Ind. Eng. Chem. Res. 2009, 48, 4127–4135
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Compositional Polydispersity in Linear Low Density Polyethylene Aleksandra Dominik† and Walter G. Chapman* Department of Chemical and Biomolecular Engineering, Rice UniVersity, 6100 S. Main, Houston, Texas 77005
Robert D. Swindoll and David Eversdyk The Dow Chemical Company, Freeport, Texas
Prasanna K. Jog and Rakesh Srivastava The Dow Chemical Company, Midland, Michigan
A study of the effect of a short-chain branching distribution (SCBD) on the phase behavior of linear low density polyethylene (LLDPE) was conducted. The perturbed chain-SAFT equation of state was the underlying thermodynamic model chosen for the study; the branched polyolefins were described using a simple modeling concept previously proposed by Dominik and Chapman.1 To isolate the effect of the SCBD on phase behavior, the copolymer systems were considered monodisperse in molecular weight. The study revealed that, in the case of low to moderate values of the polydispersity index of the SCBD, the compositional polydispersity affected the phase behavior at low polymer concentrations only. When the polydispersity index of the SCBD is high, or, in other words, when the difference in branch content between the components of the distribution is significant, additional phases appear. The formation of multiple phases results from the incompatibility of branched and linear polymers. Similar observations were made on the basis of experimental studies in the case of polymer blends. The SCBD of LLDPE was determined by the chemistry of the polymerization reaction. A broad SCBD was found to significantly affect the phase behavior of LLDPE solutions. 1. Introduction The performance of the PC-SAFT equation of state (EOS) has been extensively tested against experimental results for phase behavior of polymer and copolymer solutions.1-6 Gross and Sadowski,3 and later Tumakaka et al.,5 Gross et al.,4 Tumakaka and Sadowski,7 Dominik and Chapman,1 and Kleiner et al.6 demonstrated the applicability of PC-SAFT to a variety of polymer systems including linear, branched, polar, and associating polymers. PC-SAFT describes the phase behavior and thermophysical properties of chain fluids with an accuracy unmatched by any currently available equation of state. Smooth variation of the parameters with molecular weight, parameter transferability, flexible framework for inclusion of complex interactions (association, dipolar interactions) and possibility to rigorously account for the size and shape of the molecules are among the key advantages of this EOS based on statistical mechanics. Now that its predictive capabilities have been thoroughly verified, PC-SAFT will be used to theoretically study the effect of short-chain branching distribution in linear low density polyethylene (LLDPE). The modeling methodology for a short chain-branched copolymer like LLDPE is shown in Figure 1. LLDPE is the result of copolymerization of ethylene with a higher R-olefin, such as 1-butene, 1-hexene, 1-octene, or 4-methyl1-pentene. Therefore, it contains only short-chain branches (SCB), as opposed to low density polyethylene (LDPE), which contains both short- and long-chain branches (LCB). The polymerization reaction takes place in the gas phase or in solution, in the presence of a catalyst. The modification of the polymer structure through the addition of SCB confers specific * To whom correspondence should be addressed. Tel.: (1)713-3484900. Fax: (1)713-348-5478. E-mail:
[email protected]. † Current address: Shell Global Solutions US, Inc., Westhollow Technology Center, 3333 Highway 6 South, Houston TX, 77082-3101.
properties to the resulting material. LLDPE has higher tensile strength and higher impact and puncture resistance than LDPE; it is flexible and has a good resistance to chemicals. It is used for plastic bags and sheets, pipes, and flexible tubing among other applications. The Ziegler-Natta catalysts commonly employed in the manufacture of LLDPE have more than one active site. This feature of the catalysts gives rise to a broad, often multimodal short-chain branching distribution (SCBD).8 Consequently, LLDPE synthesized in the presence of the Ziegler-Natta catalyst can be regarded as a blend of species differing by the number of SCB on the polymer backbone. Furthermore, results obtained from temperature rising elution fractionation (TREF) experiments suggest a strong correlation between branch content and molecular weight of the copolymer species:8,9 the lowest molecular weight copolymer is characterized by the highest branching density. Commercial polymers typically have bivariate distributions, and they are polydisperse in both molecular weight and comonomer content (branch content in the case of LLDPE). Effective methods for accounting for polydispersity in molecular weight have been developed,10-13 yet little progress has been made to include information about compositional polydispersity in thermodynamic models for polymer solutions. In this work, we aim at elucidating the effect of compositional polydispersity of LLDPE on the phase behavior of LLDPE solutions. The study identifies the conditionsspressure,
Figure 1. Representation of branched polyolefins in the framework of the proposed modeling concept.
10.1021/ie800982z CCC: $40.75 2009 American Chemical Society Published on Web 03/18/2009
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temperature, solution composition, and average branch content of the copolymersat which the branching distribution in LLDPE has an impact on phase behavior of LLDPE solutions. The effect of compositional polydispersity on phase behavior will be isolated by assuming that all copolymer species in a given mixture have the same molecular weight. Phase separations in LLDPE melts, which are de facto multicomponent blends, have been studied experimentally by the small-angle neutron scattering (SANS) technique.14-18 It was found that, even for systems in which the average number of SCB was low (e.g., 1-3 branched per 100 backbone carbon atoms), the blends contain a dispersed minority phase consisting of highly branched, amorphous material, incompatible with lightly branched chains. The existence of this dispersed phase was suggested by Mirabella et al.9,19 and confirmed by Nesarikar et al.20 and Rhee and Crist21 on the basis of scanning electron microscopy observations. Wignall et al. have recently investigated the liquid-liquid phase separations in metallocene-based LLDPE15 via SANS. Such LLDPEs have a much narrower SCBD because of the structure of the metallocene catalyst and, thus, mix homogeneously in the blend, as opposed to LLDPE synthesized in the presence of the Ziegler-Natta catalyst. Theoretical studies of polymer blends were also undertaken, most of them were conducted in the framework of continuous thermodynamics. Bauer studied the effect of comonomer distribution functions on the phase behavior of blends;22 Nesarikar20 and Sollich23 performed calculations for random copolymer systems to demonstrate that multiple phases can coexist in such systems. The volume fraction of the dispersed phase containing the highly branched chains predicted from these theoretical considerations was in good agreement with experimental observations. The efforts undertaken to assess the effect of SCDB in LLDPE have been focused on polymer blends; very few studiessexperimental or theoreticalsof the effect of SCDB on the phase behavior of LLDPE solutions have been undertaken. The results of such a study would be relevant for the optimal design and operation of polymerization reactors in which the monomers (ethylene and R-olefins) are in solution (e.g., the Dow reactor technology). The synthesis of a polymeric material with specific properties, such as strength and flexibility, is contingent upon the polymerization reaction taking place in a single phase.24 On the basis of theoretical considerations in the framework of the Flory-Huggins theory, Scott demonstrated that copolymers of high molecular weight must be reasonably uniform in composition to warrant the stability of a homogeneous phase.25 Ratzsch and co-workers have studied the effect of compositional polydispersity on the phase behavior of random copolymer solutions using continuous thermodynamics.26-30 They found that the compositional polydispersity of the copolymers had an effect on the phase behavior of the copolymer solutions, but they were not able to quantify that effect because the theoretical model used for their calculations had not been used to describe the phase behavior of real systems. In this work, we will take advantage of a recently developed modeling concept for branched polyolefins in the framework of the PC-SAFT equation of state1 to quantify the effect of SCDB on the phase behavior of LLDPE solutions. To isolate the effect of SCBD, the copolymers will be considered monodisperse in molecular weight. This is an approximation since, as pointed out by Mirabella and Ford,19 the highly branched chains have the lowest molecular weight. The sensitivity of the phase behavior to the average comonomer composition has been demonstrated in previous studies, and the model was shown to
accurately capture the effect of the presence of SCB of various length. Here, the copolymer with a given average comonomer content will be represented by two or more components with different comonomer incorporation. The effect of the breadth of the distribution on the phase behavior will be assessed. 2. Modeling Concept for Branched Polyolefins The underlying thermodynamic model for this concept is the PC-SAFT equation of state proposed by Gross and Sadowski. Detailed information about the equation of state can be found in its authors’ original publication.2 Here we aim at providing the reader with essential information about the equation of state and its parameters. The molar residual Helmholtz free energy for nonassociating, nonpolar components is given in terms of a perturbation expansion: ares ) ahs + achain + adisp
(1)
where ares is the free energy residual to an ideal gas at the same temperature and density as the fluid of interest. The hard sphere contribution (ahs) is due to Carnahan and Starling,31 and the chain term was developed by Chapman and co-workers32,33 on the basis of Wertheim’s thermodynamic perturbation theory of first order.34-37 The expression for the dispersion contribution (adisp) was developed by Gross and Sadowski, who proposed the PC-SAFT equation of state.2 The molecular model underlying PC-SAFT was initially developed for chain molecules comprising spherical segments of the same type. Since the focus of this work is on copolymers, the equation of state must be applicable to heterosegmented chains. Amos and Jackson first applied the SAFT equation of state to heterosegmented trimers;38 Shukla and Chapman39 and Banaszak et al.40 modeled heterosegmented chains in a similar framework. Gross et al.4 present the expressions of the free energy in the case of heterosegmented chains; the reader is referred to their articles for further details. Each segment type in the system is characterized by three pure component parameters: the number of segments of type i in a chain (mi), the segment diameter σi, and the segment dispersion energy i/k, where i runs over all the segment types of the system. A system comprising a copolymer of the type poly(R-co-β) and a solvent (S) will be modeled using the three pure component parameters for R, β, and S. Additionally, three binary interaction parameters (kij) may be required to model the system, since the Lorentz-Berthelot mixing rules are used for the unlike pair interactions. ij ) √ij(1 - kij)
(2)
1 σij ) (σi + σj) 2
(3)
Two of these binary interaction parameters describe the interactions between each segment type in the copolymer chain and the solvent (kS-R and kS-β). The third interaction parameter (kR-β) may be used to correct the cross dispersive interactions between the different segment types in the copolymer. The values of all the model parameters for a given copolymer depend on the choice of the modeling concept. The branched LLDPE chains will be represented using the modeling concept proposed by Dominik and Chapman.1 This modeling concept distinguishes between the branch (B) and the backbone (E) segments in the copolymer chain, as shown in Figure 1. The two types of segments have the same diameter (σ) and segment number parameter (m/M, where M is the
Ind. Eng. Chem. Res., Vol. 48, No. 8, 2009 4129 Table 1. Pure Component Parameters for the Components Considered in the Study component ethyl branch hexyl branch HDPE ethylene propane n-hexane
/k [K] m/Mw [mol/g]
σ [A]
190 225 252 176.47 208.11 236.77
4.0217 Dominik and Chapman1 4.0217 4.0217 Gross and Sadowski3 3.4450 Gross and Sadowski2 3.6184 3.7983
0.0263 0.0263 0.0263 0.05679 0.04540 0.03548
ref
Table 2. Values of the Binary Interaction Parameters for all Considered Polymer Segment Types with Different Solvents segment type
solvent
kij
ref
HDPE-backbone ethylene 0.0404 fitted by Gross and Sadowski3,4 propane 0.0206 n-hexane 0.0055 ethyl-branch propane 0.0130 fitted by Dominik and Chapman1 hexyl-branch ethylene 0.0290 fitted by Dominik and Chapman1 propane -0.018 n-hexane 0
molecular weight of the polymer), which is consistent with the fact that they have the same chemical nature. They differ, however, by their dispersion energy parameter (/k) and their binary interaction parameter with the solvent (kij). All backbone segments are assigned the PC-SAFT pure component parameters of the linear HDPE published by Tumakaka et al.5 The branch segments are assigned the pure component chain length and segment diameter parameters of high density polyethylene (HDPE). The dispersion energy parameter of the branch segments was adjusted to experimental phase behavior data for polyolefins in our previous work,1 along with the binary interaction parameter between the branch segments and the solvent. The values of the pure component parameters, as well as the binary interaction parameters used in this study, are reported in Tables 1 and 2. All the binary interaction parameters between backbone segments and various solvents are assigned the values of binary interaction parameters between HDPE and solvents, previously published by Tumakaka et al.5 and Gross et al.4 The relative fractions of segments in the backbone and in the branches can be easily calculated from the comonomer incorporation, which is a known copolymer property. The proposed model for branched polyolefins is consistent with their chemical structure as previously mentioned, as well as with the experimental observations regarding their phase behaviorssee the work of Dominik and Chapman1 for more details. To assess the effect of compositional polydispersity (or SCDB), two or more components with different branch content will represent the copolymer. To simplify the computations, we show that a poly(ethylene-co-R-olefin) copolymer with a given weight fraction wcomonomer of R-olefin can be modeled as a homopolymer with averaged parameters. In the copolymer, each segment typesbackbone or branchshas its own dispersion energy i/k and its own binary interaction parameter kij with the solvent. The energy and binary interaction parameters of the copolymer, represented as a chain in which all segments are identical, can be obtained from the values of the parameters for each segment type as follows: copolymer ) backbone(1 - wbranch) + branchwbranch
(4)
kSolvent-Copolymer ) kSolvent-Backbone(1 - wbranch) + kSolvent-Branchwbranch (5) In eqs 4 and 5, backbone and branch are the dispersion energy parameters of backbone and branch segments respectively, and
ki-j are the interaction parameters between the three different types of segments in the system (branch, backbone, or solvent), whereas wbranch is the mass fraction of branch segments in the chain. It can be obtained from the mass fraction of R-olefin in the copolymer as wbranch ) wcomonomer
Mbranch Mcomonomer
(6)
where Mbranch and Mcomonomer are the molecular weights of the branch and comonomer (e.g., 1-butene or 1-octene), respectively. This simplification of the modeling concept does not affect the phase behavior calculations, which is consistent with the fact that the equation of state describes the different components based on the average parametric information. An analogous modeling concept was proposed by Jog et al.41 for polar copolymers. It was used by Ghosh to study the effect of polydispersity in polar comonomer content on the phase behavior of polar copolymers.42 The conclusion of Ghosh’s study was that the compositional polydispersity in polar comonomer content had a strong impact on the phase behavior of the copolymer solution. The simplification of modeling a copolymer as a homopolymer was introduced because the commercial phase equilibrium software used for this study (VLXE; see www.vlxe.com for more information) does not support calculations for systems in which several copolymer components are present. It is possible, however, to perform phase behavior calculations for solutions containing several homopolymer species. Consider the case of poly(ethylene-co-1-octene) with 25 wt % 1-octene in the chain. The dispersion energy parameter has the value of 252.0 K for the backbone segments and 225.0 K for the branch segments. The dispersion energy parameter of the segments in the homonuclear chain representing the copolymer in the PC-SAFT framework will take the value of 246.89 K according to eqs 4 and 6. The value of the binary interaction parameter can similarly be interpolated from the values of the binary interaction parameters between backbone segments and solvent and branch segments and solvent. The parameters needed to represent the copolymers were obtained from the work of Dominik and Chapman,1 whereas the parameters needed to model the solvents considered in this work (ethylene, propane, n-hexane) are from the work of Gross and Sadowski.2 3. Results 3.1. Bimodal Distributions. To evaluate the effect of compositional polydispersitysor SCBDsin LLDPE, a study of the phase behavior of poly(ethylene-co-1-octene) in two different solvents (ethylene and n-hexane) was conducted. A twocomponent bimodal distribution in comonomer content is first considered. In other words, the copolymer with an average branching content will be represented by two components having the same molecular weight, but containing different amounts of 1-octene. In the first part of this study, the two components representing the copolymer have a 1:1 concentration on a solvent-free basis, and copolymers with two average weight fractions of 1-octene (15 and 25 wt %) are considered. In the case of the copolymer containing an average 15 wt % 1-octene, cloud-point pressure calculations were performed for three binary, equimolar copolymer systems: 20/10, 25/5, and 30/0 wt % (or 30 wt % 1-octene/HDPE). In the case of the 25 wt % poly(ethylene-co1-octene), the following distributions were considered: 30/20,
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Figure 2. Phase boundary of poly(ethylene-co-1-octene) in ethylene in the pressure-temperature projection. Results are shown for copolymers with an average 1-octene content of 15 wt % (a) and 25 wt % (b). The numbers x/y in the figures indicate a mixture of two copolymers containing x wt % of 1-octene and y wt % of 1-octene. The overall copolymer concentration is 10 wt %; the molecular weight of all copolymer components is 124 kg/mol.
35/15, 40/10, and 50/0 wt % (or 50 wt % 1-octene/HDPE). As the distribution becomes broader, i.e. as the difference in branching content between the two copolymer components increases, the phase boundary shifts to higher pressures, as shown in Figure 2. The results presented in Figure 2 indicate that the sensitivity of the phase separation to the distribution width is largely unaffected by the average branch content of the copolymer. Similar conclusions were reached by Wignall et al.14 in the case of polymer blends. The phase behavior at high temperatures (and low pressures) is relatively unaffected by the SCBD (or the width of the distribution), except in the case of distributions containing a fraction of the linear HDPE. The phase boundary for these systems (30/0 and 50/0) is situated at higher pressures than the phase boundary of HDPE in ethylene. This occurs because a three-phase equilibrium is present at lower pressures; due to the incompatibility of the two copolymer components, the copolymer-rich phase splits into two phases. These observations are analogous to those made in the case of polymer blends.14,15,19 The pressure-composition phase diagram for such a system no longer takes the classic form observed for polydisperse polymer solutions. Multiple phases are present at low pressure; a pressure increase induces the merging of the two polymer-rich phases into one phase. The transition to a single homogeneous phase is observed at significantly higher pressures than for the single copolymer with the corresponding average branch content. Results obtained for poly(ethylene-co-1-octene) containing 15 wt % 1-octene and represented by a bimodal distribution of varying width were compared to results for solutions of the copolymers constituent of the blends. Whereas the cloud-point pressure curve for the 20/10 system practically overlaps with the cloud-point pressure curve for the copolymer containing 15 wt % 1-octene (see Figure 3), the phase boundary for the 25/5 system is significantly higher at low temperatures and is very close to the phase boundary of a copolymer containing 5 wt % 1-octene in the chain (see Figure 4). When the distribution is further broadened, the phase boundary is situated at even higher pressures, as illustrated by the example of the 30/0 system in Figure 5, whose phase boundary is situated at higher pressures than the phase boundary for HDPE in ethylene. An important observation arising from these results is that the phase boundary
Figure 3. Impact of different branch content on the position of the cloudpoint phase boundary of poly(ethylene-co-1-octene) in ethylene. Results show that the cloud-point pressure of a copolymer with a uniform 15 wt % 1-octene content is indistinguishable from that of an equimolar mixture of two polymers containing 10 wt % 1-octene and 20 wt % 1-octene. For comparison, the cloud-point pressures of copolymers with uniform 1-octene contents of 10 and 20 wt % are shown. The overall copolymer concentration is 10 wt %; the molecular weight of all copolymer components is 124 kg/ mol.
of a blend of two copolymers having a given average branch content is situated between the phase boundaries of each of the copolymers constituent of the blend only when the SCBD is rather narrow, as in the case of the 20/10 system. The results shown in Figures 2-5 represent the phase transition of the copolymer solution from a two-phase region at low pressures to a homogeneous liquid phase at high pressures. In all cases, the total copolymer concentration in the solvent was 10 wt %. From these results alone, one may infer that the high density copolymer fraction (i.e., the copolymer with fewer branches) dominates the phase behavior. This would imply that the contributions to the Gibbs free energy of mixing
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Figure 4. Impact of different branch content on the position of the cloudpoint phase boundary of poly(ethylene-co-1-octene) in ethylene. Results compare the cloud-point pressures of an equimolar mixture of two polymers containing 5 wt % 1-octene and 25 wt % 1-octene with the cloud-point pressures of copolymers with uniform 1-octene contents of 5 and 15 wt %. The overall copolymer concentration is 10 wt %; the molecular weight of all copolymer components is 124 kg/mol.
Figure 5. Impact of different branch content on the position of the cloudpoint phase boundary of poly(ethylene-co-1-octene) in ethylene. Results compare the cloud-point pressures of HDPE and of a copolymer with uniform 1-octene content of 15 wt % with the cloud-point pressures of an equimolar mixture of two polymers containing 0 wt % 1-octene (HDPE) and 30 wt % 1-octene. The overall copolymer concentration is 10 wt %; the molecular weight of all copolymer components is 124 kg/mol.
due to the difference in free volume between the polymer and the solvent, as well as to the dispersion interactions between the polymer and the solvent (both unfavorable to mixing) become disproportionately large in the case of the high-density polymer and, thus, determine the position of the phase boundary. Further calculations were carried out over a range of pressures and copolymer concentrations in the solution to obtain an indepth understanding of the phase behavior of compositionally polydisperse copolymer systems and to verify this assertion. A closer look at the pressure-composition phase diagram of these
Figure 6. Cloud- and shadow-point curves of poly(ethylene-co-1-octene) copolymers in ethylene. The dash-dotted and dotted lines are the shadow curves of the 20/10 and 21/9 systems, respectively; symbols indicate the critical points of the solutions. The copolymers have a 1:1 concentration on a solvent-free basis. The molecular weight of all copolymer components is 124 kg/mol.
systems reveals that the unstable domain, situated below the cloud-point transition curve and characterized by the presence of two or more immiscible liquid phases becomes broader as the difference in branching content between the two components representing the copolymer increases. In the case of distributions with a small to moderate difference in branch content between the two copolymer components, the pressure maximum shifts to lower polymer concentrations, whereas the critical point of the solution, situated at the intersection of the cloud and shadow curves, moves to higher copolymer concentrations, as shown in Figure 6. A stability analysis of the phase diagrams of the systems presented in Figure 6 revealed that they do not separate in more than two phases at low pressures. The two copolymer species are still compatible, and their difference in branching does not induce additional phase instability. The changes in phase behavior with increasing compositional polydispersity are very similar to those occurring when the polymer is polydisperse in molecular weight (see the work of Jog et al.,43 for example), i.e. shift of the critical point to higher polymer concentrations and curve maximum situated at lower polymer concentrations and higher pressures. The effect of polydispersity in molecular weight, as well as in comonomer content is mostly visible at low polymer concentrations, whereas the shape of the phase diagram at high polymer concentrations remains essentially unchanged. Notice how a small change in the width of the distribution (from 20/10 to 21/9) induces a significant change in the position of the phase boundary in Figure 6. The cloudpoint pressure curve for the 21/9 system is situated at higher pressures than the copolymer curve for all solution compositions. This change forewarns the imminent formation of a third phase at low pressures upon further increase of the width of the distribution, due to the instability of the polymer-rich phase. This is illustrated in Figure 7; the phase boundary between a two-phase and a homogeneous region is represented. Three phases appear at low pressure: the polymer-rich phase splits into two phases because of the incompatibility of the two copolymer components, similar to what was observed for copolymer blends. The twosor multiphase regionsbecomes
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Figure 7. Cloud-point curves of poly(ethylene-co-1-octene) copolymers in ethylene. The copolymers have a 1:1 concentration on a solvent-free basis. The molecular weight of all copolymer components is 124 kg/mol.
broader, and the pressure-composition curve indicating the transition between the two- and one-phase region is significantly flatter. Whereas the 25/5 system still exhibits a phase boundary whose shape is very similar to the one observed for the comonomer containing 15 wt % 1-octene (e.g., same slope of the pressure-composition phase boundary curve at high polymer concentrations), the phase envelope of the 27/3 system significantly differs in shape from both curves. Due to the presence of multiple phases, the shadow curve of the system does not have a defined shape. The phase transition between a threeand a two-phase equilibrium could not be mapped out due to computational difficulties inherent to such calculations, but the results of a stability analysis confirm that the transition shown in Figure 7 occurs between a two-phase region and a one-phase domain. These results indicate that the factor determining the position of the phase transition between a two-phase region and a homogeneous phase region is the presence of a three-phase equilibrium region at lower pressures, which is induced by a large difference in SCBD between the copolymer components. The transition between a two-phase and a homogeneous phase region is in fact the “absolute” boundary of the copolymer system, so labeled by analogy to the absolute boundary of gas hydrates stability. At pressures above the phase transition pressures, the copolymer solution exists in a single phase. If the polymerization reaction is to be carried out in a single phase, the pressure in the reactor must be higher than the pressure which corresponds to the transition between one homogeneous liquid phase and two liquid phases. The phase diagram of a solution of LLDPE in ethylene differs from the phase diagrams obtained for solutions of LLDPE in other alkenes, as experimentally found by Ehrlich and Krupen.44 The quadrupolar interactions between ethylene molecules are strongly favored over the nonpolar interactions between ethylene and LLDPE. This results in a sharp increase in pressure of the cloud-point curve of the system at low temperatures. As temperature increases, the strength of the quadrupolar interactions decreases, and the cloud-point pressure decreases. This behavior is also observed when the solvent is a dipolar molecule, e.g., dimethyl ether or acetone.45 When n-hexane is the solvent for LLDPE, the phase diagram obtained for such a solution differs from the phase diagram obtained for LLDPE in ethylene
Figure 8. LCST-type phase boundaries of poly(ethylene-co-1-octene) in n-hexane in the pressure-temperature projection. The copolymers have a 1:1 concentration on a solvent-free basis. The overall copolymer concentration is 10 wt %; the molecular weight of all copolymer components is 124 kg/mol.
in that the LCST and UCST boundaries do not merge but intersect with the vapor-liquid-liquid boundary at low pressures. The results for a poly(ethylene-co-1-octene) containing 15 wt % 1-octene are shown in Figure 8. The position of the LCST phase boundary remains essentially unaffected by the width of the SCDB, except in the case of the 30/0 system, for which a three-phase equilibrium at low pressures was found due to the great difference in SCB content and the resulting incompatibility of the two copolymer components. The results obtained for the phase behavior of poly(ethylene-co-1-octene) in n-hexane are affected by the width of the SCB distribution to a lesser extent than those obtained when ethylene is the solvent for the polymer. This may be due to the fact that ethylene is a poor solvent of polyolefins because of its significant quadrupole moment and low polarizability. A good solvent of polyolefins, such as n-hexane, may provide some stabilization to a system containing two incompatible copolymers, such as the 25/5 system. The presence of a good solvent reduces the effect of the incompatibility of the copolymers, thus warranting a more stable solution. On the other hand, the marked instability of the 25/5 system is further augmented in the presence of a poor solvent such as ethylene, thus leading to the formation of a three-phase equilibrium. On the basis of theoretical considerations in the framework of the Flory theory for polymers, Scott concluded that the molecular weight of the copolymers would have an effect on their compatibility in the case of compositionally heterogeneous copolymers.25 Specifically, higher molecular weight copolymers must exhibit a narrower comonomer (or branch) content distribution for the system to remain stable. Consequently, one would expect that the effect of a short-chain branching distribution would be more pronounced for higher molecular weight copolymers. This is indeed the case, as shown in Figure 9. Moreover, for the 248 kg/mol copolymer system, the results of a stability analysis indicate that a three-phase equilibrium can be found at low pressures for narrower distributions than for the 124 kg/mol copolymer system. Koningsveld and Kleintjens have reached similar conclusions based on experimental results obtained for polymer blends and solutions.46
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Figure 9. Cloud-point pressure of poly(ethylene-co-1-octene) in ethylene. The copolymers have a 1:1 concentration on a solvent-free basis and contain 15 wt % 1-octene on average. Two molecular weights are considered: 124 and 248 kg/mol. Both systems are considered monodisperse in molecular weight. The overall copolymer concentration is 10 wt %. The results for the systems containing the 248 kg/mol copolymer have been shifted to the right by 60 K for clarity. Table 3. SCBD Data on PEB from Mirabella and Ford9 LLDPE 1 2 3 4 average
mass Fraction
weight percent 1-butene NMR 13C
Mw [kg/mol]
Mw/Mn
0.276 0.365 0.202 0.157
13.4 7.8 2.4 0.0 7.0
57 75 93 124 95
4.8 4.1 2.9 3.2 3.8
3.2. Real Distributions. Thus far, only bimodal twocomponent distributions were considered in this study, and it was found that the difference in SCBD had a strong impact on the phase behavior of the LLDPE solution when the SCB density difference between the two components was large. Such distributions, the simplest from a computational stand pointsonly two copolymer components in the systemsare not representative of real copolymer systems, especially because both components are assumed to have the same molecular weight, whereas in real systems, chain length and SCB density are correlated. In fact, most distributions pertaining to real systems are bivariate, i.e. the polymer is polydisperse with respect to molecular weight as well as branching density (or comonomer content). In this section of the study, with the guidance of the conclusions reached in the first part of the work, the impact of accounting for SCBD in real polymer distributions on phase behavior will be assessed. Mirabella and Ford performed TREF experiments on poly(ethylene-co-1-butene) (PEB) samples.9 The PEB contained on average 7 wt % 1-butene and had an average molecular weight of 95 kg/mol. Detailed information about the molecular weight distribution (MWD), as well as the SCBD of this polymer sample can be found in Table 3. The molecular weight of the high density fraction is more than twice as high as the molecular weight of the most densely branched chain, and the polydispersity index (PI) of the MWD of the copolymer is 3.8. Albeit a PI of 3.8 is not considered very high, a significant influence of the polydispersity of the polymer on the phase behavior of
Figure 10. Phase boundary of poly(ethylene-co-1-butene) in propane at 450 K. All systems have an equimolar copolymer concentration on a solventfree basis (i.e., 1:1 for 14/0 (a), 1:1:1 for 14/7/0 (b), and 1:1:1:1 for 14/9/ 5/0 (c)). The molecular weight of all copolymer components is 95 kg/mol. The solid lines correspond to cloud-point curves, whereas the dotted, dashed, and dash-dotted lines correspond to shadow curves.
the polymer solution can be observed, especially at low polymer concentrations.43 The phase behavior of this PEB in propane was examined. Propane was chosen as a solvent because reliable binary interaction parameters for PEB in propane have been previously determined.1 The phase behavior of PEB containing 7 wt % 1-butene on average, but represented by distributions containing an increasing number of components having different branching density, was studied, and the results are summarized in Figure 10. In this case, all components have the same molecular weight of 95 kg/mol. Three observations can be made from Figure 10. First, as more components with intermediate branching density are added to the distribution, the cloud-point curve corresponding to the phase transition of the solution from two phases to a homogeneous phase becomes more similar to the cloud-point curve obtained for the 7 wt % PEB. This is due to the fact that upon addition of components having an intermediate SCB density, the polydispersity index of the SCBD is reduced and the system’s behavior is closer to the behavior of a monodisperse system. Second, as previously observed for systems polydisperse in molecular weight, the polydispersity in chain branching affects the phase behavior of polymer solutions only at low polymer concentrations, whereas the phase behavior at high polymer concentrations remains essentially unaffected by the SCBD. This remark is valid as long as the polymer-rich phase does not split into two separate phases; in which case, the phase behavior at high polymer concentrations is significantly affected (see Figure 7). Finally, the shape of the shadow curve is significantly affected by the SCBD, especially when the system is characterized by a large disparity in SCB density between its constituents. This is clearly visible in the case of the 14/0 PEB in Figure 10. The change in shape of the shadow curve indicates that a further broadening of the distribution would destabilize the two-phase equilibrium and lead to the formation of two polymer-rich phases. A system comprising a 3:7 distribution consisting of two PEB copolymers containing 21 and 0 wt % 1-butene, respectively, was considered, and it was found that three phases were present at low pressure, which is
4134 Ind. Eng. Chem. Res., Vol. 48, No. 8, 2009
Figure 11. Phase boundary of poly(ethylene-co-1-butene) in propane at 450 K (left) and the corresponding short-chain branching distributions representing the copolymer. The molecular weight of all copolymer species is 95 kg/mol.
consistent with the results obtained previously for poly(ethyleneco-1-octene). It is noteworthy that the PEB has a relatively low branch contentsonly 7 wt % 1-butene is present in the chain. This corroborates the assertion that the average branch content does not change the sensitivity of the phase behavior to SCBD, and a low branch content does not mitigate the effect of compositional polydispersity on the phase transitions observed. The assertion that, even in the case of multicomponent distributions, despite the compatibilization effect due to the presence of components with intermediate branch density, a large difference in branch content between the components characterized by the lowest and the highest branch content, respectively, would induce the formation of three phases was verified, and the results are summarized in Figure 11. The distributions introduced for the purpose of this study have an average 1-butene content of 7 wt %, and the molecular weight of all the components is 95 kg/mol. The first four-component distribution (a; see Figure 11) is equimolar, and the branch content of each component is adjusted so that the average branch content is 7 wt %. As the distribution is broadened, the weight fractions and branch contents are readjusted to match this average. As the width of the distribution (or the difference in branching density between the most and least densely branched components) increases, the system becomes unstable at low pressures and multiple phases appear, as shown in Figure 11 on the example of distribution c. Finally, a comparison between the impact of MWD and SCBD on phase behavior is presented in Figure 12. When the SCBD only is considered, the copolymer is represented by four pseudocomponents having a molecular weight of 95 kg/mol, and their 1-butene content is the same as that presented in Table 3, as are the relative amounts of each of the pseudocomponents. When the MWD only is considered, the copolymers contain 7 wt % 1-butene, and their molecular weights and relative amounts are as presented in Table 3. The MWD alone has a stronger impact on the phase behavior than the SCBD alone. The pressure maximum is shifted to lower concentrations and higher pressures when the MWD alone is considered, whereas the critical point is shifted to higher polymer concentrations. When a bivariate distribution is considered, i.e. both MWD and SCBD are taken into account, the effect on phase behavior at low polymer concentrations becomes very pronounced. The difference between the pressure maximum of the system represented by a
Figure 12. Phase boundary of poly(ethylene-co-1-butene) in propane. (a) SCBD. The four copolymer components have a molecular weight of 95 kg/mol, their relative fractions are as in Table 3. (b) All copolymer components containing 7 wt % 1-butene. Their relative fractions and molecular weights are as in Table 3. (c) Four-component bivariate distribution, as in Table 3.
bivariate distribution and the monodisperse copolymer system is ca. 15 bar. Again, at high polymer concentrations, the effect of neither the polydispersity in molecular weight nor the polydispersity in short-chain branching is substantial, as shown in Figure 12. 4. Conclusions This project was undertaken with the goal of qualitatively assessing the impact of a short-chain branching distribution on the phase behavior of LLDPE solutions. Real LLDPE copolymers are characterized by a bivariate distribution, they are polydisperse in molecular weight and with respect to branch content. Whereas the impact of polydispersity in molecular weight on the phase behavior is now well understood and can be effectively taken into account in the framework of models for polymer solutions, polydispersity in branch content has received less attention except for the study of polymer incom-
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patibility in blends. The results obtained in the course of this study suggest that polydispersity in chain branching has an effect on the phase behavior, which can be significant under certain conditions. The cloud-point pressure marking the transition between a two-phase domain and a homogeneous region of the phase diagram increases upon broadening of the short-chain branching distribution. This finding is of importance for polymer technology; as for most polymerization reactions, the system should be present in a homogeneous phase. Moreover, in the case of very broad distributions, the system can separate into three phases because of the incompatibility of the polymers, leading to a drastic increase of the cloud-point pressure. The effect of polydispersity in chain branching is independent of the average branch content of the system and is more pronounced in a poor solvent for the polymer, such as ethylene. Good solvents such as n-hexane seem to provide some stabilization to systems containing incompatible polymers, thus mitigating the effect of short-chain branching distribution on phase behavior. Real branch content distributions are characterized by the presence of multiple components of intermediate branch density, which compatibilize the system and thus reduce the impact of the SCBD on phase behavior. The SCBD has primarily a strong effect at low polymer concentrations. Its effect at higher polymer concentrations is visible only when the twophase region becomes unstable and three phases appear. A representation of the polymer with a bivariate distribution would improve the predictions of thermodynamic models at low polymer concentrations and, thus, yield a more accurate description of the phase diagram of the solution. In the framework of the modeling concept for branched polymers, the degree of branching of a given chain is reflected in the value of its dispersion energy parameter. Results obtained from TREF analysis of LLDPE systems suggest that the degree of branching and the molecular weight of the polymers are strongly correlated, with the most densely branched chains having the lowest molecular weight. If a function correlating the degree of branching and the molecular weight could be proposed, this function could also be used to link the dispersion energy parameter of the PC-SAFT model to the molecular weight of a given chain, thus yielding a simplistic methodology for including information about the short-chain branching distribution into the thermodynamic model. A framework for including the dependence of the dispersion energy parameter on the molecular weight of the polymer was proposed earlier by Jog and Chapman10 and should be relevant for further developments in the area of including information about the short-chain branching distribution into the equation of state. Additional TREF analyses are necessary to get a better understanding of the correlation between molecular weight and branch density. Acknowledgment The authors are grateful to the Consortium of Complex Fluids for generously providing financial support for this work. The authors also thank Torben Laursen from VLXE for providing them with the VLXE phase equilibria software. Literature Cited (1) Dominik, A.; Chapman, W. G. Macromolecules 2005, 38, 10836– 10843. (2) Gross, J.; Sadowski, G. Ind. Eng. Chem. Res. 2001, 40, 1244–1260. (3) Gross, J.; Sadowski, G. Ind. Eng. Chem. Res. 2002, 41, 1084–1093. (4) Gross, J.; Spuhl, O.; Tumakaka, F.; Sadowski, G. Ind. Eng. Chem. Res. 2003, 42, 1266–1274.
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ReceiVed for reView June 24, 2008 ReVised manuscript receiVed January 26, 2009 Accepted February 13, 2009 IE800982Z
