Topics in Transient Nucleation

The continuum approximation in nucleation theory is reconsidered. A minor change in indexing the discrete flux leads naturally to an approximation which is both simple and accurate. More complicated schemes are introduced using the formalism of spectral density (weighting) functions. Optimization of these functions produces additional approximations that minimize the errors in either the rate equation or the nucleation current. Compared to the traditional Frenkel form and to the alternatives proposed by Goodrich and Shizgal and Barrett, these new forms are more accurate. It is shown that, within the continuum approximation, nucleation is mathematically equivalent to position-dependent diffusion. Generalization to multipath kinetics (clustering or association) is also discussed. The time-lag in transient nucleation is solved exactly for both the simple monomer approximation and the more complicated multipath kinetics with detailed balancing. Only initial and steady-state quantities are required for its calculation. The time-lag within the continuum approximation is also considered. By evaluating the exact expression approximately, a simple functional form is obtained. The controversy over the "prefactor" in previous approximate time-lags is resolved. It is shown that in general none of the approximate time-lags are correct. The "relative time-lag" is also introduced. It is useful when detailed balance does not apply and when the lower cutoff cluster size is unknown. Finally, multi-component nucleation is shown to be mathematically equivalent to one-component nucleation. A global nucleation rate is proposed which at steady state has translational invariance with respect to contours of a size function. Reduction of the flow problem to one dimension allows the global nucleation rate and its time -lag to be solved exactly. The detailed vector components of the local nucleation rate and their association time -lags can also be solved exactly.