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

Mean-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 = a_r D applies. The ratio ϕ of a_r against the corresponding ring-free architecture a₀, ϕ_r = a_r/a₀ has a universal relationship applicable both for the random and nonrandom networks with ϕ_r∝r^−0.25 for large r’s, and an empirical relationship, ϕ_r = [(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.