Handbook of Nanophysics 2: Clusters and Fullerenes

30 Solid-State Structures of Small Fullerenes Gotthard Seifert Technische Universität Dresden

Andrey N. Enyashin Technische Universität Dresden and Ural Branch of the Russian Academy of Sciences

Thomas Heine Jacobs University Bremen

30.1 Introduction ...........................................................................................................................30-1 30.2 Selected Examples..................................................................................................................30-2 C20 • C28 • C36 • C50

30.3 Trends, Comparison, and Conclusions ............................................................................30-11 References .........................................................................................................................................30-12

30.1 Introduction Already in the early days of the fullerene research, there was clear evidence for the existence of stable smaller fullerene-like cage structures than C60 (Kroto, 1987; Cox et al., 1988). The C60 fullerene cage consists of 12 isolated pentagons, which are interconnected by hexagons. The pentagons cause the curvature necessary for the formation of the closed cage structure of the otherwise planar carbon structure (graphene). C60 is the smallest cage structure with no adjacent pentagons. Smaller fullerene-like cage structures cannot be constructed without adjacent pentagons in their structure. The extreme case is C20, which consists of only pentagons. As already pointed out by Kroto (1987), the suppression of structures with adjacent pentagons can simply be understood by the increase of the local strain with the number of adjacent pentagons. Therefore, cage structures with isolated pentagons should be energetically preferred—isolated pentagon rule (IPR) (Kroto, 1987). There are many theoretical investigations that support strongly the energetic prevalence of isolated pentagons in the form of this isolated pentagon rule. For small fullerenes, the IPR can be extended to a minimization of adjacent pentagons in the cage structure. A detailed study for C40 showed that for each pentagon–pentagon pair in a fullerene cage, the binding energy is decreased by about 80 kJ/mol (Albertazzi et al., 1999). On the other hand, the binding energy as a function of the cage size increases smoothly with the number of atoms for the so-called fullerene road (Fowler and Manolopoulos, 1996) (see Figure 30.1). The “magic character” of C60 can be clearly seen from the relative stability function (see Figure 30.2). The relative stability (δ) of a fullerene Cn is defined as δ = E(n + 2) +

E(n − 2) − 2E(n), where E(n) is the calculated total energy of C n. This quantity may be viewed as a second derivative (difference quotient) in a E(n) versus n plot, i.e., it visualizes better than E(n) the size trends in stability. As one can see from Figure 30.2, there are other magic peaks for cage structures with less than 60 atoms. These “magic numbers” (clusters with a number of atoms with an exceptionally high stability in comparison with clusters of similar size) obtained from calculations agree quite well with corresponding peaks in the mass spectra of carbon clusters (see, e.g., Cox et al., 1988; Kietzmann et al., 1998). Based on these observations, several attempts have been undertaken for the quantitative isolation of fullerenes with less than 60 atoms (see, e.g., Kietzmann et al., 1998). However, there was no such success for C60. One reason might be that many of the fullerenes with less than 60 atoms could be very reactive and elude the detection or isolation. Kroto and Walton (1993) argued that small fullerene cages might behave as pseudoatoms with certain reactivity patterns. This idea was further developed by Milani et al. (1996) and Fowler et al. (1999b), introducing valencies for fullerene cages. In this way, fullerene cages may polymerize by activating these valencies for intercage bonds and forming solid crystalline materials. These solids (polymers) can possibly not be extracted from the soot. The clarification as to what extent the soot of the fullerene synthesis consists of polymeric small fullerenes is still one of the open questions in fullerene science. The synthesis, and especially the isolation of small fullerenes and their polymeric forms, is a challenging task; hence, despite the nearly 20 years of intensive experimental research, the amount of unambiguously isolated small fullerenes is still very limited. On the other hand, density-functional theory (DFT)-based 30-1

30-2

Handbook of Nanophysics: Clusters and Fullerenes

and C50 as typical examples of small fullerenes, whose ability of formation of condensed structures was intensively studied— especially theoretically—over the past 15 years.

9.0

Binding energy (eV/atom)

C60 C50

30.2 Selected Examples

C36 C40

8.5

30.2.1 C20

C28

C20 8.0

30

20

40

50

60

70

80

90

Number of atoms

FIGURE 30.1 Calculated binding energies of the most stable fullerene cages from C20 to C80. 4.0

Relative stabilities (eV)

C60 2.0 C20 C50

0.0 C36 C28 C40

–2.0

–4.0 20

30

40

50

60

70

80

90

Number of atoms

FIGURE 30.2 Calculated relative stabilities (δ) of fullerene cages from C20 to C80. The relative stability (δ) of a fullerene C n is defi ned as δ = E(n + 2) + E(n − 2) − 2E(n), where E(n) is the calculated total energy of C n (see also Kietzmann et al. 1998). (From Kietzmann, H. et al. Phys. Rev. Lett., 81, 5378, 1998. With permission.)

calculations can give very accurate predictions of the geometry, and qualitatively reliable information of the energetics of fullerene structures. This holds even for the simplest approximation within the DFT, the local density approximation (LDA). For the consideration of the vast manifold of fullerene isomers, an approximate variant of the DFT, the density-functional-based tight-binding (DFTB) method (Porezag et al., 1995), has proved to be an efficient method. For the characterization of the cage structures, group theory is a useful tool. Despite the icosahedral symmetry (e.g., C60—point group Ih), stable fullerene cages frequently have high symmetries belonging to the point groups Th and Td. In addition, several other symmetries can be found, as, e.g., Dnh symmetries. Th is chapter summarizes the knowledge about polymeric structures and their properties of small fullerenes, especially from a theoretical point of view. Along the “fullerene road” a selection was made. The selection was focused on C 20, C28,

The C20 dodecahedral cage with an icosahedral symmetry consists only of condensed (adjacent) pentagons and correspondingly a very high curvature. Therefore, this enormously strained fullerene could only be detected spectroscopically, applying several synthetic “tricks” (Prinzbach et al., 2000). Other isomers, as the ring and bowl C20 structures, have been computed to be more stable than the cage (see, e.g., Jones and Seifert, 1997; Saito and Miyamoto, 2002). Some investigations even favored bicyclic rings (Yang et al., 1988; Vonhelden et al., 1993; Hunter et al., 1993a,b; Handschuh et al., 1995) and linear chains (Ott et al., 1998). However, the dodecahedral fullerene was confirmed to have the lowest energy among all the mathematically possible 20-vertex trivalent polyhedral cages (Domene et al., 1997). Due to the high curvature and the large strain, the C20 fullerene is very reactive. The strain in the cage can be reduced by a partial hydrogenation. Chen et al. (2004) discussed in detail how to identify the reactive centers of the C20 cage. The C20H8 (Th) (see Figure 30.3) molecule is a derivative of C20 with a considerably reduced strain. The presence of eight isolated sp3 carbons (C–H) can separate the dodecahedron cage into the six essentially ethylene-like C=C units in C20H8 (Th). The smaller strain energy in C20H8 (Th) can also be seen from the energy of the hydrogenation reaction: C 20 + 4H2 → C 20H8 (Th )ΔH (per H2 ) = −67.0 kcal/mol (Chen et al., 2004). The strain reduction by a partial sp2 → sp3 transformation of the carbon atoms in the C20 cage suggests the possibility of the formation of C20 oligomers, as it was extensively discussed by Chen et al. (2004) for dimers, trimers, tetramers, and linear chains. Finally,

C20

C20H8 (Td)

FIGURE 30.3 Optimized structures of C20 and C20H8 according to DFT calculations. (From Chen, Z.F. et al., Chem. Eur. J., 10, 963, 2004. With permission.)

30-3

Solid-State Structures of Small Fullerenes

xy

xz

yz

xy

xz III

yz

xy

xz IV

yz

xy

xz V

yz

xz

yz

I

xy

xz II

yz

FIGURE 30.4 Optimized cubic structures of three-dimensional C20 solids. (From Chen, Z.F. et al., Chem. Eur. J., 10, 963, 2004. With permission.)

three-dimensional solid-state topologies can be formed by aggregation of C20 units in this way. There exists a large variety of possible three-dimensional structures built from C20 monomers. A first group of related candidates was derived from the linear oligomer structures and was studied by Miyamoto and Saito (2001) and Okada et al. (2001). However, DFTB investigations (Chen et al., 2004) showed a partial instability of these structures. Only one simple cubic open structure (referred to as structure I—Figure 30.4) turned out as a stable solid state configuration. A second group is based on the extraordinarily stable C20H8 isomer, whose “C20H8-8H” cage is used as the building block. When these units are joined by transforming the C–H bonds into intercage C–C bonds, a body-centered cubic (bcc) structure can be formed (referred to as structure II—Figure 30.4). This bcc structure can be further stabilized by the introduction of additional intercage bonds, as already suggested by Okada et al. (2001). Four such structures were further studied by Chen et al. (2004). All of them have binding energies between 8.6 and 8.7 eV/atom. These binding energies are considerably larger than that of the C20 monomer (8.0 eV/atom) and only slightly smaller than the binding energy of C60 (8.9 eV/atom) (Porezag et al., 1995). At least half of their carbons are tetracoordinate, and the calculated densities are approximately 2.8–2.9 g/cm3. These densities are nearly approaching the density of diamond. Structure III is the most stable one (8.7 eV/atom). It contains both small planar sp2 areas and cage-like sp3 regions (see Figure 30.5); the tetracoordinate carbon content is relatively low (50%). Structure IV, which has been studied first by Okada et al. (2001), has so-called [2 + 2] bridges in the xz and yz layers (see Figure 30.5). Such bridges are characterized by two cage–cage C–C bonds, forming a four-membered ring. These bridges are arranged in an alternating way: for example, in the xz plane [2 + 2], bridges connect cages in the z direction in one layer. At the next layer, the [2 + 2] bridges connect cages in the x direction. The analogous arrangement is also found in the yz plane (Chen et al., 2004). The bond

xy

VI

FIGURE 30.5 Most stable structures of three-dimensional C20 solids (From Chen, Z.F. et al., Chem. Eur. J., 10, 963, 2004. With permission.)

lengths, including those of intercage connections, are rather large and range from 1.34 Å for the sp2–sp2 bonds to 1.55 Å for the sp3– sp3 bonds. Wang et al. (2000, 2001) reported a successful synthesis of caged fullerene C20 solids. The measured electron diff raction pattern is consistent with all the solid-state structures (II–VI), discussed by Chen et al. (2004), and the estimated intermolecular bond length of about 1.5 Å is also in agreement with the theoretical findings. However, no direct experimental x-ray structure determination of a C20 solid has been reported up to now. Chen et al. (2004) also calculated the electronic density of states (DOS) of the solid-state C20 modifications. Except from the metallic behavior of structure I, all modifications (II–VI) should be semiconductors with gaps in the range between ∼2 eV (III–VI) and 3.5 eV (II).

30.2.2 C28 One of the smallest fullerenes observed experimentally (Cox et al., 1988) is C28. Kroto (1987) proposed a structure with tetrahedral symmetry Td . This tetrahedral cage consists of four hexagons,

30-4


C28

C28H4 (Td)

FIGURE 30.6 Structures of C28 and C28H4 (Td).

which are symmetrically linked by four further atoms. Each of these four atoms is the center of three fused pentagons (see Figure 30.6). As already mentioned by Kroto (1987), these four atoms may be related to the central carbon atom in the triphenylmethyl radical (•C(C6H5)3). The small size with the correspondingly high strain and four unpaired electrons in each set of four triplets of pentagons cause the high reactivity of the C28 cage (Kroto and Walton, 1993). This fact was confirmed by several theoretical studies (Chen and Jiao, 2001; Makurin et al., 2001). Numerous theoretical investigations show that the C28 skeleton can be stabilized by forming endohedral fullerenes M@C28, where the metal atoms (M) are formally in the M4+ configuration (e.g., d-elements Ti, Zr, Mo, f-elements U, Pu, Ce, p-elements Si, Ge) (Dunlap et al., 1992; Pederson and Laouini, 1993; Jackson et al., 1993, 1994; Makurin et al., 2001). Such stabilization is explained by the formation of an electronic system with all paired electrons (closed electronic shell). On the other hand, it was predicted that the saturation of free valences of the C28 molecule radical can be realized through their linking with hydrogens, halogen atoms, alkyl groups, or through their joining to form polymers, films, or crystals (Choho et al., 1997, Canning et al., 1997, Makurin et al., 2001). The four unpaired electrons localized on the carbon atoms common to each set of four triplets of pentagons make the C28

Hyperdiamond C28

fullerene similar to an sp3-hybridized carbon atom. Therefore, a diamond-like crystalline phase of C28 might exist. Calculations showed (Kaxiras et al., 1994) that hyperdiamond C28 is a semiconductor with a bandgap of 1.5 eV; its lattice parameter and the elastic modulus were calculated by Bylander and Kleinman (1993). The large interest arises from possible superconducting properties of alkali metal-doped phases Mx C28, whose critical temperature is predicted to be approximately eight times higher than that of fullerides M xC60 (Breda et al., 2000). Seifert et al. (2005) studied the structural, mechanical, and electronic characteristics of hyperdiamond C28, but also hyperlonsdaleit C28 phases (see Figure 30.7). Hyperlonsdaleit is derived from the hexagonal modification of diamond, lonsdaleite. The bond lengths between the C28 cages in hyperdiamond and in hyperlonsdaleit are in accord with their expected hybridization (∼0.155 nm). Thus, it confirms the presence of covalent interactions between C28 fullerenes in crystalline state that distinguishes these crystals from “classical” fullerites (e.g., C60), where molecules are linked via weak van-der-Waals forces. The comparison of differences between the total energies from DFTB calculations of free fullerene C28 and of its crystalline modifications shows that the formation of crystals is favored for both modifications. For hyperdiamond this difference is −6.13 eV/C28, and for hyperlonsdaleit it is −6.09 eV/C28. It is important that the values of the energy of association for these structures are very close to each other. The binding energy is slightly smaller than that of the fcc-C 60 and clearly smaller than the binding energy in diamond (see Table 30.1). Table 30.1 shows that the density of hyperdiamond and hyperlonsdaleit are three times smaller than that of diamond. The bulk moduli of hyperdiamond and hyperlonsdaleit are about one order smaller than the bulk modulus of diamond. Notwithstanding the presence of sp3-hybridized carbon atoms in the lattices of hyperdiamond and hyperlonsdaleit, their hardness is smaller than expected. It can be explained by considering the repelling of π clouds of sp2-hybridized carbon atoms in the neighboring C28 cages. For example, the distance between the

Hyperlonsdaleit C28

FIGURE 30.7 Fragments of the optimized hyperdiamond and hyperlonsdaleit crystal lattices of the condensed fullerene C28. (From Seifert, G. et al., Phys. Rev. B Condens. Matter Mater. Phys., 72, 012102-4, 2005. With permission.)

30-5

Solid-State Structures of Small Fullerenes TABLE 30.1 Lattice Constants (a, c), Density (ρ), Bulk Modulus (B), Bandgap (Eg), and Binding Energies in the Free and Solid Forms (Ebf , Ebs ) of the C28 Crystalline Modifications in Comparison with Other Carbon Phases According to DFTB Calculations Crystal Diamond bcc-Fullerite C20 hyperdiamond C28 Hyperlonsdaleite C28 fcc-Fullerite C60

a (nm)

c (nm)

ρ (g/cm3)

B (GPa)

Eg (eV)

Ebf (eV/atom)

Ebs (eV/atom)

0.3576 0.6897 1.5953 1.1277 ≈1.41

— — — 1.8415 —

3.49 2.43 1.10 1.10 ≈1.7

478.20 201.87 39.26 41.20 < 0.01

7.48 3.51 2.04 1.99 1.66

— 8.01 8.29 8.29 8.85

9.22 8.07 8.50 8.50 8.85

Source: Seifert, G. et al., Phys. Rev. B Condens. Matter Mater. Phys., 72, 012102-4, 2005. With permission.

nearest carbon atoms of hexagons in the neighboring C28 cages of hyperlonsdaleit is about 0.292 nm, which is smaller than the van-der-Waals gap of graphite (0.334 nm). Both crystalline phases of C28 are semiconductors with a direct bandgap of about 2 eV (Table 30.1). Band structures and electronic DOS profiles of crystalline C28 were calculated by several authors. The calculated band structures of the crystalline modifications of C28 from DFTB calculations (Seifert et al., 2005) are shown in Figure 30.8. DFT calculations with a plane-wave basis set give very similar results (Enyashin et al., 2006). The quasicore C2s states with an admixture of the C2p states form a set of bands in the range from −16 to −13 eV below the Fermi level—not shown in here. The occupied band in the range from −11 to −2 eV consists of the C2s and C2p states responsible for σ and π bonds in the C28 fullerenes. The bands at −1 and −2 eV below Fermi level are associated with π bonds of hexagons in the skeleton of C28.

In spite of the numerous theoretical investigations suggesting the stability of solid C28, there was up to now no report on a successful synthesis of any of above-mentioned crystalline phases. However, it is believed that such crystals could be observed by chemical deposition of active C28 molecules on a respective surface. The simulations of C28 adsorption on the surfaces of semiconductors like GaAs (001) and diamond (001) or (111) were performed by several authors (Canning et al., 1997; Zhu et al., 2000; Yao S. et al., 2005). A study (Zhu et al., 2000) with an empirical force field (Brenner potential) of low-energy C28 deposition on diamond (001) surface has shown that the incident C28 clusters can be chemisorbed without breaking its cage structure and inducing surface defects. Further C28 clusters may form bonds with each other. However, the chemisorbed clusters are randomly stacked to grow a fi lm in a three-dimensional island growth mode with an amorphous fi lm. On the other hand,

1

1

0

0

–1

–1

–2

–2 E, eV

2

E, eV

2

–3

–3

–4

–4

–5

–5

–6

–6

–7

L

Γ

X

W K

Hyperdiamond C28

Γ

–7

Γ

K

M

Γ

A

H

L

A

Hyperlonsdaleit C28

FIGURE 30.8 Calculated band structure for hyperdiamond C28 and hyperlonsdaleit. (From Seifert, G. et al., Phys. Rev. B Condens. Matter Mater. Phys., 72, 012102-4, 2005. With permission.)

30-6

the results from quantum molecular dynamics simulations by Canning et al. show that under certain deposition conditions on diamond (111) surface, C28 act as building blocks on a nanometer scale to form a thin fi lm of nearly defect-free molecules and behave as carbon superatoms, with the majority of them being threefold or fourfold coordinated, similar to carbon atoms in amorphous systems. The microscopic structure of the deposited fi lm supports suggestions about the stability of the hyperdiamond solid. The low mass density and large internal surface of C28 fullerites suggest their possible applications as catalyst, nanosieve, and gas storage material. Possessing the large cavities such fullerites are attractive as the matrix for molecular absorption. Enyashin et al. (2006) have estimated the active volume accessible by molecular H2 physisorption in hyperdiamond and hyperlonsdaleit. Both structures have tubular areas where the interaction potential is attractive for light molecules, as, e.g., H2. For hyperlonsdaleit these areas are more pronounced. The potential is only attractive in the cavities between the fullerene cages and reaches well depths of about 9.5 and 13.1 kJ/mol, respectively, for hyperdiamond and for hyperlonsdaleit. These values are comparable with those of graphene bilayers or C60 intercalated graphite (Kuc et al., 2007). Also the exohedral intercalation of hyperdiamond with K was investigated (Enyashin et al., 2006). For an intercalation up to the KC28 composition, the lattice constant of the compound does not change, though it leads to the change of the electronic structure of the fullerite. The additional electrons from K, two per unit cell, populate the lowest states of the conduction band in pure hyperdiamond. Thus, the compound becomes metallic. The cavities in C28 hyperdiamond have a diameter of about 0.46 nm, which is larger than the radius of C28 cages (∼0.268 nm) constituting the C28 lattice, i.e., a self-intercalation of the C28 hyperdiamond— filling of the lattice voids with C28 fullerenes—might be possible. Based on this idea, a new three-dimensional crystalline C28 fullerene phase—self-intercalated hyperdiamond—was suggested (Enyashin and Ivanovskii, 2007). When the cavities of the initial C28 hyperdiamond lattice are completely filled, the intercalated fullerenes are arranged so that they form their “own” diamondlike lattice (Figure 30.9). Therefore, the structure of the aforementioned self-intercalated hyperdiamond can be viewed as two C28 hyperdiamond lattices inserted into each other. Since the distance between fullerene walls of these two sublattices is roughly equal to the van-der-Waals gap in graphite (∼0.3 nm), such structure is the first example of a fullerite with mixed types of interaction between its constituents—weak and covalent ones. The weak interaction between two sublattices was supported by DFTB calculations. While the density of self-intercalated hyperdiamond is two times and the bulk modulus ∼2.5 times larger than for hyperdiamond, the electronic properties do not change essentially.


FIGURE 30.9 Crystal lattice of the autointercalated hyperdiamond C28. The two sublattices are painted in black and gray. (From Enyashin, A.N. and Ivanovskii, A.L., JETP Lett., 86, 537, 2007. With permission.)

interpreted electron diff raction and solid-state NMR data of a material they had obtained in a modified Krätschmer–Huffman experiment (Kratschmer et al., 1990) as a proof for a fullerenic solid consisting of highly symmetric (D6h) C36 fullerene cages. This solid material could be sublimated and solved in organic solvents, which are similar but not as good as C60. Furthermore, it could also be doped with alkaline metals. In a further paper, results from scanning tunneling spectroscopy (STS) investigations of the material on gold and specific graphite substrates already indicated covalently bonded C36 units (Collins et al., 1999b). Fifteen “classical” fullerene cages can be constructed for C36 (Fowler and Manolopoulos, 1996). These isomers can be labeled by the number of atoms n and the place in the lexicographical order of a spiral code (spiral notation m), corresponding to a contiguous helical numbering pathway in the fullerene cage. Two of them have the minimal count of pentagon adjacencies: the isomers with D2d and D6h symmetry, No. 36:14 (Figure 30.10)

30.2.3 C36 Going further along the “fullerene road,” passing the “magic” C32, which could be prepared in small amounts (Kietzmann et al., 1998), one arrives at the “semimagic” C36. Piskoti et al. (1998)

FIGURE 30.10 Isomer No. 14 of C36 (D2d symmetry), according to the spiral notation. (From Fowler, P.W. and Manolopoulos, D.E., An Atlas of Fullerenes, Clarendon Press, Oxford, U.K., 1996.)

30-7


and No.36:15 (Figure 30.11), according to the spiral notation (No. n:m) (Fowler and Manolopoulos, 1996). Hückel calculations give a significant nonzero HOMO–LUMO gap (HOMO—Highest Occupied Molecular Orbital, LUMO—Lowest Unoccupied Molecular Orbital) for isomer 36:14, but not for isomer 36:15 (Fowler and Manolopoulos, 1996; Heath, 1998). Semiempirical calculations (Campbell et al., 1996; Slanina et al., 1998) give, in agreement with rule of the minimal number of adjacent pentagons (Albertazzi et al., 1999), the D2d isomer (36:14) as the most stable isomer of C36. DFT-based calculations (Grossman et al., 1998; Fowler et al., 1999a) predict both isomers as practically isoenergetic. The isomer 36:15 is stabilized by a second-order Jahn–Teller effect. Low-lying orbitals get close to the HOMO point at a possible stabilization by adding up to six electrons. This is consistent with the experimental C36H6 signal in the mass spectrum of the soot and with the finding of a possible doping of C36 by alkaline metals (Piskoti et al., 1998). Th is is an indication for a hexavalency of C36. Mathematically, 82,123 possible isomers of C36H6 can be constructed, based on the isomer 36:15. All these isomers were constructed and optimized with the DFTB method (Fowler et al., 1999a). The most stable isomer has D3h symmetry and a considerable HOMO–LUMO gap (2.3 eV). The hydrogen atoms occupy the three non-neighboring (1,4) positions of the equatorial hexagons in the C36 cage (Figure 30.12). The hydrogen-saturated carbon atoms of C36H6 exhibit sp3character. They fulfi ll the steric conditions to act for the formation of intercage bonds to build dimers, trimers, oligomers, and solid state structures out of C36. The stable structures obtained using the (1,4) positions of the equatorial hexagons in the C36 cage for building dimers, trimers, and tetramers are shown in Figure 30.13. The calculated binding energy (per monomer) of the D2d dimer was calculated to 152 kJ/mol by DFTB (Fowler et al., 1999a, Heine et al., 1999) and 119 kJ/mol and DFT–LDA (Collins et al., 1999b) investigations, respectively. As shown by Collins et al. (1999b), dimers linked by single bonds are less bound and they are radicals, as expected from simple valence considerations. Going along the sequence dimer → trimer → tetramer the binding energy per monomer is increasing from 152 to 239

(a)

(b)

FIGURE 30.11 Isomer No. 15 of C36 (D 6h symmetry), according to the spiral notation. (From Fowler, P.W. and Manolopoulos, D.E., An Atlas of Fullerenes, Clarendon Press, Oxford, U.K., 1996.)

FIGURE 30.12 The most stable isomer of C36H6, based on the isomer 36:15 of the C36 cage. (From Fowler, P.W. et al., Chem. Phys. Lett., 300, 369, 1999a. With permission.)

(c)

FIGURE 30.13 C36 oligomers (side and top view), dimer (a), trimer (b), and tetramer (c). (From Heine, T. et al., Solid State Commun., 111, 19, 1999. With permission.)

30-8

to 271 kJ/mol. However, the binding energy per bridge decreases from 304 to 240 to 216 kJ/mol, suggesting that successive bridges are less effective. At the same time the C–C bonds in the bridges lengthen slightly by 0.2 Å from dimer to trimer and from the trimer to the central bridge of the tetramer. The DFTB calculations suggest also a growing gap size from its small monomer value of 0.6 eV (Fowler et al., 1999a) through 0.75 eV for the dimer to an apparent limit of ∼1 eV for the trimer and the tetramer (Heine et al., 1999). Qualitatively similar gaps can be deduced from DFT–LDA calculations (Collins et al., 1999b). Further extension of a planar triangulated framework of monomers appears to lead to unstable configurations. In a pentamer, constructed by capping one edge of the tetramer, the bridged edge lengthens and then opens; in heptamers outer edges open in a similar fashion. Th is behavior is perhaps only to be expected, as a fully triangulated plane layer could not be realized with the short bridges responsible for the high binding energy of the dimer. These structural islands would therefore appear to be dead ends for the growths of a two-dimensional layer. A two-dimensional structure for (C36)∞ can be envisaged, but it is based on the less densely packed graphitic structure. This structure can be derived from a C36 hexamer, a “superbenzene” ((C36H2)6 – see Figure 30.14), based on C36, as it was proposed in Fowler et al. (1999a). In this “supergraphitic” layer (Fowler et al., 1999a), each monomer is linked by 1,4 bridges at 120° to three neighbors, with a bond length of 0.162 nm (see Figure 30.15), according to DFTB supercell calculations with eight monomers per cell (Fowler et al., 1999a, Heine et al., 1999). In contrast to graphite, this C36-based layered structure would be a semiconductor with a gap of about 2 eV, similar to the HOMO–LUMO gap of the “superbenzene” (Fowler et al., 1999a). “Supergraphite” has a binding energy of 375 kJ/mol per

FIGURE 30.14 Superbenzene (C36H 2)6. (From Heine, T. et al., Solid State Commun., 111, 19, 1999. With permission.)


FIGURE 30.15 Top and side view of the elementary cell of C36 supergraphite. (From Heine, T. et al., Solid State Commun., 111, 19, 1999. With permission.)

C36, which is considerably higher than the binding energies of the C36 oligomers. Van-der-Waals interactions between the layers would presumably further enhance the stability of supergraphite. However, the hexavalency of C36 opens also the way for three-dimensional structures. One possible three-dimensional structure was first proposed by Cote et al. (1998) (see Figure 30.16). This rhombohedral closely packed lattice with a binding energy Eb = 355 kJ/mol/C36 (Fowler et al., 1999a) is less stable than the supergraphite structure and it is metallic. Further theoretical investigations (Heine et al., 1999) predicted more stable three-dimensional structures. A further, much more stable (Eb = 522 kJ/mol/C36) (Fowler et al., 1999a) ) three-dimensional hexagonal structure has first been called (Fowler et al., 1999a) an “hcp” (hexagonal close packed) lattice (see Figure 30.17). In fact, the term “hcp” is something of a misnomer. This lattice belongs to the space group, which is also the group of hexagonal boron nitride and the off retite family of zeolites (Gard and Tait, 1972). In this structure, each C36 monomer has D3h site symmetry and is linked by single bonds (0.153 nm) to each of six neighbors, three above and three below, and by longer contacts (0.163 nm) to two axial neighbors. The lattice can be viewed as notionally derived by removal of columns of monomers from a closely packed structure, to leave channels of stacked puckered rings parallel to the c-axis (see Figure 30.17). This covalent zeolite-like structure is ideally compatible with the “hexavalency” of the monomer. The binding energy per monomer was calculated as 569 kJ/mol (Heine et al., 1999). Among the hypothetical structures considered in the literature for crystalline C36, this hexagonal form is not only the most stable one, it also clearly gives the best match to the experimental data. The center-to-center distance projected → onto the a→, b -plane of the hexagonal lattice, calculated for this structure as 0.674 nm, in accord with the d-spacing of 0.668 nm derived from the transmission electron microscopy diff raction pattern (Piskoti et al., 1998).


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FIGURE 30.16 Elementary cell of the rhombohedral modification of solid C36. (From Fowler, P.W. et al., Chem. Phys. Lett., 300, 369, 1999a. With permission.)

FIGURE 30.17 Elementary cell of the hexagonal structure (hcp) of C36. (From Fowler, P.W. et al., Chem. Phys. Lett., 300, 369, 1999a. With permission.)

Also the calculated gap (0.89 eV) (Fowler et al., 1999a) agrees with gap measured with STS (∼0.8 eV) (Collins et al., 1999b). While the supergraphite layer has a computed gap of 2.09 eV, the rhombohedral solid proposed by Cote et al. (1998) is metallic, as mentioned above. DOS profiles computed with the DFTB method for the solid-state structures of C 36 are shown in Figure 30.18. For the rhombohedral structure, a broadly similar profi le was given by Cote et al. (1998).

30.2.4 C50 Along the “fullerene road” (see Fowler and Manolopoulos, 1996), besides C 44, the C50 cage should be rather stable. Xie et al. (2004) reported about the synthesis of C50Cl10. This structure has been unambiguously characterized by measured and simulated 13C NMR. The electronic properties of this fullerene have been studied recently (Lu et al., 2004). For C50, 271 possible isomers of classical fullerenes can be constructed (Fowler and Manolopoulos, 1996). Zhechkov et al. (2004) studied the structure and the energetics of all the 271 classical fullerenes of C50. They found isomer No. 271 in the spiral nomenclature as the most stable one. It has a D5h symmetry and possesses the lowest number of pentagon–pentagon adjacencies (see Figure 30.19), which makes its high stability plausible. The chlorinated C50 cage has also D5h symmetry (Xie et al., 2004), i.e., the D5h C50 cage has

obviously a preferred valency of 10. This can simply be explained by the structure of the D5h C50 cage. This cage can be viewed as two corannulene frameworks, glued together by five “ethylenic” C2 units along the equatorial belt of the cage (see Figure 30.20). Such configuration is highly strained along the contact between the belt and the corannulene frameworks. Bonding of ten atoms (e.g., Cl) to the five C2 units can stabilize the cage in two ways. First, they decrease the strain energy of the cage by pyramidalization of the equatorial carbon atoms by sp3 hybridization, and second they remove the carbon atoms of adjacent pentagons out of the π system, and also provide in this way an electronic stabilization. The optimized structure of C50H10 is shown in Figure 30.21. Similar to C36, the valences of C50 could also act as intermolecular bonds, and in this way provide the possibility for the formation of C50 -based fullerides. There is one topological possibility for a single-bonded dimer of C50 (Figure 30.22). This dimer is expected to be instable, as two electrons are taken from the two π systems, resulting in either a biradical or a zwitterion. Among several possibilities for the connection of two C50 cages, Zhechkov et al. (2004) found a most stable double-bonded dimer, as shown in Figure 30.23. The calculated dimerization energy (Edimer = −486 kJ/mol) for the most stable dimer is about twice as high as for (C36)2 and even ∼16 times higher than for the C60 dimer. Consequently, stable oligomers are expected. Zhechkov et al. (2004) investigated the possibilities for stable oligomer structures. As the number of

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Density-of-states

Density-of-states


–4

–3

–2 –1 Energy (eV)

1

0

–5 (b)

–4

–3

0

1


0

1

Density-of-states

–5 (a)

–5 (c)

–4

–3


FIGURE 30.18 Calculated DOS profi les for supergraphite (a), rhombohedral (b), and hexagonal (c) structures of solid C36 (from top to below). The energy, E, in the DOS curves is given relative to the Fermi energy. (From Heine, T. et al., Solid State Commun., 111, 19, 1999. With permission.)

FIGURE 30.19 Most stable “classical fullerene” C50. (From Zhechkov, L. et al., J. Phys. Chem. A, 108, 11733, 2004. With permission.)

FIGURE 30.20 C50 decomposed into corannulene and ethylene units (Zhechkov et al., 2004). For clarity only the forefront of the structure is shown. That is, from the five ethylene units two can be seen completely, one partially, the other two on the back are not shown. (From Zhechkov, L. et al., J. Phys. Chem. A, 108, 11733, 2004. With permission.)

possible oligomer structures is exceedingly large, they restricted their studies to the energetically most preferable connectivity patterns of dimer structures. As for the dimer, the bare trimer with single bonds has a small HOMO–LUMO gap and hence is of either radicaloid or zwitterionic character. On the other hand, the hydrogen-saturated trimer has a large gap. Structure and electronic structure of each cage unit of the saturated single-bonded trimer do not differ considerably from C50H10, and this bonding allows polymerization in an infinite variety of structures, most likely toward amorphous

polymers. Oligomers up to the hexamer with dimer bridges, which are responsible for most fullerene solids so far observed and discussed (Menon and Richter, 1999; Collins et al., 1999a; Menon et al., 2000) were studied more in detail (Zhechkov et al., 2004). The energetically favored dimer bridge type has been found in the two most stable trimers. This trend continues with tetramers, pentamers, and hexamers, and a lowest-energy pathway from monomer to hexamer is found (see Figure 30.24). Besides being the most stable oligomers, these structures have also large HOMO–LUMO gaps and the tendency to prefer clustered

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1

3

2

FIGURE 30.21 C50H10 based on C50. (From Zhechkov, L. et al., J. Phys. Chem. A, 108, 11733, 2004. With permission.)

5

4

6

FIGURE 30.24 Minimal energy pathway from the monomer C50H10 (1), over the dimer (2), trimer (3), tetramer (4), pentamer (5) to the hexamer (C50)6H30 (6). (From Zhechkov, L. et al., J. Phys. Chem. A, 108, 11733, 2004. With permission.)

FIGURE 30.22 Single-bonded C50 dimer. (From Zhechkov, L. et al., J. Phys. Chem. A, 108, 11733, 2004. With permission.)

bonds between a C2 unit of the equatorial belt and two atoms of a corannulene unit are also possible. Hence, a much larger variety of bonding patterns is possible, but thermodynamically the formation of these bonds is less favored, as it does not saturate the highly reactive sites along the equatorial belt. This reduces the possibility of forming spontaneous, high-symmetry solids. Oligomerization and polymerization are energetically possible with single- and double-bridge links. While single-bonded links will lead to flexible intercage bonding, allowing a great manifold of possible structures, double-bonded oligomers have a lowestenergy pathway from the monomer over dimer to hexamer.

30.3 Trends, Comparison, and Conclusions

FIGURE 30.23 Most stable double-bonded C50 dimer. (From Zhechkov, L. et al., J. Phys. Chem. A, 108, 11733, 2004. With permission.)

structures. The fivefold symmetry of the monomer and the minimum energy pathway to the most stable hexamer structure suggest that it is impossible to form regular covalently bound C50 solids if only the chemically active connectivity sites along the σh plane of the molecule are used. The non-covalent interactions between the corannulene units are slightly attractive (6 kJ/mol from DFT-based calculations; Zhechkov et al. (2004). Covalent

In contrast to most of the fullerites based on C60 and higher fullerenes, those composed by small, non-IPR cages can be characterized by strong covalent interactions between the fullerene cages. The highly strained small fullerenes can be stabilized by the formation of strong intercage bonds. Th is bond formation is connected with the transformation of sp2 carbons to sp3 carbons for the atoms, which form the intercage bridges. The sp3/sp2 ratio in these fullerite structures decreases with cage size, and so does the cohesive energy and related quantities as elastic constants and also the mass density. The most strained and reactive fullerene C20 builds the crystal with carbon atoms whose hybridization is close to sp3. Such crystal has the highest density and hardness among other fullerene crystals. The bulk moduli of hyperdiamond C28, hyperlonsdaleit C28, and supergraphite C36 are much smaller than that of diamond. For a solid form of C36 cages the bulk modulus is even smaller.

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Oligomerization and polymerization give additional intercage bonds, and hence additional binding energy, but also topological constrains, which lead to unfavorable deformations of the monomer units that affect both the π system by stronger pyramidalization and the σ framework. The reaction energy of a further step in polymerization depends mainly on the type of additional intercage bond if this step does not involve strong cage deformations. For this reason, the bonding types found in dimers, oligomers, and solids are usually the same. For the total stability, the existence of a stable phase has to be examined by evaluating the trade-off of cohesive intercage energy (high for small, low for large cages) with the stability of the bare cage (low for small, higher for large fullerenes). It is not clear if fullerites can be formed directly from the soot, as the high reactivity of small cages limits their lifetime to the microsecond scale. Experimental material is scarce, but there is a unique principle for covalent intercage bonds that can stabilize the highly strained small fullerenes. Therefore, the authors feel confident that there are a number of “hidden” structures or materials whose uncovering will add to the manifold of carbonbased structures considerably.

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