Statistical theory of nucleation, condensation and coagulation

Abstract

Starting from a master-equation description of dense gases, binary mixtures, etc., we derive theories for nucleation, coagulation and droplet growth by reformulating the dynamics in terms of ‘clusters’. This treatment is shown to agree quantitatively with computer simulations of the nucleation kinetics in the lattice gas model, and also gives a much better account of recent nucleation experiments on CO2 than previous approaches. The possible definitions of clusters, the appropriate coordinates for their description and their relation to bulk equilibrium properties of the system are outlined, and the effects of fluctuations on the properties of the clusters are investigated. Near the critical point the crossover between ‘classical’ and ‘non-classical’ expressions for the droplet formation energy is described in terms of ‘scaling laws’. In contrast to the Cahn-Hilliard-Langer theories of nucleation based on the concept of a coarse-grained free energy, no significant changes of static or dynamic behaviour at a spinodal curve are found, and it is concluded that the latter does not have a physical significance. Our derivations include the use of many cluster coordinates, and they are suitable for a straightforward application to many component mixtures (‘hetero-molecular nucleation’). We find that steady-state nucleation is formally equivalent to the problem of the electric current of a point unit voltage in a conducting medium, the conductivity tensor being given by the product of the equilibrium cluster distribution and the cluster reaction tensor. The nucleation rate is then related to the electric current at great distances from the origin in the space of the cluster coordinates. Non-stationary nucleation kinetics is treated also, general ‘time-lag’-estimates are given, and a special case is solved exactly, which justifies usual steady-state assumptions. Including terms which are non-linear in the cluster concentration we find that in general there is no clear distinction between nucleation and coagulation, and—particularly at high supersaturations—both mechanisms are important at the same time. This fact is also clearly borne out by the computer experiments.