Particle coagulation and the memory of initial conditions

Particle coagulation is mathematically described by an infinite set of coupled nonlinear differential equations. A solution to these equations is derived for the case in which all particle clusters possess the same reactivity (i.e. a constant kernel) and where the initial conditions are bimodal, consisting of monomers and any sized J-mers. Properties of the solution are explored and it is shown that the scaling theory developed by Swift and Friedlander (1964 J. Colloid. Sci. 19 621) and extended by van Dongen and Ernst (1984 Phys. Rev. Lett. 54 1396) applies to all cluster sizes only in the limit t --> infinity, as reported previously by Kreer and Penrose (1944 J. Stat. Phys. 75 389). At finite times we find distinctly different scaling properties for the small and large ends of the size spectrum. Furthermore, at all times the shape of the small end of the size spectrum retains a memory of the initial conditions. These results may apply to other modes of coagulation so long as interactions between small clusters, and between small and large clusters, are as weak as the constant kernel employed here.