Exact solutions of kinetic equations in an autocatalytic growth model

Kinetic equations are introduced for the transition-metal nanocluster nucleation and growth mechanism, as proposed by Watzky and Finke. Equations of this type take the form of Smoluchowski coagulation equations supplemented with the terms responsible for the chemical reactions. In the absence of coagulation, we find complete analytical solutions of the model equations for the autocatalytic rate constant both proportional to the cluster mass, and the mass-independent one. In the former case, $\xi_{k} = s_k(\xi_{1})\propto \xi_{1}^{k}/k $ was obtained, while in the latter, the functional form of $s_k(\xi_{1})$ is more complicated. In both cases, $\xi_{1}(t) = h_{\mu}(M_{\mu}(t))$ is a function of the moments of the mass distribution. Both functions, $s_k(\xi_{1})$ and $h_{\mu}(M_{\mu})$, depend on the assumed mechanism of autocatalytic growth and monomer production, and not on other chemical reactions present in a system.