Distribution of localisation lengths in randomly crosslinked macromolecular networks

When a sufficient density of permanent random crosslinks is incorporated into a system of macromolecules, the system undergoes a continuous equilibrium phase transition from a liquid to an amorphous solid state. In this solid state, a certain fraction of monomers are entirely delocalised. The remaining fraction (i.e. the gel-fraction) are localised about random mean positions, and have random r.m.s. displacements (i.e. localisation lengths). A microscopic mean-field theory of this so-called vulcanisation transition is presented, in which the gel-fraction and statistical distribution of localisation lengths are determined self-consistently. A scaling form for the distribution of localisation lengths, valid for all near-critical crosslink densities, is obtained, and it is found that both the fraction of localised monomers and the typical inverse localisation length vanish continuously at the transition.