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Fig. 6-8 Sequence length distribution for an ideal copolymerization with r1 r2 1 and f1 f2 . After Vollmert (transl. Immergut) [1973] (by permission of Springer-Verlag New York, Inc., New York).

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Fig. 6-9 Sequence-length distribution for an ideal copolymerization with r1 5, r2 0:2 and f1 f2 . N 1 x is represented by ; N 2 x , by - - -. The plots for N 2 x are shown slightly to the left of the actual sequence length. After Vollmert (transl. Immergut) [1973] (by permission of Springer-Verlag New York, Inc., New York).

The Hamiltonian (6.1) acquires the following form 5 when we apply the unitary transformation T:





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p11 p12 p22 p21 0:50. Although the most plentiful sequence is M1 at 50%, there are considerable amounts of other sequences: 25%, 12.5%, 6.25%, 3.13%, 1.56%, and 0.78%, respectively, of dyad, triad, tetrad, pentad, hexad, and heptad sequences. The distribution of M2 sequences is exactly the same as for M1 sequences. For a feed composition other than equimolar, the distribution becomes narrower for the monomer present in lower amount and broader for the monomer in larger amount a general phenomenon observed for all sequence length distributions. It is clear that the copolymer, with an overall composition of F1 F2 0:50, has a microstructure that is very different from that of a perfectly alternating copolymer. The sequence length distribution for an ideal copolymerization with r1 5, r2 0:2 for an equimolar feed composition is shown in Fig. 6-9. This copolymerization has p11 * * p21 0:8333 and p12 p22 0:1667. Both M1 and M2 propagating centers have a 5 : 1 tendency to add M1 over M2, but M1 pentad sequences are not the most plentiful, although they are among the most plentiful. The most plentiful sequence is M1 at 16.7% with 14.0%, 11.6%, 9.7%, 8.1%, 6.7%, 5.6%, and 4.7%, respectively, of dyad, triad, tetrad, pentad, hexad, heptad, and octad M1 sequences. There are smaller amounts of longer sequences: 3.2% of 10-unit, 1.3% of 15-unit, and 0.4% of 20-unit M1 sequences. The sequence length distribution is much narrower for the less reactive M2 monomer. Single M1 units are by far the most plentiful (83.3%) with 13.9% dyads, 2.3% triads, and 0.39% tetrads. An alternating copolymerization with r1 r2 0:1 and f1 f2 has the sequence length distribution shown in Fig. 6-10 (p11 p22 0:0910; p12 p21 0:9090). The sequence

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Fig. 6-10 Sequence length distribution for an alternating copolymerization with r1 r2 0:1 and f1 f2 . After Vollmert (transl. Immergut) [1973] (by permission of Springer-Verlag New York, Inc., New York).

length distributions for both monomer units are identical. The single M1 and single M2 sequences are overwhelmingly the most plentiful the M1M2 makes up 90.9% of the copolymer structure. There are 8.3% and 0.75%, respectively, of dyad and triad sequences of both M1 and M2. Compare this distribution with that in Fig. 6-8 for the ideal copolymer having the identical overall composition. The large difference in the distributions for the two copolymers clearly indicates the difference between alternating and ideal behavior. High-resolution nuclear magnetic resonance spectroscopy, especially 13C NMR, is a powerful tool for analysis of copolymer microstructure [Bailey and Henrichs, 1978; Bovey, 1972; Cheng, 1995, 1997a; Randall, 1977, 1989; Randall and Ruff, 1988]. The predicted sequence length distributions have been veri ed in a number of comonomer systems. Copolymer microstructure also gives an alternate method for evaluation of monomer reactivity ratios [Randall, 1977]. The method follows that described in Sec. 8-16 for stereochemical microstructure. For example, for the terminal model, the mathematical equations from Sec. 8-16a-2 apply except that Pmm , Pmr , Prm and Prr are replaced by p11, p12 , p21 , and p22 . 6-2g-2 Copolymer Compositions of Different Molecules

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A second uncertainty concerning copolymer composition is the distribution of composition from one copolymer molecule to another in a sample produced at any given degree of conversion [Cheng, 1997b; Galvan and Tirrell, 1986; Tacx et al., 1988]. Stockmayer [1945] indicated that the distribution of copolymer composition due to statistical uctuations generally follows a very sharp Gaussian curve. Although the distribution is wider for

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