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X w in Eq. 6-92 is essentially the weight-average degree of polymerization that would be observed in the polymerization of monomer A in the absence of diene BB. Equation 6-92 predicts that extensive crosslinking occurs during this type of copolymerization. Thus gelation is observed at 12.5% reaction in the methyl methacrylate ethylene glycol dimethacrylate system containing 0.05 mol% EGDM [Walling, 1945; Yoshimura et al., 1978]. The use of Eq. 6-92 for the styrene p-divinylbenzene (DVB) system is shown in Table 6-13. The equation holds best for systems containing low concentrations of the diene monomer; its utility decreases as the concentration of diene increases. With increasing diene concentration, Eq. 6-92 predicts gel points at conversions that are increasingly lower than those found experimentally. This general behavior has been attributed to the wastage of the diene monomer due to intramolecular cyclization (Sec. 6-6b). Also, there is an indication that the reactivity of the second double bond in BB is decreased on reaction of the rst double bond as a consequence of its presence in a polymer chain [Hild and Okasha, 1985]. An alternate expression for pc is given by



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Introducing the frequency-independent dielectric permittivity er in terms of the change in the cavity resonance frequency due to the presence of the medium

pc 1 q af f 2 q x=2 6-93

(7.56)





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where q is the ratio of the propagation rate to the sum of the propagation rate and the rates of all termination and transfer reactions, x is given by the ratio of the rate of termination by coupling to the sum of all termination and transfer reaction, f is the functionality of the diene and is equal to 4 (i.e., each double bond is bifunctional), and a is the fraction of all functional groups in the reaction mixture belonging to the diene [Macosko and Miller, 1976; Williams and Vallo, 1988]. A second case is the copolymerization of A and BB in which the reactivities of the two groups are not equal but are, instead, r1 and r2 , respectively. In this case the critical extent of

(7.57)

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pc r1 A 2 2 A B r2 B 2 2 X w B A B r2 B A 2 6-94

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So (7.55) can be rewritten as (7.58) This is in agreement with what Glauber and Lewenstein obtained by directly quantizing the macroscopic Maxwell equations for the case of a nondispersive and nonabsorptive dielectric medium [237]. So there is no extra noise in the displacement field coming from the atoms of the medium in this case. Rosewarne, however, claimed that there is such noise in the electric field [528]. He calculated the variance of the electric field in a scalar version of the Hopfield model and found that even in the case of weak coupling (G' ,W wo), there is a noise contribution from the atoms on the order of G,2 jwow. So it is even larger than the 4G,2 jw~ contribution of the medium to the dielectric permittivity. This extra noise is not accounted for when the medium is described only in terms of a dielectric permittivity. Then how can we use the idea of dielectric permittivity in quantum electrodynamics without being inconsistent Let us examine the electric field in more detail. In terms of the displacement field, the electric field is given by

6-95

(7.59)

for A ) B . When the double bonds of the diene are more reactive than that of the other monomer r2 > r1 , crosslinking occurs in the early stages of the copolymerization. Crosslinking is delayed until the later stages if r1 > r2 . For the system where r1 > r2 and r1 > 1 crosslinking is not as extensive at a given extent of reaction as that taking place in copolymerizations of the type governed by Eq. 6-92, where r1 r2 . The third case is the copolymerization of a monomer A with the diene BC where groups A and B have equal reactivity, but group C has a much lower reactivity. An example of such a case would be methyl methacrylate-allyl methacrylate, where A and B are the two methacrylate groups and C is the allyl group. If r is the reactivity ratio between the C and B groups, then the following hold for the radicals derived from the A and B groups:

r kAC kAC kBC kBC kAA kAB kBA kBB 6-96

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