Mean-Field Critical Behavior of End-Linked Polymer Networks with Loops

The Flory-Stockmayer (F-S) theory of gelation has been a paradigm for understanding chemical gelation. In the F-S mean-field model, monomers are placed at the sites of a Bethe lattice, and the process of end-linked network formation becomes equivalent to the bond percolation process on such a lattice. Extension of F-S theory to non-ideal networks shows that bond percolation for either purely branching networks or networks containing uncorrelated loops exhibits the same classical F-S critical exponents, regardless of the exact topology of the underlying network. However, the regime where loop formation is prevalent and loops are heavily correlated has not been investigated. Here, using a mean-field kinetic Monte Carlo simulation, we demonstrate that the critical exponents deviate significantly from the classical F-S values when loops becomes strongly correlated. We show that for networks with small loop fractions, finite size scaling gives the classical critical exponents; when loop fraction is increased, the calculated critical exponents deviate from the classical values, suggesting that the introduction of loops significantly alters the topology of the network formed.