Gibbs free energies of clusters are required for predictive modeling of cluster growth during condensation of a cooling vapor. Here, we present a straightforward method of calculating free energies of cluster formation using the data from molecular dynamics (MD) simulations. We apply this method to iron clusters having from 2 to 100 atoms. The energies obtained are verified by comparing to an MD-simulated equilibrium cluster size distribution in a sub-saturated vapor. We show that these free energies differ significantly from those obtained with a commonly used spherical cluster approximation, which relies on a surface tension coefficient of a flat surface, as it is used in the classical nucleation theory (CNT). We show that the spherical cluster approximation in CNT can be improved by using a cluster-size-dependent Tolman correction for the surface tension. The Tolman length and effective surface tension values were derived for iron clusters, and they significantly differ from the commonly used experimentally measured values. This improved approximation does not account for geometric magic number effects responsible for spikes and troughs in densities of neighbor cluster sizes. Nonetheless, it allows to more accurately model cluster formation from a cooling vapor. It better reproduces the condensation timeline, overall shape of the cluster size distribution, average cluster size, and the distribution width. In contrast, using a constant surface tension coefficient (as done in CNT) resulted in incorrect condensation dynamics and cluster size distributions. The analytical expression for cluster nucleation rate from CNT was updated to account for the size-dependence of cluster surface tension.
Khrabry, Alexander, et al. "The importance of accounting for the Tolman correction to surface tension for nucleation and growth modeling of Fe clusters." Aerosol Science and Technology, vol. 59, no. 6, Feb. 2025. https://doi.org/10.1080/02786826.2025.2452895
Khrabry, Alexander, Hoffenberg, Louis E. S., Kaganovich, Igor D., Barsukov, Yuri, & Graves, David B. (2025). The importance of accounting for the Tolman correction to surface tension for nucleation and growth modeling of Fe clusters. Aerosol Science and Technology, 59(6). https://doi.org/10.1080/02786826.2025.2452895
Khrabry, Alexander, Hoffenberg, Louis E. S., Kaganovich, Igor D., et al., "The importance of accounting for the Tolman correction to surface tension for nucleation and growth modeling of Fe clusters," Aerosol Science and Technology 59, no. 6 (2025), https://doi.org/10.1080/02786826.2025.2452895
@article{osti_2520539,
author = {Khrabry, Alexander and Hoffenberg, Louis E. S. and Kaganovich, Igor D. and Barsukov, Yuri and Graves, David B.},
title = {The importance of accounting for the Tolman correction to surface tension for nucleation and growth modeling of Fe clusters},
annote = {Gibbs free energies of clusters are required for predictive modeling of cluster growth during condensation of a cooling vapor. Here, we present a straightforward method of calculating free energies of cluster formation using the data from molecular dynamics (MD) simulations. We apply this method to iron clusters having from 2 to 100 atoms. The energies obtained are verified by comparing to an MD-simulated equilibrium cluster size distribution in a sub-saturated vapor. We show that these free energies differ significantly from those obtained with a commonly used spherical cluster approximation, which relies on a surface tension coefficient of a flat surface, as it is used in the classical nucleation theory (CNT). We show that the spherical cluster approximation in CNT can be improved by using a cluster-size-dependent Tolman correction for the surface tension. The Tolman length and effective surface tension values were derived for iron clusters, and they significantly differ from the commonly used experimentally measured values. This improved approximation does not account for geometric magic number effects responsible for spikes and troughs in densities of neighbor cluster sizes. Nonetheless, it allows to more accurately model cluster formation from a cooling vapor. It better reproduces the condensation timeline, overall shape of the cluster size distribution, average cluster size, and the distribution width. In contrast, using a constant surface tension coefficient (as done in CNT) resulted in incorrect condensation dynamics and cluster size distributions. The analytical expression for cluster nucleation rate from CNT was updated to account for the size-dependence of cluster surface tension.},
doi = {10.1080/02786826.2025.2452895},
url = {https://www.osti.gov/biblio/2520539},
journal = {Aerosol Science and Technology},
issn = {ISSN 0278-6826},
number = {6},
volume = {59},
place = {United States},
publisher = {Taylor & Francis},
year = {2025},
month = {02}}
Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, Vol. 219, Issue 12https://doi.org/10.1243/095440605X8379