Instantaneous Distributions in Free-Radical Addition Polymerization

8 INSTANTANEOUS DISTRIBUTIONS IN FREE-RADICAL ADDITION POLYMERIZATION 

A growing polymer chain Px- can either propagate, terminate, or transfer. The propagation probabihty, q, is the probability that it will propagate rather than transfer or terminate and is given by 

a polymer molecule containing x units, Px, has been formed by x — 1 propagation steps (the remaining unit was incorporated in the addition step), each with a probability q, 

Example 9.7 Show the structures represented by x = 1 in Equation 9.42 for the polymerization of a vinyl monomer H2C=CHX. Assume no chain transfer and initiation according to 

Solution. Since chains here are killed only by disproportionation, dead chains of length X = 1 would be 

Rate of propagation + Rate of transfer + Rate of termination 

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Instantaneous Distributions in Free-Radical Addition Polymerization... 

FIGURE 9.4 Instantaneous number- and weight-fraction distributions of chain lengths in free-radical addition polymerization. Equations 9.47 and 9.48 are for termination by disproportionation and/or chain transfer Equations 9.58 and 9.59 are for termination by combination. 

The quantity Fi, the instantaneous copolymer composition, is analogous to x , the instantaneous number-average chain length in free-radical addition polymerization. Like x , it depends on the conditions in the reactor at a particular instant. It, too, is really an average, since not all the copolymer formed at a particular instant has exactly the same composition. However, the instantaneous distribution of compositions is normally much narrower than the instantaneous distribution of chain lengths, and because the fact that it is an average is not normally of great practical importance and cannot be controlled anyhow, the overbar is left off of Fj. 

Vicente et al. [30] used the heat of reaction and the open-loop observers developed in Section 7.2.5.3 to determine the concentration of monomer and CTA and hence to infer the instantaneous number-average molar masses during emulsion homo- and copolymerization reactions. In addition, the authors used the inferred values for online control of the molar mass distributions of copolymers with predefined distributions. They demonstrated that polymer latexes with unimodal MMD with the minimum achievable polydispersity index in free-radical polymerization (PI = 2) and bimodal distributions could be easily produced in linear polymer systems [15, 30]. 

K parameter in tydization or parameterin semibatdi control A parameter in free volume H particle volumetric growth rate or parameter in metallocene polymerization with branching parameter in fiee-volume equation Pm monomer density Pp polymer density p branching density p 6, ) CTOSS-fink density distribution pa 0, ) additional cross-Unk density distribution pj 0, ) cyclization density Pcs,a(0/ ) additional secondary cyclization density Pcp(0) primary cyclization density Paui(0) instantaneous secondary cyclization density Pt(0) instantaneous cross-link density reaction radius of the reacting species r ratio of reaction rates as defined by eqn [56] number fraction of type-i radicals monomer volume fraction polymer volume fraction chain length r number fraction < > s chain length s number fraction 0 present conversion... 

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