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

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 $0.4\mathrm{eV}$/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 $(5⩽n⩽17)$. 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 $2000\mathrm{K}$, which can explain the high abundance of ${\mathrm{C}}_{10}$ in the experimental $\mathrm{C}_n{}^{-}$ 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, ${\mathrm{C}}_{10m}+{\mathrm{C}}_{10n}$ ($m+n=2,3,4,5,6$, $m⩾n⩾1$), 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 $1500\mathrm{K}$, 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 $\left({\mathrm{C}}_{40}\right)$ 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 ${\mathrm{C}}_{2n-2}$ 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 ${\mathrm{C}}_{60}$.

Figure 1

Geometries of the even-numbered ring clusters, ${\mathrm{C}}_4$, ${\mathrm{C}}_6$, ${\mathrm{C}}_8$, and ${\mathrm{C}}_{10}$, optimized within (a) Xu TB and (b) Omata TB, respectively, compared with those within (c) the UHF (Ref. 49) and (d) the LDA. The LDA geometry of the ${\mathrm{C}}_4$ ring is from the literature (Ref. 50). Bond lengths are shown in angstroms and bond angles in degrees. In the case of Xu TB, even though the initial geometry in the optimization procedure of ${\mathrm{C}}_4$ is set to be a ring, ${\mathrm{C}}_4$ eventually takes a chain form after the optimization.

Figure 2

(Color online) Binding energies per atom of the LDA-optimized ${\mathrm{C}}_n$ chains $(3⩽n⩽17)$ and ${\mathrm{C}}_n$ rings $(5⩽n⩽24)$ with respect to that of graphene. The binding energy of a ${\mathrm{C}}_{10}$ ring is larger by $0.4\mathrm{eV}$/atom than that of a ${\mathrm{C}}_{10}$ chain, which is most prominent among the ${\mathrm{C}}_n$ clusters studied. In the binding energies of ${\mathrm{C}}_n$ rings, the periodic peaks at $n=6$, 10, 14, 18, and 22 are observed.

Figure 3

(Color online) Binding energies per atom of ${\mathrm{C}}_n$ chains $(3⩽n⩽17)$ optimized within the LDA and those within Omata TB with respect to that of graphene. Binding energies in Omata TB show good accord with those in the LDA.

Figure 4

(Color online) $⟨R⟩$ values of each small carbon cluster at $2000\mathrm{K}$ with two initial geometries, a chain and a ring. Circles represent $⟨R⟩$ in the case where the initial geometry is a chain, and squares in the case where it is a ring. Here, $R$ is the indicator, defined as formulas (1, 2), to judge whether the geometry of the cluster at the MD step is a chain or a ring; $R=0$ corresponds to a chain, and $R=1$ to a ring. The $⟨R⟩$ is the time average of $R$ calculated every $1\times {10}^3$ MD steps over $1\times {10}^5–1\times {10}^7$ MD steps.

Figure 5

Schematic picture of the initial condition in the reactions of ${\mathrm{C}}_{10m}+{\mathrm{C}}_{10n}$ where $m$ and $n$ are natural numbers, $m+n=2$, 3, 4, 5, 6, and $m⩾n$. The distance between their gravity centers of the ${\mathrm{C}}_{10m}$ and ${\mathrm{C}}_{10n}$ is set to be $20\mathrm{\AA }$. The ${\mathrm{C}}_{10n}$ has the barycentric velocity corresponding to the kinetic energy of $0.1\mathrm{eV}$ toward the gravity center of the ${\mathrm{C}}_{10m}$.

Figure 6

(Color online) Snapshots of the reaction process of ${\mathrm{C}}_{10}+{\mathrm{C}}_{10}$ in the case of $T=1500\mathrm{K}$ with Omata TBMD. Two ${\mathrm{C}}_{10}$ rings collide with each other, and then coalesce into the $s{p}^2$-hybridized ${\mathrm{C}}_{20}$ plane. Blue (dark gray) and yellow (light gray) spheres represent the atoms which initially formed two different ${\mathrm{C}}_{10}$ rings, respectively. It is interesting to note that the C atoms of the originally different ${\mathrm{C}}_{10}$ rings are already mixed in the final ${\mathrm{C}}_{20}$ plane.

Figure 7

(Color online) Snapshots of the reaction processes of ${\mathrm{C}}_{20}+{\mathrm{C}}_{10}$ at (a) $T=2000\mathrm{K}$ and (b) $2500\mathrm{K}$. Blue (dark gray) and yellow (light gray) spheres represent the atoms of the initial ${\mathrm{C}}_{20}$ plane, and those of the initial ${\mathrm{C}}_{10}$ ring, respectively. In the reaction (a), the $s{p}^2$-hybridized ${\mathrm{C}}_{30}$ plane mainly consisting of pentagons and hexagons is formed, and the C atoms of the initial ${\mathrm{C}}_{10}$ ring are distributed over the entire ${\mathrm{C}}_{30}$ cluster. In the reaction (b), the $s{p}^2$-hybridized fullerenelike ${\mathrm{C}}_{30}$ closed cage is formed.

Figure 8

(Color online) Snapshots of the reaction processes of ${\mathrm{C}}_{20}+{\mathrm{C}}_{20}$ at (a) $T=2000\mathrm{K}$ and (b) $2500\mathrm{K}$. Blue (dark gray) and yellow (light gray) spheres represent the atoms which initially formed two different ${\mathrm{C}}_{20}$ planes, respectively. In the reaction (a), the $s{p}^2$-hybridized fullerenelike ${\mathrm{C}}_{40}$ closed cage is formed. In the reaction (b), the fullerenelike ${\mathrm{C}}_{40}$ cage is formed, and then the ${\mathrm{C}}_{40}$ cage decays into ${\mathrm{C}}_{38}$ cage through ${\mathrm{C}}_2$ loss.

Figure 9

(Color online) Rotational angles of the initial ${\mathrm{C}}_{20}$ plane, $\varphi$ and $\theta$, are defined as (a) the rotational angle around the $z$ axis and (b) that around the $y$ axis, respectively. The gravity center of ${\mathrm{C}}_{20}$ is at the origin in Cartesian coordinates.

Figure 10

(Color online) Geometries of rather spherical fullerenelike (a) ${\mathrm{C}}_{50}$, (b) ${\mathrm{C}}_{52}$, (c) ${\mathrm{C}}_{60}$, (d) ${\mathrm{C}}_{62}$, (e) ${\mathrm{C}}_{70}$, and (f) ${\mathrm{C}}_{72}$ cages to shrink through the ${\mathrm{C}}_2$ loss process, which are obtained through the reactions of ${\mathrm{C}}_{40}+{\mathrm{C}}_{10}$ at $1095.34\mathrm{ps}$ $(T=2460\mathrm{K})$, ${\mathrm{C}}_{40}+{\mathrm{C}}_{12}$ at $211.81\mathrm{ps}$ $(T=2500\mathrm{K})$, ${\mathrm{C}}_{50}$ $\left(D_{5h}\right)+{\mathrm{C}}_{10}$ at $362.70\mathrm{ps}$ $(T=2510\mathrm{K})$, ${\mathrm{C}}_{50}$ $\left(D_{5h}\right)+{\mathrm{C}}_{12}$ at $109.53\mathrm{ps}$ $(T=2480\mathrm{K})$, ${\mathrm{C}}_{60}$ $\left(I_h\right)+{\mathrm{C}}_{10}$ at $362.70\mathrm{ps}$ $(T=2425\mathrm{K})$, and ${\mathrm{C}}_{60}$ $\left(I_h\right)+{\mathrm{C}}_{12}$ at $72.54\mathrm{ps}$ $(T=2420\mathrm{K})$, respectively. Blue (dark gray) and yellow (light gray) spheres represent the atoms of the original ${\mathrm{C}}_{40}$, ${\mathrm{C}}_{50}$, or ${\mathrm{C}}_{60}$ cage and those of the ${\mathrm{C}}_{10}$ or ${\mathrm{C}}_{12}$, respectively.

Figure 11

(Color online) $N_k$ values $(k=0,1,2,\dots ,40)$ of fullerenelike ${\mathrm{C}}_{50}$, ${\mathrm{C}}_{52}$, ${\mathrm{C}}_{60}$, ${\mathrm{C}}_{62}$, ${\mathrm{C}}_{70}$, and ${\mathrm{C}}_{72}$ cages, defined by formula (3), as a function of $T_k$. The straight line represents the case where ${\mathrm{C}}_2$ loss occurs at all the temperatures. In the behavior of $N_k$, only a small difference among ${\mathrm{C}}_{50}$, ${\mathrm{C}}_{52}$, ${\mathrm{C}}_{60}$, and ${\mathrm{C}}_{62}$ can be seen. However, $N_k$ values of ${\mathrm{C}}_{70}$ and ${\mathrm{C}}_{72}$ increase much more rapidly than those of other cages as a function of temperature.