Substitution of Eq. 3-31 into Eq. 3-25 yields in .NET

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Substitution of Eq. 3-31 into Eq. 3-25 yields
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Rp kp M   fkd I 1=2 kt 3-32
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Dependence of Polymerization Rate on Initiator
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Equation 3-32 describes the most common case of radical chain polymerization. It shows the polymerization rate to be dependent on the square root of the initiator concentration. This
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dependence has been abundantly con rmed for many different monomer initiator combinations over wide ranges of monomer and initiator concentrations [Eastmond, 1976a,b,c; Kamachi et al., 1978; Santee et al., 1964; Schulz and Blaschke, 1942; Vrancken and Smets, 1959]. Figure 3-1 shows typical data illustrating the square-root dependence on [I]. Deviations from this behavior are found under certain conditions. The order of dependence of Rp on [I] may be observed to be less than one-half at very high initiator concentrations. However, such an effect is not truly a deviation from Eq. 3-32. It may be due to a decrease in f with increasing initiator concentration (Sec. 3-4g-2).
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Fig. 3-1 Square root dependence of the polymerization rate Rp on the initiator concentration [I]. Methyl methacrylate, benzoyl peroxide, 50 C. After Schulz and Blaschke [1942] (by permission of Akademische Verlagsgesellschaft, Geest and Portig K.-G., Leipzig). *,* Vinyl benzoate, azobisisobutyronitrile, 60 C. After Santee et al. [1964] and Vrancken and Smets [1959] (by permission of Huthig and Wepf Verlag, Basel and Wiley-VCH, Weinheim).
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Alternately, the termination mode may change from the normal bimolecular termination between propagating radicals to primary termination, which involves propagating radicals reacting with primary radicals [Berger et al., 1977; David et al., 2001; Ito, 1980]:
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This occurs if primary radicals are produced at too high a concentration and/or in the presence of too low a monomer concentration to be completely and rapidly scavenged by monomer (by Eq. 3-14a). If termination occurs exclusively by primary termination, the polymerization rate is given by
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Rp kp ki M 2 ktp 3-33b
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Eq. 3-33b, derived by combining the rate expressions for Eqs. 3-14a, 3-15d, and 3-33a, shows that the polymerization rate becomes independent of the initiator concentration (but not ki ) and second-order in monomer concentration. Primary termination and the accompanying change in the order of dependence of Rp on [I] may also be found in the Trommsdorff polymerization region (Sec. 3-10). Situations also arise where the order of dependence of Rp on [I] will be greater than one-half. This behavior may be observed in the Trommsdorff region if the polymer radicals do not undergo termination or under certain conditions of chain transfer or inhibition (Sec. 3-7). Lower than one-half order dependence of Rp on Ri is also expected if one of the two primary radicals formed by initiator decomposition has low reactivity for initiation, but is still active in termination of propagating radicals. Modeling this situation indicates that the dependence of Rp on Ri becomes one-third order in the extreme of this situation where one of the primary radicals has no reactivity toward initiation, but still active for termination [Kaminsky et al., 2002]. 3-4a-4 Dependence of Polymerization Rate on Monomer
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The rate expression Eq. 3-32 requires a rst-order dependence of the polymerization rate on the monomer concentration and is observed for many polymerizations [Kamachi et al., 1978]. Figure 3-2 shows the rst-order relationship for the polymerization of methyl methacrylate [Sugimura and Minoura, 1966]. However, there are many polymerizations where Rp shows a higher than rst-order dependence on [M]. Thus the rate of polymerization depends on the 3-power of the monomer concentration in the polymerization of styrene in chloroben2 zene solution at 120 C initiated by t-butyl peresters [Misra and Mathiu, 1967]. The benzoyl peroxide initiated polymerization of styrene in toluene at 80 C shows an increasing order of dependence of Rp on [M] as [M] decreases [Horikx and Hermans, 1953]. The dependence is 1.18-order at M 1:8 and increases to 1.36-order at M 0:4. These effects may be caused by a dependence of the initiation rate on the monomer concentration. Equation 3-28 was derived on the assumption that Ri is independent of [M]. The initiation rate can be monomer-dependent in several ways. The initiator ef ciency f may vary directly with the monomer concentration
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f f 0 M 3-34
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which would lead (by substitution of Eq. 3-34 into Eqs. 3-31 and 3-32) to rst-order dependence of Ri on [M] and 3-order dependence of Rp on [M]. (This effect has been observed and 2
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