Spanning-tree models for Af homopolymerizations with intramolecular reactions

The spanning-tree approximation model is one of the models used for polymerizations with intramolecular reaction regardless of the size of the ring formed. We present a modification of this model that uses more accurate internal estimates of the probabilities of intramolecular reaction. This requires limited Monte Carlo simulations of some molecular structures, resulting in a hybrid probability model (a combined analytic and Monte Carlo model). We then extend the spanning-tree model so that it may be used in the post-gel region. We show three possible extensions of varying degrees of complexity. The resulting models for stepwise A_f homopolymerizations have been coded into programs that run on desktop PCs in a few seconds. The models calculate the amount of intramolecular conversion, the weight-average molecular weight, the gel point, the weight fraction of soluble material, and the weight-average molecular weight of the sol fraction. We discuss the relative merits of the modified spanning-tree model and its three post-gel extensions and show sample calculations for arbitrary homopolymerization systems.