Ring Formation and Molecular Weight Distributions in Branched‐Chain Polymers. I

The theory of molecular weight distributions in branched‐chain polymers is extended to include possible intramolecular reaction leading to ring formation. The problem is more complicated than the noncyclic case in that, in addition to the more complex combinatory and algebraic considerations, it is necessary to consider geometric restrictions on the number and relative probability of many configurations. Two approximate models for describing the extent of ring closure are investigated. The first of these, in which it is assumed that spatial restrictions may be ignored, predicts far too many structures for large molecules with many rings, and as a result always predicts gel formation. In the second approximation it is assumed that geometric restrictions limit the number of rings so that the probability of ring closure is independent of molecular weight, but only depends upon the density of available ends. This model predicts that ring closures will shift the gel point to higher extents of reaction, producing at volumes larger than a ``critical volume,'' completely soluble polymers even at complete reaction. The dependence of the molecular weight distribution on the experimental variables is discussed in some detail.