Intramolecular Cross Linking and Chain Dimensions in Amorphous Polymers

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Abstract
This paper concerns the relationship between the average dimensions of chains in a polymer solid and the subsequent frequency of intramolecular cross linking when adjacent units are joined at random. Equations are developed for calculating the relative proportion of intramolecular (loop‐forming) links prior to the gel point, and the result is specialized to the case of molecules having a Gaussian segment distribution, an approximate representation of the presumed state of affairs in amorphous polymers. An apparent deviation from Stockmayer's equation for the cross‐linking density at the gel point is predicted, the magnitude of the departure depending on the average dimensions of the random coil molecules. These deviations are shown to be small for chains with moderately large numbers of segments if the coil dimensions in the solid are similar to those in a θ solvent. Some evidence that the individual chains may, in fact, be more compact in the solid is furnished by gelation data obtained in the cross linking of several well‐characterized samples of polystyrene by high‐energy radiation.