Dimensions of Network Polymers: Universal Relationship for the Ratio between Mean‐Square Radius of Gyration and Graph Diameter

Abstract

AbstractMean‐square radius of gyration Rg2 and the graph diameter D, which describe the dimensions of polymers, are investigated for the network polymers. Both for the random and nonrandom statistical networks whose cycle rank is r, a linear relationship Rg2 = ar D applies. The ratio ϕ of ar against the corresponding ring‐free architecture a0, ϕr = ar/a0 has a universal relationship applicable both for the random and nonrandom networks with ϕr∝r−0.25 for large r’s, and an empirical relationship, ϕr = [(1 + r)−2/3 + r/2]−0.25 is proposed. For the polymer fraction having a given number of r, the nonrandom nature of crosslinking tends to make both Rg2 and D larger compared with the corresponding random networks, except for the limited cases with small values of r’s.