Branching in Batch Polymerizations

The treatment of batch polymerizations is in principle more difficult than that of continuous polymerizations in the latter, it may be permissible to assume steady-state conditions, though this has been questioned as indicated above but in the former it can never be permissible. It may, however, be permissible to make the assumption usual in treatments of the kinetics of radical polymerizations that the time-derivatives of radical concentrations may be neglected, though Kuchanov and Pismen (94) have questioned this assumption but the time-dependence of the polymer concentration must always be taken into account. 

Bamford and Tompa (93) considered the effects of branching on MWD in batch polymerizations, using Laplace Transforms to obtain analytical solutions in terms of modified Bessel functions of the first kind for a reaction scheme restricted to termination by disproportionation and mono-radicals. They also used another procedure which was to set up equations for the moments of the distribution that could be solved numerically the MWD was approximated as a sum of a number of Laguerre functions, the coefficients of which could be obtained from the moments. In some cases as many as 10 moments had to be computed in order to obtain a satisfactory representation of the MWD. The assumption that the distribution function decreases exponentially for large DP is built into this method this would not be true of the Beasley distribution (7.3), for instance. 

Graessley and his co-workers have made calculations of the effects of branching in batch polymerizations, with particular reference to vinyl acetate polymerization, and have considered the influence of reactor type on the breadth of the MWD (89, 91, 95, 96). Use was made of the Bamford and Tompa (93) method of moments to obtain the ratio MJMn, and in some cases the MWD by the Laguerre function procedure. It was found (89,91) that narrower distributions are produced in batch (or the equivalent plug-flow) systems than in continuous systems with mixing, a result referrable to the wide distribution of residence times in the latter. 

Saidel and Katz (97) have used an extension of the method of moments to calculate MJMn for a batch polymerization with transfer to polymer, and with termination exclusively by combination, but with restriction to mono-radicals. As the equations for the second and higher moments contain terms involving still higher moments it is necessary to approximate them in terms of lower moments, but this requires assumptions about the form of the distribution that may not be justified. 

Kuchanov and Pismen (94) have also applied their method to batch polymerization of a diene in which cross-linking by co-polymerization of double bonds in the polymer occurs as would be expected, the second moment diverges more rapidly than when this mode of reaction is absent. 

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