Theory of Branching Processes and Statistics of Rubber Elasticity

Abstract

Recent efforts to reformulate statistical theories of polymeric systems in terms of the theory of branching (``cascade'') processes, are here extended to calculations of statistical parameters for the theory of rubber elasticity, viz., the number and mean length of various forms of active network chains. New simple results are given for random f-functional polycondensation; it is shown that such systems are of interest in rubber elasticity studies in the region just after the gel point, where the concentration of active network chains varies rapidly with conversion, while the mean length of these chains diverges at the gel point. General exact formulas are derived for the random cross linking of arbitrary primary distributions, without assuming that the mean primary chain length is necessarily large; examples are worked out for uniform (homodisperse) chains, random, self-convoluted random, or Poisson primary distributions. Calculations are extended also to the cross-polymerization type of vulcanization reaction attributed to the cure of polybutadiene with peroxides. The results suggest reasons for differences in technical performance observed with such rubbers.