Geometries, stabilities, and reactions of carbon clusters: Towards a microscopic theory of fullerene formation

Abstract

To clarify the microscopic formation process of ${\mathrm{C}}_{60}$ and other fullerenes, we study the geometries and energetics of small carbon clusters and the reaction between carbon clusters using the long-range transferable tight-binding model parametrized by Omata et al. (Omata TB), the local-density-approximation (LDA) in the framework of the density-functional theory, and the constant-temperature molecular dynamics combined with Omata-TB (Omata TBMD). From the LDA geometry-optimization study, we find that the binding energy per atom of the ${\mathrm{C}}_{10}$ ring is $0.4\phantom{\rule{0.3em}{0ex}}\mathrm{eV}$/atom larger than that of the ${\mathrm{C}}_{10}$ chain. This energetic preference of a ring to a chain in ${\mathrm{C}}_{10}$ is most prominent among all ${\mathrm{C}}_{n}$ clusters studied $(5\ensuremath{\leqslant}n\ensuremath{\leqslant}17)$. Moreover, the study of $sp$-hybridized small carbon clusters with Omata TBMD reveals that ${\mathrm{C}}_{10}$ is the smallest stable ring at the temperature of $2000\phantom{\rule{0.3em}{0ex}}\mathrm{K}$, which can explain the high abundance of ${\mathrm{C}}_{10}$ in the experimental $\mathrm{C}_{n}{}^{\ensuremath{-}}$ mass spectra. From the remarkable stability of the ${\mathrm{C}}_{10}$ ring as well as from its high abundance, it is considered that the $sp$-hybridized ${\mathrm{C}}_{10}$ ring should play a role of major constituent units of larger clusters and fullerenes. Therefore we perform various sets of simulations of reactions between carbon clusters possessing the C atoms in multiples of 10, ${\mathrm{C}}_{10m}+{\mathrm{C}}_{10n}$ ($m+n=2,3,4,5,6$, $m\ensuremath{\geqslant}n\ensuremath{\geqslant}1$), at several temperatures with Omata TBMD. As a result, it is found that, in most cases studied, ${\mathrm{C}}_{20}$ and ${\mathrm{C}}_{30}$ clusters possess the $s{p}^{2}$-hybridized planar geometries, while ${\mathrm{C}}_{40}$, ${\mathrm{C}}_{50}$, and ${\mathrm{C}}_{60}$ take the $s{p}^{2}$-hybridized ``fullerenelike'' closed-cage geometries. These ${\mathrm{C}}_{40}$, ${\mathrm{C}}_{50}$, and ${\mathrm{C}}_{60}$ cages can be formed at as low as $1500\phantom{\rule{0.3em}{0ex}}\mathrm{K}$, which is in good accord with the experimental temperature of fullerene formation. In a few cases, even the ${\mathrm{C}}_{30}$ cluster is found to take the cage structure. These results are in good accord with the experimental results of the gas ion chromatography. The straightforward growth process from the $sp$-hybridized ring to the $s{p}^{2}$-hybridized plane and that from the $s{p}^{2}$-hybridized plane to the $s{p}^{2}$-hybridized fullerenelike cage revealed in the present study are considered to be the main road of the formation of fullerenes. Finally, the study of structural stabilities of cage geometries obtained through the reaction between a fullerenelike cage $({\mathrm{C}}_{40})$ or a symmetrical fullerene (${D}_{5h}$ ${\mathrm{C}}_{50}$ or ${I}_{h}$ ${\mathrm{C}}_{60}$) and a carbon cluster (${\mathrm{C}}_{10}$ or ${\mathrm{C}}_{12}$) at various temperatures with Omata TBMD indicates that ${\mathrm{C}}_{2n}$ fullerenelike cages larger than ${\mathrm{C}}_{60}$ tend to decay into ${\mathrm{C}}_{2n\ensuremath{-}2}$ through the ${\mathrm{C}}_{2}$ loss process at the higher rate than ${\mathrm{C}}_{60}$ or smaller fullerenelike cages. Therefore this ${\mathrm{C}}_{2}$ loss process is considered to be one of the most important processes leading to the extreme abundance of ${\mathrm{C}}_{60}$.