Rigid crystalline phases of polymerized fullerenes

Combining total energy and structure optimization calculations, we explored new possible crystalline phases of covalently bonded ${\mathrm{C}}_{60}$ fullerenes and determined their structural, elastic, and electronic properties. Motivated by reported observations that bulk structures of polymerized fullerenes may be stiffer than diamond, we have explored possible ways of fullerene polymerization and have identified 12 stable crystal structures as potential candidates. Even though all these phases are very stiff, none of them exceeds the bulk modulus of diamond. The electronic structure of three-dimensional crystals of polymerized ${\mathrm{C}}_{60}$ depends mainly on the packing structure of the system, with only minor modifications due to the specific inter-fullerene bonding.

Figure 1

Types of covalent bonding between ${\mathrm{C}}_{60}$ molecules. (a) Arrangement of polymerized ${\mathrm{C}}_{60}$ chains (C), a square (S) and a triangular (T) 2D lattice of polymerized ${\mathrm{C}}_{60}$ molecules. Polymerization in these low-dimensional structures occurs by the “cycloaddition” reaction, depicted in (b). The different inter-fullerene bonding schemes considered here are shown in (b)–(i). (b) ${\mathrm{C}}_{60}$ dimerization by the $66∕66$ $(2+2)$ cycloaddition reaction, which converts pairs of “double bonds,” facing each other in adjacent fullerenes, to single bonds, and leads to the formation of two new “single bonds” connecting the fullerenes. (c) Starting with structure (b), disruption of the two intra-fullerene bonds, affected by the cycloaddition, strengthens the inter-fullerene bonds and partly relieves structural strain. (d) Starting with the structure (c), rotation of the inter-fullerene bonds normal to the plane of the figure leads to the “open hinge” structure. (e) Compressing structure (d), the hinges may approach each other to form a four membered common ring. (f) $56∕65$ $(2+2)$ cycloaddition, related to structure (b), but involving a pair of “single bonds” at the common pentagon-hexagon edge, rather than “double bonds” at the common hexagon-hexagon edge. (g) Starting with the structure (c), rotation of the inter-fullerene bonds normal to the plane of the figure leads to a new bonding scheme, which we call the $56∕65$ four membered common ring. (h) Occurring mainly in body-centered orthorhombic fullerene lattices, the $(3+3)$ cycloaddition establishes a covalent bond along the cell diagonal between the closest atoms in adjacent fullerenes. (i) Occurring mainly in body-centered cubic fullerene lattices, $(6+6)$ cycloaddition connects two facing hexagons in adjacent fullerenes along the cell diagonal.

Figure 2

Energy change per atom $\Delta E$ vs the relative volume change $\Delta V∕V$ for the stiffest BCC and BCO structures. Closed circles are data points for BCC crystals, with bonds along the sides of the conventional unit cell formed by $(2+2)$ cycloaddition. A polynomial fit to the BCC data points, representing relaxed structures, is given by the dashed line. Open circles are the data points for the BCO lattice, containing four membered common ring (FCR) bonds between fullerenes. A polynomial fit to the BCO data points is given by the dash-dotted line.