Competing role of catalysis-coagulation and catalysis-fragmentation in kinetic aggregation behaviours

We propose a kinetic aggregation model where species A aggregates evolve by the catalysis-coagulation and the catalysis-fragmentation, while the catalyst aggregates of the same species B or C perform self-coagulation processes. By means of the generalized Smoluchowski rate equation based on the mean-field assumption, we study the kinetic behaviours of the system with the catalysis-coagulation rate kernel K(i,j;l)\propto l^\nu and the catalysis-fragmentation rate kernel F(i,j;l)\propto l^\mu , where l is the size of the catalyst aggregate, and \nu and \mu are two parameters reflecting the dependence of the catalysis reaction on the size of the catalyst aggregate. The relation between the values of parameters \nu and \mu reflects the competing roles between the two catalysis processes in the kinetic evolution of species A. It is found that the competing roles of the catalysis-coagulation and catalysis-fragmentation in the kinetic aggregation behaviours are not determined simply by the relation between the two parameters \nu and \mu, but also depend on the values of these two parameters. When \nu> \mu and \nu\geqslant 0, the kinetic evolution of species A is dominated by the catalysis-coagulation and its aggregate size distribution a_k(t) obeys the conventional or generalized scaling law; when \nu<\mu and \nu \geqslant 0 or \nu>0 but \mu \geqslant 0, the catalysis-fragmentation process may play a dominating role and a_k(t) approaches the scale-free form; and in other cases, a balance is established between the two competing processes at large times and a_k(t) obeys a modified scaling law.