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Fig. 6-14 Effect of depropagation on copolymer comosition in the anionic copolymerization of vinylmesitylene (M1) a-methylstyrene (M2) at 0 C for f2 constant at 0.91. The dashed-line plots are the calculated curves for Lowry s cases I and II (with r1 0:20 and r2 0:72); the experimental data follow the solid-line curve. After Ivin and Spensley [1967] (by permission of Marcel Dekker, New York).
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propagation steps, the four propagation steps of the individual monomers (Eqs. 6-2 through 6-5), the four propagation steps of the comonomer complex
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* M1 + M2M1
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and the equilibrium between uncomplexed and complexed monomers
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r1 r1C k11 k12 k 112 k121 k112 k11 r2 r2C k22 k21 k 221 k212 k221 k22
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where M2 M1 and M1 M2 represent the complex adding to a propagating center at the M2 and M1 ends, respectively.
The copolymer composition and propagation rate constant are given by
   F1 f1 A2 B1 r1 f1 A1 C2 f2    F2 f2 A1 B2 r2 f2 A2 C1 f1     A2 B1 r1 f1 2 A1 B2 r2 f2 2 A1 C2 A2 C1 f1 f2   A2 r1 f1 =k11 A1 r2 f2 =k22
 A1 1 r1 s1C Q f1   1  B1 1 s1C 1 Q f2 r1C   1  C1 1 r1 s1C 1 Q f1 r1c  A2 1 r2 s2C Q f2
  1  B2 1 s2c 1 Q f1 r2C   1  C2 1 r2 s2C 1 Q f2 r2C
 2Q f1 f Q f2 f1 1 2 4Q f1 g1=2 Q f1 f2 1
Q K M 1 M2 f1
M 1 M1 M2 M  1 M1 M2
M 2 M1 M2 M  2 M1 M2
Concentrations and mole fractions with superscript  refer to uncomplexed monomer. Concentrations and mole fractions without superscript  refer to the comonomer feed, speci cally, the sum of complexed and uncomplexed monomer. The complex participation model, like the depropagation model, predicts a variation of the copolymer composition with temperature and monomer concentration. The effect of temperature comes from the change in K, resulting in a decrease in the concentration of the comonomer complex with increasing temperature. Increasing monomer concentration at a constant f1 increases the comonomer complex concentration. The complex participation model has been tested in the radical copolymerizations of 1,1diphenylethylene methyl acrylate, styrene b-cyanoacrolein, vinyl acetate hexa uoroacetone, N-vinylcarbazole diethyl fumarate, N-vinylcarbazole fumaronitrile, maleic anhydride vinyl acetate, styrene maleic anhydride [Burke et al., 1994a,b, 1995; Cais et al., 1979; Coote and Davis, 2002; Coote et al., 1998; Dodgson and Ebdon, 1977; Fujimori and Craven, 1986; Georgiev and Zubov, 1978; Litt, 1971; Litt and Seiner, 1971; Yoshimura et al., 1978]. A variation of the complex participation model, referred to as the monomer complex dissociation model, involves disruption of the complex during reaction with a propagating chain end [Hill et al., 1983; Karad and Schneider, 1978]. Reaction of the propagating center with
the complex results in the addition of only one of the monomers with liberation of the unreacted monomer. The overall result is that the complex alters monomer reactivities. 6-5d Discrimination between Models
The ability to determine which copolymerization model best describes the behavior of a particular comonomer pair depends on the quality of the experimental data. There are many reports in the literature where different workers conclude that a different model describes the same comonomer pair. This occurs when the accuracy and precision of the composition data are insuf cient to easily discriminate between the different models or composition data are not obtained over a wide range of experimental conditions (feed composition, monomer concentration, temperature). There are comonomer pairs where the behavior is not suf ciently extreme in terms of depropagation or complex participation or penultimate effect such that even with the best composition data it may not be possible to conclude that only one model ts the composition data [Hill et al., 1985; Moad et al., 1989]. The sequence distributions expected for the different models have been described [Hill et al., 1982, 1983; Howell et al., 1970; Tirrell, 1986] (Sec. 6-5a). Sequence distributions obtained by 13C NMR are sometimes more useful than composition data for discriminating between different copolymerization models. For example, while composition data for the radical copolymerization of styrene acrylonitrile are consistent with either the penultimate or complex participation model, sequence distributions show the penultimate model to give the best t. The termination rate constants and molecular weights for the different copolymerization models have also been studied for purposes of discriminating between different copolymerization models [Buback and Kowollik, 1999; Landry et al., 1999]. 6-6 COPOLYMERIZATIONS INVOLVING DIENES 6-6a Crosslinking
Diene monomers are often used in copolymerizations to obtain a crosslinked structure in the nal product. The reaction is generally analogous to step polymerizations involving tri- and tetrafunctional reactants (Sec. 2-10). Crosslinking occurs early or late in the copolymerization depending on the relative reactivities of the two double bonds of the diene. The extent of crosslinking depends on the latter and on the amount of diene relative to the other monomer. There is an extensive literature on the mathematical treatment of the crosslinking process [Dotson et al., 1988; Enright and Zhu, 2000; Flory, 1947, 1953; Macosko and Miller, 1976; Matsumoto et al., 2000; Scranton and Peppas, 1990; Shultz, 1966; Szuromi et al., 2000; Tobita and Hamielec, 1989; Williams and Vallo, 1988]. Several different cases can be distinguished depending on the type of diene. In most instances it is assumed that the diene is present at low concentrations. The rst case is the copolymerization of monomer A with diene BB where all the double bonds (i.e., the A double bond and both B double bonds) have the same reactivity. Methyl methacrylate ethylene glycol dimethacrylate (EGDM), vinyl acetate divinyl adipate (DVA), and styrene p- or m-divinylbenzene (DVB) are examples of this type of copolymerization system [Landin and Macosko, 1988; Li et al., 1989; Storey, 1965; Ulbrich et al., 1977]. Since r1 r2 , F1 f1 and the extent of reaction p of A double bonds equals that of B double bonds. There are p[A] reacted A double bonds, p[B] reacted B double bonds, and p2 [BB] reacted BB monomer units. [A] and [B] are the concentrations of A and B double bonds,