Configurational Statistics of Highly Branched Polymer Systems

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Abstract
Applications of the theory of branching processes to polymer systems can be so formulated, that all statistical parameters emerge automatically in a form which applies to the soluble part of the system, i.e., to the whole system up to the gel point, and to the sol fraction beyond. In a self-contained presentation, previous work along these lines is here extended to computing configurational statistics of systems arising most directly by condensation processes. These statistics are the mean-square radii of the molecules in such systems, averaged in various ways and useful for theories of light scattering and viscosity. Numerical calculations for random f-functional polycondensation are presented as plots of configurational parameters against conversion for f=3, 4, and 6. The general equations given extend beyond the random case to condensations with ``first-shell'' substitution effect, i.e., in which the rates of making or breaking a given bond between two repeat units depends on how many other bonds these two units carry (to further units). In terms of current theories, the intrinsic viscosity of a system at its gel point is shown to be finite. The spatial averaging of molecular size is based on random flight statistics. The well-known theorem by Kramers used in this connection is rederived on a simple topological basis, and generalized to deal with copolymerization of units of different sizes.

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