Exact solution of Smoluchowski's continuous multi-component equation with an additive kernelJ. M. Fernández-Díaz and G. J. Gómez-García
Department of Physics, University of Oviedo - C/Calvo Sotelo s/n, E-33007 Oviedo, Spain
received 24 November 2006; accepted in final form 20 April 2007; published June 2007
published online 18 May 2007
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.
61.43.Hv - Fractals; macroscopic aggregates (including diffusion-limited aggregates).
45.70.-n - Granular systems.
05.90.+m - Other topics in statistical physics, thermodynamics, and nonlinear dynamical systems.
© Europhysics Letters Association 2007