Distribution of cyclic species in network formation: Microscopic theory of branching processes

Abstract

General solutions of the distribution of cyclic species are sought in irreversible A-R-B _f−1 and R-A _f branching processes. With the aid of the concept of anm tree, we find the simple explicit solutions as a function of the extent of reactionD. In the irreversible processes of a sufficiently concentrated solution,

$$[R_j ]_D \simeq \{ (k_{R_j } /k_L )[(f - 1)D_B ]^j /j for A - R - B_{f - 1} model$$

respectively. Here [R _j ] is the concentration of cyclicyj-mer,k _Rj the rate constant of ringj-mer formation, andk _L that of interconnection; the subscript B denotes the B functional unit in the A-R-B _f−1 model. For random flight chains one may replacek _Rj /k _L with the Kuhn cyclization probability (3/2π∼ r ^/2 _j 〉)^3/2 (〈r ^/2 _j 〉 is the mean square distance), which yields the known exponential law as in the case of the linear theory: [R _j] ∞j ^−5/2. Hence this theory corresponds to the generalization of the Jacobson-Stockmayer linear theory (f=2).