Coherent State Field Theory for Reversible Gelation of Star Polymers

A coherent states field theory (CST) framework for reversible gelation of end-functional f-arm stars is presented, which implicitly enumerates all cross-linking patterns and generates the correct statistical weight for each type of cluster. At the mean-field level, the CST produces only tree-like clusters, and recovers all essential predictions of the Flory–Stockmayer (FS) theory. In particular, the saddle-point (mean-field) condition reproduces the FS condition for gelation. When applied to solutions of associative polymer stars in an implicit solvent, this mean-field theory predicts a two-phase region where sol and gel phases of different composition coexist. Beyond the mean-field level, where both tree and looplike clusters are present, we develop a loop density operator by considering the spatial translation of finite clusters. The field theory is then fully developed and analyzed at the Gaussian level of fluctuations. Phantom stars, dilute solutions, and dense melts are studied to better understand the predictions of the Gaussian theory, including those for the loop density and fluctuation corrections to the cluster number density. It is shown that the contributions from the excluded volume interaction do not affect the density of loops at the Gaussian level, although they do affect the bonding probability.