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0 r1 0 r2
k211 k212
k222 r2 k221
k122 k121
and two radical reactivity ratios:
s1 s2 k211 k111 k122 k222
Each monomer is characterized by two monomer reactivity ratios. One monomer reactivity ratio represents the propagating species in which the penultimate and terminal monomer units are the same. The other represents the propagating species in which the penultimate and terminal units differ. The latter monomer reactivity ratios are signi ed by the prime notations. Each radical reactivity ratio is the ratio of the propagation rate constant for reaction of a radical in which the penultimate unit differs from the terminal unit compared to the rate constant where the penultimate and terminal units are the same. The monomer and radical reactivity ratios are used to calculate the adjusted parameters r1 , r2 , k11 , and k22 according to
0 r1 r1
f1 r1 f2 0 f1 r1 f2 f2 r2 f1 0 f2 r2 f1 f1 r1 f2 f1 r1 f2 =s1 f2 r2 f1 f2 r2 f1 =s2
0 r2 r2
k11 k111 k22 k222
The parameters r1 , r2 , k11 , and k22 are used in place of r1 , r2 , k11 , and k22 in the terminal model equations for F1 (Eq. 6-15) and kp (Eq. 6-71). The M1 centered triad monomer sequence distributions are given by
0 2 r1 r1 f1 0 2 0 2 r1 r1 f1 2r1 f1 f2 f2 0 r1 f1 f2 0 2 2r1 f1 f2 f2
112 211 212
0 2 r1 r1 f1
6-78b 6-78c
2 f2 0 2 0 2 r1 r1 f1 2r1 f1 f2 f2
The M2 centered triads (222), 221 122 , and (121) are derived from Eq. 6-78 by reversing the 1 and 2 subscripts. The implicit penultimate model was proposed for copolymerizations where the terminal model described the copolymer composition and monomer sequence distribution, but not the propagation rate and rate constant. There is no penultimate effect on the monomer reactivity ratios, which corresponds to
0 r1 r1 r1 0 r2 r2 r2
and the terminal model values of the monomer reactivity ratios are retained. There is a penultimate effect on the radical reactivity ratios and the parameters r1 , r2 , k11 , and k22 are used in place of r1 , r2 , k11 , and k22 in the terminal model equation for kp (Eq. 6-71), exactly as in the explicit penultimate model. The implicit penultimate model has a penultimate effect on reactivity (which determines propagation rate and rate constant), but not on selectivity (which determines copolymer composition and monomer sequence distribution). Penultimate effects have been observed for many comonomer pairs. Among these are the radical copolymerizations of styrene fumaronitrile, styrene diethyl fumarate, ethyl methacrylate styrene, methyl methacrylate 4-vinylpyridine, methyl acrylate 1,3-butadiene, methyl methacrylate methyl acrylate, styrene dimethyl itaconate, hexa uoroisobutylene vinyl acetate, 2,4-dicyano-1-butene isoprene, and other comonomer pairs [Barb, 1953; Brown and Fujimori, 1987; Buback et al., 2001; Burke et al., 1994a,b, 1995; Cowie et al., 1990; Davis et al., 1990; Fordyce and Ham, 1951; Fukuda et al., 2002; Guyot and Guillot, 1967; Hecht and Ojha, 1969; Hill et al., 1982, 1985; Ma et al., 2001; Motoc et al., 1978; Natansohn et al., 1978; Prementine and Tirrell, 1987; Rounsefell and Pittman, 1979; Van Der Meer et al., 1979; Wu et al., 1990; Yee et al., 2001; Zetterlund et al., 2002]. Although ionic copolymerizations have not been as extensively studied, penultimate effects have been found in some cases. Thus in the anionic polymerization of styrene 4-vinylpyridine, 4-vinylpyridine adds faster to chains ending in 4-vinylpyridine if the penultimate unit is styrene [Lee et al., 1963]. The reader is cautioned that literature references prior to 1985 1990 did not distinguish between the explicit and implicit penultimate models. The prior penultimate model did not correspond to either the explicit or implicit penultimate models. The pre-1985 1990 penultimate model contained only the four monomer reactivity ratios (Eq. 6-74) with no radical reactivity ratios. The precision and accuracy of the experimental data must be suf cient to allow one to discriminate between the terminal, explicit penultimate, and implicit penultimate models, [Burke et al., 1994a,b, 1995; Landry et al., 2000]. This has not always been the case, especially in the older literature, and the result has sometimes been contradictory reports. Penultimate effects are most easily detected in experiments carried out by including data at very low or very high f1 values. Figures 6-12 and 6-13 shows plots of copolymer composition and propagation rate constant, respectively, versus comonomer feed composition for styrene diethyl fumarate copolymerization at 40 C with AIBN [Ma et al., 2001]. The system follows well the implicit penultimate model. The copolymer composition data follow the terminal model within experimental error, which is less than 2% in this system. The propagation rate constant shows a penultimate effect, and the results conform well to the implicit penultimate model with s1 0:055, s2 0:32. 6-5b Depropagation during Copolymerization
In contrast to the kinetic approach, deviations from the terminal model have also been treated from a thermodynamic viewpoint [Kruger et al., 1987; Lowry, 1960; Palmer et al., 2000, 2001]. Altered copolymer compositions in certain copolymerizations are accounted for in this treatment in terms of the tendency of one of the monomers (M2) to depropagate. An essential difference between the kinetic and thermodynamic treatments is that the latter implies that the copolymer composition can vary with the concentrations of the monomers. If the concentration of monomer M2 falls below its equilibrium value [M]c at the particular reaction temperature, terminal M2 units will be prone to depropagate. The result would be a