Kinetics of catalytically activated aggregation–fragmentation process

We propose a catalytically activated aggregation—fragmentation model of three species, in which two clusters of species A can coagulate into a larger one under the catalysis of B clusters; otherwise, one cluster of species A will fragment into two smaller clusters under the catalysis of C clusters. By means of mean-field rate equations, we derive the asymptotic solutions of the cluster-mass distributions a_k(t) of species A, which is found to depend strongly on the competition between the catalyzed aggregation process and the catalyzed fragmentation process. When the catalyzed aggregation process dominates the system, the cluster-mass distribution a_k(t) satisfies the conventional scaling form. When the catalyzed fragmentation process dominates the system, the scaling description of a_k(t) breaks down completely and the monodisperse initial condition of species A would not be changed in the long-time limit. In the marginal case when the effects of catalyzed aggregation and catalyzed fragmentation counteract each other, a_k(t) takes the modified scaling form and the system can eventually evolve to a steady state.