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Solubility of solvents in polyethylene below the melt temperature
Fluid Phase Equilibria xxx (2017) 1e7
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Fluid Phase Equilibria j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / fl u i d
Solubility of solvents in polyethylene below the melt temperature Derek R. Sturm, Kevin J. Caputo, Siyang Liu, Ronald P. Danner* Chemical Engineering Department, The Pennsylvania State University, University Park, PA, 16802, USA
a r t i c l e i n f o
a b s t r a c t
Article history: Received 24 May 2017 Received in revised form 7 August 2017 Accepted 5 September 2017 Available online xxx
Polyethylene, the most ubiquitous polymer in use, is a semi-crystalline material. Solvents enter only the amorphous phase. The solubility in that phase, however, cannot be characterized by the amorphous solubility found above the melt temperature. The polymer chains that have ends attached to the crystalline structure but extend into the amorphous phase are referred to as tie chains, and they affect the overall solvent solubility. Six different polyethylenes covering a range of densities were studied. Below the melt temperature all the samples exhibited an increase in solubility with increasing temperature. The solubility of the penetrants, however, is not the same for different types of polyethylene. A model which incorporates an elasticity factor that accounts for the stress effects on the solubility in the tie chains was found to be valuable. When incorporated into a version of the UNIFAC free-volume model there was good correlation between of the solubilities. The present work indicates that the tie chain fraction, f, tends to increase with density. © 2017 Elsevier B.V. All rights reserved.
Keywords: Polyethylene Solubility Crystallinity Elasticity affect UNIFAC
1. Introduction Globally polyethylene (PE) is the most abundant polymer found in everyday life [1]. Some of the most common uses for PE included plastic bags, plastic film, bottles, and piping. Below its melting point polyethylene is a semi-crystalline polymer, i.e., some of the chains form an ordered dense crystalline phase while others are in an amorphous phase of lower density. In general, solvents are excluded from the crystalline phase [2]. To improve the ability of current models to predict the solubility in polyethylene the effect of the amount of crystallinity needs to be characterized. The first type of polyethylene that was commercially produced was known as low-density polyethylene (LDPE) [3]. The production of this type of PE is done without the use of a catalyst. In the absence of a catalyst, the radical site can move from a location at the end of the chain to a more stable location in the middle of the chain. This causes a new ethylene monomer to polymerize at the radical site and form a branch in the middle of the PE chain. As this process continues a non-linear, highly branched PE chain is formed. In LDPE the large amount of chain branching prevents a portion of the chains from crystallizing [4]. Since the amorphous phase of PE has a much lower density than the crystalline phase, the reduction in crystallinity results in an overall product that has a lower density
* Corresponding author. E-mail address: [email protected] (R.P. Danner).
than when the chains are completely linear. Once catalysts were introduced into PE production high-density polyethylene (HDPE) was commercially produced. PE polymerization using a catalyst, such as a Zeigler-Natta or a metallocene catalyst, ensures that the radical site stays at the end of the PE chain, resulting in a final product that is a linear chain with little to no branching. These chains have a higher tendency to crystallize resulting in a high density PE [3]. Soon after the introduction of polymerization catalysts, longer a-olefins such as 1-butene, 1-hexene, and 1-octene were added to the polymerization process. This addition of a co-monomer results in a polyethylene chain that has a linear backbone with short side chains distributed along its length. As these side chains cannot be included in the crystal structure [2], they act to reduce the crystallinity of the polyethylene. The resulting linear low-density polyethylene is sometimes divided into subgroups: mediumdensity PE (MDPE) having a density between 0.94 and 0.926 g/ cm3, linear-low-density PE (LLDPE) having a density between 0.926 and 0.915 g/cm3 and very-low-linear-density PE (VLLDPE) having a density below 0.915 g/cm3. 2. Semi-crystalline structure models There are three common models in the literature used to describe the semi-crystalline structure of PE [5]. The first was Flory's switchboard model [6] which proposes that the majority of polyethylene chains leave the crystalline surface and cross over the
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amorphous region to a different crystalline surface rather than reenter the same crystal. The second is the adjacent reentry model of Hoffman and Lauritzen [7]. They propose two forms: a smooth surface in which almost all the chains that leave the crystalline surface immediately bend and reenter adjacent to where they exited, and a rough surface model in which the majority of chains still enter adjacent to where they exited the crystalline phase, but the fold length is not uniform. The third model was originally suggested by Stamm et al. [8] and later adapted by Strobl [9]. In either version crystallization from the melt occurs by formation of an aligned, long-range, ordered state, which then proceeds to develop into the long-range ordered crystalline phase. In the past thirty years, numerous studies have been performed to study the effect of tie chains on the properties of semi-crystalline polymers. Most of these focus on mechanical properties [10,11]. A good review of the relationships between tie chains and mechanical properties in semi-crystalline polymers is provided by Seguela [12]. Only a few studies have focused on solvent solubility [13e16]. 3. Solubility in PE Unlike amorphous polymers, where the penetrant is evenly distributed throughout the polymer phase at equilibrium, semicrystalline polymers, like PE, restrict the penetrant from entering the crystalline phase due to dense chain packing [2]. While the penetrant is not evenly distributed throughout the entire polymer domain, it is assumed to be evenly distributed within the amorphous domain of the polymer. Thus, when comparing the solubility between polyethylene samples with different crystallinity, the weight fraction of penetrant should be normalized in terms of the amount of penetrant in the amorphous phase of the polymer rather than on a total polymer basis. Above the melt temperature, where PE is entirely amorphous, models like the Group Contribution Lattice Fluid-Equation of State (GCLF-EoS) [17], UNIFAC-FV [18], and UNIFAC-vdw-FV [19] have been shown to accurately predict the solubility of solvents in PE [15]. Below the melt temperature application of these models to PE needs to be evaluated. Michaels and Hausslein [20] were one of the first to propose an explanation for the reduction in solubility in the amorphous phase in semi-crystalline polymers like PE. Their model, which was originally proposed in 1965, is still the most successful model. They postulated that the cause for the large temperature dependence in penetrant solubility in polyethylene is due to the tie chains. As the penetrant enters the amorphous phase these tie chains act like springs and apply an elastic force on the amorphous region which reduces the equilibrium concentration of penetrant within the amorphous phase. The most important assumptions of their derivation are: The amorphous phase is composed of two types of tie chains: elastically effected and inelastically effected chains. The elastically effected chains extend across the amorphous region, tethering two crystalline domains. The inelastic portion consists of chains which exit and reenter the same crystal phase, as well as chain ends and other short segments excluded from the crystal. The amount of tie chains is constant in the temperature range in which the polymer's crystalline structure does not change. Thus, the temperature dependent behavior of the tie chains is reversible as long as the crystalline structure does not change. The collective properties of the tie chains, such as length, can be accurately captured by an average of those properties. The deformation is uniform in all directions, i.e., isotropic. By assuming that the force required to stretch the chains can be
approximated by Hooke's Law, the final expression lacked any terms that require the number of monomers in the tie chains, the length of the chains, or directly evaluating the Hookian spring constant. Dong and Ho [21] used Michaels and Hausslein's expression for the chemical potential that the tie chains exert on the amorphous phase and converted it into an activity.
ln aEL 1 ¼
DHf
R
ra V1
1 T
T1m
3 2f f2
f1 cf21
1
(1)
Here the aEl 1 is elastic activity, DHf is the heat of fusion for the semi crystalline polymer, ra is the density of the amorphous phase, Tm is the melt temperature of the polymer, R is the gas constant, fi is the volume fraction of component i, c is the Flory-Huggins interaction parameter, and f is the weight fraction of the elastically effected chains in the amorphous phase. This activity contribution was then added to the other contributions in the UNIFAC-FV model. EL ln a1 ¼ ln aC1 þ ln aR1 þ ln aFV 1 þ ln a1
aC1 ,
aR1 ,
(2)
aFV 1
Here and are the combinatorial, residual, and free volume activities. The predictive methods of UNIFAC-vdw-FV, UNIFAC-FV and GCLF-EoS that have been modified by the inclusion of aEL 1 will be identified with the suffix MH. The value of DHf used was the experimentally determined value, 293 J/g [22]. The density of the amorphous phase was determined using an extrapolation of the amorphous density of PE above the melt via Tait's equation [23]. The value of c was determined by both regressing the experimental data using the FloryHuggins expression and by using the relationship between the UNIFAC residual and free volume activities as described by Dong and Ho:
c¼
ln aR1 þ ln aFV 1 f22
(3)
Even though the values of c differed between the two methods, it had negligible effect on predicted activity even at the high experimental penetrant weight fractions.
4. Experimental There were six different samples of PE on which solubility studies were performed. They were all produced with Ziegler-Natta catalyst and have crystallinities that range from 40.3 to 81.3%. One is an ethylene homopolymer (no co-monomer), three are ethylene1-butene co-monomers, and two are ethylene-1-hexene copolymers. Table 1 provides the properties of these samples as supplied by the manufactures. Three different methods were used to measure the solubilities: inverse gas chromatography using packed columns (PCIGC), quartz spring gravimetric, and pressure decay. Each of these methods has been described elsewhere (as referenced below) and only pertinent details will be repeated here. In the PCIGC method the solubility of granules is determined by analyzing the elution peak [24]. In the gravimetric approach the penetrant uptake is determined by measuring the extension of a calibrated quartz spring [25]. In the pressure decay method the solubility is determined by measuring the pressure drop in a constant temperature, constant volume chamber and converting that to mass by an equation of state [26,27]. The polyethylene samples in Table 1 were pressed into films by melting the samples and then slowly cooled at a rate of 10 C/h.
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Table 1 PE sample properties. PE Sample
1
2
3
4
5
6
Density (g/cm3) Crystallinity Melt T ( C) Mn (g/mol) Comonomer Polydispersity Index
0.92 44.8 120 20679 1-Butene 3.22
0.9599 78.9 132 23591 1-Butene 2.88
0.9501 72.6 128 16118 1-Butene 2.80
0.9614 81.7 131 11656 None 12.9
0.9124 40.2 100.7/114.1 57174 1-Hexene 1.79
0.9168 43.8 107.5/116.7 56519 1-Hexene 1.80
Thus, the films were crystallized at a temperature just below their melt temperatures. Except for sample 6 the solubility of isopentane in the PE films was determined in every sample at a minimum of three different temperatures. The solubility of a different penetrant was also determined for every polymer sample, although this penetrant was not the same for each sample. For the PE samples that were polymerized with 1-hexene as the co-monomer, the second penetrant was 1-hexene. For the rest of the PE samples the second penetrant was cyclohexane. For PE samples 1e4 extruded pellets from the manufacturers were ground and packed in stainless steel columns. Using PCIGC the partition coefficients of 1-butene at infinite dilution were determined for samples 1e4 and of cyclohexane for sample 1. The finite concentration solubility data were taken with either the quartz spring, gravimetric sorption balance, or the pressure decay method. 5. Comparison of solubility in different PE samples The weight fractions of tie chains in the amorphous phase, f , were regressed using the UNIFAC-FV-MH, UNIFAV-vdw-FV-MH, and GCLF-EoS-MH models using the aEL 1 from equation (1) and the experimental amorphous weight fractions (See Table 2). Other f values reported in the literature include those by Michaels and Hausslein [20] which ranged between 0.265 and 0.363 (excluding the cold-drawn samples) depending on crystallization history, Doong and Ho's [21] 0.373, and Serna et al.’s [28] 0.363. One approach to modeling the probability that a chain will become a tie chain in PE is that of Guttman et al. [29] They provided an estimate of the chance that a polymer chain leaving the crystalline interphase on one side would cross the amorphous region to the opposite crystalline phase through the following assumptions. The amorphous phase can be treated as a cubic lattice; the PE chain is infinitely long and has no imperfections such as side chains or comonomers; the PE chain acts as a random walker in a theta solvent. If the chain does not reenter the crystal it continues to have a chance to move forward or move back towards the crystal it came from. If the chain reaches the other surface, it becomes a tie chain.
In their work, they provided relationships for the average length of chains in the bulk amorphous phase, the fraction of chains leaving the crystalline domain that a chain will become tie chains, and the average length of tie chains in the bulk amorphous phase. Through these relationships, the weight fraction of tie chains in the amorphous phase, f , is determined to be 1/3. The regressed values in Table 2 are in good agreement with these other values. The percentage error between the predicted activities and experimental activities for each PE sample across all solvents were determined for each of the three prediction models as shown in Table 2.
Avg: % error ¼
apred aexp aexp
100
(4)
Here apred is the predicted activity when evaluated at the experimental weight fraction and aexp is the experimental activity. Overall, UNIFAC-vdw-FV-MH out-preforms the other methods, as shown by the lower error in every sample except for sample 6. Sample 6 had the solubility measured for only one solvent and the difference in error between UNIFAC-vdw-FV-MH and UNIFAC-FVMH is less than 0.3%. This small improvement is much less than the difference found in the other samples. Thus, only comparison results for the UNIFAC-vdw-FV-MH model with the experimental data are shown in Figs. 1e6. Fig. 1a and b also include solid black lines which are the predicted solubility determined using UNIFAC-vdw-FV without the modification of Michaels and Hausslein, i.e., the elasticity term, at the lowest temperature of the graph. These results clearly indicate the inaccuracy in predicting activity, and therefore solubility, without considering the effect of the tie chains. For all six PE samples there is excellent agreement between the experimental data and the UNIFAC-vdw-FV-MH predictions. The biggest deviation from prediction comes in the higher crystalline samples 2, 3 and 4 when cyclohexane was the penetrant. The temperature dependence of the experimental solubility of cyclohexane in samples 2, 3, and 4 is smaller than the prediction. In contrast to the higher crystalline samples, sample 1, which had a
Table 2 Regressed f factors for films prepared by slowly cooling from the melt and errors in the predicted activity for the various models. PE Sample
1 2 3 4 5 6
Penetrants
Cyclohexane Isopentane Cyclohexane Isopentane Cyclohexane Isopentane Cyclohexane Isopentane 1-Hexene Isopentane 1-Hexene
UNIFAC-vdw-FV-MH
UNIFAC-FV-MH
f
Avg. % error
f
Avg. % error
f
Avg. % error
0.338
3.0 4.5 10.0 3.6 9.0 3.6 8.9 3.6 1.5 1.3 2.9
0.381
7.3 7.0 12.5 10.0 12.5 7.5 14.5 7.0 5.0 3.0 2.6
0.372
7.2 5.3 12.1 9.1 11.9 7.1 12.6 7.7 4.8 3.1 2.8
0.377 0.399 0.278 0.325 0.326
0.415 0.453 0.355 0.397 0.393
GCLF-EoS-MH
0.406 0.447 0.349 0.371 0.372
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Fig. 1. Solubility of cyclohexane and isopentane in sample 1. The dashed lines provide the prediction of UNIFAC-vdw-FV-MH. The solid line is the prediction of UNIFAC-vdw-FV at 34.5 C for cyclohexane and at 50 C for isopentane.
Fig. 2. Solubility of cyclohexane and isopentane in PE sample 2. The dashed lines provide the prediction of UNIFAC-vdw-FV-MH.
Fig. 3. Solubility of cyclohexane and isopentane in PE sample 3. The dashed lines provide the prediction of UNIFAC-vdw-FV-MH.
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Fig. 4. Solubility of cyclohexane and isopentane in PE sample 4. The dashed lines provide the prediction of UNIFAC-vdw-FV-MH.
Fig. 5. Solubility of 1-hexene and isopentane in sample 5. The dashed lines provide the prediction of UNIFAC-vdw-FV-MH.
Fig. 6. Solubility of 1-hexene in PE sample 6. The dashed lines provide the prediction of UNIFAC-vdw-FV-MH.
much lower amount of crystallinity, showed a greater change in cyclohexane solubility with temperature that matches more closely with prediction. To compare the predictions of UNIFAC-vdw-FV-MH with the infinite dilution experiments of cyclohexane in sample 1, the experimental data shown in Fig. 1 were converted to partition coefficients. Fig. 7 provides the experimental partition coefficients determined by PCIGC (infinite dilution) and gravimetric sorption (finite concentration) along with the UNIFAC-vdw-FV-MH predictions. As the activity goes to zero, both the predicted and experimental partition coefficient decrease. There is excellent agreement between the experimental and predicted partition coefficients. Fig. 8 provides the experimental infinite dilution amorphous partition coefficients for 1-butene in samples 1, 2, 3 and 4. Within the experimental resolution of the equipment, there is not any difference between the partition coefficients of 1-butene in these samples. The UNIFAC-vdw-FV-MH prediction for these samples does suggest a slightly larger difference than seen in the experimental partition coefficients. The prediction of the amorphous partition coefficient of 1-butene using the regressed f of sample 1, 0.338, provides the best prediction for the amorphous partition
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Fig. 7. Amorphous partition coefficient of cyclohexane in PE sample 1. The experimental data at zero activity (i.e. infinite dilution) were determined by IGC. Data at nonzero activity was determined by gravimetric sorption. The dashed lines provide the prediction of UNIFAC-vdw-FV-MH.
Fig. 8. Amorphous partition coefficient of 1-butene in PE samples 1e4. The lines are given by UNIFAC-vdw-FV-MH.
coefficient of 1-butene in all the samples as shown by the dotdashed line. The prediction using f of sample 3, 0.399, slightly under predicts the experimental partition coefficients, while f of sample 4, 0.278, over predicts the experimental partition coefficient. Fig. 9 shows that there is a general relationship between the polymer density and the optimal weight fraction of tie chains in the amorphous phase, f, determined using UNIFAC-vdw-FV-MH and the type of PE (HDPE, MDPE, LLDPE). The weight fraction of tie chains in the amorphous phase for each sample f , can be generalized for different type of polyethylene's. The lowest f (0.273) was determined for the pure polyethylene, sample 4. This sample falls under the category of HDPE. With any addition of a co-monomer, even as small as the 0.1 mol percent, the amorphous weight fraction of tie chains increases. By using 0.385 as f for the MDPE PEs (samples 2 and 3) the percent error in both samples does not increase more than 0.4%. Similarly for the LLDPE samples (1, 5, and 6) an f of 0.327 changes the error in sample 5 and 6 less than 0.1% and only changes the error in sample 1 from 3.7% to 4.2%. Unfortunately,
Fig. 9. Relationship between the weight fraction of tie chains in the amorphous phase and the type of PE.
attempting to define one f for all types of PE greatly increases the error in the predictions: using an f of 0.33 across all samples almost doubles the percent error for samples 2, 3, and 4. A direct comparison of the amount of tie chains determined through mechanical experiments and those determined by solubility experiments is difficult. The mechanical experiments determine the fraction of molecules that become tie chains, whereas the solubility experiments determine the weight fraction of tie chains in the amorphous phase. A qualitative agreement is found when comparing the effect of co-monomer on the amount of tie chains determined in our work with that of Huang and Brown [30]. They found that the addition of 1-butene co-monomer increases the percent of PE chains that become tie chains. Our work has revealed a similar trend where the addition of co-monomer increases the weight fraction of tie chains in the amorphous phase when compared to the homopolymer. Huang and Brown, however, found a different quantitative trend in the amount of co-monomer and the percent of tie chains than in our work. One thing is certain; the concentration of co-monomer does affect both the mechanical properties such as low temperature brittle fracture stress, as well as the solubility of penetrants in the amorphous phase. Two independent theories have attributed this to an increase in the amount of tie chains. Some preliminary solubility experiments were made with 1hexene in sample 5 using the virgin reactor granules, the film, and the granules annealed for 1 h at 100 C. DSC measurements indicated that the virgin reactor granules had a crystallinity about 5% less than either the film or annealed granules. This was as expected as the polymer chains were likely to organize more tightly at the higher temperature. The virgin granules unexpectedly, however, had less solubility on an amorphous basis as shown in Fig. 10. One possible explanation for this is that the amount of tie chains in the amorphous region is reduced as the crystalline region rearranges at the higher temperatures. Hedenqvist et al. [31] and Strobl and Hagedorn [32] have discussed how Raman spectroscopy can be used to identify three or four different portions in the crystallineamorphous region of PE. Efforts to use this approach to correlate the solubility in the 1-hexene-PE system have not been successful to date. 6. Conclusions The solubility of multiple solvents was determined in six different polyethylene samples. The percent crystallinity of the PE
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Fig. 10. Amorphous solubility of 1-hexene in PE sample 5 using specific crystallinities obtained by DSC.
samples ranged from 40.2 to 81.7. The experimental solubility of the solvents in the all of the polyethylene samples at temperatures below the melt temperature showed a general trend of increasing solubility with increasing temperature. This change in solubility with temperature was accurately accounted for by the force that is exerted on the amorphous region by tie chains as described by the theory of Michaels and Hausslein [20]. The weight fraction of tie chains in the amorphous phase determined in each PE sample differed. Three generalizations can be made from the results: A comparison of three different group contribution methods for modeling the solubility of penetrants in PE was performed. The method of UNIFAC-vdw-FV-MH provided a significant improvement over the other two methods. The amorphous solubility of penetrants is not the same in different types of linear PE pressed films which had similar thermal history. The results indicate that the weight fraction of tie chains in the amorphous phase for the pressed film samples increase with density. Samples where the polyethylene was polymerized with a comonomer had a higher weight fraction of tie chains in the amorphous region than the PE homopolymer. Unlike the previous works of Doong and Ho [21], and Serna et al. [28]. this work found that the weight fraction of tie chains in the amorphous phase is not universal for all types of PE and that there are distinct differences between different types of PE and different crystallization conditions. Acknowledgement Portions of this work were sponsored by NOVA Chemical Co., Calgary, Canada. References [1] World Plastics Production, Plastics europe, association of plastic manufacturers. https://committee.iso.org/files/live/sites/tc61/files/The%20Plastic% 20Industry%20Berlin%20Aug%202016%20-%20Copy.pdf, 2015. [2] R.M. Hodge, G.H. Edward, G.P. Simon, Water absorption and states of water in semicrystalline poly(vinyl alcohol) films, Polymer 37 (1996) 1371e1376.
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Fluid Phase Equilibria j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / fl u i d
Solubility of solvents in polyethylene below the melt temperature Derek R. Sturm, Kevin J. Caputo, Siyang Liu, Ronald P. Danner* Chemical Engineering Department, The Pennsylvania State University, University Park, PA, 16802, USA
a r t i c l e i n f o
a b s t r a c t
Article history: Received 24 May 2017 Received in revised form 7 August 2017 Accepted 5 September 2017 Available online xxx
Polyethylene, the most ubiquitous polymer in use, is a semi-crystalline material. Solvents enter only the amorphous phase. The solubility in that phase, however, cannot be characterized by the amorphous solubility found above the melt temperature. The polymer chains that have ends attached to the crystalline structure but extend into the amorphous phase are referred to as tie chains, and they affect the overall solvent solubility. Six different polyethylenes covering a range of densities were studied. Below the melt temperature all the samples exhibited an increase in solubility with increasing temperature. The solubility of the penetrants, however, is not the same for different types of polyethylene. A model which incorporates an elasticity factor that accounts for the stress effects on the solubility in the tie chains was found to be valuable. When incorporated into a version of the UNIFAC free-volume model there was good correlation between of the solubilities. The present work indicates that the tie chain fraction, f, tends to increase with density. © 2017 Elsevier B.V. All rights reserved.
Keywords: Polyethylene Solubility Crystallinity Elasticity affect UNIFAC
1. Introduction Globally polyethylene (PE) is the most abundant polymer found in everyday life [1]. Some of the most common uses for PE included plastic bags, plastic film, bottles, and piping. Below its melting point polyethylene is a semi-crystalline polymer, i.e., some of the chains form an ordered dense crystalline phase while others are in an amorphous phase of lower density. In general, solvents are excluded from the crystalline phase [2]. To improve the ability of current models to predict the solubility in polyethylene the effect of the amount of crystallinity needs to be characterized. The first type of polyethylene that was commercially produced was known as low-density polyethylene (LDPE) [3]. The production of this type of PE is done without the use of a catalyst. In the absence of a catalyst, the radical site can move from a location at the end of the chain to a more stable location in the middle of the chain. This causes a new ethylene monomer to polymerize at the radical site and form a branch in the middle of the PE chain. As this process continues a non-linear, highly branched PE chain is formed. In LDPE the large amount of chain branching prevents a portion of the chains from crystallizing [4]. Since the amorphous phase of PE has a much lower density than the crystalline phase, the reduction in crystallinity results in an overall product that has a lower density
* Corresponding author. E-mail address: [email protected] (R.P. Danner).
than when the chains are completely linear. Once catalysts were introduced into PE production high-density polyethylene (HDPE) was commercially produced. PE polymerization using a catalyst, such as a Zeigler-Natta or a metallocene catalyst, ensures that the radical site stays at the end of the PE chain, resulting in a final product that is a linear chain with little to no branching. These chains have a higher tendency to crystallize resulting in a high density PE [3]. Soon after the introduction of polymerization catalysts, longer a-olefins such as 1-butene, 1-hexene, and 1-octene were added to the polymerization process. This addition of a co-monomer results in a polyethylene chain that has a linear backbone with short side chains distributed along its length. As these side chains cannot be included in the crystal structure [2], they act to reduce the crystallinity of the polyethylene. The resulting linear low-density polyethylene is sometimes divided into subgroups: mediumdensity PE (MDPE) having a density between 0.94 and 0.926 g/ cm3, linear-low-density PE (LLDPE) having a density between 0.926 and 0.915 g/cm3 and very-low-linear-density PE (VLLDPE) having a density below 0.915 g/cm3. 2. Semi-crystalline structure models There are three common models in the literature used to describe the semi-crystalline structure of PE [5]. The first was Flory's switchboard model [6] which proposes that the majority of polyethylene chains leave the crystalline surface and cross over the
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amorphous region to a different crystalline surface rather than reenter the same crystal. The second is the adjacent reentry model of Hoffman and Lauritzen [7]. They propose two forms: a smooth surface in which almost all the chains that leave the crystalline surface immediately bend and reenter adjacent to where they exited, and a rough surface model in which the majority of chains still enter adjacent to where they exited the crystalline phase, but the fold length is not uniform. The third model was originally suggested by Stamm et al. [8] and later adapted by Strobl [9]. In either version crystallization from the melt occurs by formation of an aligned, long-range, ordered state, which then proceeds to develop into the long-range ordered crystalline phase. In the past thirty years, numerous studies have been performed to study the effect of tie chains on the properties of semi-crystalline polymers. Most of these focus on mechanical properties [10,11]. A good review of the relationships between tie chains and mechanical properties in semi-crystalline polymers is provided by Seguela [12]. Only a few studies have focused on solvent solubility [13e16]. 3. Solubility in PE Unlike amorphous polymers, where the penetrant is evenly distributed throughout the polymer phase at equilibrium, semicrystalline polymers, like PE, restrict the penetrant from entering the crystalline phase due to dense chain packing [2]. While the penetrant is not evenly distributed throughout the entire polymer domain, it is assumed to be evenly distributed within the amorphous domain of the polymer. Thus, when comparing the solubility between polyethylene samples with different crystallinity, the weight fraction of penetrant should be normalized in terms of the amount of penetrant in the amorphous phase of the polymer rather than on a total polymer basis. Above the melt temperature, where PE is entirely amorphous, models like the Group Contribution Lattice Fluid-Equation of State (GCLF-EoS) [17], UNIFAC-FV [18], and UNIFAC-vdw-FV [19] have been shown to accurately predict the solubility of solvents in PE [15]. Below the melt temperature application of these models to PE needs to be evaluated. Michaels and Hausslein [20] were one of the first to propose an explanation for the reduction in solubility in the amorphous phase in semi-crystalline polymers like PE. Their model, which was originally proposed in 1965, is still the most successful model. They postulated that the cause for the large temperature dependence in penetrant solubility in polyethylene is due to the tie chains. As the penetrant enters the amorphous phase these tie chains act like springs and apply an elastic force on the amorphous region which reduces the equilibrium concentration of penetrant within the amorphous phase. The most important assumptions of their derivation are: The amorphous phase is composed of two types of tie chains: elastically effected and inelastically effected chains. The elastically effected chains extend across the amorphous region, tethering two crystalline domains. The inelastic portion consists of chains which exit and reenter the same crystal phase, as well as chain ends and other short segments excluded from the crystal. The amount of tie chains is constant in the temperature range in which the polymer's crystalline structure does not change. Thus, the temperature dependent behavior of the tie chains is reversible as long as the crystalline structure does not change. The collective properties of the tie chains, such as length, can be accurately captured by an average of those properties. The deformation is uniform in all directions, i.e., isotropic. By assuming that the force required to stretch the chains can be
approximated by Hooke's Law, the final expression lacked any terms that require the number of monomers in the tie chains, the length of the chains, or directly evaluating the Hookian spring constant. Dong and Ho [21] used Michaels and Hausslein's expression for the chemical potential that the tie chains exert on the amorphous phase and converted it into an activity.
ln aEL 1 ¼
DHf
R
ra V1
1 T
T1m
3 2f f2
f1 cf21
1
(1)
Here the aEl 1 is elastic activity, DHf is the heat of fusion for the semi crystalline polymer, ra is the density of the amorphous phase, Tm is the melt temperature of the polymer, R is the gas constant, fi is the volume fraction of component i, c is the Flory-Huggins interaction parameter, and f is the weight fraction of the elastically effected chains in the amorphous phase. This activity contribution was then added to the other contributions in the UNIFAC-FV model. EL ln a1 ¼ ln aC1 þ ln aR1 þ ln aFV 1 þ ln a1
aC1 ,
aR1 ,
(2)
aFV 1
Here and are the combinatorial, residual, and free volume activities. The predictive methods of UNIFAC-vdw-FV, UNIFAC-FV and GCLF-EoS that have been modified by the inclusion of aEL 1 will be identified with the suffix MH. The value of DHf used was the experimentally determined value, 293 J/g [22]. The density of the amorphous phase was determined using an extrapolation of the amorphous density of PE above the melt via Tait's equation [23]. The value of c was determined by both regressing the experimental data using the FloryHuggins expression and by using the relationship between the UNIFAC residual and free volume activities as described by Dong and Ho:
c¼
ln aR1 þ ln aFV 1 f22
(3)
Even though the values of c differed between the two methods, it had negligible effect on predicted activity even at the high experimental penetrant weight fractions.
4. Experimental There were six different samples of PE on which solubility studies were performed. They were all produced with Ziegler-Natta catalyst and have crystallinities that range from 40.3 to 81.3%. One is an ethylene homopolymer (no co-monomer), three are ethylene1-butene co-monomers, and two are ethylene-1-hexene copolymers. Table 1 provides the properties of these samples as supplied by the manufactures. Three different methods were used to measure the solubilities: inverse gas chromatography using packed columns (PCIGC), quartz spring gravimetric, and pressure decay. Each of these methods has been described elsewhere (as referenced below) and only pertinent details will be repeated here. In the PCIGC method the solubility of granules is determined by analyzing the elution peak [24]. In the gravimetric approach the penetrant uptake is determined by measuring the extension of a calibrated quartz spring [25]. In the pressure decay method the solubility is determined by measuring the pressure drop in a constant temperature, constant volume chamber and converting that to mass by an equation of state [26,27]. The polyethylene samples in Table 1 were pressed into films by melting the samples and then slowly cooled at a rate of 10 C/h.
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Table 1 PE sample properties. PE Sample
1
2
3
4
5
6
Density (g/cm3) Crystallinity Melt T ( C) Mn (g/mol) Comonomer Polydispersity Index
0.92 44.8 120 20679 1-Butene 3.22
0.9599 78.9 132 23591 1-Butene 2.88
0.9501 72.6 128 16118 1-Butene 2.80
0.9614 81.7 131 11656 None 12.9
0.9124 40.2 100.7/114.1 57174 1-Hexene 1.79
0.9168 43.8 107.5/116.7 56519 1-Hexene 1.80
Thus, the films were crystallized at a temperature just below their melt temperatures. Except for sample 6 the solubility of isopentane in the PE films was determined in every sample at a minimum of three different temperatures. The solubility of a different penetrant was also determined for every polymer sample, although this penetrant was not the same for each sample. For the PE samples that were polymerized with 1-hexene as the co-monomer, the second penetrant was 1-hexene. For the rest of the PE samples the second penetrant was cyclohexane. For PE samples 1e4 extruded pellets from the manufacturers were ground and packed in stainless steel columns. Using PCIGC the partition coefficients of 1-butene at infinite dilution were determined for samples 1e4 and of cyclohexane for sample 1. The finite concentration solubility data were taken with either the quartz spring, gravimetric sorption balance, or the pressure decay method. 5. Comparison of solubility in different PE samples The weight fractions of tie chains in the amorphous phase, f , were regressed using the UNIFAC-FV-MH, UNIFAV-vdw-FV-MH, and GCLF-EoS-MH models using the aEL 1 from equation (1) and the experimental amorphous weight fractions (See Table 2). Other f values reported in the literature include those by Michaels and Hausslein [20] which ranged between 0.265 and 0.363 (excluding the cold-drawn samples) depending on crystallization history, Doong and Ho's [21] 0.373, and Serna et al.’s [28] 0.363. One approach to modeling the probability that a chain will become a tie chain in PE is that of Guttman et al. [29] They provided an estimate of the chance that a polymer chain leaving the crystalline interphase on one side would cross the amorphous region to the opposite crystalline phase through the following assumptions. The amorphous phase can be treated as a cubic lattice; the PE chain is infinitely long and has no imperfections such as side chains or comonomers; the PE chain acts as a random walker in a theta solvent. If the chain does not reenter the crystal it continues to have a chance to move forward or move back towards the crystal it came from. If the chain reaches the other surface, it becomes a tie chain.
In their work, they provided relationships for the average length of chains in the bulk amorphous phase, the fraction of chains leaving the crystalline domain that a chain will become tie chains, and the average length of tie chains in the bulk amorphous phase. Through these relationships, the weight fraction of tie chains in the amorphous phase, f , is determined to be 1/3. The regressed values in Table 2 are in good agreement with these other values. The percentage error between the predicted activities and experimental activities for each PE sample across all solvents were determined for each of the three prediction models as shown in Table 2.
Avg: % error ¼
apred aexp aexp
100
(4)
Here apred is the predicted activity when evaluated at the experimental weight fraction and aexp is the experimental activity. Overall, UNIFAC-vdw-FV-MH out-preforms the other methods, as shown by the lower error in every sample except for sample 6. Sample 6 had the solubility measured for only one solvent and the difference in error between UNIFAC-vdw-FV-MH and UNIFAC-FVMH is less than 0.3%. This small improvement is much less than the difference found in the other samples. Thus, only comparison results for the UNIFAC-vdw-FV-MH model with the experimental data are shown in Figs. 1e6. Fig. 1a and b also include solid black lines which are the predicted solubility determined using UNIFAC-vdw-FV without the modification of Michaels and Hausslein, i.e., the elasticity term, at the lowest temperature of the graph. These results clearly indicate the inaccuracy in predicting activity, and therefore solubility, without considering the effect of the tie chains. For all six PE samples there is excellent agreement between the experimental data and the UNIFAC-vdw-FV-MH predictions. The biggest deviation from prediction comes in the higher crystalline samples 2, 3 and 4 when cyclohexane was the penetrant. The temperature dependence of the experimental solubility of cyclohexane in samples 2, 3, and 4 is smaller than the prediction. In contrast to the higher crystalline samples, sample 1, which had a
Table 2 Regressed f factors for films prepared by slowly cooling from the melt and errors in the predicted activity for the various models. PE Sample
1 2 3 4 5 6
Penetrants
Cyclohexane Isopentane Cyclohexane Isopentane Cyclohexane Isopentane Cyclohexane Isopentane 1-Hexene Isopentane 1-Hexene
UNIFAC-vdw-FV-MH
UNIFAC-FV-MH
f
Avg. % error
f
Avg. % error
f
Avg. % error
0.338
3.0 4.5 10.0 3.6 9.0 3.6 8.9 3.6 1.5 1.3 2.9
0.381
7.3 7.0 12.5 10.0 12.5 7.5 14.5 7.0 5.0 3.0 2.6
0.372
7.2 5.3 12.1 9.1 11.9 7.1 12.6 7.7 4.8 3.1 2.8
0.377 0.399 0.278 0.325 0.326
0.415 0.453 0.355 0.397 0.393
GCLF-EoS-MH
0.406 0.447 0.349 0.371 0.372
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Fig. 1. Solubility of cyclohexane and isopentane in sample 1. The dashed lines provide the prediction of UNIFAC-vdw-FV-MH. The solid line is the prediction of UNIFAC-vdw-FV at 34.5 C for cyclohexane and at 50 C for isopentane.
Fig. 2. Solubility of cyclohexane and isopentane in PE sample 2. The dashed lines provide the prediction of UNIFAC-vdw-FV-MH.
Fig. 3. Solubility of cyclohexane and isopentane in PE sample 3. The dashed lines provide the prediction of UNIFAC-vdw-FV-MH.
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Fig. 4. Solubility of cyclohexane and isopentane in PE sample 4. The dashed lines provide the prediction of UNIFAC-vdw-FV-MH.
Fig. 5. Solubility of 1-hexene and isopentane in sample 5. The dashed lines provide the prediction of UNIFAC-vdw-FV-MH.
Fig. 6. Solubility of 1-hexene in PE sample 6. The dashed lines provide the prediction of UNIFAC-vdw-FV-MH.
much lower amount of crystallinity, showed a greater change in cyclohexane solubility with temperature that matches more closely with prediction. To compare the predictions of UNIFAC-vdw-FV-MH with the infinite dilution experiments of cyclohexane in sample 1, the experimental data shown in Fig. 1 were converted to partition coefficients. Fig. 7 provides the experimental partition coefficients determined by PCIGC (infinite dilution) and gravimetric sorption (finite concentration) along with the UNIFAC-vdw-FV-MH predictions. As the activity goes to zero, both the predicted and experimental partition coefficient decrease. There is excellent agreement between the experimental and predicted partition coefficients. Fig. 8 provides the experimental infinite dilution amorphous partition coefficients for 1-butene in samples 1, 2, 3 and 4. Within the experimental resolution of the equipment, there is not any difference between the partition coefficients of 1-butene in these samples. The UNIFAC-vdw-FV-MH prediction for these samples does suggest a slightly larger difference than seen in the experimental partition coefficients. The prediction of the amorphous partition coefficient of 1-butene using the regressed f of sample 1, 0.338, provides the best prediction for the amorphous partition
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Fig. 7. Amorphous partition coefficient of cyclohexane in PE sample 1. The experimental data at zero activity (i.e. infinite dilution) were determined by IGC. Data at nonzero activity was determined by gravimetric sorption. The dashed lines provide the prediction of UNIFAC-vdw-FV-MH.
Fig. 8. Amorphous partition coefficient of 1-butene in PE samples 1e4. The lines are given by UNIFAC-vdw-FV-MH.
coefficient of 1-butene in all the samples as shown by the dotdashed line. The prediction using f of sample 3, 0.399, slightly under predicts the experimental partition coefficients, while f of sample 4, 0.278, over predicts the experimental partition coefficient. Fig. 9 shows that there is a general relationship between the polymer density and the optimal weight fraction of tie chains in the amorphous phase, f, determined using UNIFAC-vdw-FV-MH and the type of PE (HDPE, MDPE, LLDPE). The weight fraction of tie chains in the amorphous phase for each sample f , can be generalized for different type of polyethylene's. The lowest f (0.273) was determined for the pure polyethylene, sample 4. This sample falls under the category of HDPE. With any addition of a co-monomer, even as small as the 0.1 mol percent, the amorphous weight fraction of tie chains increases. By using 0.385 as f for the MDPE PEs (samples 2 and 3) the percent error in both samples does not increase more than 0.4%. Similarly for the LLDPE samples (1, 5, and 6) an f of 0.327 changes the error in sample 5 and 6 less than 0.1% and only changes the error in sample 1 from 3.7% to 4.2%. Unfortunately,
Fig. 9. Relationship between the weight fraction of tie chains in the amorphous phase and the type of PE.
attempting to define one f for all types of PE greatly increases the error in the predictions: using an f of 0.33 across all samples almost doubles the percent error for samples 2, 3, and 4. A direct comparison of the amount of tie chains determined through mechanical experiments and those determined by solubility experiments is difficult. The mechanical experiments determine the fraction of molecules that become tie chains, whereas the solubility experiments determine the weight fraction of tie chains in the amorphous phase. A qualitative agreement is found when comparing the effect of co-monomer on the amount of tie chains determined in our work with that of Huang and Brown [30]. They found that the addition of 1-butene co-monomer increases the percent of PE chains that become tie chains. Our work has revealed a similar trend where the addition of co-monomer increases the weight fraction of tie chains in the amorphous phase when compared to the homopolymer. Huang and Brown, however, found a different quantitative trend in the amount of co-monomer and the percent of tie chains than in our work. One thing is certain; the concentration of co-monomer does affect both the mechanical properties such as low temperature brittle fracture stress, as well as the solubility of penetrants in the amorphous phase. Two independent theories have attributed this to an increase in the amount of tie chains. Some preliminary solubility experiments were made with 1hexene in sample 5 using the virgin reactor granules, the film, and the granules annealed for 1 h at 100 C. DSC measurements indicated that the virgin reactor granules had a crystallinity about 5% less than either the film or annealed granules. This was as expected as the polymer chains were likely to organize more tightly at the higher temperature. The virgin granules unexpectedly, however, had less solubility on an amorphous basis as shown in Fig. 10. One possible explanation for this is that the amount of tie chains in the amorphous region is reduced as the crystalline region rearranges at the higher temperatures. Hedenqvist et al. [31] and Strobl and Hagedorn [32] have discussed how Raman spectroscopy can be used to identify three or four different portions in the crystallineamorphous region of PE. Efforts to use this approach to correlate the solubility in the 1-hexene-PE system have not been successful to date. 6. Conclusions The solubility of multiple solvents was determined in six different polyethylene samples. The percent crystallinity of the PE
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Fig. 10. Amorphous solubility of 1-hexene in PE sample 5 using specific crystallinities obtained by DSC.
samples ranged from 40.2 to 81.7. The experimental solubility of the solvents in the all of the polyethylene samples at temperatures below the melt temperature showed a general trend of increasing solubility with increasing temperature. This change in solubility with temperature was accurately accounted for by the force that is exerted on the amorphous region by tie chains as described by the theory of Michaels and Hausslein [20]. The weight fraction of tie chains in the amorphous phase determined in each PE sample differed. Three generalizations can be made from the results: A comparison of three different group contribution methods for modeling the solubility of penetrants in PE was performed. The method of UNIFAC-vdw-FV-MH provided a significant improvement over the other two methods. The amorphous solubility of penetrants is not the same in different types of linear PE pressed films which had similar thermal history. The results indicate that the weight fraction of tie chains in the amorphous phase for the pressed film samples increase with density. Samples where the polyethylene was polymerized with a comonomer had a higher weight fraction of tie chains in the amorphous region than the PE homopolymer. Unlike the previous works of Doong and Ho [21], and Serna et al. [28]. this work found that the weight fraction of tie chains in the amorphous phase is not universal for all types of PE and that there are distinct differences between different types of PE and different crystallization conditions. Acknowledgement Portions of this work were sponsored by NOVA Chemical Co., Calgary, Canada. References [1] World Plastics Production, Plastics europe, association of plastic manufacturers. https://committee.iso.org/files/live/sites/tc61/files/The%20Plastic% 20Industry%20Berlin%20Aug%202016%20-%20Copy.pdf, 2015. [2] R.M. Hodge, G.H. Edward, G.P. Simon, Water absorption and states of water in semicrystalline poly(vinyl alcohol) films, Polymer 37 (1996) 1371e1376.
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