Dimensions of Network Polymers

Abstract

AbstractThe mean‐square radius of gyration Rg2 and the graph diameter D of the network polymers formed through the random crosslinking of predetermined primary chains with a fixed number of intermolecular and intramolecular crosslinks are investigated by the numerical calculation based on the graph theory. In the case of ring‐free polymers, D is the longest end‐to‐end path length, sometimes referred to as the maximum span length. The expected Rg2‐value for a given D is found to follow a linear relationship, Rg2 = a D + b, similarly as in the cases of the ring‐free polymers. The proportional coefficient, a decreases approximately linearly with the number of intramolecular crosslinks kc, or the cycle rank in the graph theory, essentially independent of the total number k of crosslinks and the primary chain length distribution. The contraction factor g of the average Rg2 of the whole network polymer system is also governed by kc, and decreases with kc. One may be able to design and control the Rg2‐values of crosslinked polymers through the magnitude of D, which is usually easier to imagine than the complex 3D architecture.