In irreversible aggregation processes without a gelation transition the cluster size distribution approaches a scaling form, $c_k\mathrm{}\left(t\right)\mathrm{}\sim s^{-2\mathrm{}}\phi \left(\frac{k}{s}\right)$. Usking Smoluchowski's coagulation equation we determine the exponents in the mean cluster size $s\left(t\right)\mathrm{}\sim t^z$ ($t\to \infty$) and in the small- and large-$x$ behavior of the scaling function $\phi \mathrm{}\left(x\right)\mathrm{}$. Depending on certain characteristics of the coagulation coefficients, $\phi \mathrm{}\left(x\right)\mathrm{}\sim x^{-\tau }$ ($x\to 0$) or $\phi \mathrm{}\left(x\right)\mathrm{}\sim \exp (-x^{\mu })$ ($x\to 0$) with $\mu$ some negative constant. In aggregation processes with gelation a similar scaling form is obtained as $t$ approaches the gel point.

• Received 13 December 1984

DOI:https://doi.org/10.1103/PhysRevLett.54.1396