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Equations 6-45 and 6-47 and other variations have been useful for predicting the composition of a terpolymer from the reactivity ratios in the two component systems M1/M2, M1/ M3, and M2/M3 [Alfrey and Gold nger, 1944; Braun and Cei; 1987, Disselhoff, 1978; Ham, 1964; Hocking and Klimchuk, 1996; Janovic et al., 1983; Roland and Cheng, 1991; Tomescu and Simionescu, 1976; Valvassori and Sartori, 1967]. The two equations have been successfully extended to multicomponent copolymerizations of four or more monomers. Both equations yield essentially the same results, and there is good agreement with the experimental terpolymer compositions. Table 6-1 shows the calculated and experimental compositions in several systems. The terpolymerization and multicomponent composition equations are generally valid only when all the monomer reactivity ratios have nite values. When one or more of the



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5Frornthe factthat T can bewrittenasexp (iq/cn) d3 k r A(k) - h.c.l and the factthat A. andII. are canonical conjugate variables. a straightlorward generalization ofthe resufts presented in the preceding paragraph yields T II. Tt = II. - qT We also notice that T affects neither A nor r.

TABLE 6-1 Predicted and Experimental Compositions in Radical Terpolymerizationa Feed Composition Mole Percent 31.24 31.12 37.64 35.10 28.24 36.66 34.03 34.49 31.48 35.92 36.03 28.05 20.00 20.00 60.00 25.21 25.48 25.40 23.91 Terpolymer (mol%) Calculated from Found Eq. 6-45 Eq. 6-47 43.4 39.4 17.2 50.8 28.3 20.9 52.8 36.7 10.5 44.7 26.1 29.2 55.2 40.3 4.5 40.7 25.5 25.8 8.0 44.3 41.2 14.5 54.3 29.7 16.0 52.4 40.5 7.1 43.6 29.2 26.2 55.8 41.3 2.9 41.0 27.3 24.8 6.9 44.3 42.7 13.0 56.6 23.5 19.9 53.8 36.6 9.6 45.2 33.8 21.0 55.8 41.4 2.8 41.0 29.3 22.8 6.9





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The problem can be simplified even further if we assume that there are only two atomic levels involved. This is the famous two-level atom approximation. Within this approximation, the Hamiltonian (6.46) becomes

Monomer Styrene Methyl methacrylate Vinylidene chloride Methyl methacrylate Acrylonitrile Vinylidene chloride Styrene Acrylonitrile Vinylidene chloride Styrene Methyl methacrylate Acrylonitrile Styrene Acrylonitrile Vinyl chloride Styrene Methyl methacrylate Acrylonitrile Vinylidene chloride

~ [a!(k)a!(k) +~]

Data and calculations from Valvassori and Sartori [1967] and Walling and Briggs [1945].

a!(k)] ge(k),

monomers is incapable of homopolymerization the equations generally become indeterminate. Various modi ed expressions based on both the conventional and simpli ed equations have been derived for these and other special cases [Braun and Elsasser, 2000; Chien and Finkenaur, 1985; Quella, 1989].

(6.49)

6-3 RADICAL COPOLYMERIZATION The discussions thus far have been quite general without any speci cation as to whether copolymerization occurs by radical or ionic propagation. Consider now some of the speci c characteristics of radical copolymerization.

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where we have dropped the primes and expressed the displacement field in terms of the creation and annihilation operators defined by equation (5.117). The coupling constant ge (k) is obtained from equation (6.46) when the atomic dipole is expressed in terms of the two-level atom operators [i.e.,d = -iliJ-L (O"t - 0") x). We have shifted the zero of energy to the average of the energies of the two atomic levels as in 5. Now we want to obtain a nonperturbative expression for the expectation value of the atomic energy 0" zliwo/2 as a function of time for the case where the field is initially in the vacuum state. We can solve for the field operators as we have done before:

6-3a 6-3a-1

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Monomer reactivity ratios are generally but not always independent of the reaction medium in radical copolymerization. There is a real problem here in that the accuracy of r values is often insuf cient to allow one to reasonably conclude whether r1 or r2 varies with changes in reaction media. The more recent determinations of r values by high-resolution NMR are much more reliable than previous data for this purpose. It has been observed that the

ae(k, t)

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