Simulating the structural diversity of carbon clusters across the planar-to-fullerene transition

Together with the second generation REBO reactive potential, replica-exchange molecular dynamics simulations coupled with systematic quenching were used to generate a broad set of isomers for neutral ${\mathrm{C}}_n$ clusters with $n=24$, 42, and 60. All the minima were sorted in energy and analyzed using different order parameters to monitor the evolution of their structural and chemical properties. The structural diversity measured by the fluctuations in these various indicators is found to increase significantly with energy, the number of carbon rings, especially six-membered, exhibiting a monotonic decrease in favor of low-coordinated chains and branched structures. A systematic statistical analysis between the various parameters indicates that energetic stability is mainly driven by the amount of ${sp}^2$ hybridization, more than any geometrical parameter. The astrophysical relevance of these results is discussed in the light of the recent detection of ${\mathrm{C}}_{60}$ and ${\mathrm{C}}_{60}^{+}$ fullerenes in the interstellar medium.

Figure 1

Energy distributions of quenched structures for ${\mathrm{C}}_{24},{\mathrm{C}}_{42}$, and ${\mathrm{C}}_{60}$ relative to the corresponding reference structure (lowest isomer). The vertical blue arrows locate the highest-energy isomers found in the REMD simulations.

Figure 2

Distribution of quenched structures for ${\mathrm{C}}_{24},{\mathrm{C}}_{42}$, and ${\mathrm{C}}_{60}$ as a function of the squared gyration radius and the fraction of ${sp}^2$ carbon atoms.

Figure 3

Selection of quenched structures obtained for ${\mathrm{C}}_{24},{\mathrm{C}}_{42}$, and ${\mathrm{C}}_{60}$. Carbon atoms are colored accordingly with their identified hybridization state, $sp$ and ${sp}^2$ atoms being shown in red and blue, respectively, and ambiguous atoms in green. No ${sp}^3$ atom is present in these structures.

Figure 4

Distributions of the squared radius of gyration $R_g^2$, asphericity $A_3$, and prolateness $S$ as a function of the energy per carbon atom for ${\mathrm{C}}_{24},{\mathrm{C}}_{42}$, and ${\mathrm{C}}_{60}$. The blue circles correspond to reference structures for ${\mathrm{C}}_{24}$, to the 45 fullerene isomers of ${\mathrm{C}}_{42}$, and to the 1812 fullerene isomers of ${\mathrm{C}}_{60}$.

Figure 5

Radial densities of the structures as a function of the distance from the center of mass, for increasing isomer energy of ${\mathrm{C}}_{24},{\mathrm{C}}_{42}$, and ${\mathrm{C}}_{60}$. The blue circles locate the corresponding peaks in the radial densities of the highly symmetric reference structures.

Figure 6

Angular distribution for carbon atom with two neighbors (left panel) or three neighbors (right panel) in ${\mathrm{C}}_{60}$. The horizontal dashed lines represent the angles for ideal ${\mathrm{C}}_3$ rings $\left({60}^{\circ }\right), {\mathrm{R}}_5$ rings $\left({108}^{\circ }\right), {\mathrm{R}}_6\left({120}^{\circ }\right)$, and ${\mathrm{R}}_7$ rings $\left(128.6^{\circ }\right)$.

Figure 7

Probabilities of carbon atoms showing $sp,{sp}^2,{sp}^3$ or ambiguous hybridization states in the various configurations of ${\mathrm{C}}_{24},{\mathrm{C}}_{42}$, and ${\mathrm{C}}_{60}$, as a function of their energy.

Figure 8

Average numbers of ${\mathrm{R}}_3,{\mathrm{R}}_5,{\mathrm{R}}_6$, and ${\mathrm{R}}_7$ rings as a function of isomer energy in ${\mathrm{C}}_{24},{\mathrm{C}}_{42}$, and ${\mathrm{C}}_{60}$. The solid circles at zero energy correspond to the values in the reference structures.

Figure 9

Pair distribution functions as a function of the carbon-carbon distance for ${\mathrm{C}}_{24},{\mathrm{C}}_{42}$, and ${\mathrm{C}}_{60}$ at various energies.

Figure 10

Correlations between fitted energy as a function of the REBO energy (eV/atom) for the databases of minima of ${\mathrm{C}}_{24}$ (upper panel), ${\mathrm{C}}_{42}$ (middle panel), and ${\mathrm{C}}_{60}$ (lower panel). For ${\mathrm{C}}_{42}$ and ${\mathrm{C}}_{60}$ the smaller sets of fullerene cages are superimposed as scatter plots with blue symbols.