Monte Carlo Simulation of Long-Chain Branched Polyolefins Made

Ind. Eng. Chem. Res. 2005, 44, 2461-2468

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Monte Carlo Simulation of Long-Chain Branched Polyolefins Made with Dual Catalysts: A Classification of Chain Structures in Topological Branching Families Leonardo C. Simon* and Joa˜ o B. P. Soares* Department of Chemical Engineering, University of Waterloo, 200 University Avenue West, Waterloo, Ontario, Canada N2L 3G1

The production of polyolefins by use of two single-site catalysts, where one of the catalysts forms linear chains only (linear catalyst) and the other makes linear and long-chain branched chains (LCB catalyst), is an attractive route to control the molecular architecture of branched polyolefins. For modeling purposes, these chains can be conveniently divided into families containing different numbers of long-chain branches per chain. However, when the number of long-chain branches per chain is higher than three, there is more than one possible chain topology for each family; that is, highly branched families have several family members. In this paper, we developed a Monte Carlo model to describe how the fraction and molecular weight distribution of these family members vary as a function of the ratio of linear to LCB catalyst used during polymerization. Introduction Controlling the formation of long-chain branches (LCB) in polyolefins has significant practical interest because of the marked impact of small amounts of LCBs on shear thinning and melt strength of these polymers.1-3 Polyethylene with LCBs can be produced by mono- and dicyclopentadienylmetallocenes in solution, slurry, and gas-phase reactors.4-9 Simulation studies have shown that these catalysts produce polymer chains with a welldefined branched structure that can be described with several mathematical modeling approaches.10-16 The use of dual catalyst systems, where one catalyst produces chains containing LCBs (LCB catalyst) while the other produces only linear chains (linear catalyst), has been proposed as a way to control the LCB distribution of these polymers and to maximize the total number of LCBs per chain.17-26 A recent review on the state-ofart modeling of LCB formation with metallocene catalysts has been published by Soares.27 Simulations and experimental studies of polyethylene produced by a dual catalyst, LCB and linear, have shown that the branching frequency for the polymer is maximized at a certain ratio of the two catalysts.13,18,19,21 In one recent study, Monte Carlo simulation was used to classify the chains made with dual catalysts into linear, comb-branched, and hyperbranched.20 Combbranched chains have only linear branches, while hyperbranched chains have branches that also contain long-chain branches. In that particular investigation, the ratio between hyperbranched and comb-branched chains was maximized when 80 wt % of the polymer chains were produced by the LCB catalyst. Subsequently, Haag et al.22 extended this modeling approach to describe the molecular structure of graft-block thermoplastic elastomers. The model proposed in this paper describes the topology of branched polyolefin chains made with dual * To whom correspondence should be addressed. Tel.: (519) 888-4567. Fax: (519) 746-4979. E-mail: jsoares@ cape.uwaterloo.ca (J.B.P.S.); [email protected] (L.C.S.).

LCB/linear catalyst systems in more detail than the previously published models. Similarly to previous models, a Monte Carlo model was developed to classify the polymer chains into different families: linear, 1 LCB/ chain, 2 LCBs/chain, etc. However, in addition to the existing models, the members of each family were classified according to their branching topologies. It will be shown that families with more than 3 LCBs/chain have two or more members with distinct topologies. Full topology distributions are described for families with up to 9 LCBs/chain. For families with higher LCBs per chain, the topology is described in terms of inner segments and free arms. Additionally, the distribution of priorities and seniorities, as proposed by Read and McLeish,2 is also predicted by our Monte Carlo model. Mechanism for LCB Formation. The mechanism for LCB formation for ethylene polymerization with metallocene catalysts is macromonomer incorporation. In other words, LCBs are formed because dead chains containing terminal vinyl groups (macromonomers) are copolymerized with ethylene during the polymerization. In the case of polyethylene, macromonomers are produced in situ when a living polymer chain is terminated by β-hydride elimination or transfer to ethylene reactions. The mechanism of LCB formation has been discussed extensively in the literature.27 Figure 1 is a graphical representation of the mechanism of LCB formation with metallocene catalysts. Model Description. The Monte Carlo algorithm for the generation of linear and branched chains used in this paper is similar to the one described by Simon and Soares.20 Each polymer chain is generated individually from a set of propagation and termination probabilities. For the linear catalyst, a chain can either propagate by monomer addition or terminate, which naturally generates chains that follow Flory’s most probable chainlength distribution. For the LCB catalyst, the propagation step is subdivided into propagation of monomer or macromonomer, the last one leading to LCB formation. Only a few model parameters are needed in the simulation: nc is the total number of polymer chains used in the simulation, FLCB is the molar fraction of

10.1021/ie049615l CCC: $30.25 © 2005 American Chemical Society Published on Web 08/04/2004

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Ind. Eng. Chem. Res., Vol. 44, No. 8, 2005

Figure 1. Mechanism of macromonomer incorporation leading to long-chain branch formation during ethylene polymerization with a coordination catalyst.

Figure 2. Monte Carlo flowchart used to simulate polymerization with the LCB catalyst (ran is a random number generated between 0 and 1; a different random number is generated for each event).

chains made with the LCB catalyst, PpLin is the probability of propagation of the linear catalyst, PpLCB is the probability of propagation of the LCB catalyst, PLCB is the probability of LCB formation by the LCB catalyst, and PmLin is the probability of incorporation of macromonomers made by the linear catalyst. The probabilities of termination for both catalysts are simply PtLin ) 1 - PpLin and PtLCB ) 1 - PpLCB, and hence these parameters do not need to be defined independently from PpLin and PpLCB. Similarly, the probability of incorporation of macromonomers made by the LCB catalyst is simply PmLCB ) 1 - PmLin and is not defined independently of PmLin. Figure 2 shows the flowchart used to generate the polymer chains made with the LCB catalyst in our Monte Carlo simulation. As mentioned

above, the chain-length distribution of polymers made with the linear catalyst is simply given by Flory’s most probable distribution. These probabilities depend on reactor conditions and polymerization kinetic parameters. A detailed description of each model probability and their correlation to polymerization kinetics was published previously.20 Long-Chain Branch Topology: Branching Families and Family Members. Table 1 describes the branching families and family members considered in our simulations up to 7 LCBs/chain. Topologies of chains with 8 and 9 LCBs were also developed by the authors but are omitted herein for brevity. They are available from the authors upon request. Chains in the same family have the same number of LCBs per chain. Families with more than 3 LCBs/chain have two or more members. The two simplest topologies are called Y (1 LCB/chain) and H (2 LCBs/chain). There is only one synthetic route for the production of the Y topology: the incorporation of one linear macromonomer onto a living polymer chain. The H topology may be produced by two pathways: the reaction of a growing chain with either two linear macromonomers or one macromonomer already containing 1 LCB, i.e., a Y macromonomer. Polymers with higher numbers of LCBs per chain can be made by increasingly complex synthetic routes to generate the several family members depicted in Table 1. These fine differences in topology are important if one is concerned with the rheological behavior of these macromolecules or simply for a better understanding of the topology of these complex materials. Another important structural aspect of these polymers is the number and length of their free arms (FA) and internal segments (IS), as illustrated in Figure 3. Wood-Adams and co-workers28 have shown how these structural characteristics could be used to describe the rheological responses of branched polyolefins. Our Monte Carlo model is also capable to predict the distribution of free arms and internal segments as a function of the LCB/linear catalyst ratio used during polymerization. When the functionality of the macromonomers is one, that is, when there is only one terminal vinyl group per chain available for branch-formation reactions, the numbers of free arms, inner segments, and long-chain branches per chain are correlated by the equations:

FA ) LCB + 2

(1)

IS ) LCB - 1

(2)

Ind. Eng. Chem. Res., Vol. 44, No. 8, 2005 2463 Table 1. Classification of Chain Topology in Families According to Their Number of Long-Chain Branches or Free Arms

Table 2. Parameters for Monte Carlo Simulation of Ethylene Polymerization with a Mixture of LCB and Linear Catalysts20 FLCB

PLCBa

PmLin

FLCB

PLCBa

PmLin

1 0.95 0.90 0.80 0.75

0.000 200 0.000 267 0.000 333 0.000 466 0.000 529

0 0.114 0.213 0.379 0.442

0.65 0.50 0.35 0.20 0

0.000 654 0.000 842 0.000 434 0.000 300 0

0.568 0.710 0.819 0.907 1

a

Figure 3. Definition of free arms, internal segments, priorities, and seniorities.

FA ) IS + 3

(3)

The concepts of seniority and priority are also very useful to describe rheological and mechanical properties of polyolefins. Seniority and priority are calculated as follows: first, the seniority of a segment is computed by counting the number of segments to the furthest chain end on each side of the segment and then taking the smaller of the two values; second, the priority of a segment is obtained by counting the free arms attached to each side of the segment and taking the smaller of the two values. Free arms have seniority and priority equal to 1 and internal segments have seniority and priority equal to 2 or higher. Note that seniority and priority are not necessarily the same for a certain segment, especially on chains with high numbers of internal segments. Figure 3 shows an example of the priority and seniority distributions for two members of family g. To illustrate how seniorities and priorities are calculated, consider the segment labeled (3,4) in family g5: first, for seniority, count the number of segments, including

PpLCB ) PpLin ) 0.999 25.

the segment in analysis itself, to the furthest left (4) and right (3) chain ends and select the smaller value (3); second, for priority, count the number of chain ends to the left (5) and right (4) of the segment, and select the smaller value (4). The model builds each polymer chain individually, insertion by insertion step, following the catalytic mechanism adopted for olefin polymerization in the presence of terminal branching.20 A C++ program was designed to keep track of all details necessary to describe topology: number of free arms and internal segments, number of monomers in each segment, number of branching points, type of termination, and relative position of each segment in relation to the other segments in the chain, i.e., seniority and priority. For calculating seniority and priority for internal segments, the algorithm has to search upstream and downstream of the chain growth direction, comparing values of neighboring segments. Every time a macromonomer is incorporated into the growing polymer chain the seniority and priority must be recalculated to account for the change in topology. Results and Discussion Table 2 lists the model probabilities used in the simulations. Notice that the probabilities of propagation

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Figure 4. Number-average chain length and polydispersity index (PDI) of polyethylene made with mixtures of LCB catalyst and linear catalyst.

for the linear and LCB catalysts are the same. Consequently, in the absence of LCB formation, both catalysts would produce polymer with the same chain-length distribution and polydispersity of 2.0. However, because the LCB catalyst forms polymer with LCBs, the average chain length will be higher and the chain-length distribution will be broader for polymers made with the LCB catalyst than for polymers made with the linear catalyst. Figure 4 shows how the number-average chain length and polydispersity index (PDI) vary as a function of the fraction of LCB catalyst in the reactor. Curves O describe the number-average chain length and PDI of the polymer mixture made by both catalysts. Curves 0 show the number-average chain length and PDI of the polymer synthesized only by the LCB catalyst site in the catalyst mixture, obtained by subtracting the chains made by the linear catalyst. The number-average chain length and PDI for the polymer made by the linear catalysts are constant and are equal to 1336 and 2.0, respectively. For the whole polymer (i.e., curves O), the PDI of the polymer produced when only the LCB catalyst is present equals approximately 2.8 and increases as the fraction of LCB catalyst is reduced from 1.0 to approximately 0.5. The PDI decreases for higher fractions of linear catalyst to the theoretical value of 2.0 when no LCB catalyst is present in the reactor. The number-average chain length passes through a maximum when the fraction of LCB catalyst is approximately equal to 0.75. The curves for the polymer made exclusively by the LCB catalyst (i.e., curves 0) show similar trends but have higher number-average chain length and generally lower PDI than the equivalent values for the mixture. Note that the maximum in the PDI curve is observed at a higher system composition when the polymer chains produced by linear catalysts were subtracted. This behavior is expected since both the linear and LCB catalysts have the same probabilities of propagation and termination and, consequently, the polymer made by the LCB catalyst will always have higher number-average chain length because of the long-chain branching reactions. The PDI for the chains made with the LCB catalyst is lower than the PDI for the whole polymer for fractions of LCB catalyst higher than approximately 0.35, but it becomes higher for lower fractions. This is caused by the fact that, at lower LCB catalyst fractions, most of the polymer is made by the linear catalyst and therefore the PDI for the mixture is approximately 2.0.

Figure 5. Fraction of chains (calculated with respect to the overall polymer population) with different number of free arms (FA) for different fractions of LCB catalyst.

Figure 6. Fraction of different topology families as a function of catalyst mixture composition. The linear chains were excluded from the calculation of the fractions.

On the other hand, the PDI of the polymer made with the LCB catalyst is always higher than 2.0 because of the long-chain branching reactions. When more of the LCB catalyst is added to the reactor, the PDI of the binary polymer population exceeds the PDI of the polymer made with the LCB catalyst alone, as expected due to the differences in their chain lengths. Figure 5 describes how the fraction of chains containing a different number of free arms (FAs) depends on the fraction of LCB catalyst in the reactor when the entire chain population is considered. A maximum in LCB frequency is observed when the fraction of LCB catalyst equals 0.8 for this particular combination of model catalysts. Note that the fractions of all LCB families are maximized at this particular catalyst ratio. The inset in Figure 5 shows that the fraction of chains with 11 FAs is very small: for the case when the fraction of LCB catalyst is 0.5, the fraction of chains with 11 FAs is 0.0037. Chains with 12 FAs or more are lumped in a single curve since their fractions are very small. Figure 6 shows how the fractions of each topology family vary as a function of LCB catalyst fraction. Branched chains with 3 FAs (family a) are always dominant but reach a minimum value when the fraction of LCB catalyst is 0.5. When the fraction of LCB catalyst

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Figure 7. Chain length distributions for different topology families.

Figure 8. Distribution of the members of family d as a function of LCB catalyst fraction.

is small, most of the macromonomers are linear and, consequently, the fraction of branched chains containing 3 FAs is maximized. The fraction of chains with 4 FAs (family b) is rather stable, regardless of the catalyst composition. Interestingly, the fractions of chains with 5 FAs or more reach a maximum when the fraction of the LCB catalyst is 0.5, in detriment of chains with 3 FAs. Evidently, this result cannot be generalized to other catalyst combinations because it reflects the particular choice of polymerization probabilities used in Table 2. However, this example shows how the Monte Carlo model can be used to investigate catalyst combinations to produce polymers with particular branching topologies. The chain-length distribution of several topology families is shown in Figure 7 for topologies with up to 11 FAs (family i). As expected, the more highly branched families have higher chain-length averages and narrower chain-length distributions. These results are in agreement with previous publications by our and other research groups.10,27 More interestingly, Figure 8 shows some of the unique aspects of the Monte Carlo simulation proposed in this investigation: the fractions of the two configurations for chains with 6 FAs (family d) are illustrated as a function of the fraction of the LCB catalyst. Interest-

Figure 9. Chain length distributions for family members d1 and d2 for three fractions of LCB catalyst.

ingly, a decrease in the fraction of the LCB catalyst will favor the formation of family member d1, which has a comb structure. This was expected since, in this case, an increasing fraction of the macromonomers is made by the linear catalyst. Hence, Figure 8 is a very vivid representation of how one can affect the topology of these branched chains by use of two single-site catalysts. Figure 9 compares the chain-length distributions of family members d1 and d2 made with different fractions (0.38, 0.82, and 0.91) of the LCB catalyst. As expected, for a given fraction of LCB catalyst, the chain-length distributions for d1 and d2 are the same and do not depend on the branching structure but only on the number of branches per chain.10 Increasing the fraction of LCB catalyst from 0.38 to 0.91 shifts the chain-length distribution of chains d1 and d2 to higher chain lengths. The inset in Figure 9 shows that the difference between the distributions of family members d1 and d2 for the same LCB catalyst composition is just random noise due to the nature of Monte Carlo simulation. However, although the number of monomer units in each slice of the chain-length distribution of topologies d1 or d2 is the same, it is expected that in solution these two topologies would have different radii of gyration and therefore different exclusion volumes during gel-permeation chromatographic analysis. Since these highly branched chains are generally present in a rather small fraction (cf. Figure 6), it is likely that their impact on the shape of gel-permeation chromatograms will not be very large. Figure 10 shows similar results for family e containing 7 FAs. Once again, the addition of linear catalyst will favor the synthesis of family member e1, with a comb structure, at the expense of the family member e2, which is hyperbranched. Similarly, Figure 11 extends the same observations to family f (8 FAs). Interestingly, for this family the fractions of family members f2 and f3 are always the same. The fraction of the hyperbranched family member f4 is very small even in the presence of pure LCB catalyst and rapidly dwindles to zero with the introduction of the linear catalyst. Read and McLeish2 have shown how the rheological properties of branched chains could be understood in terms of priority and seniority distributions. According to Read and McLeish, the seniority distribution controls the linear rheological response and the priority distribution controls nonlinear properties such as the damping function in strong step shear and strain hardening in

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Figure 10. Distribution of the members of family e as a function of LCB catalyst fraction.

Figure 12. Priority distribution as a function of LCB catalyst fraction: (a, top) priorities 2-11; (b, bottom) priorities 11-20.

Figure 11. Distribution of the members of family f as a function of LCB catalyst fraction.

extension.2 Figure 12 shows how the distribution of priorities varies as a function of LCB catalyst fraction in the reactor for segments with priorities from 2 to 19 (the fraction of segments with priority equal to 20 or higher are lumped as p20). Segments of high priority are maximized when the fraction of LCB catalyst in the reactor falls in the range from 0.5 to 0.8. This is in agreement with the distribution of topologies shown in Figure 6, since when the population of chains with 3 FAs (family a) is minimized in relation to other topologies, the fraction of segments with priority equal to 2 or higher (internal segments) should increase. Figures 6 and 12 illustrate very well how the ratio of LCB to linear catalyst can be used to control long-chain branch topologies. Likewise, Figure 13 shows how the distribution of seniorities varies as a function of LCB catalyst fraction in the reactor for segments with seniorities from 2 to 19 (the fraction of segments with seniority equal to 20 or higher are lumped as s20). Upon comparing Figures 12b and 13b, it is clear that there is a LCB catalyst fraction around 0.5 that will maximize the

presence of segments with high priorities and seniorities for this particular modeling example. Finally, Figure 14 shows the distribution of segment lengths for segments with different seniorities and priorities for LCB catalyst fractions of 0.75 and 1.0. Both panels show the size distribution of segments with seniorities or priorities up to 7 (higher seniorities and priorities followed the same trends and were not plotted for clarity). The left side of the distributions is interrupted at chain length 50 because 50 monomer units was the smallest segment size computed in our simulations. A comparison of the panels in Figure 14 shows that the composition of the catalyst mixture will affect not only the seniority and priority distributions but also the distributions of their segment lengths. When the fraction of the LCB catalyst equals 0.75, the average length of the high priority and seniority segments is much higher than when only the LCB catalyst is used. Longer segments generally have lower seniority and priorities (1 and 2), as expected. The inset in Figure 14 a compares the segment length distribution of the normalized populations of segments with same seniorities (s1-s19) and priorities (p1-p19). The noise is produced by the small number of chains generated during Monte Carlo simulation for populations with high seniorities and priorities. Regardless of the priority

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Figure 14. Length distribution for segments with different priorities and seniorities: (a, top) LCB catalyst fraction ) 0.75; (b, bottom) LCB catalyst fraction ) 1.0. Figure 13. Seniority distribution as a function of LCB catalyst fraction: (a, top) seniorities 2-11; (b, bottom) seniorities 11-20.

illustrates very well how to use combined metallocenes to design the microstructure of polyolefins.

or seniority of the segment, they all have the same length distribution, as already observed by other researchers.2

Literature Cited

Conclusions The Monte Carlo model proposed in this paper is able to predict the detailed topology of branched polyolefin chains made with two single-site catalysts where one catalyst makes only linear chains (linear catalyst) and the other makes linear and branched chains (LCB catalyst). The polymer chains are classified into families with different number of long-chain branches per chain. For chains with more than 3 long-chain branches (5 free arms) per chain, two or more chain configurations are possible. These different configurations within each family are classified as separate family members. The model is also able to keep track of the chain-length distribution of each polymer family member, its number of free arms and inner segments, and the seniority and priority of its segments. It was also shown that by varying the ratio of LCB to linear catalyst it is possible to control not only the overall level of long-chain branching but also the relative proportion of the distinct members of a family, as well as their seniorities and priorities. The model

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Received for review May 10, 2004 Revised manuscript received June 24, 2004 Accepted June 25, 2004 IE049615L

Monte Carlo Simulation of Long-Chain Branched Polyolefins Made

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