Distribution of cyclic species in network formation: Microscopic theory of branching process in A_g-R-B _f−g model

Abstract

The distribution of cyclic species is explored for an irreversible A_g-R-B_f-g model on the basis of the concept of the “m tree” which was introduced in a preceding report by the authors. On the assumption of equal reactivity, the explicit solution is derived; i.e., for a sufficiently concentrated solution the concentration of cyclicj-mers can be expressed as$\left[ {R_j } \right] = \left( {k_{Rj} /k_L } \right)\left[ {\left( {f - g} \right)D_B } \right]^j \omega _j /j$, wherek _Rj andk _L are the rate constants of cyclicj-mer formation and interconnection, respectively, and

$$\omega _j = \sum\limits_{k = 0}^{[j/2]} {\left( {_{2k}^j } \right)} \alpha ^k $$

where α=(g − 1)(f − g − 1)/g(f − g) and [j/2] is the Gauss' symbol. Forg → 1, ω_j → 1, so that the solution reduces to the A-R-B_f−1 case. At a critical point one observes the strong divergence of the chances ∑ φ_j of cyclization.