W. H. Stockmayer

The most probable distributions of molecular sizes are calculated for certain types of branched-chain polymers. The results represent an extension of the previous work of Flory, who showed that very large polymeric molecules appear suddenly at a critical extent of reaction, which is predicted to occur very nearly at the experimentally observed gel point. This transition from liquid to gel is shown to be analagous to the condensation of a saturated vapor. It is believed that the size distributions obtained herein will aid in a study of viscosity-molecular weight relationships in branched-chain polymers.

Formulas for the mean square radii of various branched and ringed polymer molecules are developed under the usual assumptions regarding the statistics of chain configuration. For branched molecules, the mean square radii vary less rapidly with molecular weight than for strictly linear molecules, while for systems containing only rings and unbranched chains the variation is more rapid than for the linear case. These results show that in principle the quantity of branches or of rings can be determined from light-scattering measurements.

For three types of linear polycondensing systems, equilibrium molecular size distributions, including rings as allowable species, are derived. Average molecular weights and amounts of ring and chain fractions are calculated therefrom. The fractions of rings are shown to increase with dilution, and with molecular weight. It is shown that beyond a critical dilution it is sometimes possible to obtain 100 percent yield of rings by driving the condensation to completion. Detailed calculations are made for two important cases corresponding to condensations of the decamethylene glycol-adipic acid type: (1) for equimolar amounts of the two monomers, and (2) unequal amounts, with one monomer type completely reacted.

A statistical calculation of the gel point and of the molecular size distribution is made for cross-linked high polymers of arbitrary initial size distribution. The weight-average polymerization degree of the cross-linked polymer depends only on the weight-average polymerization degree of the initial polymer and the degree of cross linking. The results obtained also apply to certain vinyl-divinyl copolymers. Polymerization degrees can be estimated from observations of the gel point.