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respectively, because {~(h ~ii2 + ~ii2 ~iil) vanishes. This, however, does not explain why we can obtain an exact result for {( ~n )2) by discarding terms involving products of fluctuations. To understand this point, let us look into the quantum term corresponding to the 1/2 in the last line of (4.114). Using (4.116), we write the number operator as (4.119) The last two terms in (4.119) correspond to the operator ~ndescribing the fluctuations in the photon number. The reader can easily verify that this operator can also be written as



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The sign and value of ER vary from one monomer to another Even for the same monomer, the value of ER may vary considerably depending on the initiator, coinitiator, and solvent [Biswas and Kabir, 1978, 1979; Biswas and Kamannarayana, 1976; Brown and Mathieson, 1957, 1958; Plesch, 1953] Table 5-9 shows that ER may vary from 355 to 586 kJ mol 1 for styrene polymerization The variations in ER are a consequence of the differences in Ei , Ep , and Et caused by the differences in the initiator, coinitiator, and the solvating power of the reaction medium It should be noted that, irrespective of sign, the values of ER are generally smaller than in radical polymerizations The rates of most cationic polymerizations do not change with temperature as much as those of radical polymerizations.

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~n = 2A~iil(8)

A negative value of ER does not mean a negative activation energy for some reactant(s) going to a transition state No simple, single step in any reaction pathway will have a negative activation energy A negative ER is simply a consequence of Et being larger than Ei Ep Increased temperature decreases Rp because termination becomes more favored relative to initiation and propagation What appears at rst glance to be even more anomolous are negative values of Ep observed in a few polymerizations [Cotrel et al, 1976; Villesange et al, 1977] The negative activation energy indicates that the measured Ep includes not only the.

+ [~iil(8) - i ~ii2(8)1 [~iil(8) + i~ii2(8)1.

TABLE 5-9 Activation Energy for Rate of Cationic Polymerization of Styrene Initiating System TiCl4 -H2 Oa TiCl4 -CCl3 CO2 Ha CCl3 CO2 H SnCl4 -H2 O CCl3 CO2 H CCl3 CO2 H

(4.120)

ER (kJ mol 1 ) 35.5 6.3 12.6 23.0 33.5 58.6

Data from Plesch [1953]; all other data from Brown and Mathieson [1957, 1958].

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The term [~iil(8) - i~ii2(8)1 [~iil(8) +i~ii2(8)1 corresponds to the product of the two noise ellipse projections, (1/2 - i/2)(1/2 + i/2), that gave rise to the 1/2 in the last line of (4.114). Caves's diagrams cannot reproduce any quantum correlations between these two noise operators. However, if the average amplitude A is large enough, this term will be negligible in comparison with the first term on the righthand side of (4.120). Then the calculation in terms of Caves's diagrams will be a good approximation to the exact result. The coherent-state example we have considered is the best of both worlds, because for a coherent state the last term in (4.120) does not contribute to the final result. The variance of n is given by

activation energy for propagation but also a term for some other process. Since increased temperature decreases the dielectric constant of a liquid, the negative Ep may be indicative of a change in the relative concentrations of different propagating species (free ion, ion pair) or the effect of solvent polarity of kp . The phenomenon may also be responsible for the changes in ER with temperature that are sometimes observed [Biswas and Kabir, 1978, 1979; Plesch, 1971]. The composite activation energy for the degree of polymerization EXn is almost always negative because Et or Etr is generally greater than Ep irrespective of the mode of termination or transfer [Biswas and Kabir 1978, 1979; DiMaina et al., 1977; Kennedy and Feinberg, 1978; Kennedy et al., 1977]. This means that the degree of polymerization decreases as the temperature increases. EXn in more negative when transfer reactions predominate over termination by combination or spontaneous termination since the transfer reactions have larger activation energies. As the polymerization temperature is increased, the principal mode of chain breaking shifts from termination to transfer [Biswas and Kamannarayana, 1976; Kennedy and Squires, 1965]. Further, there may be a shift from one transfer mode to another as the temperature is changed. Figure 5-3 shows a plot of log EX n versus 1=T for the polymerization of isobutylene by AlCl3 in methylene chloride solution. There is a change in the

((~n)2)

Fig. 5-3 Temperature dependence of X n for the aluminum chloride polymerization of isobutylene. After Kennedy and Squires [1965] (by permission of Butterworth, London and Elsevier, Oxford).

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