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Increased reactivity toward propagation by complexation of the propagating radical and/or monomer with the Lewis acid appears to be the predominant effect in other systems Simultaneous effects on both initiation and propagation (and possibly also termination) are indicated in many systems, since the polymer molecular weight often shows a maximum and a minimum with increasing concentration of Lewis acid Some reports of complexation between a monomer or propagating radical with solvent leading to changes in reactivity have also been reported [Ghosh and Mukhopadhyay, 1980; Kamachi et al, 1979] Some effects of solvent on reactivity have been observed in copolymerization (Sec 6-3a-1)..



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J~rS{wt(in:tw)+/12q t(in:tq )},

The molecular weight distribution in radical chain polymerizations is more complex than those in step polymerization. Radical chain polymerization involves several possible modes

(3.81)





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by which propagation is terminated disproportionation, coupling, and various transfer reactions. The situation is further complicated, since the molecular weight of the polymer produced at any instant varies with the overall percent conversion because of changes in the monomer and catalyst concentrations and the propagation and termination rate constants. Molecular weight distributions can be easily calculated for polymerizations restricted to low conversions where all the kinetic parameters ([M], [I], kd , kp , kp ) are approximately constant [Clay and Gilbert, 1995; Flory, 1953; Macosko and Miller, 1976; Peebles, 1971; Tobita, 1995; Tompa, 1976; Vollmert, 1973]. Under these conditions, the polymer molecular weight does not change with conversion. Consider the situation where one polymer molecule is produced from each kinetic chain. This is the case for termination by disproportionation or chain transfer or a combination of the two, but without combination. The molecular weight distributions are derived in this case in exactly the same manner as for linear step polymerization (Sec. 2-7). Equations 2-86, 2-88, 2-89, 2-27, 2-96, and 2-97 describe the number-fraction, number, and weight-fraction

2 x 1

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which has the proper form that we promised above. What about the momentum The kth component of the field's three-momentum 'Pk is given by (3.82) Now using

2-86 2-88 2-89 2-27 2-96 2-97

(3.83)

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2 x 1

S {( -ik):FtF}

distributions, the number- and weight-average degrees of polymerization, and the breadth of the distribution, respectively. One difference in the use of these equations for radical chain polymerizations compared to step polymerizations is the rede nition of p as the probability that a propagating radical will continue to propagate instead of terminating. The value of p is given as the rate of propagation divided by the sum of the rates of all reactions that a propagating radical may undergo

Moreover, using S{ZlZ:2}

p Rp Rp Rt Rtr 3-185

S {P ikF} .

where Rp , Rt , and Rtr are the rates of propagation, termination by disproportionation, and chain transfer, respectively. There is a very important second difference in the use of the equations above for radical chain polymerization compared to step polymerizations. The equations apply to the whole reaction mixture for a step polymerization but only to the polymer fraction of the reaction mixture for a radical chain polymerization. A consideration of the preceding equations indicates that high polymer (i.e., large values of X n and X w ) will be produced only if p is close to unity. This is certainly what one expects from the previous discussions in Sec. 3-5. The distributions described by Eqs. 2-86, 2-88, and 2-89 have been shown in Figs. 2-9 and 2-10. The breadth of the size distribution X w =X n [also referred to as the polydispersity index (PDI)] has a limiting value of two as p approaches unity. For termination by coupling (Rtr 0 and Rt is the rate of coupling) where a polymer arises from the combination of two kinetic chains, the size distribution is narrower. The

(3.84)

situation is analogous to that for step polymerizations with branching (Sec. 2-9). Polymer molecules of sizes much different from the average are less likely, since the probability for coupling between same-sized propagating radicals is the same as that between differentsized propagating radicals. The size distributions can be derived in a manner analogous to that used in Sec. 2-7a. Consider the probability N y z of formation of an x-sized polymer molecule by the coupling of y- and z-sized propagating radicals. N y z is the product of the probabilities, N y and N z , of forming the y- and z-sized propagating radicals. N y and N z are given by

(3.85)

N y 1 p p y 1 N z 1 p pz 1 3-186 3-187

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