Dimensions of Network Polymers

The mean-square radius of gyration Rg(2) 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 Rg(2)-value for a given D is found to follow a linear relationship, Rg(2) = 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 k(c), 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 Rg(2) of the whole network polymer system is also governed by k(c), and decreases with k(c). One may be able to design and control the Rg(2)-values of crosslinked polymers through the magnitude of D, which is usually easier to imagine than the complex 3D architecture.