The crystal structure of hexam ethylenetetramine I. X -ray studies at 298, 100 and 34 °K

The theory of branching processes allows various weight, size, configuration and network statistics of polymer systems to be derived in a unified and routine manner, fairly free from ad hoc combinatorial investigations. The method is briefly summarized for use in copolymer systems, of which homopolymer systems form a degenerate case. The calculations of statistical parameters are readily restricted to the sol fraction of a system after the gel point, and derivations from randomness of reaction due to chemical substitution effects are easily allowed for. The general randomly branched system of Zimm & Stockmayer is considered in detail. This arises from the vulcanization or radiation crosslinking of randomly distributed primary chains, from ideal copolymerization of monovinyl and polyvinyl monomers, or (most simply) from random condensation of 2-functional and f-functional monomers. Known formulae from several sources are rederived more simply in forms which are variously generalized. In particular, approximations assuming long linear sequences in the system are eliminated, and effects of the different sizes of the 2-functional and f-functional units on the mean square radii are allowed for to a good approximation. Complicated configurational statistics, such as the mean square radius $(\overline{R^2_x})$ of the isomer distribution of the $\mathbf{x}$-mer fraction, are evaluated explicitly. The method of Lagrange expansion of functions of several variables (Poincare, I. J. Good) is found to be especially useful.