Dynamic Scaling in the Kinetics of Clustering

In irreversible aggregation processes without a gelation transition the cluster size distribution approaches a scaling form, cₖ(t)~s^-2φ(k/s). Usking Smoluchowski's coagulation equation we determine the exponents in the mean cluster size s(t)~tᶻ (t→∞) and in the small- and large-x behavior of the scaling function φ(x). Depending on certain characteristics of the coagulation coefficients, φ(x)~x^-τ (x→0) or φ(x)~exp(-x^μ) (x→0) with μ some negative constant. In aggregation processes with gelation a similar scaling form is obtained as t approaches the gel point.