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 intrinsicviscosity 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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