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There are few studies of the effect of temperature on monomer reactivity ratios [Morton, 1983]. For styrene 1,3-butadiene copolymerization by s-butyllithium in n-hexane, there is negligible change in r values with temperature with r1 0:03, r2 13:3 at 0 C and r1 0:04, r2 11:8 at 50 C. There is, however, a signi cant effect of temperature for copolymerization in tetrahydrofuran with r1 11:0, r2 0:04 at 78 C and r1 4:00, r2 0:30 at 25 C. The difference between copolymerization in polar and nonpolar solvents is attributed to preferential complexing of propagating centers and counterion by 1,3-butadiene as described previously. The change in r values in polar solvent is attributed to the same phenomenon. The extent of solvation decreases with increasing temperature, and this results in 1,3-butadiene participating in the solvation process at the higher reaction temperature. 6-5 DEVIATIONS FROM TERMINAL COPOLYMERIZATION MODEL The derivation of the terminal (or rst-order Markov) copolymer composition equation (Eq. 6-12 or 6-15) rests on two important assumptions one of a kinetic nature and the other of a thermodynamic nature. The rst is that the reactivity of the propagating species is independent of the identity of the monomer unit, which precedes the terminal unit. The second is the irreversibility of the various propagation reactions. Deviations from the quantitative behavior predicted by the copolymer composition equation under certain reaction conditions have been ascribed to the failure of one or the other of these two assumptions or the presence of a comonomer complex which undergoes propagation.



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We will now show that the Hamiltonian of the medium remains diagonal (i.e .. that L~l bjbj = L~=l BkBk). This is more conveniently demonstrated by induction. Induction proofs are suited to hypotheses that depend on an arbitrary integer parameter, such as our number N of atoms in the material media. One first has to show that the hypothesis holds for N = 1. Then, if it can also be shown that when it holds for N - 1, it will hold for N, the hypothesis will have been proved for arbitrary N. In our case, it is trivial to see that it holds for N = I, as in that case there is only one "collective" annihilation operator, Bl. and according to (7.13), Bl = bl. The other collective operators given by (7.14) do not even come into play in the N = 1 case. So let us consider the next case, N = 2. For N = 2, (7.13) and (7.14) yield and Then





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The behavior of some comonomer systems indicates that the reactivity of the propagating species is affected by the next-to-last (penultimate) monomer unit This behavior, referred to as second-order Markov or penultimate behavior, manifests itself in a particular copolymerization by giving inconsistent values of the monomer reactivity ratios for different comonomer feed compositions More speci cally, plots of F1 , kp , and comonomer sequence versus feed composition do not t Eqs 6-15, 6-71, and 6-72, respectively, well over the entire range of feed compositions The ability to discern the failure of the terminal model has not always been apparent The failure of the terminal model is clear when the deviation is relatively large as it is in radical copolymerizations where the monomers contain highly bulky or polar substituents.

(7.16)

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For example, in the copolymerization of styrene and fumaronitrile, chains rich in fumaronitrile and having styrene as ther last added unit show greatly decreased reactivity with fumaronitrile due to steric and polar repulsions between the penultimate fumaronitrile unit in the propagating chain and the incoming fumaronitrile monomer [Fordyce and Ham, 1951] The deviation from the terminal model is most apparent for feeds rich in fumaronitrile In systems where the deviation from the terminal model is smaller, the experimental limitations in F1 and kp measurements may mask the deviations However, more recently, the improvements in F1 and kp measurements indicate that most comonomer systems deviate from the terminal model The original mathematical treatment of the penultimate effect was presented by Merz and coworkers [Barb, 1953; Ham, 1964; Merz et al, 1946].

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Fukuda and coworkers developed a more extensive treatment, which distinguished between two penultimate models of copolymerization behavior the explicit penultimate and implicit penultimate models [Coote and Davis, 1999, 2000; Davis, 2001; Fukuda et al, 1985, 1987, 1992, 2002; Ma et al, 2001] The explicit penultimate model for copolymerization involves the use of eight propagation reactions.

(7.17) (7.18)

k111

(7.19)

k112

Now suppose that our hypothesis holds for N = n, that is,

k221

(7.20)

k222

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