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using (9.38). However, we must not forget that the perfect intracavity and free-space modes do not describe correctly, beyond first order in E, the field configuration on the boundary formed by the semitransparent mirror. Because of this single point, the perfect mode projections that define an, a~, b(k), and bt(k) will miss some of the global a(k) and at(k). In other words, the an, a~, b(k), and bt(k) are not enough to cover the entire continuum of Fock spaces spanned by the global operators a(k) and at (k). Something is missing. Thus, (9.39) does not hold in general, only up to first order in E. As the breakdown of (9.39) beyond first order in E is a key point in our firstprinciples derivation of the Gardiner-Collett Hamiltonian (9.1), let us examine it a bit more closely. Let us call

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Again, the situation is dif cult to predict with some comonomer systems showing altered r values for different initiators and others showing no effects Thus the isobutylene p-chlorostyrene system (Table 6 10) shows different r1 and r2 for AlBr3 and SnCl4 The interdependence of the effects of solvent and counterion are shown in Table 6 11 for the copolymerization of styrene and p-methylstyrene The initiators are listed in order of their strength as measured by their effectiveness in homopolymerization studies Antimony pentachloride is the strongest initiator and iodine the weakest The order is that based on the relative concentrations of different types of propagating centers The data in Table 6 11 show the copolymer composition to be insensitive to the initiator for solvents of high polarity (1,2-dichloroethane and nitrobenzene) and also insensitive to solvent polarity for any initiator except the strongest (SbCl5).

an(k) ==

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The styrene content of the copolymer decreases with increasing solvent polarity when SbCl5 is the initiator The styrene content also decreases with decreasing initiator strength for the low-polarity solvent.

'E {a nl(k)an +an2(k)a~}

TABLE 6-11 Effects of Solvent and Counterion on Copolymer Composition in Styrene p-Methylstyrene Cationic Copolymerizationa % Styrene in Copolymerb Toluene 1,2-Dichloroethane Nitrobenzene (E 2.4) (E 9.7) (E 36) 46 34 28 25 34 27 27 17 28 28 27 30

[aD(k), ab(k')]

(toluene). These results can be interpreted in terms of the effect of solvent and counterion on the identity of the propagating centers and on the extent of preferential solvation of propagating centers by one of the monomers. In the styrene p-methylstyrene system pmethylstyrene is both the more polar and the more reactive of the two monomers. In the poor solvent (toluene) the monomers compete, against the solvent, with each other to solvate the propagating centers. The more polar p-methylstyrene preferentially solvates the propagating centers and is preferentially incorporated into the copolymer. The selectivity increases in proceeding from SbCl5 to AlCl3 to the other initiators, which corresponds to increases in the amount of ion pairs relative to free ions. For the better solvents the counterion does not appreciably in uence the reaction, since the monomers cannot compete with the solvent. In the SbCl5 initiated copolymerization increasing the solvent power of the reaction medium also decreases the ability of the monomers to compete with the solvent to complex with propagating centers. The copolymer composition is then determined primarily by the chemical reactivities of the monomers. 6-4a-3 Effect of Temperature

(9.45)

Temperature has a greater in uence on monomer reactivity ratios in cationic copolymerization than in radical copolymerization because of the greater spread of propagation activation energies for the ionic process. The ratio of any two rate constants is expected to tend toward unity with increasing temperature since the smaller rate constant (larger activation energy) will increase faster with increasing temperature than the larger rate constant (smaller activation energy). However, there is no general trend of r values tending toward unity (i.e., less selective reaction) in cationic copolymerization with increasing temperature as there is in radical copolymerization. Some r values increase with temperature and others decrease. Various combinations of effects have been observed for different comonomer pairs [Kennedy and Marechal, 1983]. There are comonomer systems where both r1 and r2 tend toward unity as expected, but there are also many systems where an r value decreases below or increases above unity with increasing temperature. This unexpected behavior is probably the result of changes in the identities and relative amounts of different propagating species (free ion, ion pair, covalent) either directly as a result of a change in temperature, or indirectly by the effect of temperature on solvent polarity.

(continued)

the discrete part. Then if (9.39) holds,

6-4b 6-4b-1

[a(k), at(k')]

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