## Abstract

The Stone–Wales (SW) transformation, or carbon-bond rotation, has been fundamental to understanding fullerene growth and stability, and *ab initio* calculations show it to be a high-energy process. The nature and topology of the fullerene energy landscape shows how the *I*_{h}-C_{60} must be the final product, if SW transformations are fast enough, and various mechanisms for their catalysis have been proposed. We review SW transformations in fullerenes and then discuss the analogous transformation in graphite, where they form the Dienes defect, originally posited to be a transition state in the direct exchange of a bonded atom pair. On the basis of density functional theory calculations in the local density approximation, we propose that non-equilibrium concentrations of the Dienes defect arising from displacing radiation are rapidly healed by point defects and that equilibrium concentrations of Dienes defects are responsible for the divergent ultra-high-temperature heat capacity of graphite.

This article is part of the themed issue ‘Fullerenes: past, present and future, celebrating the 30th anniversary of Buckminster Fullerene’.

## 1. Introduction

Within a year of fullerenes being discovered, Kroto *et al.* [1] was the famous theory paper that discussed how C_{60} isomers could be interconverted through rotations of bonded atom pairs about the centre of their bond [2] in what has become known as the Stone–Wales (SW) transformation. The paper underlined the problem that the process is forbidden by the Woodward–Hoffman rules [3] for pericyclic reactions if treated on a par with standard organic reaction chemistry, and would be difficult even if allowance is made for the extreme conditions of fullerene synthesis and condensation. This was an obstacle for understanding the experimental observation of the *I*_{h} isomer and none other, as proved by the first C_{60}^{13}C NMR [4], which comprised a single line arising from a single chemical environment shared by all atoms. The fact that the *I*_{h} isomer has global thermodynamic stability for C_{60} was understood in terms of the isolated pentagon rule (IPR), although a strong indication was given in [2] that many other isomers are energetically similar to the *I*_{h} one and should be observed.

The isomers of C_{60} were enumerated [5] and it was apparent that the transformation from an arbitrary isomer to the *I*_{h} isomer must proceed through the *C*_{2v} isomer, so most calculations focused on this step. The *ab initio* calculations gave two different saddle points: symmetrical and asymmetrical involving sp^{3} carbon [6], but both still with very high activation barriers (respectively, 8.4 and 6.9 eV, taking the highest level of theory at the time: 4s2p1d/BLYP and the largest C_{60} fragment). The asymmetrical transition state was later dismissed as an artefact of lower levels of theory in a comprehensive review and revision [7], which concluded that the activation barrier was 6.9–7.3 eV through the symmetrical state.

Early work [8,9] by two of us (M.I.H. and C.D.L.) with others based on density functional theory (DFT) in the local density approximation (LDA) within the cluster AIMPRO code [10] showed that trace hydrogen could in principle be responsible for lowering the barrier and facilitate removal of IPR-violating isomers. The reason is that the ground state of C_{60}H is radical [11], pushing up the chemical potential of the molecule, while the transition state is relatively stabilized by a stronger C–H bond than in the ground state. There is also a binding between the H atom and more defective regions of C_{60} isomers, such as neighbouring pentagons, which attract it to these regions prior to catalysing the transformation.

This idea of a species which makes the ground state less stable and the transition state more stable led to studies of the structure and effects of the carbon adatom [8,12,13]. Using DFT and LDA, these confirmed that the *C*_{2v}–*I*_{h} transition of C_{60} is catalysed by an adatom, lowering the barrier from 4.7 to 2.9 eV [12], and that the adatom is attracted to regions which violated the IPR. The carbon growth path through various IPR-violating carbon cages, which resulted ultimately in *I*_{h}-C_{60}, could thus be regarded as *autocatalytic* [12]. Later inspection of another path found an even greater catalytic effect, lowering the barrier to 1.1 eV [13], and importantly carried forward the effect into graphene, which we will return to later.

While our group and others focused on the catalysis, Wales and co-workers [14] developed an elegant theory of the connectivity of energy valleys, which revealed why *I*_{h}-C_{60} is the sole C_{60} product. As an aside, we remark that the same theory, and the relative ease of the SW transformation when autocatalysed, has been invoked to explain why the hypothetical material pentagraphene is not likely to be stable [15].

Taken with the work on energy landscapes [14], there seems now to be little problem in reconciling the high barrier to the original SW transformation with the lack of appearance of non-*I*_{h} isomers. This has been shown very strongly by the most recent work in the area by Professor Sir Harry Kroto and co-workers [16], combining DFT and mass spectroscopy measurements. This has shown that fullerenes grow from atomic and clustered carbon, primarily C and C_{2}, through ‘a closed network growth (CNG) mechanism’ demonstrably for C_{60}, C_{70}, C_{76}, C_{78}, *C*_{84} and some endohedral metallofullerenes. The atomic carbon catalyses bond rearrangements after inclusion in the fullerene shell and the Endo–Kroto mechanism [17], whereby addition of C_{2} units across opposing sides of a hexagon inherently moves pentagons around concomitant with its inclusion. This work identifies the importance of *I*_{h}-C_{60} as being the ‘bottleneck’ in the growth of larger fullerenes, which nevertheless could grow without upper limit. Their approach used a hydrogen atmosphere as a mechanistic probe and showed that fullerene growth could still occur, thus ruling out any mechanism that did not proceed through closed cages. In addition, it showed that adatom SW catalysis could still proceed in the case of a partially hydrogenated adatom (CH), which is distinct from the early calculations of catalysis of SW transformation by H [8].

Also, in recent times, there have been several works confirming, theoretically, that the SW transformation in fullerenes is catalysed by hydrogen [18] and by adatoms [19].

Now, we return to graphene. Almost all early defect calculations claiming to represent graphite were, in fact, based on graphene, and this goes for nearly all calculations of the vacancy [20–23] and the analogue of the SW transformation in graphite [9,13,24]. In the case of finite graphene segments, these generally had hydrogenated edges and were thus polycyclic aromatic hydrocarbons (PAHs).

Initial theory work by Dienes in 1952 on diffusion in graphite considered vacancy diffusion and diffusion by direct exchange of neighbouring atoms [25]. Nowadays, hardly any of this work is regarded as quantitative, given the limited computer resources available and the empirical nature of the approximations used. In Dienes’ article, the saddle point (transition state) of direct exchange has the same bonding topology (four carbon cycles, alternately pentagon, heptagon, pentagon and heptagon) as what has become known as the SW defect in graphene. Since the first and landmark DFT study of diffusion [24] at the end of the decade that saw the dawn of reliable DFT/pseudo-potential calculations, it was apparent this was a metastable state and not a saddle point. Nevertheless, there has recently been a rationalization of nomenclature proposed [26], which recommends that this structure be known as a Dienes defect, which we follow here, while retaining the SW nomenclature for the transformation process.

There is now good experimental evidence for the existence of Dienes or SW defects in graphene from aberration-corrected electron microscopy [27,28] and theory confirms that they have properties and activation problems similar to fullerenes [13,29]. Note that this includes theory up to the quantum Monte Carlo level [29]. Furthermore, theory also confirms the same catalytic effect in graphene of adatoms [13,30] and of hydrogen [31–33]. One important difference in graphene compared with fullerenes is the buckling of the carbon network that it causes [30]—this is something that was missed in [13], which employed a flat PAH as its model of graphene.

It is striking how the understanding of graphene in the electron microscope has evolved from simple evidence of existence, to the beginnings of quantitative understanding of defect motion and lifetimes [34]. So far, it appears that the catalytic mechanisms of Dienes defect destruction in the electron microscope are not sufficient to explain their unexpectedly low lifetime, and furthermore, it is enigmatic that their creation at accelerating voltages below the displacement threshold (60 keV) appears little different from that at the threshold (80 keV). For both of these, electronic excitation mechanisms have been suggested [34].

There is a great deal of interest in the radiation damage of graphite, because of its application as a moderator in gas-cooled reactors, particularly the advanced gas-cooled reactors which are operational in the UK [35]. Neutron moderation proceeds by the absorption of kinetic energy in atomic displacement followed by diffusion of vacancies and interstitials [36], but creation of the Dienes defect can also arise, as has been clearly shown in molecular dynamics simulations [37]. Therefore, the question arises as to what the mechanism is for Dienes defect annihilation, given that on its own it is metastable with a large activation barrier against elimination, as discussed earlier.

Clearly, the catalytic effect of the carbon adatom can substantially lower the barrier for the removal of a Dienes defect in fullerene and graphitic systems. Therefore, the stability of a population of Dienes defects will depend on both the concentration and mobility of this intrinsic defect. According to a number of *ab initio* studies, in suspended single-layer graphene, an adatom (interstitial) can migrate through the structure with relatively low barriers (0.3–0.5 eV [28,38,39]). However, in the case of the three-dimensional graphite structure, the lower-energy ‘spiro’ configuration of the self-interstitial in between adjacent graphene layers [9,40] significantly decreases the interstitial mobility: the activation energy to leave this state is calculated to be in the range 1.2–2 eV [41,42].

It may be the case that in the graphite structure the lattice vacancy is more mobile than the interstitial, contrary to historical inferences [43]: both DFT calculations and experimental atomic resolution imaging determine the activation energy for vacancy migration to be in the range 1.0–1.2 eV [27,28,44,45] as opposed to the historical inferred value 3.1 eV [43]. Recent calculations have also revealed that the activation energy for monovacancy migration is extremely sensitive to the in-layer strain, with the barrier being halved or doubled with uniaxial strains of just 1–2% [46], and it was shown that this has a profound effect on vacancy aggregation [47].

In the light of the questions concerning the lifetime of Dienes defects under irradiation, we performed calculations to investigate the interaction of diffusing lattice vacancies with Dienes defects. These calculations demonstrate that the strain field created by the Dienes defect has a substantial effect on the mobility of a monovacancy in its vicinity, and that a mobile vacancy is attracted towards the areas of compressive strain that surround the Dienes defect. After a vacancy has interacted with a Dienes defect and formed a Dienes–vacancy complex, the barrier for the removal of the Dienes defect via a bond rotation is lowered to 1.2 eV, leaving only the isolated vacancy, which can then migrate. In this way, a single diffusing vacancy can ‘heal’ an unlimited number of Dienes defects.

Finally, we shall consider the role of the SW transformation in thermal energy storage: graphite has a very high heat capacity, which was first modelled in DFT in 2005 [48]. It is the basis of commercial heat storage systems operating up to 1000°C and, remarkably, above 2000°C graphite’s heat capacity has been reported to grow strongly [49–51]. In the third reference, the measurement is ultrafast, so the divergence temperature is much enhanced (around 3200°C). Speculations to date are that this is due to the reversible formation of lattice vacancies [51,52], but it has been suggested [52] that this is not compatible with the direct measurement of vacancy concentrations by etching, which gives below 10^{−10} per atomic site [53]. In addition, unless surfaces are involved, increasing the vacancy concentration must be achieved by thermal generation of Frenkel pairs, which have an activation energy greater than the Frenkel pair energy of 14 eV [43,44]. For this reason, we do not proceed with the vacancy here.

Here, we provide an alternative thermal energy storage mechanism, which is the reversible formation of Dienes defects. The barrier to their formation, which is 9.7 eV [44], is more accessible than that for Frenkel defects and there are reports that heating single-walled carbon nanotubes to 2000°C creates a measurable concentration of these defects [54], as well as indirect evidence, from reduced interlayer separation in multi-walled nanotubes, that they are being created where the temperature exceeds 2800 K [55].

These heat capacity effects involve the Dienes defect in thermal equilibrium, where, as the lowest-energy structural defect in graphite, they should be expected to dominate in relative isolation with little or no interaction with the higher-energy and much rarer Frenkel defects. Contrarily, in radiation damage, the creation of Dienes defects is rapid and they are very far from thermal equilibrium concentrations. Where the radiation is above the displacement threshold, as in a nuclear reactor, a strong flux of Frenkel pairs can be expected, so the interaction of Dienes defects with vacancies and interstitials is important. Thus, after describing the computational models we use, we present results and discussion separately for the two cases: Dienes defect out of thermal equilibrium and Dienes defect in thermal equilibrium. We finish with concluding remarks.

## 2. Material and methods

The DFT–LDA studies reported here use the AIMPRO supercell code [56–58], which expands the charge density in plane waves (up to an energy cut-off, *E*_{cut}). The LDA exchange correlation energy functional was parametrized by Perdew & Wang [59]. Gaussian basis functions centred at atom sites are used to construct the Kohn–Sham wave function. Each radial Gaussian is multiplied by polynomials *x*^{α}*y*^{β}*z*^{γ}, where *α*+*β*+*γ*≤*n*; so *n*=0 gives one function, *n*=1 gives four and *n*=2 gives 10. By analogy with atomic angular momentum labels, these are represented by symbols s, p and d, respectively. Typically, four or five exponents are used with different angular variations, and here pdpp is used, unless otherwise specified. Norm-conserving pseudo-potentials are based on the Hartwigsen–Goedecker–Hutter scheme [60] and a Monkhorst–Pack (MP) [61] *k*-point grid is used. Oscillations of charge density in part-filled degenerate orbitals during the self-consistency cycle are damped using a Fermi occupation function of *k*_{B}*T*=0.04 eV typically, where *k*_{B} is the Boltzmann constant and *T* is absolute temperature. Minimum-energy paths between states are determined using the climbing image nudged elastic band method (NEB) [62]. All cells were hexagonal AB-stacked graphite.

Vibrational modes are calculated for the fully optimized structures from the energetic double derivatives, with respect to displacement of each of the atoms. Each atom *a* is displaced a distance *ε* (usually 0.02 atomic units) along each axis *l*, then the charge density is recalculated, giving the forces on each atom *b* in each direction *m*. The dynamical matrix in this quasi-harmonic approximation is given by
2.1where *m*_{C} is the atomic mass of carbon, and this matrix yields the square of the vibrational frequencies, *ω*_{j}, after diagonalization.

The total lattice heat capacity at constant volume, *C*_{V}, at temperature *T* from all the modes is given by the sum
2.2where [63]. As the cell is 18 times the primitive cell, only Gamma point sampling of the phonon Brillouin zone is used.

The total vibrational entropy of a supercell is given by the expression
2.3The entropy due to the formation of a defect *d* in a supercell containing *n* atoms with total entropy *S*_{total} is then *S*_{f}=*S*_{total}−*nS*_{C}, where *S*_{C} is the entropy per atom for graphite, calculated using a defect-free supercell of comparable size to the one containing *d*.

The heat capacity contribution from reversible formation of defects of formation energy *E*_{f} and entropy *S*_{f} is obtained from (eqn 4.66 in [50] has a small typographical error which is corrected in our equation (2.3) and the definition has been altered from specific to molar heat capacity) [64]
2.4where *N*_{0} is the number of defect sites per mole, which for the Dienes defect is the number of bond centres, i.e. 1.5 times Avogadro’s number.

For the heat capacity of graphite, 72-atom cells are used with *E*_{cut}=200 Ha. These have been checked with a 288-atom 6×3×2 orthorhombic supercell, *E*_{cut}=248.8 Ha, an MP mesh of 6×6×3 *k*-points and *k*_{B}*T*=0.001 eV using the formalism set out in [58] and similarly for point defects in graphite.

A hexagonal 256-atom (8×8) graphite cell is used for the Dienes defect with full geometry optimization. The vibrational frequencies are obtained for 80 atoms, in the defect and up to third neighbours of the defect, using only the point (0.5, 0.5, 0.25) in relative coordinates of the electronic Brillouin zone, pppp basis and *E*_{cut}=175 Ha.

The stability of Dienes defects in graphene in the presence of point defects is studied in cells of 12×6 times the four-atom orthorhombic unit cell (288 atoms) with a vacuum gap of 1.3 nm between periodic images in the direction perpendicular to the plane. The MP mesh was 4×4×1. Energy minima were determined when the maximum force falls below 0.1 eV nm^{−1} and the NEB chains consisted of nine images. Activation energies for migration are determined by the height of the saddle point relative to the energy minimum.

## 3. Dienes defect out of thermal equilibrium

### (a) Results

The two-dimensionally optimized structure of the Dienes defect in a planar graphene supercell is shown in figure 1, coloured according to the in-layer strain. For the purposes of illustration, the out-of-plane displacements were excluded, enhancing the in-layer strain and making it closer to the case of graphite, where neighbouring layers inhibit out-of-plane displacements. Here, it can be seen that the darker areas emanating from the two five-membered rings either side of the defect are under compressive strain, while the lighter-coloured areas emanating from the seven-membered rings above and below the defect are under tensile strain. As shown in [46,47], the activation energy for vacancy migration is very sensitive to the strain. Its dependence on the strain tensor is complex, but it suffices to note here that mobile monovacancies will be more mobile in, and hence preferentially accumulate in, areas of compressive strain. In this case, it is then expected that, when a mobile monovacancy diffuses to the vicinity of the Dienes defect, it will be rapidly channelled to the five-membered rings. To demonstrate this, we performed a series of DFT calculations for the migration of a monovacancy towards the five-membered ring of a Dienes defect, as shown in figure 2, where the minimum-energy path and barriers between each state were determined with the NEB method and full geometry optimization was performed to include out-of-plane distortions.

As the vacancy moves from configuration 1 to configuration 5 (where it removes one of the atoms forming the fivefold ring), the energy monotonically decreases and each barrier to be crossed is less than 0.8 eV (substantially less than in the perfect lattice). The energies of the potential minima and the barriers are depicted in figure 3.

When the vacancy has reached configuration 5, the Dienes defect can be healed via a bond rotation (of the atom pair highlighted). This transition is shown in figure 4 and is calculated to have a barrier of 1.2 eV, substantially lower than the 5 eV barrier to remove the isolated defect.

The NEB reaction paths for the catalysed and uncatalysed paths are shown in figure 5, showing directly the strong catalytic effect. In these energy profiles, the formation energy of the vacancy is not included, because overall it is unchanged in the reaction. Before elimination of the Dienes defect, the Dienes defect–vacancy complex (figure 5, left) is strongly bound, by approximately 3 eV.

### (b) Discussion

Given a non-equilibrium concentration of Dienes defects at room temperature or even reactor temperatures, how quickly should it fall back to its normally negligible equilibrium concentration? In the case of the electron beams below the displacement threshold, the annihilation barrier should be in the region of 7 eV [13]. This would make the destruction rate negligible and the concentration stable, but the observation is that this is not the case for graphene, and an unspecified electronic mechanism has been invoked [34]. Above the displacement threshold, it is already known that the adatom on graphene will catalyse removal: it is a highly mobile species even at room temperature and is attracted to, binds with and annihilates the Dienes defect [13,30].

The vacancy in graphene is also mobile, but not at room temperature except in regions of high compressive strain [47], so it is not expected to make an important contribution to the annihilation of the Dienes defect.

Nevertheless, in the case of graphite, the interstitial atom is relatively immobile compared with the vacancy and here vacancy-catalysed destruction should be important.

The calculations presented are based on graphene: large-area cells are needed because of the Dienes defect strain field, and a full calculation for graphite of say a four-layer cell would require more than a thousand atoms. The weak interlayer interactions are not expected to affect the results strongly, and the reduced out-of-plane distortions in graphite will only serve to enhance in-layer strains (to approach those of figure 1) and hence increase the speed of vacancy–Dienes defect complexing and annihilation.

What is clear from the strain map (figure 1) and the strain sensitivity of vacancy migration is that vacancies in the vicinity of the Dienes defect should be expected to drift towards the defect along the axis of the two rotating atoms (i.e. the horizontal direction of figure 1) and then catalyse its destruction. Figure 2 shows this, where the axis is parallel to the bond between the two red atoms and no longer horizontal. The vacancy swings between alpha and beta sites as it approaches the Dienes defect, culminating at a pentagon site (configuration 5, figure 3). The energy landscape is such that the highest barrier traversed is approximately 0.8 eV (from configuration 2 to 3) giving mobility around room temperature. Then it takes an activation of 1.2 eV to annihilate the Dienes defect. The last, rate-limiting step should be occurring at a rate of 1 s^{−1} around 160°C. In regions where there is an overall compressive strain field, these energies are expected to reduce strongly.

These data for graphene should not be dissimilar to those in graphite, and they suggest that Dienes defects should not survive during radiation damage of graphite at typical reactor temperatures, which are in the region of 400°C. This is contrary to the expectation if destruction were not catalysed by point defects.

## 4. Dienes defect in thermal equilibrium

### (a) Results

Heat capacity calculations relate to isolated Dienes defects in thermal equilibrium. The original calculations [64] of heat capacity, based on a 72-atom cell with numerical differentiation as in equation (2.1) for the dynamical matrix and then diagonalization to produce vibrational frequencies at the Gamma point of the Brillouin zone, gave the saturating curve of figure 6 for the lattice heat capacity from equation (2.2). The agreement with measurement below 2000 K is clearly excellent and the 288-atom cell calculations gave results which fall within the line width of the curve of this plot.

The AIMPRO DFT formation energy for the Dienes defect was found to be 5.1 eV and its entropy rises to 7.6*k*_{B} at 3500 K. Employing equation (2.4) gave the additional contribution from reversible formation of Dienes defects, which produced the ‘total’ heat capacity of figure 6. The experimental data for figure 6 were taken from Sheindli *et al*. [49] and the agreement is good.

Further checks on this calculation included checking the formation energy of the Dienes defect in 288-atom cells, obtaining 5.14 eV, very close to the value used in figure 6. The uniqueness of the Dienes defect proposal was examined by calculating the properties of the spiro-interstitial, which yielded *E*_{f}=5.85 eV and *S*_{f}=5.8*k*_{B} at 3500 K. The full temperature dependence of *S*_{f} is given in figure 7, justifying the use of equation (2.4) with temperature-independent entropy for temperatures above 2000 K. The associated contribution to heat capacity is given as a broken line in figure 6 and it does not account for the experimental data.

### (b) Discussion

Looking at the heat capacity measurements, if the activation entropy is the same as the formation entropy, and the activation energy is 9 eV, then the rate of formation of Dienes defects is about 1 s^{−1} at 2700 K, given an attempt frequency of 10^{13} Hz. This does appear consistent with at least the possibility of achieving an equilibrium concentration for the temperatures at which the divergent heat capacity is reported. The heat capacity results would be dependent on the duration of measurement. In this context, it should be noted that not all experimental reports exhibit this ultra-high-temperature feature, and sometimes it appears at a higher temperature.

Our calculation estimates the Dienes defect concentration to be 10^{−7} per atom at 2500 K, rising to 10^{−3} at 3600 K, and so, for this interpretation of heat capacity to hold, it would require that the etching of Dienes defects does not occur in the same way as vacancies, otherwise they would be detected by the technique used in [53], which sets an upper limit on their concentration which is not compatible with these high values.

The thermodynamic parameters for the spiro-interstitial do not appear close enough to model the divergent heat capacity and, as with the vacancy, production in the bulk would have to proceed through Frenkel pair formation with an activation of at least 14 eV, so, unless there is a strong contribution of graphene edges, it will be ruled out kinetically.

## 5. Conclusion

The impact of the discovery of fullerenes is still being felt in the nanoscience of carbons and in revivifying the science of graphite. Here, we have reviewed, briefly and selectively according to our purpose, the development of thinking about fullerene isomerization, including catalysis by trace hydrogen and by carbon atoms. The concepts appear to be transferrable from fullerenes, through carbon nanotubes to graphene and graphite, although nanotubes have not been the focus of this work. It is this unifying thread through carbon materials that illustrates the profound nature of the discovery of fullerenes [1] and its ramifications.

In this article, we have shown that catalysis concepts offer explanations for the absence of evidence for Dienes or SW defects in graphite irradiated by neutrons at reactor temperatures. In addition, we have provided new data on the mobilization of vacancies around these defects and their catalytic effect on defect removal.

The final section challenged the accepted wisdom that the divergent heat capacity of graphite at ultra-high temperature arises from reversible vacancy formation, and gave evidence for the reversible formation of Dienes defects instead, supported by the direct observations of these defects in carbon nanotubes.

## Authors' contributions

M.I.H. drafted the introduction, discussion and conclusion and compiled the results and methodology sections, with contributions from G.L.H., who performed the original heat capacity and entropy calculations; C.D.L. confirmed, improved and added to these calculations, including the spiro-interstitial, and T.T. performed the vacancy-catalysed destruction of Dienes defect calculations. All authors commented on and approved the subsequent and final revisions.

## Competing interests

We have no competing interests.

## Funding

M.I.H., G.L.H. and C.D.L. were funded by British Energy Generation and M.I.H. and C.D.L. are funded by EDF Energy Generation. T.T. is funded by Innovate UK.

## Disclaimer

The views expressed in this article are those of the authors and do not necessarily represent those of the sponsors.

## Acknowledgements

The authors are grateful to Prof. P. R. Briddon and Dr M. J. Rayson of the University of Newcastle for the provision of the AIMPRO code. M.I.H. thanks Prof. C. P. Ewels for discussions and Prof. Sir Harry Kroto F.R.S. for friendship, inspiration and encouragement over the last twenty years.

## Footnotes

One contribution of 12 to a theme issue ‘Fullerenes: past, present and future, celebrating the 30th anniversary of Buckminster Fullerene’.

- Accepted July 6, 2016.

- © 2016 The Author(s)

Published by the Royal Society. All rights reserved.