Dimensions of Network Polymers: Universal Relationship for the Ratio between Mean‐Square Radius of Gyration and Graph Diameter
Hidetaka Tobita
Abstract
Abstract Mean‐square radius of gyration Rg 2 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 Rg 2 = a r D applies. The ratio ϕ of a r against the corresponding ring‐free architecture a 0 , ϕ r = a r / a 0 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 Rg 2 and D larger compared with the corresponding random networks, except for the limited cases with small values of r ’s.