Self-consistent rate equation theory of cluster size distribution in aggregation phenomena

Cluster nucleation and growth by aggregation is the central feature of many physical processes, from polymerization and gelation in polymer science, flocculation and coagulation in aerosol and colloidal chemistry, percolation and coarsening in phase transitions and critical phenomena, agglutination and cell adhesion in biology, to island nucleation and thin-film growth in materials science. Detailed information about the kinetics of aggregation is provided by the time dependent cluster size-distribution, a quantity which can be measured experimentally. While the standard Smoluchowski rate-equation approach has been in general successful in predicting average quantities like the total cluster density, it fails to account for spatial fluctuations and correlations and thus predicts size distributions that are in significant disagreement with both experiments and kinetic Monte Carlo simulations. In this work we outline a new method which takes into account such correlations. We show that by coupling a set of evolution equations for the capture-zone distributions with a set of rate-equations for the island densities one may obtain accurate predictions for the time- and size-dependent rates of monomer capture. In particular, by using this method we obtain excellent results for the capture numbers and island-size distributions in irreversible growth on both one- and two-dimensional substrates.