Theory of Irreversible Branching Process: A- R- B f-1 Model

With the aid of the enumeration method, a rigorous solution which involves ring formation is derived for a particular branching model A - R - B f -1 . The solution provides an unexpected result of ring size distribution: The system yields less abundance of rings than a linear system. The bigger the ring size and the larger the functionality, the lesser the yields. In a concentrated solution, however, the concentrations of smaller x -sized rings can approximately be expressed as [ R x ]=( k R x / k L )· D x A / x , where k R x denotes the cyclization rate constant of an x -sized chain, k L an interconnection rate constant and D A the extent of reaction of A functional units, respectively. The result is closely related to the shape of clusters ( L x ) which still possesses the classical form under strong ring formation. However, the numerical solution shows that the classical Flory formula of size distribution does not work in general.