Irreversible aggregation kinetics: Power-law exponents from series

The kinetics of irreversible aggregation are studied using power series in time for general sum and product kernels of the type (${\mathit{i}}^{\mathrm{\alpha }}$+${\mathit{j}}^{\mathrm{\alpha }}$) and (ij${)}^{\mathrm{\alpha }}$. Assuming a power-law form for the asymptotic behavior of the first moment of the cluster distribution, ${\mathit{M}}_0$∼${\mathit{t}}^{\mathrm{-}\gamma }$, we are able to determine the exponent γ by inverting the series to give t as a function of the first moment. Exact solutions are known for α=0 and 1. Our numerical method gives γ as a function of α for α in the range from 0 to 1 for the sum kernel. For the product kernel we are able to determine γ near α=0 and 1, but a power-law form does not seem to fit the series well in the midrange near α=1/2. For the product kernel we clearly detect the onset of the gelation transition at α=1/2.