Statistical thermodynamics of the formation of an infinite cluster of thermally reversible chemical bonds

A systematic "mean- field'' treatment of the thermodynamic equilibrium formation of an infinite cluster of bonds in a system of identical monomers capable of forming from n = 0 to n > 2 reversible chemical bonds with one another is proposed within the Cayley-tree approximation. For this purpose the difference between the symmetry of the monomers appearing in "point-to-point'' and closed bond paths, respectively, is taken into account on the basis of an analysis of the structure of the infinite cluster. Minimization with respect to the distribution of such monomers yields a nontrivial solution corresponding to a lower free energy than the classical solution, which does not allow for the symmetry difference indicated. In addition, it is shown that the classical solution corresponds to the free-energy maximum when the infinite cluster is formed and that the formation of the infinite cluster is a first-order phase transition. The possible form of the phase diagrams of the systems considered is analyzed. (C) 1999 American Institute of Physics. [S1063-7761(99)01503-6].