Structure of randomly cross-linked networks. A theoretical relationship between the molecular weight between cross links and the cycle rank, and its comparison with experiment

Abstract

Randomly cross-linked networks have a complex topology. The determination of their cross-link density is still a challenge since no definitive relationships have been established between Mc, the molecular weight between junctions, and macroscopic characteristics such as the modulus or volume fraction of polymer present at swelling equilibrium. It is shown here that a simple relationship exists between Mc, Mn (the number average molecular weight of the primary chains), and ξ, the cycle rank for regular networks having no defects other than chain ends. Specifically: ξ/V0=(ρ/2Mc)(1−3Mc/Mn), where V0 and ρ are the network volume and density, respectively. Since the cycle rank is directly related to the phantom modulus, it is also shown that the later Flory–Erman theory treating entanglements as restrictions on junction fluctuations can be successfully used to determine Mc. Some experimental results are given. The Flory–Erman proportionality constant I is calculated and found to be in good agreement with values reported in the literature.