Journal of Statistical Physics, Vol. 43, Nos. 1/2, 1986, p. 143-196.
This paper delivers a flexible formalism for handling equilibrium ring formation. Based on graphical models of polymerization, it includes as special cases the Flory Stockmayer RA_f model, the Flory A_fRB_g model, and Gordon's branching process formalism. When simple ring formation occurs in equireactive systems, it also includes the Jacobson-Stockmayer RA_2 and Hoeve RA_f models. The formalism is built from first principles in statistical mechanics and all assumptions are clearly stated. All parameters are given in terms of thermodynamic variables. With ring weights generalizing the Jacobson-Stockmayer Gaussian random walk, the formalism yields results for branching RA_f, A_fRB_g, and RA_f- RB_g polymer models. Equireactivity then gives explicit solutions. The equireactive RA_f- RB_g model compares favorably with data from gel-point vs. dilution experiments. With the exception of the Spanning Tree Approximation, graphical models of polymerization suffer from combinations of the following defects: equireactivity assumptions, restrictions to one type of monomer or bond, absence of rings, or absence of fused rings. This paper provides a promising "exact" approach to handling all of these problems simultaneously.