Rubber elasticity for incomplete polymer networks.

We investigated the relationship between the elastic modulus, G and the reaction probability, p for polymer networks. First, we pointed out that the elastic modulus is expressed by G = {(fp∕2 - 1) + O((p - 1)(2))} Nk(B)T∕V (percolated network law), which does not depend on the local topology of the network structure or the existence of the loops. Here, N is the number of lattice point, V is the system volume, f is the functionality of the cross-link, k(B) is the Boltzmann constant, and T is the absolute temperature. We also conducted simulations for polymer networks with triangular and diamond lattices, and mechanical testing experiments on tetra-poly(ethylene glycol) (PEG) gel with systematically tuning the reaction probability. Here, the tetra-PEG gel was confirmed to be a potential candidate for ideal polymer networks consisting of unimodal strands free from defects and entanglements. From the results of simulations and experiments, it was revealed, for the first time, that the elastic modulus obeys this law in the wide range of p (p(c) ≪ p ≤ 1), where p(c) is the reaction probability at gelation threshold.