Exact solution of Smoluchowski's continuous multi-component equation with an additive kernel

Department of Physics, University of Oviedo - C/Calvo Sotelo s/n, E-33007 Oviedo, Spain

Corresponding author: julio@uniovi.es

Received: 24 November 2006
Accepted: 20 April 2007

Abstract

Smoluchowski's equation is used to analyse the dynamics of particulate systems under aggregation processes in aerosol physics, atmospheric physics, astrophysics, polymer chemistry, colloidal chemistry, etc. Here we provide an exact analytical solution for Smoluchowski's general, continuous, multi-component equation with additive kernel, for any initial particle size distribution (PSD). Once obtained the general solution, we apply it to a case with initial gamma PSD, which can be used to test numerical methods developed for solving more general cases. We have analysed the behaviour for large sizes and time, and a scaling approximation has been obtained as Vigil and Ziff conjectured. For bi-component mixtures we prove that as time increases, for the additive kernel, we cannot use the scaling solution to describe the behaviour of the number PSD on the whole. This fact contradicts a recent affirmation on the subject done by Matsoukas et al.