Homogeneous nucleation of clusters that exhibit magic numbers is studied numerically, using as an example aluminum at 2000 K, based on recent calculations of free energies [Li, J. Phys. Chem. C 111, 16227 (2007)] and condensation rate constants [Li and Truhlar, J. Phys. Chem. C 112, 11109 (2008)] that provide a database for Ali up to i=60. The nucleation behavior for saturation ratios greater than about 4.5 is found to be dominated by a peak in the free energy change associated with the reaction iAl→ Ali at i=55, making it the critical size over a wide range of saturation ratios. Calculated steady-state nucleation rates are many orders of magnitude lower than predicted by classical nucleation theory (CNT). The onset of nucleation is predicted to occur at a saturation ratio of about 13.3, compared to about 5.1 in CNT, while for saturation ratios greater than about 25 the abundance of magic-numbered clusters becomes high enough to invalidate the assumption that cluster growth occurs solely by monomer addition. Transient nucleation is also predicted to be substantially different than predicted by CNT, with a much longer time required to reach steady state: about 10 -4 s at a saturation ratio of 20, compared to about 10-7 s from CNT. Magic numbers are seen to play an important role in transient nucleation, as the nucleation currents for clusters of adjacent sizes become equal to each other in temporally successive groups, where the largest cluster in each group is the magic-numbered one.
Bibliographical noteFunding Information:
This work was partially supported by the U.S. National Science Foundation under Grant No. CBET-0756315, by the U. S. Department of Energy, Office of Basic Energy Sciences, under Grant No. DE-FG02-86ER13579, and by the Minnesota Supercomputing Institute. The authors thank Dr. Zhen Hua Li, currently at the Department of Chemistry, Fudan University, Shanghai, for providing tabular summaries of results not tabulated in Refs. 23 and 35 , and Dr. David Porter of the Minnesota Supercomputing Institute for helpful discussions on numerical methods for the transient calculations.