DISTRIBUTION OF LOCALIZATION LENGTHS IN RANDOMLY CROSS-LINKED MACROMOLECULAR NETWORKS

When a sufficient density of permanent random crosslinks is incorporat ed 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 del ocalised. The remaining fraction (i.e. the gel-fraction) are localised about random mean position;, and have random r.m.s. displacements (i. e. localisation lengths). A microscopic mean-field theory of this so-c alled vulcanisation transition is presented, in which the gel-fraction and statistical distribution of localisation lengths are determined s elf-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 transit ion.