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average. Whatever comes out of an average will never have any trace of this order lower than dt because averaged paths have no kinks. This means that as long as we do it consistently throughout the calculations, we are free to approximate X(tk-l) by any arbitrary interpolation of X (t) and X (t + dt), such as (8.46) where Pb and Pe are positive and Pb + Pe = 1. The error of order lower than dt does not alter the statistical information we can extract from the stochastic process. Therefore,



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Certain combinations of nucleophiles and electrophiles undergo polymerization without the need for initiator [Culbertson, 2002; Kobayashi and Saegusa, 1985; Kobayashi and Uyama, 2002; Saegusa, 1977, 1979, 1981]. The polymerization, referred to as zwitterion polymerization, proceeds via zwitterion intermediates. For example, polymerization between 2-oxazoline and b-propiolactone involves nucleophilic attack of 2-oxazoline on b-propiolactone to form the dimer zwitterion LXXXIX, which reacts with itself to form the tetramer zwitterion LXXXX. The latter reacts with itself and with LXXXIX to form octamer and hexamer

T(dt, X(t), t)

7-125

= LT(dtjs, X(t), t).

zwitterions respectively. Propagation continues in this manner as various-sized zwitterion react with each other to form larger zwitterions. Among the nucleophilic monomers studied are 5,6-dihydro-4H-1,3-oxazines, cyclic phosphites and phosphonites, iminodioxolanes, and imines; electrophilic monomers include

(8.47)





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cyclic anhydrides, lactones and sultones as well as acyclic compounds such as acrylic acid and acrylamide. With only a very few exceptions, these polymerizations do not yield polymer molecular weights higher than 1000 3000. Low molecular weights are a consequence of the low concentration of the reacting species, the zwitterions, coupled with facile termination of the zwitterion and centers by reaction with the nucleophilic and electrophilic monomers [Odian and Gunatillake, 1984; Odian et al., 1990]. High molecular weights are obtained in zwitterion polymerizations only when the zwitterion concentrations are high relative to other materials present that can act as terminating agents. This has been successfully achieved in several systems. One of the systems (Eq. 7-91) has a low concentration of zwitterion intermediate, but there are no materials present (such as the electrophilic and nucleophilic monomers) to act as terminating agents. The other system (Eq. 7-93) contains only zwitterion since the zwitterion is isolated and is stable. Polymerization occurs when the zwitterion is heated. REFERENCES

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Taking s to be arbitrarily large, it follows from the central limit theorem [224] that T(dt, X(t), t) is a normal (Gaussian) random variable. Normal random variables are characterized completely by their means and their variances. Using (8.47), we find the following expressions for the mean and variance of T (dt, X (t), t): mean {T(dt, X(t), t)}

Afshar-Taromi, F, M Scheer, P Rempp, and E Franta, Makromol Chem, 179, 849 (1978) Aida, T and S Inoue, Acc Chem Res, 29, 39 (1996) Aida, T, K Sanuki, and S Inoue, Macromolecules, 18, 1049 (1985) Aida, T, M Ishikawa, and S Inoue, Macromolecules, 19, 8 (1986) Aida, T, K Kawaguchi, and S Inoue, Macromolecules, 23, 3887 (1990) Aida, T, Y Maekawa, S Asano, and S Inoue, Macromolecules, 21, 1195 (1988) Allcock, H R, J Macromol Sci Revs Macromol Chem, C4, 149 (1970); Science, 193, 1214 (1976); Makromol Chem Macromol Symp, 6, 101 (1986) Allcock, H R, Polyphosphazenes, pp 31 42 in Encyclopedia of Polymer Science and Engineering, 2nd ed, Vol 13, H F Mark, N M Bikales, C G Overberger, and G Menges, eds, Wiley-Interscience, New York, 1988 Allcock, H R.

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s mean {T(dtjs, X(t), t)} ,

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