An Equation of State-Based Modeling Approach for Estimating the

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An Equation of State-Based Modeling Approach for Estimating the Partial Molar Volume of Penetrants and Polymers in Binary Mixtures Muhammad Ahsan Bashir,*,†,‡ Vincent Monteil,† Vasileios Kanellopoulos,§ Mohammad Al-haj Ali,§ and Timothy F. L. McKenna*,† †

Université de Lyon, Univ. Lyon 1, CPE Lyon, CNRS, UMR 5265, Laboratoire de Chimie Catalyse Polymères et Procédés (C2P2), LCPP team, Bat 308F, 43 Bd du 11 novembre 1918, F-69616 Villeurbanne, France ‡ Dutch Polymer Institute DPI, P.O. Box 902, 5600 AX Eindhoven, The Netherlands § Process Development Group − Innovation Process Technology, Borealis Polymers PO PDO, Borealis Polymers Oy, P.O. Box330, Porvoo, Finland S Supporting Information *

ABSTRACT: Partial molar volumes (PMVs) of highly soluble hydrocarbon gases and polymers in macromolecular mixtures have been estimated by using a model-based methodology that utilizes binary gas/polymer sorption data. It has been shown that the PMVs of the penetrants and polymers as well as other volumetric properties (e.g., molar volume per repeating unit and volumetric thermal expansion coefficients) can be accurately calculated over a wide range of temperatures and pressures. The advantages of the proposed methodology are that it relies on available experimental sorption data of gases in polymers and an equation of state, provides an attractive alternative to the more onerous experimental procedures of densimetery and sorptive dilation for obtaining volumetric data of polymeric mixtures, and it can be extended to multicomponent systems.

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INTRODUCTION Thermodynamic properties of chemical species in their pure form and in mixtures with other chemically similar or dissimilar species are invaluable and inevitable in chemical processes design. Polymers are more complex in structure compared to light compounds, exhibiting nonideal thermodynamic behavior, both in their pure and mixture forms. This complexity of macromolecular structure necessitates the knowledge of their volumetric properties, especially in polymeric solutions, commonly known as the partial molar volumes or partial specific volumes. The development of modern analytical techniques for the measurement of macromolecular properties requires the precise knowledge of such volumetric properties (e.g., ultracentrifugation, small-angle scattering and supercritical fluid chromatography).1,2 In other areas, the modeling of chemical processes involving mixtures of polymers and other compounds necessitates the use of numerous equations of states (EoS) that rely on the chemical potential of the species for predicting various physical and thermodynamic properties of the mixtures. The partial molar volume (PMV) is also required to calculate the pressure coefficient of the chemical potential of all mixture compounds.3 The pressure dependence of the solubility of a given compound in polymers (normally known as the penetrant solubility) can also be obtained by PMV values. The Flory−Huggins (FH) equation,4 Michaels-Haussleix equation,5 and Banaszak’s modified Michaels-Haussleix equation6 also need the precise PMV values of substances in the polymeric mixture. In addition, a correct representation of a component’s PMV in the mixture is a rigorous test for a model predicting the solubility of such a component in a solvent. Moreover, the PMV of a penetrant in a mixture with a polymer © 2013 American Chemical Society

also provides information on the molecular interactions between penetrant molecules and polymer chains.2,7 In the open literature, various experimental techniques have been employed for measuring the PMV of various components in polymers. Densimetery1,8 and sorption-swelling or dilation data4,9−15 of penetrant/polymer mixtures have been mainly used for calculating the PMV of various penetrants (i.e., monomers and light inert gases) in polymers, under different conditions. Densimetery is used for liquid phase polymer/ solvent systems, whereas in the sorptive dilation technique a gaseous component or a mixture of gaseous components is sorbed in a solid polymer phase (i.e., film or powder).1,4,9−12,14,15 Simultaneous sorption and dilation data has been extensively used in the past since it takes into account the polymer compressibility due to the development of the gas pressure. Polymer compression by the gas pressure is significant in the case where less soluble gases (e.g., O2, N2 etc.) are involved, whereas it is negligible for highly soluble gases (e.g., C2H4, C2H6, C3H8 etc.).4,9,11,14,15 Kamiya et al.,4,13,14 carried out sorption and dilation measurements for various penetrant/polymer mixtures and calculated the PMV of various penetrants in different polymers. They observed no concentration dependence of PMV for a number of organic and inorganic gases in poly(dimethylsiloxane), polyethylene, polypropylene, 1,2-polybutadiene, and poly(ethylene-co-vinyl acetate) at 25 °C. On the other hand, a linear relationship between the critical volume, Received: Revised: Accepted: Published: 16491

August 12, 2013 October 21, 2013 October 22, 2013 October 22, 2013 dx.doi.org/10.1021/ie4025193 | Ind. Eng. Chem. Res. 2013, 52, 16491−16505

Industrial & Engineering Chemistry Research

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It is shown that the results obtained by this methodology are in excellent agreement with those obtained by the other more onerous, complex, and expensive experimental methods (i.e., densimetery, sorption, and swelling measurements). It is important to point out that the established methodology is applicable for highly soluble gases in polymers for which the polymer compressibility due to system pressure caused by the accumulation of gases is negligible. The proposed methodology can also be considered as an alternative to the use of density and dilation experiments for the calculation of penetrants and polymers volumetric properties. Finally, it can be used to study the PMV behavior of penetrants and polymers in multicomponent systems where more than two components are sorbed in a polymer.

van der Waals volume, and PMV was observed for all of the penetrants studied. It should be pointed out that the authors did not report the effect of polymer crystallinity on the PMV. De Angelis et al.,9 reported the PMV of N2, O2, CO2, CH4, C2H6, C3H8, CF4, C2F6, and C3F8 in poly(dimethylsiloxane) (PDMS) at 35 °C and pressure values up to 27 bar using sorption and dilation data. They observed that the PMV values of hydrocarbons and fluorocarbons were similar in all of the studied polymers and organic liquids. However, the PMV of ethane and propane showed a concentration dependence, but the authors provided no further information. Following the same approach, Bonavoglia et al.,10−12 studied the evolution of the PMV of CO2 under supercritical conditions in poly(methylmethacrylate) (PMMA), poly(vinylidene fluoride) (PVDF), poly(tetrafluoroethylene) (PTFE), and tetrafluoroethylene-perfluoromethylvinyl ether (MFA) and observed larger values in semicrystalline polymers (e.g., PVDF and PTFE) than in the amorphous polymers (e.g., PMMA and MFA). They attributed the unexpectedly large observed values of the CO2 PMV in semicrystalline polymers to the effect of crystallinity on the restriction of the mobility of polymer chains (i.e., elastic effect).6 It was also concluded that polymer crystallinity induces a nonequilibrium phenomena and, thus, the nonequilibrium version of Sanchez−Lacombe EoS (SL EoS) is more suitable in predicting the polymer swelling as compared to its equilibrium versions (i.e., the original SL EoS and the Sanchez−Lacombe network theory (SLNT)). Recently, Riberio et al.,15,16 have shown that the PMV of CO2 and C2H6 in rubbery polymers depends on the polymer structure. It can be concluded from this discussion that both the methods (i.e., densimetery and sorption-swelling experiments) for estimating PMV of components in penetrant/polymer mixtures require corresponding experimental volumetric/ density data. However, it also appears that the equilibrium Lattice Fluid model of Sanchez and Lacombe,17commonly used for macromolecular systems, can be used to obtain reliable volumetric or density data for different mixtures, provided that experimental solubility data is available for the mixture under consideration. It should be pointed out that other EoS models have been used to estimate the PMV of the low molecular weight chemical components,2,18−20 however, to the best of our knowledge, no application of any EoS model (similar to what is proposed in this work) for estimating PMVs is available for penetrants/polymer mixtures. In the present work, a model-based methodology is established with the objective of predicting the penetrant/ polymer mixture density by employing the Sanchez−Lacombe equation of state (SL EoS). According to the proposed approach the PMV of various hydrocarbon penetrants in different polymers can be accurately estimated by using the SL EoS predicted binary mixture density data at different temperatures and pressures. More specifically, a number of important volumetric properties (i.e., PMV of α-olefins and polyolefins, molar volume of polyolefins per repeating unit and volumetric thermal expansion coefficients of polyolefins) have been precisely estimated with the proposed methodology at conditions of industrial importance. The proposed methodology involves: (i) fitting the binary mixture (i.e., penetrantpolymer) solubility data with SL EoS at the given temperatures and pressures; (ii) calculating the mixture density from the reduced mixture density predicted by the SL EoS; (iii) calculating the PMV of penetrants(s) and polymers by considering the mixture density derived in the second step.

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MODEL DEVELOPMENT Phase behavior and other thermo-physical properties of the mixtures containing one or more macromolecular components are different from nonpolymeric fluids/mixtures. To precisely describe thermodynamic properties of such mixtures involving molecules of arbitrary size, Sanchez and Lacombe17 developed the lattice-fluid model, commonly known as the Sanchez− Lacombe Equation of State (SL EoS). This equation of state (EoS) has its basis in statistical mechanics, according to which a partition function is used to estimate the thermodynamic properties.17,21,22 In its generalized form, the SL EoS can be written as follows for a pure component: ⎡ ⎛ 1⎞ ⎤ ρi2 + Pi + Ti ⎢ln(1 − ρi ) + ⎜1 − ⎟ρi ⎥ = 0 ⎢⎣ ri ⎠ ⎥⎦ ⎝

(1)

where P̅ , T̅ , and ρ̅ are the reduced pressure, reduced temperature, and reduced density of the ith pure component, respectively. The relationship between the absolute and reduced properties of the ith component can be written as17,23 Pi =

Pv* P = i , Pi* εi*

ρi =

ρi rv* = ii , vi ρi*

T RT = , Ti* εi* MiPi* ri = RTi*ρ *

Ti =

i

(2)

where Pi* is the characteristic pressure, vi* is the closed packed mer volume, εi* is the interaction energy per mer, ri is the segment number, T*i is the characteristic temperature and ρ*i is the characteristic density of ith component. Further details about the derivation, fitting of the model equations to solubility data, and obtaining the penetrant mass fraction in the mixture from SL EoS can be found in refs 23 and 24. The mass fraction of each component, ωi, at a specified temperature and gas phase pressure provides the penetrant solubility, S. The difference between calculated monomer solubility, Scal, and the experimental solubility, Sexp, is calculated by the following objective function equation. ⎛ |S exp − S calc| ⎞ ΔS = ⎜ ⎟100 S exp ⎝ ⎠

(3)

The partial molar volume of the ith component in a binary mixture,V̅ i, is defined as ⎛ ∂V ⎞ Vi = ⎜ ⎟ ⎝ ∂ni ⎠T , P , n

j

16492

(4)

dx.doi.org/10.1021/ie4025193 | Ind. Eng. Chem. Res. 2013, 52, 16491−16505


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where V is total volume of the system, ni is the number of moles of the ith component added into the system and nj is the number of moles of the other mixture component. As a norm, the change in total volume of the penetrant/polymer system is measured in terms of mixture density by using densimetery1,8 or by dilation/swelling experiments.4,9,12 In general, penetrant sorption in a polymer matrix is experimentally measured by keeping the amount of polymer constant and changing only the gas phase pressure (i.e., the number of moles of penetrant, ni, are increased stepwise in the mixture, whereas, the number of moles of polymer, nj, are kept constant). Therefore, if the solubility data of a penetrant can be fitted with an appropriate EoS, the penetrant/polymer mixture density can also be predicted precisely and thus used to calculate V̅ i at various conditions. It is important to point out that the SL EoS considers the polymer phase to be completely amorphous, and includes the assumption that it behaves like a macromolecular liquid.9,17,21,22,25,26 Therefore, a liquid mixture or solution is assumed to exist at the interface of penetrant/polymer mixture below the polymer melting temperature. The reduced penetrant/polymer mixture density, ρ̅, can be obtained by solving SL EoS explained in detail elsewhere.23 Finally, the mixture density can be obtained by the following relationship:17,21,22,26,27

ρ = ρ ̅ × ρ*

is not available are simulated. On the basis of that analysis, a relationship between the PMV and critical volumes of C2 to C8 α-olefins in binary mixtures with polyolefins is proposed. Finally, the SL EoS fitted binary interaction parameter, kij, values obtained for various α-olefin/polyolefin binary mixtures are correlated with the calculated PMVs of C2 to C8 α-olefins. 1. PMV of Alkanes in 1,2-Poly(butadiene). Sorption studies performed in the past have revealed that the solubilities of hydrocarbon gases in polymers exhibit similar behavior to their corresponding solubilities in liquid n-alkanes.4,14 Sorption and dilation isotherms of various hydrocarbon penetrants in 1,2 poly(butadiene) (PB) have been measured experimentally by Kamiya et al.,14 at 25 °C and the PMVs were estimated using that data. Among the gases analyzed by Kamiya et al.,14 ibutane, n-butane, and n-pentane have the highest solubility in PB. Therefore, the sorption data of these gases was used to calculate the penetrant/polymer binary mixture densities with SL EoS at 25 °C, and thus the PMV of the penetrant. Figure 1 shows an excellent agreement between the experimental and SL EoS fitted sorption data of i-butane, n-butane, and n-pentane in PB. It should be noted that the relative error between the experimental data and fitted model predictions is 2%. It is important to point out that the relative error between the experimental data and fitted SL EoS predictions should be as minimum as possible because the same error will be approximately transmitted to the mixture densities. Figure 2 shows the mixture densities calculated by the SL EoS for the three binary mixtures mentioned above. It should be noted that the same binary interaction parameter values (kij) were used to predict the penetrant solubility and mixture density. As can be seen, mixture densities decrease with an increase in penetrant solubility and molecular size, indicating a high degree of polymer dilation-swelling. Once the mixture densities are calculated from SL EoS, the PMV of the penetrants in the polymers can be calculated by following the same methodology for low molecular weight compound mixtures;3 note that sample calculation is shown in Table S1 of the Supporting Information for i-butane/PB mixture. Table 2 depicts the comparison between the calculated PMVs of ibutane, n-butane, and n-pentane with the ones reported by Kamiya et al.14 As it can be seen, there is an excellent agreement between the values obtained by both methods. It is important to point out that the PMV values obtained by the conventional method of using sorption and dilation data and the one proposed in this work which uses only the sorption data in conjunction with the SL EoS show a maximum deviation of 1.9% for n-butane. In addition, the PMV of the penetrants were independent of their concentration in PB, which is in full agreement with the experimental work done by Kamiya et al.14 The above results indicate that the proposed PMV calculation approach can provide excellent PMV estimates, provided that the compression of the polymer phase, caused by the gas pressure during sorption experiments, is negligible. 2. The PMV of Alkanes and Alkenes in Poly(dimethylsiloxane). In a separate work, Kamiya et al.,4 used the experimental sorption and dilation data of various hydrocarbons in amorphous poly(dimethylsiloxane) (PDMS) to calculate their PMV at 25 °C. Experimental sorption data of five different binary hydrocarbon gas/PDMS systems studied by Kamiya et al.,4 were used to estimate the PMVs with the combined sorption data/model-based methodology proposed in this work. The selected binary systems involved gases for

(5)

After the mixture density and penetrant mass fraction, ω1, in the mixture is calculated, V̅ i can be calculated numerically or graphically by following the same approach for the mixtures of light compounds.3 Table 1 shows the SL EoS pure component characteristic parameters for all the penetrants and polymers studied in this work. Table 1. SL EoS Pure Component Characteristic Parameters Used in This Work component

T* (K)

P* (bar)

ρ* (kg/m3)

ref

ethane ethylene propane i-butane n-butane n-pentane propylene 1-butene 1-hexene 1-octene PDMS 1,2-poly(butadiene) HDPE LLDPE-1-butene LLDPE-1-hexene polypropylene impact polypropylene

320 283 375 398 403 441 346 410 450 487 498 615 650 667 653 692 689

3300 3395 3200 2880 3220 3100 3790 3350 3252 3300 2925 3620 4250 4370 4360 3007 3175

640 680 690 720 736 755 755 770 814 842 1080 956 905 900 903 890 890

9 26 9 28 28 28 26 26 26 26 9 29 26 26 26 26 26

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RESULTS AND DISCUSSION To prove the concept of the proposed model-based methodology for calculating the PMVs of penetrants and polymers, a number of penetrant/polymer systems have been considered. First, the binary systems for which the experimental values of PMVs (especially that of the penetrant) are available are analyzed and then the binary mixtures for which the PMV data 16493



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Figure 1. Experimental and fitted SL EoS solubility predictions of i-butane/PB, n-butane/PB and n-pentane/PB binary systems at 25 °C:14 i-butane (⧫) with kij = 0.010 (); n-butane (▲) with kij = 0.0115 (···); n-pentane (□) with kij = 0.0115 (---). S* is solubility in grams of solvent per gram of amorphous polymer.

Figure 2. Mixture density predictions obtained with SL EoS at 25 °C. i-butane/PB with kij = 0.010 (), n-butane/PB with kij = 0.0115 (···), npentane/PB with kij = 0.0115 (---).

used for all calculations performed regarding solubility and mixture density predictions. In Table 3 the calculated PMVs of the five penetrants in PDMS at 25 °C are compared with the corresponding ones measured by Kamiya et al.,4 using sorption and dilation data. As can be seen, the theoretical predictions are in very good agreement with the available experimental data derived by using the conventional sorption-swelling measurement-based method.4,9,14 It should be noted that for all studied systems, the theoretical predictions show less than 10% deviation from the experimental measurements. Such difference can be either attributed to experimental errors in sorption or in dilation measurements (i.e., Kamiya et al.,4 reported an error of ±2% in their PMV calculations). In addition, uncertainties in estimating the pure component characteristic parameters (i.e., T*, P*, ρ*) used in the SL EoS can also be the reason for such differences between the two methods.23 For all studied systems it was found that PMVs were almost independent of penetrant concentration in the mixture (not shown here), which is in-line with the results of Kamiya et al.,4 indicating the consistency with volume additivity rule in the amorphous polymer mixtures.

Table 2. Comparison between Penetrant PMV in PB Calculated by This Work and Reported by Kamiya et al. PMV this work

PMV Kamiya et al.,14

penetrant

cm3/mol

cm3/mol

i-butane n-butane n-pentane

95.2 93.4 107.6

96.1 95.2 106.1

which the polymer compression due to applied gas pressure can be neglected, since their solubility in the polymer phase is high. Figure 3 illustrates the comparison of experimental data and the fitted SL EoS solubility predictions for the five binary systems, namely, (i) ethane/PDMS, (ii) ethylene/PDMS, (iii) propane/ PDMS, (iv) propylene/PDMS and (v) 1-butene/PDMS.4 It is apparent that fitted SL EoS predictions are in excellent agreement with the experimental sorption data. As it can be seen, mixture density of each binary system calculated by the SL EoS decreases with the increase in pressure and the penetrant molecular size (see Figure 4). It should be noted that the same value for the binary interaction parameter (kij) was 16494



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Figure 3. Experimental and fitted SL EoS solubility predictions of five binary systems at 25 °C.4 Comparison of data (points) and SL EoS with fitted kij: ethane (◇) with kij = 0.036 (---); ethylene (●) with kij = 0.013 (−·−); propane (▲) with kij = 0.0252 (···); propylene (□) with kij = 0.035 (−··−); 1-butene (⧫) with kij = 0.035 ().

Figure 4. Binary mixture densities predicted by SL EoS at 25 °C: C2H4/PDMS with kij = 0.013 (---); C2H6/PDMS with kij = 0.036 (−·−); C3H6/ PDMS with kij = 0.0252 (···); C3H8/PDMS with kij = 0.035 (−··−); 1-C4H8/PDMS with kij = 0.035 ().

sities at three different temperatures, predicted by the proposed methodology using the sorption data of Kiparissides et al.,30 and Banaszak et al.,6 respectively. It can be seen that mixture density decreases with the increase in temperature and gas pressure following a similar linear trend for all studied systems. It should be emphasized that mixture density predictions for ethylene/LLDPE-1-hexene obtained in this work are in good agreement with those derived by Banaszak et al.,6 using PCSAFT. The calculated average PMV of ethylene in its mixture with HDPE was 52.5 cm3/mol at 50 °C, 53.5 cm3/mol at 60 °C and 54.1 cm3/mol at 80 °C. Similarly, in ethylene/LLDPE-1hexene binary mixture, the average PMV of ethylene was found to be 53 cm3/mol at 70 °C, 55 cm3/mol at 90 °C, and 59 cm3/ mol at 150 °C. Almost no concentration dependence of PMV of ethylene was observed for both the mixture types. In this temperature range, getting similar values of PMVs in both polymer types indicates that interaction between ethylene molecules and polymer chains are almost the same, with a slightly higher interaction energy in HDPE (as it is also indicated by the kij values). The highest PMV values were observed at 150 °C in LLDPE1-hexene (i.e., above the melting temperature), and were found to be independent of the penetrant concentration. It can also be noticed that the PMV of ethylene in both systems is a function

Table 3. Comparison of PMVs Calculated in This Work and Kamiya et al. PMV, this work

PMV, Kamiya et al.4

penetrant

cm3/mol

cm3/mol

ethane ethylene propylene propane 1-butene

65 59 76 84 92

70 63 78 85 96

3. PMV of α-Olefins in Polyolefins. 3.1. PMV of Ethylene in PE. The solubility of ethylene in HDPE has been measured experimentally by Kiparissides et al.,30 at 50, 60, and 80 °C and in LLDPE-1-hexene copolymer by Banaszak et al.,6 at 70, 90, and 150 °C, respectively. The latter group also estimated the ethylene PMV in linear low density polyethylene using the hyper-parallel tempering osmotic ensemble technique. In the simulations of this technique, for a given temperature, pressure and chemical potential of the gases, the gases are assumed to sorb into a constant number of polymer chains in a simulation cell. The details of this technique can be found elsewhere.31,32 Figure 5 and Figure 6 show the effect of pressure on ethylene/HDPE and ethylene/LLDPE-1-hexene mixture den16495



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Figure 5. Ethylene/HDPE binary mixture densities predicted by SL EoS at 50 °C and kij = 0.008 (···), at 60 °C with kij = 0.008 (---), at 80 °C with kij = 0.008 ().

Figure 6. Ethylene/LLDPE-1-hexene mixture density predicted by SL EoS at 70 °C with kij = 0.025 (), at 90 °C with kij = 0.025 (---), and at 150 °C with kij = −0.061 (···).

Figure 7. HDPE PMV temperature dependence in the binary mixture of ethylene/HDPE.

ing experimental data of Prausnitz et al.,33 at high temperatures (i.e., 130 and 300 °C for liquid low density polyethylene) can be attributed to the fact that SL EoS assumes the polymer as completely amorphous. Banaszak et al.,6 reported a constant value for the PMV of ethylene (i.e., 65 cm3/mol) over the whole range of temperatures (i.e., 70 to 150 °C), whereas, in the present work it is shown that PMV of ethylene depends on

of temperature, and increases with the increase in temperature. It should be noted that the PMV values of ethylene obtained in this work for the ethylene/HDPE mixture are in good agreement with those obtained by Prausnitz et al.,33 and comparable with the value calculated by Banaszak et al.,6 for an ethylene/LLDPE-1-hexene mixture at 150 °C. The good agreement between the model predictions and the correspond16496



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Figure 8. Propylene solubility in ICP modeled with SL EoS at different temperatures. Comparison of data (points) and fitted SL EoS predictions: T = 14 °C ( × ) and kij = 0.092 (---) ; T = 36 °C (⧫) and kij = 0.084 (−··−) ; T = 50 °C (o) and kij = 0.078 (−·−) ; T = 71 °C (▲) and kij = 0.078 ; T = 90 °C (□) and kij = 0.078 (···). Data from Kamiya et al.13

Figure 9. SL EoS predicted propylene/ICP mixture densities at different temperatures. T = 14 °C, kij =0.092 (), T = 36 °C, kij = 0.084 (−··−), T = 50 °C kij = 0.078 (−·−), T = 71 °C kij = 0.078 (---), T = 90 °C kij =0.078 (···).

of the HDPE thermal expansion coefficient (α) calculated by eq 6 (i.e., 6.28 × 10−4 °C−1) is in agreement with the value reported in literature for amorphous HDPE (i.e., 7.15 × 10−4 °C−1).35 The above findings strongly support the PMV calculation strategy established in the present study.

temperature. That deviation can be attributed to the inability of the SL EoS to consider the elastic effects caused by the polymer crystallinity. Bonavoglia et al.,10,12 showed that in semicrystalline polymers, the PMV of the penetrants increases due to a reduction in the ability of polymer chains to recover the free volume created by the incoming penetrant pressure. Contrary to this, a very close agreement between the PMV of 1-hexene calculated in this work and the one estimated by Banaszak et al. in LLDPE-1-hexene has been found (see next section). In Figure 7 the PMV of HDPE in an ethylene/HDPE mixture calculated by the proposed approach is shown at different temperatures, and a linear relationship between polymer PMV and temperature is observed. The molar volume of HDPE per repeating unit is calculated at 25 °C by extrapolating the correlation depicted in Figure 7. The calculated value is equal to 32.7 cm3/mol which is in excellent agreement with the corresponding value reported by Van Krevelen34 (i.e., 32.8 cm3/mol at 25 °C). Moreover, the value

α=

1 ⎛ d VP ⎞ ⎟ ⎜ VP ⎝ dT ⎠

(6)

where, VP is the partial molar volume of the polymer in cm3/ mol. 3.2. PMV of Propylene in Polypropylene and Impact Polypropylene. Despite the commercial importance of polypropylene (isotactic semicrystalline polypropylene (PP) and heterobiphasic impact polypropylene (ICP)), studies related to the estimation of the PMV of propylene in polymer mixtures are scarce. Sato et al.36 experimentally studied the sorption behavior of propylene in semicrystalline polypropylene (PP) at 50 and 75 °C. Akio et al.,13 estimated the PMV of 16497



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Figure 10. Comparison of the PMV of propylene in ICP calculated in this work (---) with the values of PMV calculated by Akio et al.13 (■). Solid line shows propylene PMV in PP for which the solubility data was taken from Sato et al.36

Figure 11. Temperature dependence of polymer PMV in propylene/PP mixture.

propylene as well as a number of other hydrocarbons in PP and in impact polypropylene (ICP) by using the experimental sorption and dilation data over a wide range of temperature (i.e., 14 to 90 °C). Among the ICP samples used by Akio et al., the one considered in this study had 84% PP content and only 16% EPR (rubbery copolymer of ethylene and propylene). Notice that the crystallinity of PP samples used by Sato et al.,36 and Akio et al.,13 was approximately same, that is, 50%. The experimental solubility data of propylene/ICP binary mixture was fitted with the SL EoS, and the mixture density values were calculated at different temperatures and pressures. Note that this is not a true binary system since the EPR and homo-PP are thermodynamically incompatible, and thus form separate phases in the polymer matrix. Thus, this is more correctly a ternary system with one penetrant and two polymer phases. However, we will use this opportunity to evaluate the importance of this distinction by treating the amorphous fractions of each type of polymer as a single “pseudo-phase”. Figure 8 shows the experimental and predicted solubilities of propylene in ICP, and the predicted mixture densities are shown in Figure 9. It can be observed that propylene solubility in ICP, as well as the mixture density decrease with increasing temperature. In addition, the binary interaction parameter becomes temperature dependent below 50 °C. Similar behavior was also observed for propylene solubility in PP (not shown here).

The PMV of propylene in PP and ICP calculated on the basis of SL EoS mixture density predictions show a good agreement with the values reported by Akio et al.,13 (see Figure 10). As it can be observed, the PMV of propylene increases with increase in temperature in both the polymers. Akio et al. calculated a value of 68.4 cm3/mol for the PMV of propylene in PP at 50 °C, whereas the corresponding value calculated by the present approach is 71.5 cm3/mol. The PMV of propylene in the propylene/ICP mixture also increases as the temperature increases. It should be noted that at temperatures lower than 30 °C a relative difference of approximately 20% is observed between the PMV of propylene calculated by the present work and by Akio et al.’s experimental results. However, for temperatures above 50 °C the model predictions are in much better agreement with the data of Akio et al.13 The deviations observed at the lower temperatures can most likely be attributed to the different solubility behavior of propylene in EPR domains and amorphous PP domains (50% of PP) of ICP sample. Akio et al.,13 showed that propylene solubility in EPR domains is high as compared to amorphous PP domains. Therefore, it appears that SL EoS cannot take into account completely this behavior of propylene solubility in different domains of ICP at lower temperatures. Also, this behavior of SL EoS can be attributed to the uncertainties in the estimation of pure component characteristic parameters23 and, to some extent, to the experimental errors involved in sorption 16498



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Figure 12. PMV of 1-butene in HDPE (···) and LLDPE (---) at different temperatures.

Figure 13. 1-Hexene/LLDPE-1-hexene mixture density predicted by SL EoS at 70 °C with kij = 0.027 (), at 90 °C with kij = 0.016 (---) and at 150 °C with kij = −0.030 (···).

partially attributed to the enhancement of ethylene solubility due to a cosolubility effect, and partially to an increase of the intrinsic activity of the catalyst.37 The PMV of 1-butene in binary 1-butene/polyolefin mixtures is also an important volumetric property which provides information about the thermodynamic environment experienced by 1-butene during its dynamic sorption in polyolefins. However, studies on the calculation of the PMV of 1-butene in polyolefins are scarce in the open literature. On the basis of the model-based methodology proposed in this work, the PMV of 1-butene in two binary mixtures (i.e., 1-butene/HDPE and 1-butene/ LLDPE-1-butene) was calculated using the sorption measurements of Moore et al.37 (see Figure 12). No significant effect of the polyolefin structure on the PMV of 1-butene is observed at all studied temperatures indicating that the thermodynamic environment is similar in both mixture types. It is also found that an increase in temperature leads to an increase in the PMV of 1-butene, whereas, although not shown here, a fairly constant PMV was predicted at various 1-butene concentrations for a constant temperature. 3.4. PMV of 1-Hexene in LLDPE. 1-Hexene sorption in polyethylene and the swelling of polyethylene caused by it has been studied at different process conditions.37−40 However, the studies related to the estimation of its PMV in polyolefins are few. Banaszak et al.,31,32 estimated the PMV of 1-hexene in linear and branched polyethylene samples using a hyper-parallel tempering osmotic ensemble technique, and did not observe

and dilation experiments, especially at low temperatures. It is important to point out that polymer crystallinity causes an increase in penetrant PMV, as discussed above; therefore, these deviations cannot be attributed to the elastic effects caused by the crystalline domains. Nevertheless, according to Figure 10, the overlapping of the propylene PMV curves in the PP homopolymer and in the ICP, as well as the similar temperature dependence indicates that propylene exhibits very similar thermodynamic behavior in both polymers (i.e., PP and ICP) despite the different characteristics of the amorphous phases. This can be also confirmed by the similarity of the binary interaction parameter values for the two mixtures at the same temperatures (i.e., for propylene/PP mixture at 50 °C kij = 0.063 and for propylene/ ICP at 50 °C kij = 0.078). Figure 11 demonstrates the temperature dependence of polypropylene PMV. It should be pointed out that the molar volume per repeating unit of the polymer calculated at 25 °C is 49.4 cm3/mol and the corresponding value reported by Van Krevelen34 is 49.5 cm3/mol. In addition, the volumetric thermal expansion coefficient for polypropylene calculated in this work is 5.53 × 10−4 K−1, which is in good agreement with the reported literature value (i.e., 6.6 × 10−4 K−1).35 3.3. 1- PMV of 1-Butene in PE. LLDPE-1-butene is an important olefin copolymer for which 1-butene is a major raw material. During the polymerization, the solubility of 1-butene in LLDPE results in enhanced reaction rates. This can be 16499



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Figure 14. 1-Hexene PMV in LLDPE-1-hexene at T = 70 °C (), T = 90 °C (···) and T = 150 °C (---) calculated in this work. ω1 is the mass fraction of penetrant in the mixture.

Figure 15. 1-Hexene/LLDPE-1-hexene mixture density data predicted by SL EoS at different temperatures. T = 50 °C and kij = 0.012 (---), T = 60 °C and kij = 0.012 (···), T = 70 °C and kij = 0.010 (−·−).

energy between 1-hexene and LLDPE-1-hexene. The maximum interaction energy is observed at 150 °C (i.e., at lowest kij) where the polyethylene is in molten form and the chains have higher mobility as compared to lower temperatures. The PMV of 1-hexene calculated on the basis of SL EoS predicted mixture densities, shown in Figure 14, are in good agreement with the corresponding value reported by Banaszak et al.,6 (i.e.,130 cm3/mol). Note that Banaszak et al.,6 estimated a constant PMV value of 130 cm3/mol for 1-hexene at all studied temperatures and pressures. However, the findings presented in the present work indicate that using a constant PMV value can lead to erroneous results since it appears that the PMV of 1-hexene is a strong function of temperature and weak function of concentration. Comparison of the PMV of ethylene and of 1-hexene in LLDPE-1-hexene at the same temperatures shows that both penetrants exhibit different thermodynamic interactions with the substrate during their sorption in the same polymer (it is also depicted by the different kij values). The temperature and concentration dependences of the PMV of 1-hexene are stronger than the same dependences in ethylene due to the larger size of the 1hexene molecules.

any considerable effect of branching on the PMV of this penetrant. In the present work, the PMV of 1-hexene in different 1-hexene/LLDPE-1-hexene mixtures was estimated by employing the proposed methodology in order to analyze the thermodynamic interactions experienced by 1-hexene in the binary mixtures. Figure 13 shows the effect of pressure on the binary 1hexene/LLDPE-1-hexene mixture densities predicted by the SL EoS at different temperatures (i.e., 70, 90, and 150 °C), using the experimental solubility data of Banaszak et al.6,39 The mixture density values predicted by SL EoS are in good agreement with those predicted by PC-SAFT EoS.6 As it can be seen, the reduction in the density of 1-hexene/LLDPE-1hexene binary mixture is larger compared to ethylene/LLDPE1-hexene at the same temperature (see Figure 13 and Figure 6). This is due to larger sorption rate of 1-hexene and as well as to higher degree of polymer swelling caused by 1-hexene. Contrary to what was observed for ethylene/LLDPE-1hexene, kij for 1-hexene/LLDPE-1-hexene decreases with respect to temperature, indicating a change in the thermodynamic interactions taking place during the sorption process. It should be noticed that the decreasing value of kij with increasing temperature represents an increase in interaction 16500



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Figure 16. Comparison of 1-octene solubility in LLDPE-1-butene with fitted SL EoS predictions. Data points taken from Yoon et al.40 T = 70 °C (⧫) with SL EoS predictions at kij = 0.006 (), T = 80 °C (▲) with SL EoS predictions at kij = 0.006 (−··−), T = 85 °C (●) with SL EoS predictions at kij = 0.0058 (···).

Figure 17. T = 70 °C SL EoS mixture density predictions at kij = 0.006 (), T = 80 °C SL EoS mixture density predictions at kij = 0.006 (−··−), T = 85 °C SL EoS mixture density predictions at kij = 0.0058 (···).

decreases with increase in penetrant concentration and temperature (see Figure 15), whereas the calculated PMV of 1-hexene remained almost constant with respect to temperature and 1-hexene concentration with a value of approximately 122 cm3/mol. In addition, the volumetric thermal expansion coefficient of LLDPE-1-hexene, calculated by using eq 6, is in the range of 6.86 to 9.50 × 10−4 °C−1 for temperatures between 50 to 150 °C. These values are of the same order of magnitude as the linear thermal expansion coefficient values of 1.6 to 2 × 10−4 °C−1 reported in the literature for the same polymer.35 The experimental measurements by Jin et al.,38 of the solubility of 1-hexene in LLDPE-1-hexene copolymers of different co-monomer content (i.e., up to 12 mol %) were also used to study the effect of 1-hexene content on its PMV in the binary mixture. By increasing the comonomer content from 3.5 to 12 mol % the PMV of 1-hexene remained fairly constant with the average value of 121 cm3/mol at 50 °C, 122 cm3/mol at 60 °C, and 123 cm3/mol at 70 °C. Therefore, it can be concluded that the comonomer content in LLDPE does not appear to influence the 1-hexene PMV. In itself this is perhaps not surprising as increase the co-monomer content of the LLDPE will simply increase the amorphous fraction in the polymer and as we have already seen above, the crystallinity of

In addition, the PMV of LLDPE-1-hexene calculated by using the SL EoS predicted mixture densities and penetrant mass fractions is in very good agreement with the constant value of 17000 cm3/mol estimated by Banaszak et al.6 The PMV of LLDPE-1-hexene increased with temperature (i.e., at 70 °C, an average PMV of LLDPE-1-hexene was found to be 16837 cm3/mol, at 90 °C the average value was found to be 17051 cm3/mol, and at 150 °C the average calculated value was 17775 cm3/mol) and decreased slightly with increase in penetrant concentration. However, the decrease in polymer PMV with respect to penetrant concentration was not equal to the increase in penetrant PMV which could be either due to experimental error in solubility measurements or small relative error (i.e., less than 8%) in solubility predictions by SL EoS that also affected the density predictions. To further study the PMV behavior of 1-hexene in 1-hexene/ LLDPE-1-hexene mixture, the experimental measurements of Jin et al.,38 were used to predict the mixture densities along with 1-hexene mass fractions in various 1-hexene/LLDPE-1hexene mixtures at different temperatures and pressures. Note that the 1-hexene incorporation in LLDPE is 3.5 mol % which is comparable with the 1-hexene content used by Banaszak et al.,6 (i.e., 3.8 mol %). It is evident that mixture density 16501



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Figure 18. 1-Octene PMV in LLDPE-1-butene at T = 70 °C (), T = 80 °C (···), T = 85 °C (---). ω1 is the mass fraction of penetrant in the mixture.

Figure 19. Correlation between PMV and Vc of α-olefins in α-olefin/polyolefin mixtures at T = 50 °C (●), T = 70 °C (□) and T = 90 °C (▲). Vc values take from ref 42.

the polyethylene does not appear to have a noticeable impact on the PMV of small penetrants. 3.5. PMV of 1-Octene in LLDPE-1-Butene. In polyolefin manufacturing, the production of very low density polymer grades (i.e., density below 910 kg/m3) can be achieved by utilizing high molecular weight α-olefins (i.e., 1-octene) as comonomers. 1-Octene is the heaviest α-olefin commonly used in the polyolefin production industry, and while it is typically not used in the gas phase, understanding LLDPE swelling behavior during 1-octene sorption allows us to establish a trend of the effect of penetrant size molecules on PMV. Experimental solubility data on 1-octene sorption in polyethylene as well as polymer swelling measurements caused by 1-octene mass uptake are few in the literature, and no data are available concerning the PMV of 1-octene in olefin/polyolefin mixtures. The experimental measurements of Yoon et al.40 on the solubility of 1-octene in LLDPE-1-butene were compared with model predictions obtained by the SL EoS. In Figure 16 the experimental and theoretical solubilities of 1-octene in LLDPE are depicted for different temperature and pressures. Figure 17 shows a mixture density for different temperature and pressures that have been calculated by the proposed methodology using the solubility measurements presented in Figure 16. It can be seen that the mixture density differs from the other α-olefin/

polyolefin mixtures due to the larger size of the penetrant molecule (i.e., 1-octene), as well as to higher mass uptake compared with the lighter α-olefins (i.e., 1-butene, 1-hexene). According to the findings illustrated in Figure 17, at 70 °C, the SL EoS predicts an abrupt mixture density decrease at 0.13 bar, and the mixture density attains even lower values compared to those at 80 and 85 °C under the same pressure range. Similar behavior can be seen at 80 °C after 0.2 bar . The high solubility of 1-octene in the polymer, especially, at low temperatures, as well as the large molecular size of the penetrant are the two reasons for this behavior which causes large volume changes in the binary mixture. It is important to point out that the PMV of 1-octene depends strongly on the temperature and composition of the mixture (see Figure 18). In addition to possible experimental errors during the sorption measurements, a sudden increase in 1-octene PMV in the binary mixture can be attributed to the low ability of polymer chains to recover the large strain/volume changes caused by the large size of the incoming penetrant. It should be noted that similar behavior has also been observed in the case of the PMV of 1-hexene during its sorption in LLDPE-1-hexene copolymer (see Figure 14). 3.6. Correlation between PMV and Critical Volume of αOlefins in Polyolefins. Establishing a relationship between the 16502



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Table 4. Fitted kij for Olefin/Polyolefin Mixtures Studied in This Work kij ethylene/HDPE ethylene/LLDPE-1-hexene propylene/HDPE* 1-butene/HDPE 1-butene/LLDPE-1-butene 1-hexene/LLDPE-1-hexene 1-hexene/LLDPE-1-hexene 1-octene/LLDPE-1-butene

T = 30 °C

T = 50 °C

T = 60 °C

0.008

0.008

0.015 0.012

T = 75 °C

T = 80 °C

T = 90 °C

T = 30 °C

0.025 0.024* 0.018 −0.014

−0.061

0.008 0.025 0.025 0.023 0.00 0.011 0.027 0.006

0.030 0.039 0.022

T = 70 °C

0.012

0.006

0.016 0.0056

−0.03

*

Solubility data taken from ref 45. At T = 90 °C, no experimental data are available, therefore this values was estimated by a linear fit of the kij values at 50 and 70 °C.

Figure 20. Correlation of SL EoS kij with PMV of C2 to C8 α-olefins in polyethylene at T = 50 °C (●), T = 70 °C (△), and T = 90 °C (⧫). Solid line is a polynomial fit between PMV and kij.

PMV of α-olefins and a commonly available characteristic property (e.g., critical volume (Vc)4,41) of the penetrants at different temperatures would be of great importance for αolefin/polyolefin mixture thermodynamic studies. Figure 19 presents the dependence of α-olefins PMV (i.e., C2 to C8) on their critical volume over a range of temperatures (i.e., 50 to 90 °C) at which polyolefins reactors operate. It should be emphasized that the proposed relationship does not depend on the polyolefin type (see sections above). On the basis of the findings of Figure 19, one can derive an analytical expression describing the dependence of PMV, V̅ i (cm3/mol), of α-olefins on their critical volume Vc (cm3/mol) during their sorption in polyolefins: Vi̅ = 14.394 + 0.3094Vc

EoS model used to calculate the binary mixture density, it would be of great interest to analyze the variation in PMV with kij at different temperatures. Table 4 summarizes the kij values used for all α-olefin/ polyolefin binary systems studied in this work. It can be seen that, except for 1-butene at 90 °C, positive kij values are used for all the penetrants during their sorption in polyolefins at temperatures below the polymer melting point. In fact, interaction energy (ε*) is low for a positive value of kij, whereas it is high for a negative kij. Low interaction energies predicted for most of the studied mixtures can be attributed to low penetrant pressures (especially in case of propylene, 1butene, 1-hexene, and 1-octene). In general, strong penetrant/ polymer interactions are expected at high pressures (e.g., in the case of ethylene where system pressure can be above 40 bar). Negative kij near or above the polymer melting temperature indicates high interaction energy due to the increased mobility of polymer chains. The above findings are in excellent agreement with the ones presented by Costa et al.,43 who used the PC-SAFT EoS to estimate the binary interaction parameters for a multiphase system during high and low pressure separation stages in the low density polyethylene (LDPE) production process. For measuring the nonideal behavior of mixtures, two representative quantities are mainly used: (i) molecular asymmetry of the penetrants; and (ii) intermolecular forces between penetrant molecules and polymer chains. Because of the similar nonpolar chemical nature of α-olefins and polyolefins, intermolecular forces do not play an important role.43 On the basis of molecular asymmetry, low kij (i.e.,

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Linear relationship between the PMV and Vc indicates that the thermodynamic state of gaseous α-olefins in polyolefins is similar to their state when they are dissolved in liquids of the same chemical nature.4,9,13 3.7. Correlation of Binary Interaction Parameter with Partial Molar Volumes. The binary interaction parameter (kij) used in the SL EoS is a dimensionless quantity used to analyze the deviation of real (nonideal) mixtures from the ideal ones. It is a measure of deviation from the geometric mean of interaction energies of the mixture species, and hence describes the magnitude of interactions between the penetrant molecules and the polymer chains during the sorption process.4,9,43 In addition to ionization potentials of the chemical components in a mixture, kij is also a function of molecular asymmetry in a mixture.44 Since kij is the only adjustable parameter in the SL 16503



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volumetric thermal expansion coefficients of polyolefins) have been precisely estimated with the proposed methodology at conditions of industrial importance. Penetrant/polymer mixture densities predicted by SL EoS were in good agreement with those obtained by PC-SAFT EoS. Among the series of C2 to C8 α-olefins, a stronger temperature dependence of PMV was observed for higher α-olefins. In addition, molar volumes per unit repeating unit of polyethylene and polypropylene samples calculated at 25 °C were found to be in excellent agreement with the literature values. Moreover, thermal expansion coefficients calculated in this work for HDPE, LLDPE-1-hexene, and polypropylene also showed a close agreement with the literature values. Hence, the method proposed in the present work toward calculating penetrants and polymer PMVs in their binary mixtures provides an alternative to the current state of the art experimental and theoretical approaches, i.e., densimetery, dilation experiments and Monte Carlo simulations. In the case of α-olefins/polyolefin mixtures, it was found that the PMVs of α-olefins increased proportionately with their critical molar volumes at temperatures of industrial importance (i.e., at 50, 70, and 90 °C). The binary interaction parameter (kij) used in SL EoS was correlated with the PMV of the αolefins in polyethylene over a wide range of temperature. It was illustrated that when the molecular size of the penetrants increases, kij decreases, and the interaction energy between penetrant and polymer molecules (ε*) increases. Finally, an analytical relationship between the SL EoS kij and PMV of αolefins in polyethylene was established.

ethylene/HDPE binary mixture) indicates less deviation from the ideal mixing behavior as a consequence of restricted molecular movements/arrangements due to high pressures. On the other hand, large molecular movements/arrangements are expected during sorption of high molecular weight α-olefins in polyolefins, resulting in significant deviations from ideal mixing.43 Since penetrant partial molar volumes and binary interaction parameters describe the extent of nonideality in binary mixtures, a correlation between them would be of high importance. In the open literature, there are a number of papers dealing with the establishment of a correlation between the PMV of a given penetrant and the interaction parameter (χ) used in the Flory−Huggins (FH) equation. A linear relationship exists between χ and the PMV of various hydrocarbons, light gases, and fluorocarbons.4,13−15 χ was also found to be an increasing function of penetrant concentration and a decreasing function of temperature for the ethylene/ICP, propylene/PP, and propylene/ICP binary mixtures.13 The SL EoS binary interaction parameter is independent of penetrant concentration and decreases with increase in temperature (see Table 4). As was discussed in the previous sections, the PMV of various α-olefins are independent of polyolefin type, thus at a certain temperature the binary interaction parameter value of the same olefin in different polyolefin types can be used in the case where experimental data of the mixture of interest is not available. Figure 20 presents the relationship between the PMV of α-olefins considered in the present study (i.e., C2 to C8) and the corresponding kij fitted by the SL EoS model at various temperatures. It can be seen that kij decreases as the size of the penetrant (i.e., α-olefin) increases over a wide range of temperatures. It is apparent that the kij-PMV dependence is not influenced by the temperature over a wide range (i.e., 50 to 90 °C), thus, an analytical expression describing the relationship between kij and PMV can be established (see eq 8). It is important to point out that the predicted relationship between kij and α-olefins PMV is in very good agreement with the one developed for the FH interaction parameter for propylene, 1butene, and 1-hexene in polyethylene.4,13 However, the applicability of eq 8 is limited due to the dependence of SL EoS performance on the kij values which in turn depends upon the fitting procedure and pure component characteristic parameters of SL EoS. The error bars in Figure 20 represents the uncertainty in the obtained kij and PMV values. It should be mentioned that in the present work the maximum allowable relative difference between experimental and SL EoS fitted solubility values is 10%. kij = 0.0181 + 0.0002Vi̅ − 2 × 10−6Vi̅ 2

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ASSOCIATED CONTENT

S Supporting Information *

The procedure for the calculation of partial molar volume of penetrant and polymer by using the Sanchez−Lacombe Equation of State predicted mixture density has been elaborated. This information is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]; mu.ahsanbashir@ gmail.com. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS Financial support by the Dutch Polymer Institute is gratefully acknowledged. This work is part of the Research Programme of the Dutch Polymer Institute (DPI, Eindhoven,The Netherlands), Project no. 753. The authors would also like to thank Borealis Polymer Oy for supporting this work.

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CONCLUSION In the present work, a model-based methodology is established with the objective of predicting the penetrant/polymer mixture density by employing the Sanchez−Lacombe equation of state (SL EoS) and the solubility data. According to the proposed approach the PMV of various hydrocarbon penetrants in different polymers can be accurately estimated by using the SL EoS predicted binary mixture density data at different temperatures and pressures. More specifically, a number of important volumetric properties (i.e., PMV of α-olefins and polyolefins, molar volume of polyolefins per repeating unit and

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