This chapter reviews models for the growth, contraction, and fragmentation of fullerenes. The models for the growth process that have been proposed for fullerenes are discussed in §6.1. Models for the addition and subtraction of a single hexagon C2 to and from a fullerene C„c are considered. Because of the great importance of mass spectrometry to growth, contraction, and fragmentation studies of fullerenes, a brief review of mass spectrometry techniques is given in §6.2. Special attention is given to the experimentally observed stability of C60 and C70 in §6.3. Experiments showing the accretion and contraction of fullerenes using photofragmentation and ion collision techniques are reviewed in §6.4. Finally, some discussion is given in §6.5 of molecular dynamics models for fullerene growth.

Referring to the mass spectra for fullerenes (see Fig. 1.4), we see that the C60 and C70 species have the highest relative abundances and therefore are observed to have the greatest relative stability of the fullerenes. Other fullerenes with relatively high stability are C76, C7g and C84 [6.1-3], but C7g and Cg4 have many isomers that obey the isolated pentagon rule (see §3.2). On the basis of Euler's theorem, a fullerene shell has 12 pentagons and an arbitrary number of hexagons (see §3.1). The addition of one extra hexagon to a fullerene requires two additional carbon atoms. Thus, all closed-shell fullerenes have an even number of carbon atoms (see §3.1), and therefore it would seem logical to go from one fullerene to another by the addition of a C2 unit. However, to go from one highly stable fullerene, such as C60, to the next highly stable fullerene, such as C70 by adding C2 clusters, it is necessary to overcome a large potential barrier [6.4], Therefore it has been suggested that, during synthesis at high temperature, stable fullerenes form from the coalescence of larger units such as C]3 or C24 or C30, perhaps in the form of rings, which then add or emit a few C2 clusters to reach the highly stable fullerenes [6.5,6],

To achieve the most stable molecular configurations for a given number of carbon atoms nc, fullerenes must rearrange the locations of their carbon atoms to separate the pentagons from one another in order to minimize the local curvature of the fullerenes [6.7]. In §6.1.1 we discuss first the most common mechanism that has been proposed for the rearrangement of carbon atoms on a fullerene molecule, the Stone-Wales model [6.8], followed by a generalization of the Stone-Wales mechanism for the rearrangement of pentagons (5 edges), hexagons (6 edges), and heptagons (7 edges) and for the formation and movement of pentagon-heptagon (5,7) pairs on the surface of fullerenes. Models for the growth mechanism for fullerenes are then discussed.

In describing the growth of fullerenes, the formation of various fullerene isomers (see §3.2), or the growth of carbon nanotubes or nanospheres, the Stone-Wales transformation is often discussed [6.8,9]. According to this model, the pyracylene motif [see Figs. 3.2(c) and 6.1] is, for example, transformed from a vertical orientation to a horizontal orientation, by breaking the bond connecting two hexagons and converting the two vertices on the horizontal line to two vertices on the vertical line, without modifying the bonds on any but the four polygons of the pyracylene structure, as illustrated in Fig. 6.1 [6.10],

The Stone-Wales transformation has been used by many workers as a method for moving pentagonal faces around the surface of fullerene structures, thereby providing a mechanism for achieving the maximum separations between all pentagonal faces and stabilizing the structure. The most stable structure is one for which the polygon arrangement best satisfies the isolated pentagon rule (see §3.1), or more precisely, the arrangement with minimal energy. The Stone-Wales model was first used to maximize the separation between the pentagonal faces in C28 [6.10]. Although the initial and final states for the Stone-Wales transformation may be relatively stable, corresponding to local energy minima, the transition between these states involves a large potential barrier [6.10] estimated to be ~7 eV, on the basis of cutting the two a bonds [6.11] required to move a pentagon. This value

Fig. 6.1. (a) Stone-Wales transformation of the pyracylene motif between the vertical and horizontal orientations, (b) The embedding of the pyracylene motif (shaded) into a fullerene shell [6.10], for the potential barrier is considerably higher than what a physical system can surmount under actual synthesis conditions (3000-3500°C or ~0.3 eV).

Even though the Stone-Wales mechanism may not be the actual physical mechanism that conveniently interchanges pentagons and hexagons on fullerene surfaces, the model is useful geometrically to generate all fullerene isomers obeying the isolated pentagon rule for fixed nc [6.10, 12]. The Stone-Wales mechanism thus provides a mechanism for moving specific polygons around a fullerene surface, as illustrated below in the discussion of growth mechanisms. Figure 6.2 shows an extension of the Stone-Wales model for the basic interchange of pentagons, hexagons, and heptagons among four polygons without changing the number of carbon atoms. Although fullerenes contain only hexagons and pentagons, as discussed in connection with Euler's theorem, heptagons are observed experimentally in connection with the formation of a concave surface (see §3.1) and in the formation of bill-like structures for carbon nanotubes (see §19.2.3). In the growth process, heptagons can be used to move pentagons around the surface of fullerenes, by forming a (7,5) heptagon-pentagon pair, moving the pentagon, and then annihilating the (7,5) pair.

Each pentagon introduces long-range positive convex curvature to the fullerene shell corresponding to an angle of tt/6, while a heptagon introduces a similar long-range concave curvature. On the other hand, a (7,5) pair produces only local curvatures, since the long-range effects of the pentagon and heptagon cancel. Thus, the energy to create a (7,5) pair should be relatively low. Furthermore, the introduction of the (7,5) pair does not

Fig. 6.2. Diagrams describing basic steps for the Stone-Wales transformation and showing interchanges between polygons within a four-polygon motif that includes pentagons (5), hexagons (6), and heptagons (7), where the number in parentheses indicates the number of edges of a polygon, (a) Two pentagon movements as in the standard Stone-Wales transformation, (5,6,5,6)->(6,5,6,5) without creation of a (7,5) pair, (b) Creation of two (7,5) pairs according to the (6,6,6,6)-» (7,5,7,5) transformation, (c) Creation of a single (7,5) pair, based on (5,6,6,6)-s-(6,5,7,5). (d) Motion of a (7,5) pair, based on (5,7,6,6)-+ (6,6,7,5) [6.13],

violate Euler's theorem, so that a fullerene C„c containing / faces including a (7,5) pair would in addition have 12 pentagonal faces and / — 14 hexagonal faces. Figure 6.2(a) shows the conventional Stone-Wales transformation for the rearrangement of the pentagons in the pyracylene motif [see Fig. 3.2(c)] [6.13]. Figure 6.2(b) shows the creation of two (7,5) pairs from four hexagons in the arrangement shown in the figure, or equivalently the annihilation of two (7,5) pairs into four hexagons. Figure 6.2(c) shows the creation (or annihilation) of a (7,5) pair in the presence of a pentagon by the extended Stone-Wales mechanism introduced in Fig. 6.2(b). Correspondingly, Fig. 6.2(d) shows a method for moving a (7,5) pair around the surface of a fullerene, thereby also moving a pentagon around the surface. The (7,5) pair mechanisms shown in Figs. 6.2(c) and (d) assume no dangling bond formation [6.13], Although there is no experimental evidence for heptagons on fullerene surfaces, examples of heptagons have been reported to be responsible for introducing bends and diameter changes in

sorption of a C2 cluster near two pentagons without generating any new pentagons or heptagons. The new hexagon is denoted by 6. The relative distance between the two pentagons, expressed as a linear combination of unit vectors of the honeycomb lattice, becomes closer after C2 absorption in (a) (2,0)—>(1,0)

formed to accommodate the C2

and (b) (2,l)-+(2,0), respectively. Dotted lines indicate new bonds unit, and A denotes bonds that are

sorption of a C2 cluster near two pentagons without generating any new pentagons or heptagons. The new hexagon is denoted by 6. The relative distance between the two pentagons, expressed as a linear combination of unit vectors of the honeycomb lattice, becomes closer after C2 absorption in (a) (2,0)—>(1,0)

formed to accommodate the C2

and (b) (2,l)-+(2,0), respectively. Dotted lines indicate new bonds broken [6.13], unit, and A denotes bonds that are

carbon nanotubes, as observed in transmission electron microscopy (TEM) experiments (see §19.2.3).

One model for the growth of fullerenes is based on the absorption of a C2 cluster, thereby adding one hexagon in accordance with Euler's theorem (see §3.1). Two possible methods for absorbing one C2 cluster into a carbon shell, restricted to contain only pentagonal and hexagonal faces, are shown in Fig. 6.3 [6.13]. The absorption takes place in the vicinity of a pentagon, where bonds are broken (indicated on the figure by A) and new bonds are formed (dotted lines). In order to preserve the three-bond connectivity at each carbon atom, the numbers of new bonds and deleted bonds for each vertex must be equal. The chemical bonds denoted by a,b,c,... (which connect carbon atoms within the cluster to carbon atoms outside the cluster) do not change during the C2 absorption (desorption) and thus only local bonds are changed by the C2 absorption process.

The pathway for the absorption of a C2 dimer by a C„c fullerene shown in Fig. 6.3(a) leads to two adjacent pentagons on the fullerene surface. This arrangement is energetically unfavorable because it does not satisfy the isolated pentagon rule. However, the absorption pathway shown in Fig. 6.3(b) results in a configuration C„c+2 that is consistent with the isolated pentagon rule and therefore could be used more effectively to model the growth of fullerenes. In Fig. 6.4, we show a fullerene hemisphere with six pentagons, with the pentagon labeled 6 in a central position. The relative separation between pentagons 2 and 3 in Fig. 6.4(a) corresponds to the (n, m) = (2,1) distance (see §3.2), which is active for absorbing a C2 unit in accordance

Fig. 6.4. A C2 cluster is added to two hexagons, each of which is bounded by a pentagon. The process adds one hexagonal face (dark shading), two carbon vertices, and three edges, thereby satisfying Eu-ler's theorem and yielding the fullerene shown in (b) [6.13], The dashed lines represent the six new bonds that form to replace the three broken bonds [6.14], with the mechanism shown in Fig. 6.3(b). When we introduce a C2 unit between two pentagons according to the process of Fig. 6.3(b), a new hexagon is introduced [dark shading in Fig. 6.4(b)]. If we rotate the picture in Fig. 6.4(b) by « 72° around the axis of the hemisphere (i.e., an axis normal to pentagon 6), it is seen that the resulting shape of the hemispherical cap is identical to that of Fig. 6.4(a), except for the symmetric addition of five hexagons, symmetrically placed with respect to pentagon 6, indicative of a screw-type crystal growth. Repeating the process shown in Fig. 6.4 five times, we return to Fig. 6.4(a), but with the addition of five carbon hexagons to the fullerene. The application of this growth mechanism to carbon tubules is further discussed in §19.3.

Energetically, the absorption of a C2 unit is exothermic because each carbon atom in the C2 cluster, having two sp-hybridized cr bonds and two sp-hybridized tt bonds, has a relatively high energy before C2 absorption. However, after C2 absorption, each of the two carbon atoms in C2 has three <j bonds and one 7t bond, representing a decrease in energy of ~4 eV per carbon atom [6.15]. In terms of the initial and final states, the absorption of a C2 dimer is therefore expected to be energetically favorable, as shown in Fig. 6.5, although the transition between C„c and C„c+2 requires surmounting a high (~7 eV) potential barrier [6.10] (not shown in Fig. 6.5), corresponding physically to the rearrangements of bonds that are required to make the transition, such as the bonds indicated schematically by Fig. 6.3(b).

It has been suggested that a corannulene-like carbon cluster (i.e., the same carbon atom organization as for C20H10 shown in Fig. 6.3) could be the nucleation site for the formation of a fullerene in a carbon plasma [6.17,18].

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