Combinatorial solutions to coagulation kernel for linear chains

Abstract

For this paper, we studied the evolution of the cluster size distribution for a system of coagulating particles under a linear-chain (LC) kernel, Ki,j=1i+1jα, and monodisperse initial conditions. We used a combinatorial framework in which time (i.e., time steps counted by the subsequent system states) and cluster sizes were discrete and the binary aggregation governed the evolution of the system. We modified a previously-known solution for the constant kernel to cover the LC kernel and used it in the framework to obtain the exact expression for the cluster size distribution (the average number of particles of a given size) and its standard deviation. Our theoretical solution is validated by a comparison to numerically simulated results for several values of α and to the experimental data of coagulating polystyrene particles. Theoretical predictions were accurate for any stage of the aggregation process and for a wide range of α.