Before the discovery of fullerenes, the known allotropes of carbon were diamond and graphite. Both have crystalline structures, in which carbon atoms are neutral atoms with a tetravalent bonding structure [1]. In 1970, E. Osawa predicted the existence of a sphere formed with 60 carbons, the first fullerene [2]. These were later experimentally produced in complex conditions involving vaporizing graphite with laser irradiation [3]. Fullerenes are composed of carbon atoms forming nearly spherical molecules, with a solid surface around an empty volume [4] [5] [6]. This surface has nearly the same bonding structure as graphene [7] resembling a rolled sheet of graphene. Numerous characterizations where performed to understand the exact arrangement of carbons, bring to light hexagonal rings of carbon with a few pentagon rings [8] [9] [10] [11] [12]. In graphene, fullerenes, and carbon nanotubes alike, each carbon atom has three neighbors. The generally accepted bonding rules impose two single bonds and one double bond [13] [14].

In previous work, G. Auvert has proposed that only electrically charged carbon atoms can erect three bonds, whereas neutral carbon atoms may only have 2 or 4 bonds [15] [16] [17]. The present paper focuses on the consequences of carbon bonding capacities on the bonding pattern in fullerenes, carbon nanotubes and closed-end carbon nanotubes. It will be considered that the most stable configurations imply that most carbons bear an electrical charge with three bonds and sometimes the possibility that some bonds are broken leaving some neutral carbons with two single bonds.

For the sake of example, the authors have chosen to focus on fullerenes having a regular repartition of pentagons, using the Euler characteristic [4] [18]. The present article hence starts by describing the geometrical repartition of carbon atoms in fullerenes that can be inscribed in an icosahedron. Then, connections rules between neutral and charged carbons are applied, leading to bonding patterns and their two or three-dimensional representations. A focus is later made onto carbon nanotubes and caped nanotubes. In the discussion, classical and recent connection rules are compared, as well as implication on endohedral species. The special case of C20 is also addressed.

In this paper, C20 or C120 represent molecules of fullerenes with carbon atoms.

Note: To draw fullerenes and nanotubes, two software are used [19] [20].

Graphene, fullerene, and carbon nanotubes are composed of equidistant carbon atoms, measured to be between 0.134 and 0.154 nm [14] [21]. Moreover, in graphene layers and in nanotubes, carbons form regular hexagonal structures [13]. In spherical fullerenes however, carbons form two different types of rings: pentagonal and hexagonal rings [5]. We know from Euler’s theorem that only twelve pentagons are always present in fullerenes (the 12 Pentagon Theorem) [4] [5] [6] [18]. To obtain a regular repartition of these twelve pentagons, carbon atoms can be positioned on the facets of a virtual convex regular icosahedron [22]. As illustrated in Figure 1, this specific icosahedron has twelve vertices forming twenty equilateral triangles.

To maintain the regularity of the atom structure, carbon atoms must be regularly disposed over the twenty triangular facets.

In scientific literature, C60 is known to be a truncated icosahedron [3].

IUPAC define the suffix -ene as designing structures containing carbons atoms that are bonded with three atoms, over four covalent bonds that are not fully distinct [23] [24]. This is usually represented as two single bonds and one double bond. With three neighbors to each carbon atom, fullerenes belong to the -ene group and the classical bonding structure prevails in literature as shown for C60 in [3] [4] [25].

In this chapter, the authors use an alternative rule previously published under the name “even-odd” rule [15]. Applied to carbon atoms, it implies that:

· two neighboring carbon atoms can only be connected through a single covalent bond,

· carbon atoms bearing a charge can have only three covalent bonds,

· neutral carbon atoms on the contrary can only erect two or four distinct covalent bonds. In fullerenes, because carbon atoms are organized in planes (hollow molecules) neutral carbon atoms have two bonds,

· connected charged carbons bear opposite charges.

In fullerenes, carbons atoms are mainly organized in hexagonal rings, each with three neighbors. Following the above rule, each hexagon is an alternating pattern of positively and negatively charged atoms, with each atom bonded to their three neighbors. This alternating pattern is interrupted by the presence of pentagons, solved by a neutral carbon atom with two bonds only. Each neutral carbon atoms faces another neutral atom, resulting in a void, a missing bond. The alternance of charges ensures the overall neutrality of the fullerene compound. For a better understanding, Figure 4 illustrates the “Triple-series” fullerene bonding pattern with the C60, the smallest fullerene of the “Triple-series”.

C60 in Figure 4 is very stable, thanks to the uniform charge repartition and the bonding mesh in hexagons. Said differently, a continuous path with alternating positively and negatively charged atoms can be followed without interruption all around the circumference of the fullerene.

Figure 5 shows three-dimensional representations of the three smallest fullerenes in the Triple-series. Uncharged carbons have two bonds and form facing pairs along some of the edges of the icosahedron.

It is possible to calculate the number of unbonded carbon atoms in fullerenes as a multiple of n: 6 * n. In Table 2, the number of unbonded carbon pairs, i.e. pair of uncharged atoms, in the first fullerenes of the “Triple-series” is reported. Uncharged carbon atoms thus represent (20/n)% of the overall number of atoms in a fullerene of the “Triple series”. This decreasing factor favors stability of the compound even for large values of n.

We will now turn our attention to the bonding pattern of nanotubes and caped nanotube.

Nanotubes have been intensively studied since 1952 [26] [27] [28]. They are empty cylinders [29] with an external surface composed of carbon atoms organized like in graphene [13]. Ends of nanotubes have been experimentally observed, with evidence that nanotubes are sometimes caped [30] [31]. We will explain how semi-hemispheric fullerenes can close a nanotube of the same width and how the pattern observed earlier in fullerenes is still applicable. The usual bonding structure involves two single-bonds and one double-bond [30].

Let us concentrate on nanotubes for which the total number of atoms can be expressed as C(10 * n * m), where m is the number of layers, 10 * n the number of carbon atoms in one layer and nthe same number as used above in fullerenes. Views from the tip and side of a tube with ten-atom transverse layers (n = 1) and eight planes (m = 8) are drawn in Figure 6.

In his 1987 article, H. W. Kroto has pointed out that C70, the other most stable fullerene, is formed by separating two halves of a C60 by a ring of ten extra carbon atoms [25]. These extra atoms can be likened to a short nanotube. Extending the rationale, the number of layers could be of any size, forming longer nanotubes (C80 with m = 2, C100 with m = 4 …). In other words, nanotubes of chiral indices (5, 5) are naturally caped by half C60. Having observed this, the bonding pattern described in the above paragraphs can be used to close nanotubes, as illustrated in Figure 7. The body of the nanotube is composed of alternating positively and negatively charged carbons, connected with three bonds. At each end, six neutral carbon atoms are bonded only twice.

Discussions focus on four topics. First why a neutral atom is the only solution for carbon pentagon rings; then a quick note on the structure of C20 fullerenes, which doesn’t belong to the “Triple-series”; next a thought on endohedral species and last with a comparison of different representations of the C540 fullerenes.

The present article has proposed a bonding pattern for regular fullerenes like C60 and bigger fullerenes of type C(20 * 3 * n²), obtained by applying the recently published even-odd rule. As an alternative to the classical model, the pattern involves alternating positively and negatively charged carbon atoms, then a few uncharged and partially bonded atoms close to the pentagon rings. First, we have modelled C60 and other regular fullerenes as being inscribed in an icosahedron form. Then, we derived a family of fullerenes belonging to a specific geometrical pattern that preserves equidistance between carbon atoms, which we named the Triple-Series, of which C60 is the smallest. The same bonding pattern applies to all fullerenes of this family, satisfying both overall compound neutrality and local stability, with a maximum of 20% of atoms partially bonded. Next, we used the notion that nanotubes can be capped with half-fullerenes of the Triple-Series to derive their bonding pattern. C70 is a special case of this sort and the bonding pattern points toward good stability. Each elongation by ten carbon atoms is also stable. In contrast, the same bonding rules were applied to C20, not in the Triple-Series, and the resulting pattern displays many missing bonds, indicating compound instability. We conclude that the even-odd rule can be used to explain why C60 and its derivative are the most common fullerenes and why the C20 is unstable unless stabilized.

The authors declare no conflicts of interest regarding the publication of this paper.

Auvert, G. and Auvert, M. (2020) Single Covalent Bonding Structure in Fullerenes, Carbon Nanotubes and Closed Nanotubes. Open Journal of Physical Chemistry, 10, 183-195. https://doi.org/10.4236/ojpc.2020.103011