STATISTICAL MODELS OF PROPAGATION

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Catalyst (Initiator) Site Control

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Isoselective polymerization by traditional Ziegler Natta, metallocene, and other initiators is described by the catalyst (initiator) site control model. The initiator has enantiomeric propagation sites, R and S sites, at which propagation occurs through the re and si faces of monomer (R and S monomer, respectively, if the monomer is optically active). The model is described in terms of the single parameter s [Doi, 1979a,b, 1982; Farina, 1987; Inoue et al., 1984; Le Borgne et al., 1988; Shelden et al., 1965, 1969]. Parameter s is the probability of an R or re monomer unit adding at the R site; s is also the probability of an S or si monomer unit adding at the S site. The dyad fractions are given by

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m s2 1 s 2 r 2s 1 s 8-100

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The triad fractions are given by

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mm 1 3s 1 s mr 2s 1 s rr s 1 s 8-101

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The tetrad fractions are given by

mmm 2s4 4s3 6s2 4s 1 mmr mrr 4s4 8s3 6s2 2s mrm rmr rrr 2s 4s 2s

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4 3 2

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The pentad fractions are given by

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mmmm 5s4 10s3 10s2 5s 1 mrrm 3s4 6s3 4s2 s mmmr mmrr 6s4 12s3 8s2 2s rmmr rrrr s 2s s

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8-103

mmrm rmrr rmrm mrrr 2s4 4s3 2s2

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Syndioselective polymerization by a Cs metallocene such as Me2 C(Cp)(Flu)ZrCl2 proceeds by catalyst site control. A statistical model for syndioselective catalyst site control has been described in terms of the parameter r [Resconi et al., 2000]. Parameter r is the probability of a monomer with a given enantioface inserting at one site of the initiator; r is also the probability of the monomer with the opposite enantioface inserting at the other site of the initiator. The pentad fractions are given by

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mmmm r4 2r3 r2 rmmm mmrm rmrr rmrm 2r4 4r3 2r2 rmmr 3r4 6r3 4r4 r mmrr 6r4 12r3 8r2 2r rrrr 5r4 10r3 10r2 5r 1 rrrm 6r4 12r3 8r2 2r mrrm r4 2r3 r2 8-104

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STEREOCHEMISTRY OF POLYMERIZATION

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Application of Propagation Statistics

Experimentally obtained sequence distributions in a particular polymerization are used to determine whether the polymerization follows the Bernoullian, Markov, or catalyst site control models. The general approach is to t data on the dyad, triad, and higher sequence fractions to the appropriate equations for the different models. One needs to recognize the difference between tting and testing of data. Fitting of data involves calculating the value(s) of the appropriate probability term(s) from the sequence distributions. Testing involves determining that the sequence data are consistent or inconsistent with a particular model. All the models can be tted but not tested with dyad data. The Bernoullian and catalyst site control models require triad data as a minimum for testing; the rst-order Markov model requires tetrad data. The appropriate level of sequence data tests a model by showing the consistency or inconsistency of the Pm values (Bernoullian model) or Pmr and Prm values ( rst-order Markov), or s (or r) values (catalyst site control models). There are alternate criteria for testing the different models [Bovey, 1972; Chujo, 1967; Inoue et al., 1971, 1984]. The Bernoullian model requires

4 mm rr mr 2

8-105

The term on the left is extremely sensitive, and this criterion should be used only with suf ciently accurate triad data. This is especially important if the polymer is very highly isotactic or syndiotactic, that is, with very small value of either (rr) or (mm). The term 4(mm)(rr)/ (mr)2 is considerably larger than one for the Markov and catalyst site control models. The rst-order Markov model requires

4 mmm rmr mmr 2 1 and 4 mrm rrr mrr 2 1 8-106

while the catalyst site control models have the criteria

2 rr 4 1 1 and 1 1 mr mr 2 rr rr 8-107

Having established that a particular polymerization follows Bernoullian or rst-order Markov or catalyst site control behavior tells us about the mechanism by which polymer stereochemistry is determined. The Bernoullian model describes those polymerizations in which the chain end determines stereochemistry, due to interactions between either the last two units in the chain or the last unit in the chain and the entering monomer. This corresponds to the generally accepted mechanism for polymerizations proceeding in a noncoordinated manner to give mostly atactic polymer ionic polymerizations in polar solvents and free-radical polymerizations. Highly isoselective and syndioselective polymerizations follow the catalyst site control model as expected. Some syndioselective polymerizations follow Markov behavior, which is indicative of a more complex form of chain end control. More sophisticated experimental and theoretical analysis of isoselective polymerizations have been performed by using a two-site model for propagation [Inoue et al., 1984; Wu et al., 1990]. The polymer product is fractionated into the highly isotactic, insoluble and atactic,