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For polymerizations where termination occurs by a combination of coupling, disproportionation, and chain transfer, one can obtain the size distribution by a weighted combination of the two sets of distribution functions presented above. Thus the weight distribution can be obtained as
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where A is the fraction of polymer molecules formed by disproportionation and chaintransfer reactions [Smith et al., 1966].
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High-Conversion Polymerization
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For many practical radical chain polymerizations, the reaction is carried out to high or complete conversion. The size distributions for high-conversion polymerizations become much broader than those described above or those in step polymerizations. Both the monomer and initiator concentrations decrease as a polymerization proceeds. The polymer molecular weight depends on the ratio [M]/ I 1=2 according to Eq. 3-98. In the usual situation [I] decreases faster than [M] and the molecular weight of the polymer produced at any instant increases with the conversion. The overall molecular-weight distributions for high or complete conversion polymerizations are quite broad, with the X w =X n ratio being in the range 2-5. Broad molecular weight distributions are usually not desirable from the practical viewpoint, since most polymer properties show optimum values at speci c molecular weights. Commercial polymerization processes often involve the addition of multiple charges of initiator and/or monomer during the course of the reaction to minimize the molecular weight broadening. When autoacceleration occurs in polymerization there is even larger broadening of the 1=2 size distribution. The large increases in the kp =kt ratio (Table 3-17) that accompany the gel effect lead to large increases in the sizes of the polymers being produced as the polymerization proceeds. X w =X n values as high as 5 10 may be observed when the gel effect is present. Excessive molecular-weight broadening occurs when branched polymers are produced by chain transfer to polymer. Chain transfer to polymer can lead to X w =X n ratios as high as 20 50. This very extensive size broadening occurs because chain transfer to polymer increases as the polymer size increases. Thus branching leads to even more branching as a polymerization proceeds. One usually attempts to minimize the molecular weight broadening due to the get effect and chain transfer to polymer, but it is quite dif cult to do so successfully. Thus, for example, low temperatures minimize chain transfer to polymer but maximize the gel effect (Sec. 3-10a).
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The effect of pressure on polymerization, although not extensively studied, is important from the practical viewpoint since several monomers are polymerized at pressures above atmospheric. Pressure affects polymerization through changes in concentrations, rate constants, and equilibrium constants [Ogo, 1984; Weale, 1974; Zutty and Burkhart, 1962]. The commercial polymerizations of most gaseous monomers (e.g., vinyl chloride, vinylidene chloride, tetra uoroethylene, vinyl uoride) are carried out at very moderate pressures of about 5 10 MPa (1 MPa 145 psi), where the primary effect is one of increased
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concentration (often in a solvent) leading to increased rates. Signi cant changes in rate and equilibrium constants occur only at higher pressures such as the 100 300-MPa pressures used to produce the large-volume polymer low-density polyethylene (Sec. 3-13b-1).
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Effect on Rate Constants
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High pressure can have appreciable effects on polymerization rates and polymer molecular weights. Increased pressure usually results in increased polymerization rates and molecular weights. Figure 3-17 shows these effects for the radiation-initiated polymerization of styrene [Moore et al., 1977].
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The effect of pressure on Rp and X n , like that of temperature, manifests itself by changes in the three rate constants initiation, propagation, and termination. The quantitative effect of P (at constant temperature) on a rate constant is given by
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where V z is the volume of activation, that is, the change in volume (cm3 mol 1 ) in going from the reactant(s) to the transition state [Asano and LeNoble, 1978]. A negative value of
Fig. 3-17 Effect of pressure on the polymerization rate (*) and polymer molecular weight (4) of radiation-initiated polymerization of styrene at 25 C. After Moore et al. [1977] (by permission of WileyInterscience, New York.)