Classical Theory of Homogeneous Nucleation Kinetic Approach

To develop the rate equation for clusters it is assumed that clusters grow and shrink via the acquisition or loss of single molecules. Cluster-cluster collision events are so rare that they can be ignored, as can those in which a cluster fissions into two or more clusters. Moreover, from the principle of microscopic reversibility, that is, that at equilibrium every forward process has to be matched by its corresponding reverse process, it follows that if clusters grow only by the addition of single molecules, evaporation also occurs one molecule at a time. 

Equation (10.2) provides the basis for studying transient nucleation. For example, if the monomer concentration is abruptly increased at / = 0, what is the time-dependent development of the cluster distribution Physically, in such a case there is a transient period over which the cluster concentrations adjust to the perturbation in monomer concentration, followed eventually by the establishment of a pseudo-steady-state cluster distribution. Since the characteristic time needed to establish the steady-state cluster distribution is generally short compared to the time scale over which typical monomer concentrations might be changing in the atmosphere, we can assume that the distribution of clusters is always at a steady state corresponding to the instantaneous monomer concentration. There are instances, generally in liquid-to-solid phase transitions, where transient nucleation can be quite important (Shi et al., 1990), although we do not pursue this aspect here. 

We define as the net rate at which clusters of size i become clusters of size i+l. This net rate is given by 

for a given monomer concentration (or saturation ratio S), the cluster concentrations can be assumed to be in a steady state, then the left hand side equals zero in (10.2). At steady state from (10.2) we see that all the fluxes must be equal to a single, constant flux J, 

setting f = 1 and summing (10.6) from t = 1 to some maximum value, /max, gives 

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CLASSICAL THEORY OF HOMOGENEOUS NUCLEATION KINETIC APPROACH 493... 

The classical theory of homogeneous nucleation dates back to pioneering work by Volmer and Weber (1926), Farkas (1927), Becker and Doring (1935), Frenkel (1955), and Zeldovich (1942). The expression for the constrained equilibrium concentration of clusters (11.57) dates back to Frenkel. The classical theory is based on a blend of statistical and thermodynamic arguments and can be approached from a kinetic viewpoint (Section 11.1) or that of constrained equilibrium cluster distributions (Section 11.2). In either case, the defining crux of the classical thoery is reliance on the capillarity approximation wherein bulk thermodynamic properties are used for clusters of all sizes. 

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