Accepted Manuscript

Title: Modelling of Multicomponent Olefins Solubility inpolyolefins using Sanchez-Lacombe Equations of State

Author: Muhammad Ahsan Bashir Mohammad Al-haj AliVasileios Kanellopoulos Jukka Seppala

PII: S0378-3812(13)00436-6DOI: http://dx.doi.org/doi:10.1016/j.fluid.2013.08.009Reference: FLUID 9716

To appear in: Fluid Phase Equilibria

Received date: 5-2-2013Revised date: 5-8-2013Accepted date: 6-8-2013

Please cite this article as: M.A. Bashir, M.A.-h. Ali, V. Kanellopoulos,J. Seppala, Modelling of Multicomponent Olefins Solubility in polyolefinsusing Sanchez-Lacombe Equations of State, Fluid Phase Equilibria (2013),http://dx.doi.org/10.1016/j.fluid.2013.08.009

This is a PDF file of an unedited manuscript that has been accepted for publication.As a service to our customers we are providing this early version of the manuscript.The manuscript will undergo copyediting, typesetting, and review of the resulting proofbefore it is published in its final form. Please note that during the production processerrors may be discovered which could affect the content, and all legal disclaimers thatapply to the journal pertain.

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Modelling of Multicomponent Olefins Solubility in polyolefins using Sanchez-

Lacombe Equations of State

Muhammad Ahsan Bashir1,4

, Mohammad Al-haj Ali2*

, Vasileios Kanellopoulos2, Jukka

Seppälä1,

1. Department of Chemical and Biotechnology, Aalto University, 02150, Espoo, Finland.

2 Innotech Process Technology, Borealis Polymers PO PDO, Borealis Polymers Oy – P. O.

Box330 – Porvoo – Finland.

* Corresponding author.E-mail: mohammad.al-hajali@borealisgroup.com

4. Current address: Laboratoire de Chimie, Catalyse, Polymères et Procédés (C2P2) – LCPP

group, CPE Lyon, Université de Lyon, CNRS UMR 5265, 43 Boulevard du 11 Novembre

1918, 69616 Villeurbanne Cedex, France

*Revised Manuscript

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Abstract

The Sanchez-Lacombe Equations of State (SL EoS) was employed to predict the solubility of

different α-olefins in their corresponding mixtures with polyolefins. Its predictive capabilities

depend on binary α-olefin/polyolefin interaction parameters, which are extracted from the

corresponding binary systems. These parameters can be used directly unless polymer

crystallinity is high where fine adjustment is needed. It was found that model predictions were

in excellent agreement with the available experimental data for different multicomponent

systems over a wide range of pressure, temperature and mixture composition. An accurate

quantitative knowledge of olefins solubility in polyolefins is essential for the detailed design

and modelling of polymerization reactors. Nevertheless, there are a limited number of studies

that deal with modelling of multicomponent α-olefins/polyolefin systems. This work can be

considered as a part of a comprehensive plant-scale modelling approach for industrial

polyolefin manufacturing processes.

Keywords: Sanchez-Lacombe Equations of State, Multicomponent solubility,

Thermodynamics, Polyolefins, Olefins, Crystallinity, Modelling

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1. Introduction

Polyolefins are the most widely used class of polymers due to their wide range of applications

(e.g., packaging, building and construction, transportation, electrics and electronics, furniture,

etc.), low production costs and less environmental impact. The share of polyethylene (i.e.,

HDPE, LDPE and LLDPE) in total production of polyolefins is around 60% and that of

polypropylene is nearly 40%. With the exception of LDPE, polyolefins are industrially

produced in low pressure processes (bulk, slurry and gas phase) that utilize coordination

catalysts, as Z-N, chromium and metallocenes, [1-3].

In commercial catalytic olefin polymerization reactors, a higher α-olefin, typically propylene,

1-butene, 1-hexene, or 1-octene, is used as a co-monomer. Thus, a multicomponent mixture of

α-olefins, diluents, inert and polyolefin coexist in the reactor during polymerization. To

develop a sound understanding of the kinetics of polymerization processes, and enhance the

quality of the produced resins, precise knowledge of the simultaneous and competitive

solubility of α-olefins in polymer phase is required at certain reaction conditions. Therefore,

these properties are required for designing polymerization reactors as well as devices for safe

handling of the reaction products [4-6].

In gas phase polyolefin production processes, the sorption of α-olefins in polyolefins is a

function of temperature, pressure, composition of the gaseous feed and polymer morphology

in terms of crystallinity. For binary α-olefin/polyolefin systems, the solubility of a single

component (e.g., ethylene, propylene, 1-butene, 1-hexene, 1-octene etc.) in polyolefins has

been experimentally measured and modelled by employing various activity coefficient and

equations of state (EoS) models [7-22].

Experimental studies of multicomponent gas mixtures sorption in polyolefins under reactor

conditions are expensive and time-consuming. Furthermore, safety considerations can

significantly add to the cost of such work. Practical industrial applications have traditionally

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relied on empirical correlations and semi-theoretical equations-of-state to generate solubility

data on such solutes and their mixtures in relevant polymeric systems. This makes it

impractical to extend such relationships to different operating conditions due to the large

errors in model predictions. Consequently, it is necessary to understand the thermodynamic

behaviour of such systems under conditions close to that used industrially to improve the

predictability and accuracy of exciting models or to develop more accurate models.

The sorption process of low molecular weight penetrants in semi-crystalline polyolefins

during gas-phase catalytic olefin polymerization has extensively studied by many

investigators. On the other hand, only a few theoretical and experimental studies are available

for the determination of multicomponent sorption processes of such systems. It has been

observed that the presence of a higher α-olefin (e.g., 1-butene, 1-hexene etc) in the reaction

mixture can increase the polymerization rate up to four times, depending upon the reactor

conditions [5,23,24]. This effect is known as the co-monomer effect which had not been

studied for many α-olefins/polyolefin systems over a wide range of operating conditions. In

literature, the co-monomer effect was attributed to one or more of the following reasons: (i)

increased swelling of the polyolefin matrix due to higher-molecular weight component

sorption, which in turn causes an increase in lower-molecular weight component solubility

and diffusivity [5] and/or (ii) changes in chemical nature of the catalyst due to the presence of

higher-molecular weight component.

Robeson and Smith [25] studied the co-sorption (or co-solubility) of different ethane/butane

mixtures in low density polyethylene (LDPE), at atmospheric pressure within a temperature

range from 30 ºC to 60 ºC. They found that the sorption of ethane was mainly affected by the

presence of butane at temperatures above 30ºC. Li and Long [26] found that the presence of

methane enhances the solubility of ethylene in LDPE at 25ºC. Co-solubility of

ethylene/propylene mixture in poly(ethylene-co-propylene) copolymer containing 48.4mol%

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of ethylene was experimentally measured by Yoon et. al., [14] at three different temperatures

(i.e., 50 ºC, 70 ºC and 90 ºC). The authors found that the mixture solubility was higher than

the corresponding ethylene or propylene solubility in the polymer. McKenna [27] measured

experimentally the co-solubility of ethylene/1-butene/nitrogen mixture in LDPE. The author

concluded that the presence of 1-butene or nitrogen in the mixture did not affect the solubility

behaviour of ethylene in LDPE, which was attributed to a very low 1-butene partial pressure

in the mixture.

Moore and Wanke [22] measured the co-solubility of ethylene/1-hexene mixture in linear low

density polyethylene (LLDPE) at 69 ºC and they found that 1-hexene solubility in LLDPE

was reduced as compared to the binary solubility of 1-hexene in LLDPE. It should be noted

that equilibrium was not reached in that work. Novak et. al.,[6] measured ethylene/1-hexene

multicomponent solubility in LLDPE-1-hexene copolymer for various mixture compositions

above and below the melting temperature of LLDPE-1-hexene. It was reported that the

solubility of the ternary mixture was less than the summation of the individual component

solubilities in its corresponding binary mixture. The authors also studied the co-solubility

effect using Perturbed Chain Statistical Associating Fluid Theory (PC-SAFT) EoS

simulations. They found that the presence of 1-hexene in the mixture enhances ethylene

solubility in the polymer (i.e. 1-hexene acts as a co-solvent for ethylene); whereas, the

presence of ethylene in the mixture reduces 1-hexene solubility in the polymer (i.e. ethylene

acts as anti-solvent for 1-hexene). Banaszak et. al., [5] used the experimental data presented

by Novak et. al., [6] for binary systems to determine the required parameters for PC-SAFT

EoS that will be used to predict the solubility of ethylene and 1-hexene in a ternary mixture

consisting of ethylene, 1-hexene, and LLDPE-1-hexene. It was found that PC-SAFT over-

predicted the solubility of different penetrants at temperatures below 100ºC. Moreover, the

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authors found that model predictability can be improved by including the elastic effects of

polymer crystallinity on monomer sorption in the semicrystalline polymer matrix.

Paricaud et. al., [28] investigated the effect of molecular interactions in 1-butene/1-

hexene/polyethylene system on individual solubilities, using Statistical Associating Fluid

Theory-Variable Range (SAFT-VR) EoS. In this work, the co-monomer effect was explained

in terms of the interactions between gas molecules and polyethylene chains. Yao et. al., [4]

measured and modelled the multicomponent sorption of ethylene/iso-pentane mixture and

ethylene/n-hexane mixture in polyethylene below polymer melting temperature using the

Universal Functional Activity Coefficient-Free Volume (UNIFAC-FV). Yao and coworkers

concluded that the presence of iso-pentane or n-hexane increases the solubility of ethylene in

polyethylene.

Phase equilibrium thermodynamic models are essential tools for modelling the sorption of α-

olefins/polyolefins binary and multicomponent mixtures. Developing such models has been

mostly performed by employing group contribution activity coefficient models (e.g., UNIFAC

[4]) or equations of state models (e.g., PC-SAFT and SAFT-VR [5,6,28,29]). Sanchez-

Lacombe equations of state (SL EoS) [30] has excellent predictive capabilities to binary α-

olefins/polyolefins systems [7-10,17,21,31-33]. However, to our best of knowledge, it has

never been applied for modelling multicomponent systems of α-olefins/polyolefins as well as

for analysing the related phenomena like co-solubility effect at different temperatures and

compositions.

The objective of the present work is threefold. First, extending the use of SL EoS for

predicting the solubility of various components in the corresponding α-olefins/polyolefin

mixtures. Second, studying the effect of polymer crystallinity in solubility predictions made

by SL EoS. Third, quantifying the effect of the presence of a higher α-olefin in the ternary

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mixture on individual component’s solubility at various operating conditions and mixture

compositions.

3. Results and Discussion

In this work, SL EoS had been used to predict the solubility of different compounds in

the corresponding mixtures. The derivation of model equations, fitting model parameters as

well as the solution methodology are described in detail in [34]. SL EoS equations are briefly

described in Appendix A.

Predicting the solubilites of different components in a ternary mixture requires

information about the interaction parameters between the penetrating molecules and polymer.

As an example, consider a system comprising of penetrant 1 (1), penetrant 2 (2) and polymer

(3); the corresponding required binary interaction parameters for solving SL EoS are: k12, k13,

and k23. Generally, k13 and k23 are estimated by fitting SL EoS using available experimental

data for corresponding binary systems while k12 is equal to 0 since interaction between small

olefin molecules in such systems can be neglected.

Ethylene/1-hexene/LLDPE-1-hexene is considered as a first system to be studied in

this work. Binary interaction parameters were obtained by fitting SL EoS using the

experimental data given by Novak et. al.,[6] at 70˚C, 90˚C and 150˚C, respectively.

In Figures 1 and 2, the experimentally measured solubilites of the above mentioned binary

systems at two temperatures; namely 70°C and 90

°C, are compared with the corresponding

model predictions obtained by SL EoS. As can be seen, the theoretical predictions agree very

well with the available experimental data.

Note that the binary interaction parameter for ethylene/LLDPE-1-hexene system is

temperature independent; whereas, the binary interaction parameter for 1-hexene/LLDPE-1-

hexene system decreases with increasing temperature.

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Figure 1. Comparison of experimental ethylene solubility in ethylene/LLDPE-1-hexene at

70˚C (), at 90˚C () [6] with SL EoS predictions at 70˚C () and 90˚C ( • ), kij =

0.038 for both temperatures.

Figure 2. Comparison of experimental 1-hexene solubility in 1-hexene/LLDPE-1-hexene at

70˚C (), 90˚C () [6] with SL EoS predictions at 70˚C (), kij = 0.027 and at 90˚C ( •

) kij = 0.016.

0.000

0.002

0.004

0.006

0.0 5.0 10.0 15.0 20.0 25.0 30.0

S

(g/g.PE)

P (bar)

0.000

0.010

0.020

0.030

0.040

0.050

0.060

0.0 0.4 0.8 1.2 1.6 2.0

S

(g/g.PE)

P (bar)

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Novak et. al.,[6] measured the multicomponent solubility of ethylene/1-hexene/LLDPE-1-

hexene mixture consisting of 95.7 mol% ethylene and 4.3 mol% 1-hexene, at 70˚C, 90˚C and

150˚C respectively. The overall mixture solubility predicted by the SL EoS (i.e., the sum of

individual ethylene and 1-hexene solubilities in LLDPE-1-hexene) is shown in Figures 3 and

4. Note that the binary interaction parameters are used to estimate the individual solubility of

different components in the ternary mixture, see Table 1.

Table 1. Fitted binary interaction parameters for binary solubility data of Novak et. al.[6]

T(°C) Ethylene/1-hexene,

k12

Ethylene/LLDPE-1-

hexene, k13

1-hexene/LLDPE-1-

hexene, k23

70 0.00 0.038 0.027

90 0.00 0.038 0.016

150 0.00 -0.05 -0.03

Figure 3 shows that SL EoS cannot predict the overall solubility of ethylene and 1-hexene in

LLDPE-1-hexene when the corresponding binary interaction parameters are used. More

specifically, SL EoS over predicts the experimentally obtained overall solubility, especially at

pressures above 5 bar. Such deviation can be attributed to the fact that SL EoS does not take

into account the elastic effects imposed by the crystalline fraction of the semicrystalline

polyolefin matrix. Generally, there are two methods to minimize these deviations: (i) tuning

the corresponding binary interaction parameters [10,35], or (ii) considering the elastic effects

using either Micheal Hausslein model [36,37] or Sanchez-Lacombe Network Theory [38]. In

this paper, the first approach is considered; meanwhile, the second approach will be discussed

in a separate publication. It is worth mentioning that α-olefin/polyolefin mixtures are complex

systems that involve polymer phase consisting of amorphous and crystalline domains. The

presence of crystalline phase significantly affects on the interaction energy between penetrant

molecules and polymer chains that is visualized as a variation in kij value with respect to the

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system temperature (below polymer melting point, especially when multicomponent systems

are considered). The variation in kij value as a function of temperature can be mitigated by

considering the elastic effects imposed by the crystalline phase. Therefore, combing the SL

EoS with elastic constraint models (e.g. Micheal Hausslein model, Banaszak’s method and

Sanchez-Lacombe Network Theory) allows using a single-value for kij in different α-

olefin/polyolefins systems without taking temperature effect into account as described by

Bashir et. al., [39]. It should also be noted that similar kij dependency on temperature was

reported for different olefins/polyolefins systems using more rigorous approaches like

combined SL EoS/molecular dynamics models [1]. Finally, it is important to emphasize that

for some α-olefins/polyolefins systems, it was found that both interaction parameters used in

SL EoS and PC-SAFT are temperature-dependent as shown in different publications [40-43].

To improve SL EoS solubility predictions for this ternary system, the binary interaction

parameters were further adjusted. It was found that SL EoS predictive capabilities are affected

more by changing k23 than making changes in k13. By modifying k23 value from 0.027 to

0.045 at 70°C and from 0.016 to 0.035 at 90

°C, a very good agreement between experimental

mixture solubility and SL EoS predictions is observed, see Figure 3.

Figure 4 illustrates a comparison between experimental and theoretically calculated overall

mixture solubility in LLDPE-1-hexene, at T=150ºC. It can be seen that SL EoS over predicts

the overall mixture solubility data when kij values of the corresponding binary systems are

used, see Table 2. However, the errors in model predictions are less compared to that depicted

in Figure 3. Thus, k23 was slightly modified to capture the mixture solubility behaviour, see

Figure 4. These findings could be attributed to the fact that the fraction of crystalline phase is

low since solubility experiments were conducted at temperatures above LLDPE-1-hexene

melting point. The findings of Yoon et al. [14] confirm our observation regarding the effect of

polymer crystallinity on mixture solubility predicted by SL EoS.

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Figure 3. Comparison of experimental and SL EoS predicted overall ethylene and 1-hexene

mixture solubility in 1-LLDPE-1-hexene. (♦) experimental overall mixture solubility at T =

70ºC [6] and () SL EoS predictions with k13 = 0.038, k23 = 0.027 and ( • • ) SL EoS

predictions with k13 = 0.038, k23 = 0.045 at T = 70ºC. () experimental overall mixture

solubility at T = 90ºC [6] and ( ) shows SL EoS predictions with k13 = 0.038, k23 =

0.016 and (•••••) SL EoS predictions with k13 = 0.038, k23 = 0.035 at T = 90ºC.

0.000

0.010

0.020

0.030

0.040

0.050

0.0 5.0 10.0 15.0 20.0 25.0 30.0

Overall

solubility

(g/gPE)

P (bar)

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Figure 4. Experimental overall ethylene and 1-hexene mixture solubility in LLDPE-1-hexene

(♦) [6] and SL EoS predictions at T=150ºC with; (i) k13 = -0.05, k23 = -0.03 (), (ii) k13 = -

0.05, k23 = -0.012 ( ).

Ethylene and 1-hexene copolymers with varying co-monomer content are widely produced in

polyolefin manufacturing industry for different applications. During such process, gaseous

feeds having different ethylene to 1-hexene molar ratios are injected to the reactor;

consequently, a multicomponent system of reactants and polymer co-exists in the

polymerization reactor. Figures 5 and 6 show the effect of varying 1-hexene content in the

gaseous feed on overall solubilities in a ternary mixture of ethylene, 1-hexene and LLDPE-1-

hexene at 70ºC and 90ºC, respectively. It can be seen that the overall solubility is

temperature-dependent; nevertheless, this dependency is highly affected by gas mixture feed

composition. For example, by considering 20 bar as system pressure, which is the typical

operating pressure in industrial gas-phase olefin polymerization reactors, increasing the

temperature by 20°C (i.e., from 70°C to 90°C) decreases the overall solubility by

0.000

0.005

0.010

0.015

0.020

0.025

0 5 10 15 20 25 30

Overall

Solubility

(g/g PE)

P(bar)

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approximately 15 %, Figures 5-6, when 3% 1-hexene in feed is selected. However, dramatic

decrease in solubility, ~ 70%, is predicted when 8% 1-hexene is fed to the reactor.

Both figures show also that the overall solubility increases lineally with system pressure for

all compositions at 90°C. On the other hand, nonlinear behaviour is noticed at high

concentration (8 mol %) of 1-hexene in the feed.

Figure 5. Ternary mixture overall solubility predictions with SL EoS at T=70˚C for different

compositions of ethylene/1-hexene feed. (♦)experimental solubility data [6], () SL EoS

predictions for 3mol% of 1-hexene in feed, ( •• ) SL EoS predictions for 4.3mol% of 1-

hexene in feed and ( ) SL EoS predictions for 8mol% of 1-hexene in feed with k13 =

0.038 and k23 = 0.045.

0.000

0.010

0.020

0.030

0.040

0.050

0.060

0.070

0.080

0.090

0 5 10 15 20 25 30

Overall

solubility

(g/gPE)

P (bar)

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Figure 6. Ternary mixture overall solubility predictions with SL EoS at T=90˚C for different

compositions of ethylene/1-hexene mixture. (♦) experimental solubility data [6], () SL

EoS predictions for 3mol% of 1-hexene in feed, ( •• ) SL EoS predictions for 4.3mol% of

1-hexene in feed and ( ) SL EoS predictions for 8mol% of 1-hexene in feed with k13 =

0.038 and k23 = 0.035.

SL EoS not only predicts the overall mixture solubility but also the individual solubility of

different components in the mixture. Figures 7 and 8 compare ethylene solubility in the binary

system of ethylene/LLDPE-1-hexene [6] with the SL EoS predicted ethylene solubility in the

ternary mixture of ethylene/1-hexene/LLDPE-1-hexene at 70˚C and 90˚C, respectively.

Figures 9 and 10 compare 1-hexene solubility in binary and ternary systems at 70˚C and 90˚C,

respectively.

The solubility of ethylene and 1-hexene exhibits different behaviour depending on both

mixture composition and system temperature. It can be seen that, for pressure below 10 bar,

ethylene solubility in the ternary mixture is almost equal to its binary mixture

0.000

0.005

0.010

0.015

0.020

0.025

0.030

0.035

0.040

0 5 10 15 20 25 30

Overall

solubility

(g/gPE)

P (bar)

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(ethylene/LLDPE-1-hexene) solubility at the studied temperatures. On the other hand, by

increasing pressure above 10 bar, ethylene solubility in the ternary mixture is higher than its

corresponding binary mixture solubility due to co-solvent effect of 1-hexene (see Figure 7).

Moreover, 1-hexene solubility in the ternary mixture is lower than its corresponding binary

mixture solubility for pressure values up to 15 bar in all studied ternary mixtures

compositions (see Figure 9). For pressure values above 15 bar and 8mol% 1-hexene ternary

mixture, SL EoS predicts an exponential increase in the individual solubility of both studied

gases.

Figure 7. Comparison of ethylene solubility predicted by SL EoS in the ternary mixture of

ethylene/1-hexene/LLDPE-1-hexene with the experimental binary mixture (ethylene/LLDPE-

1-hexene) solubility at T=70ºC. (♦) binary solubility data [6]. Lines have the same meaning as

that of Figure 5 with same kij values. Note gamor PE represents the mass of amorphous PE

phase.

0.000

0.010

0.020

0.030

0 5 10 15 20 25 30

S

(g/gamor PE)

P (bar)

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Figure 8. Comparison of ethylene solubility predicted by SL EoS in the ternary mixture of

ethylene/1-hexene/LLDPE-1-hexene with the experimental binary mixture (ethylene/LLDPE-

1-hexene) solubility at T= 90ºC. (♦) binary solubility data [6]. Lines have the same meaning

as that of Figure 6 with same kij values.

Figure 9. Comparison of 1-hexene solubility in the ternary mixture of ethylene/1-

hexene/LLDPE-1-hexene with the experimental binary mixture (1-hexene/LLDPE-1-hexene)

solubility at T= 70ºC. (♦) binary solubility data [6]. Lines have the same meaning as that of

Figure 5 with same kij values.

0.000

0.010

0.020

0 5 10 15 20 25 30

S

(g/gamor PE)

P (bar)

0.000

0.040

0.080

0.120

0.160

0.200

0 5 10 15 20 25 30

S

(g/gamor PE)

P (bar)

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Figure 10. Comparison of 1-hexene solubility in the ternary mixture of ethylene/1-

hexene/LLDPE-1-hexene with the experimental binary mixture (1-hexene/LLDPE-1-hexene)

solubility at T= 90ºC. (♦) binary solubility data [6]. Lines have the same meaning as that of

Figure 6 with same kij values.

Contrary to these observations, at 90ºC the co-solvent effect of 1-hexene on ethylene

solubility in the ternary mixture is manifested at pressure values above 10 bar only for the gas

mixture of 8mol% 1-hexene (see Figure 8). Similarly, 1-hexene solubility in the ternary

mixture also shows a linear behaviour at all pressures and it is lower than its binary mixture

solubility (see Figure 10). The linear behaviour of ethylene and 1-hexene solubility with

respect to pressure at 90ºC suggests that the anti-solvent effect of ethylene dominates the co-

solvent effect of 1-hexene. These results are supported by Kumkaew et. al.,[24] who showed

that by increasing both 1-hexene content (above 20 mol/m3) in the gas phase and

polymerization temperature above 80°C, the polymerization rate decreases significantly.

The above mentioned experimental and theoretical results demonstrate that temperature,

pressure and gas phase composition in the ethylene/1-hexene mixture can significantly affect

0.000

0.020

0.040

0.060

0.080

0.100

0.120

0.140

0 5 10 15 20 25 30

S

(g/gamor PE)

P (bar)

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the multicomponent solubility behaviour of α-olefins/polyolefin mixtures. In particular, the

co-solvent effect of 1-hexene dominates the anti-solvent behaviour of ethylene at high

pressures and low temperatures, whereas, the anti-solvent effect of ethylene dominates the co-

solvent effect of 1-hexene at higher temperatures and lower pressures.

McKenna [27] measured the overall solubility of a mixture consisting of nitrogen, 1-butene

and ethylene in low density polyethylene (LDPE) with co-monomer content of 3.3mol% at

three different temperatures (i.e., 70˚C, 80˚C and 90˚C). It should be mentioned that reactor

pressure was kept constant at 21 bar by adding 12 bar of nitrogen. Since nitrogen solubility in

polyolefins is very low [21], the solubility data was modelled without considering the

presence of nitrogen. In addition, during these experiments ethylene and 1-butene were

supplied at a constant flow rate, therefore, a constant gas phase composition can be assumed.

McKenna reported the individual solubilities of ethylene and 1-butene in LDPE obtained from

the mixture solubility experiments. These individual solubilities were summed up to get the

overall solubility of the mixture in LDPE.

The experimental measurements of McKenna [27] of the individual solubility of 1-butene,

ethylene as well as the overall solubility of the mixture in LDPE were also used to verify the

predictive capabilities of SL EoS at different temperature as shown in Figure 11. The

variation of the interaction parameters i.e., k13 for ethylene/LDPE and k23 for 1-butene/LDPE,

with respect to temperature is depicted in Table 2. It is important to point out that both

interaction parameters decrease nonlinearly with temperature; however, the dependency is

different.

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Figure 11. Experimental and SL EoS predicted ethylene, 1-butene and overall solubility of the

mixture in LDPE at different temperatures and a total pressure of 11.85 bar(g). ()

experimental ethylene solubility and () SL EoS predicted ethylene solubility, (♦)

experimental 1-butene solubility and ( ) SL EoS predicted 1-butene solubility, ()

experimental overall mixture solubility and ( • • ) SL EoS predicted overall mixture

solubility. Experimental solubility data taken from McKenna et. al., [27].

Table 2: Temperature dependence of kij for different α-olefin/polyolefin systems

System kij

Ethylene(1) /1-butene(2)/LDPE(3) k13 = - 4.5 10

-5 T

2 + 0.03 T - 6.2

k23 = 5 10-5T

2 - 0.034 T + 5.75

Ethylene(1)/propylene(2)/ICP-PP(3) k13 = - 8 10

-4 T + 0.0408

k23 = - 6 10-4 T + 0.0594

Yoon et. al.,[14] measured the overall solubility of ethylene and propylene mixture in random

ethylene/propylene copolymers at 50ºC, 70ºC and 90ºC respectively. The random copolymers

studied were of low crystallinity (i.e., less than 20%). To model the solubility of this ternary

mixture by employing SL EoS, the binary interaction parameters were obtained by fitting the

binary experimental solubility data of Sato et. al., [17] who measured the solubility of

ethylene and propylene in impact polypropylene (ICP-PP) at 50.2ºC, 70.2ºC and 90.2ºC,

0.000

0.010

0.020

0.030

0.040

0.050

0.060

0.070

65 70 75 80 85 90 95

S

(g/g. amor PE)

T(ºC)

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respectively. Figures 12 and 13 show the comparison between ethylene/ICP-PP and

propylene/ICP-PP experimental solubility data with SL EoS model predictions. It can be seen

that there is excellent agreement between model predictions and the experimental solubility

data for both binary systems. In addition, it should be noted that the binary interaction

parameters for both binary systems depend linearly on temperature as illustrated in Table 2.

Figure 12. Experimental and SL EoS predicted binary solubility of ethylene in ICP-PP. (♦)

experimental solubility, () SL EoS predictions with kij = 0.00 at T = 50.2°C, ()

experimental solubility, ( ) SL EoS predictions with kij = -0.014 at T = 70.2°C, ()

experimental solubility, ( • • ) SL EoS predictions with kij = -0.032 at T=90.2°C.

0

5

10

15

20

25

0 5 10 15 20 25 30 35

S

(kg/kg.Polymer)

P(bar)

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Figure 13. Experimental and SL EoS predicted binary solubility of propylene in ICP-PP. (♦)

experimental solubility, () SL EoS predictions with kij = 0.031 at T = 50.2°C, ()

experimental solubility, ( ) SL EoS predictions with kij = 0.018 at T = 70.2°C, ()

experimental solubility, ( • • ) SL EoS predictions with kij = 0.008 at T=90.2°C.

The obtained binary interaction parameters are used in SL EoS to predict the overall solubility

of ethylene/propylene mixture in ethylene-propylene copolymer at various temperatures, see

Figures 14 and 15, which are reported by Yoon et .al. [14]. Interestingly, due to the

amorphous nature of the copolymer (1.3 wt % crystallinity), SL EoS captured the overall

solubility of the ternary mixture at both temperatures although binary interaction parameters

were used in the thermodynamic model.

The figures show that SL EoS can predict the solubility of ethylene/propylene mixture in

ethylene-propylene copolymer (EPC) at 50°C and 90°C with maximum deviation of 20 % at

0.2 atm propylene pressure.

In binary systems of olefin/polyolefin, the solubility of penetrant (i.e. olefin) in polyolefin

exhibits linear behaviour at low system pressure; this behaviour deviates from linearity as the

pressure increases. The range over which such linearity can be seen depends upon the

0

20

40

60

80

100

120

0 5 10 15 20 25 30 35

S

(kg/kg.Polymer)

P(bar)

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molecular weight of olefin molecules [1]. The presence of ethylene and propylene in a

mixture affects the linear behaviour of overall solubility with respect to system pressure.

Consequently, ethylene solubility deviates from linearity although it exhibits linear behaviour

over the wide range of system pressure, 1-30 bar [1]. Note that although the fraction of

ethylene in the system, Figures 14 and 15, approaches 1.0 (i.e. close to binary system) the

overall solubility deviates from linearity. The deviation of overall solubility of a mixture,

which contains ethylene and α-olefin, is expected to be more pronounced as the molecular

weight of α-olefin increases, propylene < 1-butene < 1-hexene < 1-octene. This observation is

of paramount importance for the industrial operation of catalytic olefin polymerization

reactors. Generally, when copolymer grades are produced, relatively low fractions of co-

monomers are injected to the reactor resulting in ethylene solubility enhancement that has to

be taken into account to ensure smooth reactor performance.

As the copolymer was almost amorphous (i.e., 1.3% crystalline) the SL EoS co-solubility

predictions are in close agreement with the experimental values and there is no need to further

adjust the binary interaction parameters in order to fit the co-solubility data with minimum

error.

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Figure 14. Comparison of experimental and SL EoS predicted co-solubility for the ternary

system of Yoon et. al.,[14] at T=50ºC. (♦)experimental overall mixture solubility at propylene

pressure = 0.3atm, ( ) SL EoS predictions with k13 = 0.00, k23 = 0.031, ()

experimental overall mixture solubility at propylene pressure = 0.2atm, () SL EoS

predictions with k13 = 0.00, k23 = 0.031, () experimental overall mixture solubility at

propylene pressure = 0.1atm, (• • •) SL EoS predictions with k13 = 0.00, k23 = 0.031

0.0000

0.0005

0.0010

0.0015

0.0020

0.0025

0.0030

0.0035

0.0040

0.350 0.450 0.550 0.650 0.750 0.850 0.950

Overall

solubility

(g/g PE)

Mole fraction of Ethylene

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Figure 15. Comparison of experimental and SL EoS predicted co-solubility for the ternary

system of Yoon et. al.,[14] at T= 90ºC. (♦) experimental overall mixture solubility at

propylene pressure = 0.3atm, ( ) SL EoS predictions with k13 = -0.032, k23 = 0.008, ()

experimental overall mixture solubility at propylene pressure = 0.2atm, () SL EoS

predictions with k13 = -0.032, k23 = 0.008, () experimental overall mixture solubility at

propylene pressure = 0.1atm, (• • •) SL EoS predictions with k13 = -0.032, k23 = 0.008.

0.0000

0.0005

0.0010

0.0015

0.0020

0.0025

0.35 0.45 0.55 0.65 0.75 0.85 0.95

Overall

solubility

(g/g PE)

Mole fraction of Ethylene

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Conclusion:

Sanchez-Lacombe Equations of State (SL EoS) had been used to model the solubility of

different ternary mixtures containing α-olefins and a polyolefin at various temperatures and

pressures. Temperature, pressure and mixture composition affect significantly the solubility

behaviour of multicomponent α-olefin mixtures in polyolefins. SL EoS predictions show that

the presence of co-monomer enhances the overall solubility of the mixture in the polymer

phase; moreover it was found that this enhancement is co-monomer-type dependent. SL EoS

model predictions are in full agreement with the available experimental data. The

experimental and theoretical results demonstrate that temperature, pressure and gas phase

composition in the ethylene/1-hexene mixture can significantly affect the multicomponent

solubility behaviour of α-olefins/polyolefin mixtures. In particular, the co-solvent effect of 1-

hexene (i.e., 1-hexene acts as a co-solvent for ethylene) dominates the anti-solvent behaviour

of ethylene (i.e., ethylene acts as an anti-solvent for 1-hexene) at high pressures and low

temperatures whereas the anti-solvent effect of ethylene dominates the co-solvent effect of 1-

hexene at higher temperatures and lower pressures.

The predictive capabilities of SL EoS depend on binary interaction parameters which are

found to be temperature-dependent for the studied systems. These parameters can be used to

predict the solubility of multicomponent systems; some adjustment to one of them was found

to be necessary to improve model predictions unless polymer crystallinity is low.

Model results confirm the effect of co-monomer on the linear behaviour of ethylene solubility

in polyolefins. By increasing the molecular weight of co-monomer, ethylene individual

solubility in polyolefins deviates more from its well-known solubility behaviour with respect

to pressure.

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Solubility of α-olefins in polyolefins is one of the most important thermodynamic properties

which have been used as a design parameter in industrial catalytic olefin polymerization

reactors. The present work brings a tool that can be used to precisely estimate the solubility of

different components of interest in polyolefins under wide range of operating conditions

commonly used in industry.

Acknowledgment

The authors gratefully acknowledge Borealis Polymer Oy for supporting this work.

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Appendix A. Sanchez-Lacombe Equation of State (SL EoS)

SL EoS is one of the simplest statistical thermodynamics models capable of describing the

phase behaviour of monomer(s)/polymer binary and multicomponent systems [1,7-10,12,44].

Following the original developments of Sanchez and Lacombe, the general expression of the

SL EoS can be written as:

(A.1)

Where , and are the reduced density, the reduced pressure and the reduced temperature

of a pure component, respectively. These reduced properties are related to the corresponding

absolute properties as follows:

(A.2)

Where ρ*, P

* and T

* are the scale factors known as characteristic density, characteristic

pressure and characteristic temperature, respectively, and are used to characterize each pure

component in the mixture. According to McHugh and Krukonis [45] the chemical potential of

the ith

component in a multicomponent system can be expressed as:

(A.3)

Where, is the volume fraction of the ith

and the jth

component in the mixture, rmix is the

number of lattice sites a fluid molecule occupies in the mixture, v*

mix is the mass based volume

of empty lattice sites in the mixture in m3.mole

-1, ε

*mix

is the characteristic interaction energy

per mole of segments in the mixture in J.mole-1

, mix is the reduced density of the mixture,

0)1

1()1ln(2

rTP

,*

**

1

*

1 1

****

*

2ln1ln1

21ln

mixij

Nc

j

jmixmix

i

mixmixmixmixi

mix

Nc

j

Nc

j

ijjmixijijj

mix

mixi

mix

iii

vvvPr

vRTr

vvv

rr

rRT

,*P

PP

*T

TT

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mixv is reduced volume of mixture calculated from the reciprocal of the reduced mixture

density ( mix ), R is the universal gas constant in m3.bar.mole

-1.K

-1 and Nc denotes the

number of components in the mixture.

At equilibrium, the chemical potential, i , of each component in a two-phase multicomponent

system will be equal. In α-olefin polymerization, one phase consists of components

(monomer, comonomer, diluted, chain transfer agents, etc) and the other phase consists of

polymer and sorbed components. Therefore, the equality of chemical potential of each species

in both phases can be written as:

(A.4)

In the fore-coming discussion, (co)-monomer(s) will be represented by 1and 2 and the

polymer will be represented by 3. Therefore, by solving the Equation A.4 using a non-linear

algebraic equation solver, (e.g., GRG in MS Excel and FSOLVE in MATLAB) for each

monomer in the ternary system, the volume fraction of each sorbed monomer in the

amorphous polymer, 1 and 2 can be calculated. Finally, the equilibrium solubility of the

each monomer per gram of polymer, S, can be calculated by Equation A.5.

(A.5)

where, ωi is the mass fraction of ith

component in the polymer phase, ω3 is the polymer phase

volume fraction and αc is the amorphous mass fraction in the semi-crystalline polymer.

Kanellopoulos et. al.,[9,34] used molecular dynamics to determine pure components’

characteristic parameters that are used in the present work, see Table A.1.

...3,2,1, ipolymer

i

gas

i

ciS

3

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Table A.1. Pure Component Characteristic Parameters used in SL EoS model.

Pure component T* (K) P*(bar) ρ* (kg/m3)

Ethylene 283 3395 680

1-Butene 410 3350 770

1-Hexene 450 3252 814

Propylene 692 3007 890

LLDPE-1-Hexene 653 4360 903

Impact polypropylene 689 3175 890

LDPE [32] 655 4399 900