Smoluchowski Equations for Agglomeration in Conditions of Variable Temperature and Pressure and a New Scaling of Rate Constants: Application to Nozzle-Beam Expansion

The Smoluchowski equations provide a rigorous and efficient means for including multiple kinetic pathways when modeling coalescence growth systems. Originally written for a constant temperature and volume system, the equations must be modified if temperature and pressure vary during the coalescence time. In this paper, the equations are generalized, and adaptations appropriate to the situation presented by supersonic nozzle beam expansions are described. Given rate constants for all the cluster-cluster reactions, solution of the Smoluchowski equations would yield the abundances of clusters of all sizes at all times. This is unlikely, but we show that if these rate constants scale with the sizes of the reacting partners, the asymptotic (large size and large time) form of the cluster size distribution can be predicted. Experimentally determined distributions for He fit the predicted asymptotic distribution very well. Deviations between predicted and observed distributions allow identification of special cluster sizes that is, magic numbers. Furthermore, fitting an observed distribution to the theoretical form yields the base agglomeration cross section, from which all cluster-cluster rate constants may be obtained by scaling. Comparing the base cross section to measures of size and reactivity gives information about the coalescence process.