2

 2 Crystallinity results via DSC for ASFR polymers.

	χ (%)

Acknowledgements

The authors gratefully acknowledge the financial support Asahi Glass Company (AGC Si-Tech, Tokyo, Japan), and thank MI for discussions and his availability. FM thanks Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) and Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES). All authors kindly thank Dr. Niyi B. Ishola for the help given with the MATLAB code used for the ASFR activity calculations, and Pierre-Yves Dugas with the help given during the SEM-EDX samples preparation.

Author Information

* Corresponding Author: E-mail: timothy.mckenna@univ-lyon1.fr


Niyi B Ishola

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

Estimation of Reactivity Ratios for Olefin Polymerization Catalysts -On the importance of thermodynamics

A systematic study of the impact of gas phase composition on the estimation of the reactivity ratios of a Ziegler-Natta catalyst during the gas phase copolymerization of ethylene with 1butene and 1-hexene has been carried out. The results of the study show that if one uses a realistic equation of state to estimate the co-and anti-solubility effects of multiple species in the gas phase, one can obtain a unique value of the reactivity ratio pair from any number of experiments. However, it was found that using only binary solubility data and ignoring the impact of chemically inert species on solubility will lead to the estimate of compositiondependent reactivity ratio pairs.

Introduction

The microstructure of polyolefins is the key to the diversity end-use properties and to the success of polyolefins as a choice for many modern materials. It is well accepted polymer microstructure (here meaning the molecular weight and copolymer composition distributions -MWD, CCD -and comonomer sequence length distribution, CSLD) made using coordination catalysts is determined by the temperature and by the concentrations of active species at the catalytic sites. Very commonly, one controls the density of linear low-density polyethylene (LLDPE; a copolymer of ethylene and an alkene such as 1-butene or 1-hexene) by carefully controlling the amount of comonomer incorporated in the polymer chain. The quantity of alkene, as well as the distribution of alkenes along the chain contribute greatly to defining the final physical properties of LLDPE. Very often kinetic models are used in conjunction with reactor models of varying degrees of detail to predict the influence of process conditions on the polymer composition, to control the process and so forth. In the case of copolymerization, the reactivity ratios are important kinetic parameters because they provide important information about copolymer microstructural indicators. While the concept of reactivity ratios was originally developed for free radical polymerization, it is also a convenient means of describing the tendency of a catalytic site to insert a molecule between the active site and a growing polymer chain as a function of the nature of the last monomer unit in the chain that is attached to said site.

For a system of one monomer (M) and one comonomer (C) these ratios are defined as:

(1) (2) where and are the homopolymerization rate constants for monomer and comonomer respectively (i.e. they represent the tendency of a molecule to insert itself between the active site and a chain beginning with same type of molecule), and and are the cross propagation constants (i.e. the tendency of a molecule to insert itself between the site and a chain ending in the other type of molecule). Thus, if a reactivity ratio is greater than one, the active site "prefers" to homopolymerize the species in question, otherwise, it prefers inserting the opposite molecule into the chain.

There are different ways of estimating these parameters, but a survey of the literature shows that the most common ways are based on using the Mayo-Lewis equation [1]which is given by:

(

where is the mole fraction of ethylene (assuming ethylene is the principal monomer) in the copolymer, and and are the fractions of monomer and comonomer at the active sites. These last parameters are defined as:

, (4a, 4b)

where [E] and [C] are the concentrations of ethylene and comonomer at the active sites. In the rest of this work, the comonomers are either 1-butene (B), or 1-hexene (H). In the specific case of interest here, this means the concentration of monomer and comonomer in the amorphous phase of the polymer just above the active sites. If one knows these concentrations, 13 C NMR can be used to estimate the composition of the polymer, and if the polymer is produced in conditions with no composition drift, equation ( 3) can be used to fit the data and obtain the true reactivity ratios. There are of course other methods, such as the Fineman-Ross [1] or Kelen-Tudos [2] methods, that take slightly different approaches, but that also rely on values of the copolymer composition, and the fractions of monomer at the active sites. In other words, regardless of how one estimates the true reactivity ratios, the reliability of the values will depend greatly on the precision of the NMR measures and on the accuracy of the estimation of the monomer and comonomer fractions at the active sites.

Knowing the composition of the monomers at the active sites might be less problematic in a solution polymerization, but in the case of supported catalysts the accumulation of polymer between the bulk phase of the reactor and the sites themselves can make the sorption of the reactive species difficult to calculate. In fact, when two or more species are present in the reactor, cosorption effects can influence the solubility of each of the species in question. The sorption of one or more penetrants in the amorphous phase of semicrystalline polyethylene provokes a change in the free volume of the amorphous polymer, which in turn leads to changes in penetrant solubility and diffusivity [3,4,5].

This means that the solubility of a mixture of penetrants in amorphous PE is different from the sum of the individual solubilities of these penetrants, and the solubility of a single species in the presence of others is different from the solubility of the same species when it dissolves in the polymer as a pure component. For instance, Novak et al. [6] demonstrated very clearly that ethylene is more soluble in LLDPE in the presence of 1-hexene than it is when present as a pure component at the same temperature (T) and pressure (P). Conversely the same authors also showed that the solubility of 1-hexene decreased in the same polymer when ethylene is added to the gas phase. Furthermore, the composition of the gas phase for LLDPE production always contains more than 2 species. Hydrogen is used to control the molecular weight distribution, and chemically inert alkanes are often used in gas phase processes to enhance temperature control [7]. Several studies have demonstrated the importance of these co-and antisolublity effects. [6,8,9,10,11,12,13], including on both the rate of polymerization [3], and on the physical properties of the copolymers. [14] For instance, Ishola et al. [14] showed that adding a chemically inert compound like pentane to the copolymerization of ethylene and 1-butene leads to a higher crystallinity in the final polymer than one finds for a simple mixture of the same two reactive species.

This signifies that kinetic parameters estimated using simplifying assumptions that the ratio of the monomer and comonomers in the continuous phase of an experiment is the same as at the active site, [15,16,17,18,19,20,21,22,23] or assuming the solubility of individual species in a mixture is simply the sum of the pure component solubilities [24,25,26,27] will contain biases due to the aforementioned thermodynamic effects. In other words, one could find cases where the reactivity ratios appear to be a function of the composition of the species in the reactor even though this is physically unrealistic. Prediction of the solubility of the reactive species should therefore be done with an equation of state since this approach allows one to account for thermodynamic interactions between the penetrating species and the polymer. While Henry's law can be used to predict the solubility of light components such as methane, ethane, and ethylene over a range of T moderate P, it fails at high pressures of light components, and for heavier components such as alkanes and alkenes. [11,28,29] It is therefore preferred to use an equation of state when trying to predict the solubilities of gas mixtures. The Sanchez-Lacombe (SL-EOS) [30,31] and perturbed-chain statistical associating fluid theory (PC-SAFT) [32] are widely used in for polyolefins, and the advantages of the two approaches have been discussed elsewhere [33]. In the current work we will apply the SL-EOS to calculate the concentration of one, two, or three penetrants in the amorphous phase of PE based on its (relative) simplicity and predictive capability.

SL-EOS treats the polymer chain as a set of beads which are in networked to each other on a lattice. Like a Flory-Huggins model, the polymer chain is assumed to be randomly mixed with penetrant molecules (and thus assumed to be completely amorphous). However, unlike Flory-Huggins model, SL-EOS allows vacant sites in the lattice structure of the polymer to permit change of volume when penetrants and polymer molecules are mixed [31,34] It is important to underline the fact that both the SL-EOS and the PC-SAFT equations of state require semi-empirical interaction parameters that depend on both the composition of the penetrating vapour mixture and the nature of the polymer. Since they were both developed for purely amorphous polymers, but are applied to semi-crystalline systems, the structure of the macromolecular chains can also have an impact on the values of the interaction parameters, so care must be taken to measure their values on polymers like the ones to which one will apply the thermodynamic model.

With these points in mind, the objective of the current paper is to demonstrate that it is still possible to get a set of reactivity ratios that describe the incorporation of a comonomer under a wide range of circumstances when the thermodynamics of sorption deviate from ideal solution thermodynamics by using an equation of state with the right interaction parameters.

It will also be shown that using pure component solubilities in a multicomponent system leads to erroneous estimates of the reactivity ratios. To do this for some realistic gas phase compositions, it is also necessary to have a set of interaction parameters for the different mixtures. We will therefore use the pressure decay method developed by M'Rad et al. [9] to measure the solubilities of the different mixtures and use the data to fit the interaction parameters of the SL-EOS.

In the current paper we will focus on the estimation of the reactivity ratio of ethylene (C2) and 1-hexene (1-C6) and of C2 and 1-butene (1-C4) in different systems in a gas phase polymerization on a supported Ziegler-Natta catalyst. We will demonstrate that it is imperative to account for the composition of the gas phase, and in particular the impact of the presence of chemically inert alkanes when estimating the reactivity ratios.

Materials and Methods

Materials

Argon, hydrogen, ethyene (C2), propane (C3), 1-butene (1-C4), isobutane (iC4) all with a minimum purity of 99.5%, were procured from Air Liquide (Paris, France). Purification columns of zeolite and active carbon were used to purify ethylene before being use. 1-hexene

(1-C6) and n-pentane (nC5) with a minimum purity of 99%, was obtained from Sigma-Aldrich ICN (Germany) and was purified through distillation over CaH 2 . The co-catalyst (Triethylaluminium) was obtained from Witco (Germany). Two commercial Zeigler-Natta catalysts were used during this work, catalyst 1 was used for the C2 -1-C6 copolymerization and the C2 -1-C4 copolymerizations, with and without iC4. Catalyst 2 was used for the other runs.

Solubility Measurements

The sorption experiments for single and multicomponent vapour mixtures were carried out using the pressure decay method and apparatus described by Ben M'Rad et al. [9], as was the procedure for identifying the interaction parameters for the SL-EOS. The different vapour mixtures considered in this work, as well as the temperatures, pressure ranges and compositions are given in Table S1. The results of the solubility measurements are shown in the Supplemental Information, Figures S2-S5.

Polymerization procedure

Polymerizations were conducted in a spherical stirred bed semi-batch reactor, using an NaCl bed (40 g) according to the procedure outlined in reference [3]. All polymerization experiments were performed at constant temperature of 70 o C, a constant hydrogen pressure of 1 bar and a time of 15 minutes to keep the conversion of 1-hexene or 1-butene below 10%.

At the completion of the reaction, the reactor was depressurized and cooled to room temperature. The product was collected and washed by water and then dried under vacuum at 70 °C. For the copolymerization involving ethylene/1-hexene, 12 LLDPE copolymers of polyethylene-co-1-hexene were made by varying ethylene (4, 7, 9 bars) and 1-hexene pressure (0.2, 0.4, 0.6, and 0.8 bars) in the absence of any induced condensing agents (ICA).

The C2/1-C4 copolymerizations were performed with and without isobutane as ICA. 12 LLDPE copolymers were made at constant ethylene and isobutane pressure of 7 and 3 bar, respectively, while 1-butene pressure was varied from 0.6 -2.1 bar. For the copolymerization involving ethylene/1-butene in the presence of propane or n-pentane, the reaction condition includes constant pressure of propane (5 bar) or n-pentane (1 bar) and varying ethylene pressure (4 and 7 bar) and 1-butene pressure (0.6 -1.8 bar). The polymerization rates of these experiments are present in the Supplemental Information, Figure S6-S10.

Copolymer characterization

Each of the polymer samples was dissolved in a mixture of benzene-d6/tetrachloroethylene (C 6 D 6 /TCE) with volume ratio of 2:1 under heating with concentration of 80 mg/mL in a 10 mm tube. Bruker AVANCE II spectrometer operating at 400 MHz for 13 C and for 1 H at 90 °C. 1 H spectra were recorded under the following operating conditions: zg30 sequence, an acquisition time of 4.09 s, a relaxation delay of 3 s, and 256 scans. 13 C spectra were recorded under the following operating conditions: zgig70 sequence, an acquisition time of 1.36 s, a relaxation delay of 10 s, and 4000 scans. Residual protons (δ7.15 ppm) of benzene and backbone carbon Sδ+δ+ (δ 29.58 ppm) of polyethylene were used as an internal reference for 1 H and 13 C NMR spectra, respectively. The method of Randall [35] was adopted to calculate the copolymer compositions.

Calculations

Estimation of reactivity ratios

As mentioned above, the reactivity ratio can be estimated from the Mayo-Lewis equation which relates the mole fraction of ethylene in the copolymer ( to ethylene mole fraction in the amorphous phase ( of the polymer. A Levenberg-Marquardt nonlinear least squares algorithm will be used to fit monomer composition obtained from SL-EOS and copolymer composition obtained fron 13 C NMR to Mayo-Lewis equation. This approach takes into account the non-linearity of the Mayo-Lewis equation, and is preferred to using the Fineman-Ross method which is based on linearizing the kinetic model.

Sanchez Lacombe Equation of State

The following terms will be used to describe the different systems under consideration:

-binary refers to a system with one polymer and one penetrant. This is slightly different from a "pure component" system. In a pure component system, we will calculate the binary solubilities of the species present in the vapour phase and assume that the other species present do not influence the solubility;

-ternary refers 2 penetrants and one polymer where we account for the interaction parameters;

-quaternary refers to 3 penetrants and 1 polymer where we account for the interaction parameters.

The mole fractions of the monomer and comonomer in the amorphous polymer were calculated using the SL-EOS for binary [9,30], ternary [31] and quaternary [36] systems. The SL-EOS also requires characteristic parameters for the species of interest. These are known à priori and are given in Table S2.

3 Results and discussion

Solubility Data

The interaction parameters found from the solubility experiments shown in the supplemental material and used in the current work are given in Table 1. Except for the ethylene co-1-hexene polymerization systems, all of the interaction parameters were found by fitting the data shown in the Supplementary Material. It is not unexpected that the value of k 12 for the ethylene co-1-hexene system be different from the other binary data as the polymer is not the same as these values were taken from the literature. For this reason, we decided use both the binary and ternary data estimated in reference [38] so that it would be coherent. We will treat this case separately from the others.

As we will see below this does not change the conclusions on the role of solubility in estimating the reactivity ratios. *The number of the polymer in the system is always one more than the last penetrant.

Number increases from lightest to heaviest component. ** see supplementary material for experimental data

The interaction parameters between the penetrant molecules in the mixture are taken as being equal to zero. In other words, it is assumed that mixtures of ethylene comonomer and ICA behave as ideal mixtures conforming to Lorentz-Berthelot combining rules. [39] 3.2 Reactivity Ratios

Two component mixtures: polyethylene-co-1-hexene

12 LLDPE copolymers of polyethylene-co-1-hexene were made by varying ethylene (4, 7, 9bar) and 1-hexene pressure (0.2 -0.8 bar). The different copolymer composition distributions measured using 13 C NMR are given in Table 2. Note that in this table the meaning of the names of each run is as follows: E x H y refers to a run with ethylene (E) and 1-hexene (H) where the ethylene pressure in the gas phase is x bars and that of 1-hexene y bars. In this Table, the terms binary and ternary refer to the definitions given above. In other words, f E (binary) refers to a copolymer composition calculated using pure component solubilities, and f E (ternary) refers to a copolymer composition calculated using ternary solubilities (i.e. that take into consideration the cosolubility effect). The estimation of the reactivity ratios was done using Levenberg-Marquardt nonlinear least squares algorithm by fitting an experimental data set to the Mayo-Lewis equation (Equation 3). The estimates were obtained by minimizing the sum of squares of the residuals between model predictions and experimental data. The reactivity ratios estimated using binary and ternary solubilities are given in Table 3 along with the values of the confidence interval as well as the sum of squares of the residuals. It can be seen from Figure 1 that the joint confidence region for the values calculated from the binary solubility data includes a value of 0 for r H, indicating that this value is statistically insignificant. The data fit is slightly better when the ternary data are used (R 2 is slightly higher), and that the values from both reactivity ratios are statistically significant at 95% confidence when the ternary solubility data is used.

Nevertheless, it can be seen from the Mayo-Lewis plots in Figure 2 that the estimates of copolymer composition obtained using both binary solubilities and ternary solubilities are acceptable for all practical purposes if one needs to calculate copolymer compositions over the range of pressures studied. However, as we shall see below one cannot generalize this conclusion to all systems of interest. In the case of the ethylene co-1-hexene polymerizations presented in this section, only 2 gas phase components have been considered. However, it is more common to find several penetrants in a commercial scale plant, especially if one is running a gas phase polymerization in condensed mode where one can find one or more alkanes in the reactor in addition to monomer and comonomers. For this reason, this same analysis will be extended to multicomponent systems in the next section. The difference between the two cases (binary vs ternary solubilities) shown in this example shows that it is preferential to consider a more accurate thermodynamic model, but for all intents and purposes, the difference between the two is not significant. The reason for this is while there is a cosolubility effect of hexene on ethylene, Figure 3 shows that this is not particularly pronounced. However, if the gas phase composition leads to more significant cosolubility effects, the choice of thermodynamic model can become important.

Figure 3. Cosolubility effect of hexene on ethylene (7 bars ethylene, 70°C) represented by the Ternary model. The Binary example does not account for the cosolubility.

Multicomponent mixtures: ethylene co-1-butene + alkanes

In this section, we will evaluate the reactivity ratios for ethylene co-1-butene in the case of 2 and different 3 component gas mixtures. As one can see from the results in section S2.3 of the Supplementary material, the solubility of ethylene in the amorphous phase of LLDPE can vary over a wide range when an alkane is present in addition to the comonomer. The copolymer composition for the different runs is shown in Table 4 and the ethylene mol fraction in the amorphous phase of the polymer is given in Table 5 using different methods of calculation. In this last table, Ternary refers to the ternary SL-EOS in the absence of any induced condensing agent (no alkane) that accounts for the cosolubility effects between ethylene and 1-butene. The quaternary models include the alkane in addition to the other gaseous species. Let us first consider the estimation of the reactivity ratio the ethylene co-1-butene polymerization in the presence of isobutane as ICA at an ethylene pressure of 7 bars

(experiments E 7 B 0.6 -E 7 B 2.1 ). The estimated reactivity ratios for copolymerizations at 7 bars of ethylene, the upper and lower bounds of the 95% confidence intervals, and the coefficient of determination (R 2 ) are given in Table 6. If we compare reactivity ratios based on pure component solubility (binary) with those estimated using ethylene mol fractions calculated using a SL-EOS that accounts for penetrant mixture composition, two main observations can be made. First, if one does not account for the thermodynamic interactions of the penetrating species (i.e., one uses pure component or binary solubility models), then predicted pair of reactivity ratios is 131 for C2 and 0.13 for butene. Furthermore, the latter is not statistically different from zero with this limited data set. These values are very different from those in the second column (Ternary Solubility), where the reactivity ratios for ethylene and for butene are both statistically significant, and quite different from the binary values. The second important observation is that when one uses the correct thermodynamic representation, the estimated reactivity ratios are very close for the cases where one does and does not include iC4 as an ICA. This is important since it means that even if we change the mixture composition, and thus the ethylene concentration in the amorphous phase (see Table 5), we will find essentially the same values for the reactivity ratio. It can be seen from the graphs in Figure 4 that the JCR found with thermodynamic models that account for the solubility effect are very close. Ideally, they would overlap of course, but with the limited data sets it was possible to generate, such small differences do not appear to be important with respect to the values calculated using the pure component solubilities. If we enlarge the data set to include 4 and 7 bars of ethylene, it can be seen from Table 7 that the reactivity ratios calculated using binary (pure component) solubilities are different from those in Table 6. In other words, changing the compositions of the mixtures changes the reactivity ratioswhich clearly it should not! On the other hand if thermodynamic interactions are accounted for (Ternary Solubility), then we find a value of the reactivity ratio pair this is very close to that of those shown in Table 6. Finally, repeating this analysis for ethylene co-1-butene polymerizations in the presence of 5 bars of propane or 1 bar of npentane and accounting for the interactions of the penetrant species (c.f. Table 8, Quaternary The reason for the importance of correctly accounting for the cosolubility effects when estimating the reactivity ratios can be seen in Figure 5 for the system C2/1-C4/C3. It can be seen here that the compostions of ethylene and of butene in the amorphous polymer with no propane are quite different from those estimated as one adds propane to the gas phase mixture. In this particular instance, propane acts as a cosolvent for ethylene, enhancing the solubility of the latter in the amorphous phase of the polymer, and as an anti-solvent for 1butene. Further examination of Figure 5 also shows that while there is an antisolvent effect, it is weak and the main contribution to the change in local monomer concentration will be the cosolubility effect. The impact of this effect increases as the pressure of propane increases.

One can observe similar effects for the other systems considered here. This demonstrates that relative amounts of ethylene and of 1-butene at the active sites will be a function of the concentration of inert species in the gas phase mixtures, and if one did not account for this when estimating the reactivity ratios (and other kinetic constants!), then the values of the reactivity ratios could in fact appear to be a function of the concentration of inter species in the reactor, which is nonsense.

Figure 5. Example of the cosolubility effect of propane on a mixture of 7 bars of ethylene and 1 bar of 1-butene at 70°C.

To summarize the importance of this approach, one can consider the plot in Figure 6. This graph shows two Mayo-Lewis plots: one using an average value of the "correct" reactivity ratios estimated using the SL-EOS from Tables 6-8 (rE=10.5 and rB=0.53), and one using the reactivity ratios using the binary calculations shown in Table 7. Clearly, the use of a thermodynamic approach that allows for co-and antisolubility effects allows us to the reactivity ratios calculated from one data set to successfully model any other data set (provided we always use the same thermodynamic approach). On the other had, the use of reactivity ratios that fit a given data set, but that are calculated with binary reactivity ratios is not reliable at all. (data points), with an average value of the reactivity ratios estimated using the SL-EOS accounting for interactions (solid line, r E =10.5, r B =0.53) and the copolymer compositions calculated using the binary reactivity ratios in Table7.

Conclusion

The importance of accounting for cosolubility effects when estimated the reactivity ratios of gas phase olefin copolymerizations has been demonstrated. In certain simple cases, such as the copolymerization of ethylene and 1-hexene, accounting for the cosolubility effects over a narrow composition range does not appear to be critical. However, if one were to add a typical amount of induced condensing agent to this system (as would be typical in an industrial context), this is no longer true.

In the case of ethylene and 1-butene copolymerizations, the cosolubility effect corresponding to the mixture compositions make it important to consider the thermodynamic model used to

Figure 1 .Figure 2 .

 12 Figure 1. Plot of 95% joint confidence region for ethylene/1-hexene copolymerization; (a) binary and (b) ternary systems.

Figure 4 .

 4 Figure 4. Plot of 95% joint confidence region for ethylene/1-butene copolymerization in the presence of isobutane as ICA for (a) pure component solubility and (b) accounting for the cosolubility effect. 7 bars of ethylene, 1 bar 1-C4, 0 and 3 bars of iC4.

  Co-solubility effectSolubility), leads once again to reactivity ratios that are very close to those shown in the previous two Tables.

Figure 6 .

 6 Figure 6. Mayo-Lewis plot for the ensemble of ternary and quaternary data from Tables6-8(data points), with an average value of the reactivity ratios estimated using the SL-EOS accounting for interactions (solid line, r E =10.5, r B =0.53) and the copolymer compositions calculated using the binary reactivity ratios in Table7.

Table 1 .

 1 Interaction parameters at 70°C for the different polymerizations in the current work.

	System	Species (subscript no.)*	k ij	Reference
		C2 (1)	k 12 = 0.038	[37]
	C2 + 1-C6 + LLDPE	1-C6 (1) C2( 1) + 1-C6 (2)	k 12 = 0.027 k 13 = 0.038	[38] [38]
			k 23 = 0.045
		C2 (1)	k 12 = -0.01796	This work**
	C2 + 1-C4 + LLDPE	1-C4 (1)	k 12 = -0.0019 k 13 = -0.1401	This work This work
		C2 (1) + 1-C4 (2)	k 23 = 0.0508
	C2+ C3+ 1-C4 + LLDPE	C2 (1) + C3 (2) + 1-C4 (3)	k 14 = -0.1414	This work
			k 24 = -0.0034
			k 34 = 0.0484
	C2+ iC4+ 1-C4 + LLDPE	C2 (1) + iC4(2) + 1-C4 (3)	k 14 = -0.0959	This work
			k 24 = 0.0443
			k 34 = 0.0663
	C2+ nC5+ 1-C4 + LLDPE	nC5 (1)	k 12 =0.0171	This work
		C2 (1) + nC5 (2) + 1-C4 (3)	k 14 = -0.3719	This work
			k 24 = 0.1595
			k 34 = -0.1128

Table 2 .

 2 Copolymer composition, monomer fraction at the active sites using binary and ternary SL-EOS

	Run	F E ( 13 C NMR)	f E (binary)	f E (ternary)
	E 4 H 0.2	0.98	0.16	0.18
	E 4 H 0.4	0.95	0.084	0.11
	E 4 H 0.6	0.93	0.054	0.076
	E 4 H 0.8	0.91	0.038	0.061
	E 7 H 0.2	0.99	0.25	0.29
	E 7 H 0.4	0.98	0.14	0.17
	E 7 H 0.6	0.97	0.09	0.12
	E 7 H 0.8	0.96	0.064	0.098
	E 9 H 0.2	0.99	0.30	0.35
	E 9 H 0.4	0.99	0.17	0.21
	E 9 H 0.6	0.98	0.11	0.15
	E 9 H 0.8	0.96	0.081	0.12

Table 3 .

 3 Reactivity ratios for Ethylene/1-hexene copolymerization and the JCR result.

		Binary system			Ternary system
	Parameter	Confidence interval		Confidence interval
	values	lower	higher	values	lower	higher
	293	198	387	265.16	182.72	347.59
	0.009	-0.01	0.03	0.05	0.009	0.092
	0.9392			0.9719

Table 4 .

 4 Copolymer compositions for the different multicomponent mixtures during the polymerization of ethylene co-1-butene.

	Run		Copolymer Composition (F E ) -NMR
		0 alkane	3 bar iC4	5 bar C3	1 bar nC5
	E 4 B 0.6	0.97	-	0.97	0.98
	E 4 B 0.9	0.93	-	0.95	0.96
	E 4 B 1.2	0.92	-	0.94	0.93
	E 4 B 1.5	0.91	-	0.92	0.92
	E 4 B 1.8	0.86	-	0.89	0.89
	E 7 B 0.6	0.99	0.99	0.98	0.99
	E 7 B 0.9	0.98	0.98	0.98	0.98
	E 7 B 1.2	0.96	0.97	0.96	0.97
	E 7 B 1.5	0.95	0.96	0.96	0.97
	E 7 B 1.8	0.94	0.94	0.95	0.96
	E 7 B 2.1	0.93	0.93	-	-

Table 5 .

 5 Ethylene mol fraction in the amorphous phase of the polymer the different multicomponent mixtures during the polymerization of ethylene co-1-butene predicted using different thermodynamic models.

			Ethylene Fraction in Amorphous Polymer (f E )
	Run	Pure Component (binary)	Ternary (no alkane)	Quaternary 3 bar iC4	Quaternary 5 bar C3	Quaternary 1 bar nC5
	E 4 B 0.6	0.26	0.75	-	0.79	0.82
	E 4 B 0.9	0.19	0.67	-	0.70	0.69
	E 4 B 1.2	0.15	0.61	-	0.63	0.60
	E 4 B 1.5	0.12	0.58	-	0.58	0.53
	E 4 B 1.8	0.10	0.50	-	0.53	0.48
	E 7 B 0.6	0.38	0.89	0.91	0.87	0.91
	E 7 B 0.9	0.29	0.80	0.84	0.81	0.83
	E 7 B 1.2	0.23	0.73	0.77	0.75	0.76
	E 7 B 1.5	0.19	0.66	0.72	0.72	0.71
	E 7 B 1.8	0.16	0.64	0.67	0.70	0.70
	E 7 B 2.1	0.15	0.60	0.62	-	-

Table 6 .

 6 Reactivity ratios and Joint Confidence Region for Ethylene co-1-butene polymerization at 7 bars of ethylene, with and without isobutane.

	Binary Solubility	Ternary Solubility (no iC4)	Quaternary Solubility (3 bars iC4)
		95% JCR		95% JCR		95% JCR
		Lower Upper		Lower Upper		Lower Upper
	131	74	188	11.4	9.4	13.3	10	8.8	11.9
	0.13	-0.01	0.28	0.52	0.14	0.89	0.58	0.21	0.95
	0.9891			0.9975			0.9980

Table 7 .

 7 Reactivity ratios and Joint Confidence Region for Ethylene co-1-butene

	polymerization at 4 and 7 bars of ethylene
	Binary Solubility	Ternary Solubility (no alkane)
		95% JCR		95% JCR
		Lower Upper		Lower Upper
	75	54	96	9.2	7.7	10.8
	0.04	-0.01	0.09	0.62	0.30	0.98
	0.9825			0.9905

Table 8 .

 8 Reactivity ratios and Joint Confidence Region for Ethylene co-1-butene polymerization at 4 and 7 bars of ethylene for C3 and nC5 alkanes.

	Quaternary Solubility	Quaternary Solubility
		(5 bar C3)			(1 bars C3)
		95% JCR		95% JCR
		Lower Upper		Lower Upper
	10.7	8.5	13	11.4	9.3	13
	0.6	0.15	0.98	0.32	0.06	0.57
	0.9839			0.9884

  estimate component solubilities for the calculation of reactivity ratios. Furthermore, as the complexity of the mixture increases, it becomes very difficult to ignore co-and anti-solubility effects. It is therefore important to have a good thermodynamic representation of the solubilities of the reactivity species of interest in multicomponent systems.

	1
	0.96	Mayo Lewis with binary ethylene co-1-butene
		reactivity ratios
	0.92
	F E
			Mayo Lewis with average "correct" reactivity ratios
	0.88
			Combined ternary and quaternary data
	0.84
	0.8
	0.4	0.5	0.6	0.7	0.8	0.9	1
				f E