The distributions of chain lengths in a crosslinked polyisoprene network

A fundament of classical rubber elasticity theory is the Gaussian chain approximation formula, P(n,r) for the probability distribution of end-to-end distances of a polymer chain composed of n beads. It is considered to provide a realistic distribution of end-to-end distances, r, provided that the length of the polymer chain is much greater than its average end-to-end distance. By considering the number of beads (n) to be the independent variable, we can use P(n,r) to construct the probability distributions of network chain lengths, for fixed r. Since the network crosslinks reduce the probability for the occurrence of longer chains, the formula must be modified by a correction factor that takes this effect into account. We find that, both the shape of the n-probability distribution, its height, and the position of the peak vary significantly with r. We provide a numerical procedure for constructing networks that respect these distributions. The algorithm was implemented in a three-dimensional, random polymer-and-node network model to construct polyisoprene networks at two common crosslink densities. Although the procedure does not constrain the density, we find that the networks constructed have densities very close to the measured bulk density.