If a charged particle species of a collisionless plasma possesses a non-Maxwellian velocity distribution function, a short wavelength normal mode of the system may grow in amplitude. This is a microinstability; its theory is well described by the Vlasov equation. The purpose of this monograph is to describe in an accurate way the theory of damped normal modes and a limited number of microinstabilities that may arise in various space plasma environments.

The two words that best characterize the work described in this book are “limited” and “accurate.” In order to keep the discussion limited, I have chosen idealized, not observed, distribution functions. Many spacecraft have provided excellent observations of electron and ion distributions in the Earth's magnetosphere and nearby solar wind. The tremendous variety of these distributions makes it difficult to select a few for special representation. My choice here has been to use Maxwellian or bi-Maxwellian distributions with field-aligned drifts to represent some of the more important general free energy sources. Although the resulting instabilities may not correspond to any particular data set, I hope that each one represents the general properties of a very broad class of data.

To provide accuracy, I have followed the same procedure for each distribution function and plasma model. After assuming a zeroth-order distribution, I derive (or at least explicitly state) the associated dispersion equation without approximation. Because I deal with linear theory throughout this book, it is always straightforward to do this, although the algebra gets tiresome at times.

The structural form of geodesic domes, composed of pentagons and hexagons, played an important role in understanding the structure of carbon clusters. In this paper an analogy between geodesic domes and fullerenes is investigated. A brief survey is given of the geometry of geodesic domes applied in engineering practice, in particular of the geodesic domes bounded by pentagons and hexagons. A connection is also made between these sorts of geodesic domes and the mathematical problem of the determination of the smallest diameter of n equal circles by which the surface of a sphere can be covered without gaps. It is shown that the conjectured solutions to the sphere-covering problem provide topologically the same configurations as fullerene polyhedra for some values of n. Mechanical models of fullerenes, composed of equal rigid nodes and equal elastic bars are also investigated, and the equilibrium shapes of the space frames that model C28, C60 and C240 are presented.

Introduction

From visual inspection one can easily discover an analogy between the structure of C60 and the inner layer of the structure of the great U.S. pavilion of R. B. Fuller at the 1967 Montreal Expo. This analogy and other geodesic structures of Fuller were responsible for the name of C60: Buckminsterfullerene (Kroto et al. 1985). This is not the first time that Fuller's geodesic domes have helped researchers to understand the structure of matter. In the early 1960s Fuller's geodesic domes, especially his tensegrity spheres, inspired Caspar & Klug (1962) to develop the principle of quasiequivalence in virus research.

Small carbon grains are assumed to be the carrier of the prominent interstellar ultra violet absorption at 217 nm. To investigate this hypothesis, we produced small carbon particles by evaporating graphite in an inert quenching gas atmosphere, collected the grains on substrates, and measured their optical spectra. In the course of this work – which in the decisive final phase was carried out with the help of K. Fostiropoulos and L. D. Lamb – we showed that the smoke samples contained substantial quantities of C60. The fullerene C60 (with small admixtures of C70) was successfully separated from the sooty particles and, for the first time, characterized as a solid. We suggested the name ‘fullerite’ for this new form of crystalline carbon.

Introduction

The production of laboratory analogues of interstellar grains was the initial aim of our research. In the autumn of 1982 while one of us (D.R.H.) was a Humboldt Fellow at the Max Planck Institute of Nuclear Physics in Heidelberg we decided to study the optical spectra of carbon grains. We felt challenged by the intense, strong interstellar ultra violet (uv) absorption at 217 nm which it had been proposed was due to graphitic grains (see, for example, Stecher 1969). The arguments in favour of such carriers are based primarily on calculations of the absorption of small, almost spherical, particles which exhibit the dielectric functions of graphite (for more recent literature see, for example, Draine 1988). There had already been very early experimental attempts to produce graphitic smoke particles by almost the same technique that we later applied to C60 production (see, for example, Day & Huffman 1973).

The chemistry by which the closed-cage carbon clusters, C60 and C70, can be formed in high yield out of the chaos of condensing carbon vapour is considered. Several mechanisms for this process that have been proposed are critically discussed. The two most attractive are the ‘pentagon road’ where open sheets grow following the alternating pentagon rule and the ‘fullerene road’ where smaller fullerenes grow in small steps in a process which finds the buckminsterfullerene (C60) local deep energy minimum and to a lesser extent the C70 (D5h) minimum. A clear choice between the two does not seem possible with available information.

Introduction

The observation (Kroto et al. 1985) that the truncated icosahedron molecule, CBF60 (buckminsterfullerene), is formed spontaneously in condensing carbon vapour was greeted by some in the chemical community with some doubt. It seemed incredible that this highly symmetrical, closed, low entropy molecule was forming spontaneously out of the chaos of condensing high-temperature carbon vapour. We still believe that the formation of CBF60 in supersonic cluster beam sources must be a relatively minor channel, probably accounting for less than 1% of the total carbon. Thus when CBF60F was finally isolated from graphitic soot (Krätschmer et al. 1990), it came as a surprise that the CBF60 plus C70 yields were as large as 5%. Later yields have improved substantially, for example Parker et al. (1991) obtained a total yield of CBF60 of about 20% with total extractable fullerene yields totalling 44% from a carbon arc soot. Thus conditions can be found where CBF60 and fullerene formation in carbon condensation can hardly be called a minor channel.

C60 has not yet been detected in primitive meteorites, a finding that could demonstrate its existence in the early solar nebular or as a component of presolar dust. However, other allotropes of carbon, diamond and graphite, have been isolated from numerous chondritic samples. Studies of the isotopic composition and trace element content and these forms of carbon suggest that they condensed in circumstellar environments. Diamond may also have been produced in the early solar nebula and meteorite parent bodies by both low-temperature-low–pressure processes and shock events. Evidence for the occurrence of another carbon allotrope, with sp hybridized bonding, commonly known as carbyne, is presented.

Introduction

At the same time that buckminsterfullerene was being conceived as a molecule of possible astrophysical significance, a number of much older forms of carbon were about to enjoy a new lease of life because of their discovery as presolar grains in primitive meteorites. Ever since the 1960s, it has been recognized that carbonaceous chondrites were a host for noble gases of anomalous isotopic composition (Anders 1981). The carriers of a litany of components, enjoying names such as Xe(HL) (also called CCF-Xe), s-Xe, Ne-E(L), Ne-E(H), etc., were believed to be unidentified carbon species called C∂, Cβ, Cα and C∈ respectively, themselves exhibiting unusual or exotic isotopic compositions (Swart et al. 1983a; Carr et al. 1983). In 1987, C∂ was shown to be diamond (Lewis et al. 1987) the meteorite mineral which contained Xe(HL) and nitrogen whose isotopic composition was greatly enriched in the light isotope 14N (Lewis et al. 1983).

The early prediction of hollow graphite molecules suggested that they should be supercritical under ambient conditions. This is not true of C60, but might still be true of higher fullerenes and graphite nanotubes of large diameter.

Introduction

My title refers to the celebrated vision of Kekulé, one of the founders of the concept of chemical structure. In 1865, staring drowsily one evening into the fire, he saw in a dream the cyclic structure for benzene, that fundamental unit of all aromatic molecules, and of graphite and the fullerenes. In his reverie, he imagined the atoms gambolling before his eyes… ‘one of the snakes had seized hold of its own tail, and the form whirled mockingly before my eyes’ (Kekulé 1890). In this paper I deal, not so much with the recent triumphs of the identification and bulk preparation of buckminsterfullerene, as with its imaginative prehistory. This begins with Dalton's atomic theory, elaborated from 1803 onwards. Despite a very promising start, atomic theory languished for decades as merely a sort of useful metaphor. One good reason was its failure to come up with consistent atomic weights for the elements and formulae for their compounds. Whether, for example, the atomic weight of oxygen was 8 and water was HO, or whether it was 16 with water as H2O, remained uncertain for half a century.

And yet shortly after Dalton proposed his theory, the whole problem had been solved (Avogadro 1811).

By chance in 1970, we conjectured the possibility of the football-shaped C60 molecule, now known as buckminsterfullerene, while considering superaromatic molecules having three-dimensional π-electron delocalization. A translation of the original description, initially written in Japanese, is given. The processes leading to scientific discoveries are analysed in the light of our missed opportunity.

Introduction

The timescale of scientific and technological advance is becoming shorter and shorter in modern society, partly as a consequence of the rapid advance of technology and improving information transfer. It is no wonder then that the time has come to look back and discuss the future of fullerene science after less than a decade since its discovery by Kroto et al. (1985) and after only two years since it was isolated by Kratchmer et al. (1990). The purpose of this paper is to recount the story of original early proposal of the football-shaped C60 molecule back in 1970, and refer to other interesting ‘prehistoric’ events and analyse the process of scientific discovery.

Background

In the 1960s and 1970s, non-benzenoid aromatics were favourite targets for organic chemists. There was a prevailing dogma that aromaticity, due to the delocalization of π-electrons, is best realized in planar molecules. Everyone wished to constrain their molecules to be as planar as possible and for this reason aromaticity tacitly remained a two-dimensional concept. [18]Annulene (1), synthesized by Sondheimer et al. (1962), can be regarded as the masterpiece of planar aromaticity for its symmetric beauty (D6h) and high level of π-electron delocalization. In view of the wide availability and its perfect aromaticity, however, benzene remains the archetypal superstar of aromatic molecules.

Within the neighbourhood of the Sun, a number of highly evolved stars are carbonrich in the sense that they have more carbon than oxygen so their outer atmospheres contain molecules such as CN, CH and C2H2. These stars are cool with atmospheric temperatures near 3000 K and they are also luminous, typically 104 times more powerful than the Sun. The outer envelopes of these stars are tenuously bound, and they all are losing mass at a very high rate, in some cases more than 10−5MOa a−1 (where MO denotes the mass of the Sun). These high luminosity carbon stars remain in this phase for a time, very approximately, near 105 years. They exhibit a large amount of carbon in their atmospheres because the products of the nuclear burning that occurs in the very centre of the star, including the synthesis of carbon, appear on the surface.

In the extended envelopes around these stars, there is a very active chemistry, and the gas is sufficiently cool that nucleation of solid dust grains occurs. These solid particles may grow to sizes as large as 1 μm although a more typical size is near 0.05 μm. We therefore can identify both relatively small carbon-bearing molecules (for example HC7N) and much larger carbon-containing dust grains in the outflows. The amount of intermediate size particles or molecules, such as C60, and their possible role in the circumstellar chemistry is not yet well understood. At least in the envelope of the well studied carbon star IRC+10216, there appears to be more carbon in CO and solid grains than in poly cyclic aromatic hydrocarbons.

Fullerenes and icosahedral virus particles share the underlying geometry applied by Buckminster Fuller in his geodesic dome designs. The basic plan involves the construction of polyhedra from 12 pentagons together with some number of hexagons, or the symmetrically equivalent construction of triangular faceted surface lattices (deltahedra) with 12 five-fold vertices and some number of six-fold vertices. All the possible designs for icosahedral viruses built according to this plan were enumerated according to the triangulation number T = (h2+hk+k2) of icosadeltahedra formed by folding equilateral triangular nets with lattice vectors of indices h, k connecting neighbouring five-fold vertices. Lower symmetry deltahedra can be constructed in which the vectors connecting five-fold vertices are not all identical. Applying the pentagon isolation rule, the possible designs for fullerenes with more than 20 hexagonal facets can be defined by the set of vectors in the surface lattice net of the corresponding deltahedra. Surface lattice symmetry and geometrical relations among fullerene isomers can be displayed more directly in unfolded deltahedral nets than in projected views of the deltahedra or their hexagonally and pentagonally facted dual polyhedra.

Introduction

Buckminster Fuller (1963) called his discipline ‘comprehensive anticipatory design science’. Anticipatory science involves recognizing evident answers to questions that have not yet been asked. Fuller's dymaxion geometry (cf. Marks 1960) started with his rediscovery of the cuboctahedron as the coordination polyhedron in cubic close packing, which he renamed the ‘vector equilibrium’. Visualizing this figure not as a solid but as a framework of edges connected at the vertices, he transformed the square faces into pairs of triangles to form an icosahedron; and subtriangulation of the spherical icosahedron led to his frequency modulated geodesic domes.

We consider the geometries of hypothetical structures, derived from a graphite net by the inclusion of rings of seven or eight bonds, which may be periodic in three dimensions. Just as the positive curvature of fullerene sheets is produced by the presence of pentagons, so negative curvature appears with a mean ring size of more than six. These structures are based on coverings of periodic minimal surfaces, and surfaces parallel to these, which are known as exactly defined mathematical objects. In the same way that the cylindrical and conical structures can be generated (geometrically) by curving flat sheets so that the perimeter of a ring can be identified with a vector in the two-dimensional planar lattice, so these structures can be related to tessellations of the hyperbolic plane. The geometry of transformations at constant curvature relates various surfaces. Some of the proposed structures, which are reviewed here, promise to have lower energies than those of the convex fullerenes.

Introduction

The characteristics of the process of X-ray crystal structure analysis have led to an undue emphasis on classically crystalline materials to the neglect of organized structures which do not give simple diffraction patterns with sharp spots.

Gradually, even in the inorganic field, curved layers have become recognized as essential structural components. These were first recognized in asbestos and halloysite (Whittaker 1957; Yada 1971), where concentric cylinders and spiral windings of silicate sheets were disclosed. We can now begin to assemble the basic geometry of such curved structures under the rubric of ‘ flexi-crystallography’. This might be part of what de Gennes (1992) has called the study of ‘soft matter’, the main characteristics of which are complexity and flexibility.

Conjugation in C60 is not as extensive as was originally anticipated because, for various reasons, the pentagon rings avoid containing double bonds. As a consequence, there is extensive bond localization and the molecule, which is quite reactive, and displays superalkene rather than superaromatic properties. C70 behaves in a similar fashion; other fullerenes may follow suit. Additions predominate and C60 is particularly susceptible to nucleophilic attack. Added groups may also be readily replaced by nucleophiles, although the reaction mechanism is uncertain at present. The functionalized molecule tends to revert to the parent fullerene at moderate temperatures, and characterization of reaction products by mass spectrometry is thus particularly difficult. This fact, coupled with the complexity of the addition products, makes work with fullerenes exacting. A selection of reactions studied to date and the progress made towards identifying various patterns of addition are described.

Introduction

When spectroscopic evidence for the existence of C60 was first obtained, a view prevailed that it would be a very unreactive molecule. This conclusion was based on the assumption that with a possible 12500 resonance structures (Klein et al. 1986), C60 would be superaromatic. However, the earlier molecular orbital calculations (Bochvar & G'alpern 1973) correctly predicted that there would be substantial bond fixation in the molecule; more recently, the bond lengths have been determined by neutron diffraction studies to be 1.391 Å and 1.455 Å (David et al. 1991); other methods give similar values.

Thus C60 is a superalkene rather than a superarofnatic and readily undergoes additions. Bond fixation arises because structures with double bonds in pentagonal rings are unfavourable in chemistry, due probably to the increase in strain that would result from bdnd-shortening; this phenomenon…

The synthesis and microwave study of linear cyanopolyynes, HC5N and HC7N, in the mid-1970s was followed by the unanticipated detection of these, and longer chains (HC9N and HC11N), in space. To gain insight into the way in which such species and carbon clusters in general might form, an experiment was devised in 1985 to simulate conditions in carbon stars, involving the laser vaporization of graphite in a supersonic nozzle and detection of the resulting carbon species by mass spectrometry. This initiative resulted in the serendipitious discovery of an entirely new allotrope of carbon, C60, named buckminsterfullerene after the inventor of the geodesic dome.

Introduction

Acetylenes continue to provide a seemingly inexhaustible reservoir of novel materials, with extended conjugated systems embodied in polyynes, polyenes (polyacetylenes), enynes, cumulenes and various combinations thereof featuring prominently. Researches into conducting polymers are a case in point (Masuda & Higashimura 1984; Wegner 1981), as is the quest for natural products (Bohlmann et al. 1973; Jones & Thaller 1978) and their derivatives, some of which display high levels of pharmacological activity. A set of unique circumstances, which augur well for synthesis is partly responsible for this situation; notably the relatively high acidity of the alkynyl hydrogen (facilitating substitution and oxidative coupling) and the ease with which the triple bond can be induced to polymerize or participate in cycloadditions. The ubiquitous C2 unit also acts as a focus for combustion studies and, for example, for investigations into the nature of soot. The results of such work often turn out to be wholly unexpected and this is particularly true in the way fullerenes were discovered.

Some recent trends in industrial basic research are considered. They are driven by a highly competitive global marketplace. The discussion focusses on two recent major scientific advances which have been pursued in industry: high-temperature superconductivity and C60 with its chemical family. Fullerenes appear to be appropriate candidates for basic research in industry.

This paper might be subtitled ‘Research on a Globe, Chemistry on a Sphere’, referring to two very different aspects of the chemical community involved with fullerenes. Research worldwide is undergoing major transformations. Although noticeable in academic and government laboratories, the changes are larger in the industrial sector, especially in companies that have been major centres of basic research for decades. Here we shall consider some of the more prominent developments in industry.

The second aspect is the extraordinary chemistry of C60 and the rapidly increasing family of carbon structures. The chemistry will be that developed at Du Pont; the work of Dr Fagan, Dr Krusic and Dr Tebbe with others. Taylor's contribution in this collection provides an extended consideration of some of the topics and will not be repeated here. Rather we focus on a few features of chemistry on a sphere as opposed to the planar or linear compounds with which we are more familiar.

The bridge between these two parts, and a conclusion, is that basic research on fullerenes may well be an appropriate activity for a modern company dealing with chemicals or materials.

Changes in research philosophy and practice have occurred throughout technologically based industry in recent years. As companies position themselves for a competitive, fast-moving marketplace the fundamental importance of research and technology is rarely questioned.

Qualitative theoretical treatments of the fullerene family of molecules can be used to count possible isomers and predict their geometric shapes, point groups, electronic structures, vibrational and NMR spectroscopic signatures. Isomers are generated by the ring-spiral algorithm due to D. E. Manolopoulos. Geometrically based magic number rules devised by the present author account for all electronically closed π shells within the Hückel approximation and these ‘leapfrog’ and ‘cylinder’ rules apply to the wider class of ‘fulleroid’ structures constructed with rings of other sizes. Extrapolations from the theory of carbon clusters are described for doped fullerenes, metallocarbohedrenes, fully substituted boron–nitrogen heterofullerenes and decorated-fullerene models for water clusters.

Introduction

The fullerenes offer a challenge to theoretical chemistry. They are large molecules and, even with modern computational methods, it would be expensive and often uninformative to perform full ab initio calculations on them cage by cage. In the first steps towards understanding these new materials a more qualitative approach is necessary and desirable. Methods based on little more than topology and elementary valence theory can provide information on isomerism, geometric and electronic structure, spectroscopic signatures, stability and selection rules for interconversion. This paper touches on developments in these areas but, in line with the ‘postbuckminsterfullerene’ theme of the meeting, concentrates on new ideas in the theory of exotic fullerenes, heterofullerenes and related heteroatom and molecular clusters.

Fullerenes: isomers and electronic structure

The discovery of C60 (Kroto et al. 1985) carried with it the implication of the existence of a whole series of similar molecules. In the original experiments strong C60 and C70 signals were found in association, and intensity was distributed over a wide range of even-numbered clusters.

The gaseous, solution and solid state experimental evidence for electron addition to the fullerenes is reviewed and it is shown that this class of molecules function as powerful electron acceptors. The topological character of C60 as described by Hiickel molecular orbital theory suggests that the molecule will undergo facile reduction, but comparisons with planar conjugated hydrocarbons show that this feature alone cannot account for the very low half-wave reduction potential of C60. Because of the curvature of the surface, fullerene hybridization falls between graphite (sp2) and diamond (sp3) and these new carbon allotropes are therefore of intermediate, and perhaps variable hybridization. According to POAVI theory the carbon atoms in C60 are of sp2.28 hybridization. It is concluded that rehybridization plays an important role in determining the electronic structure of the fullerenes and it is the combination of topology and rehybridization that together account for the extraordinary ability of C60 to accept electrons.

Introduction

The ability of the fullerenes to function as electron acceptors has been recognized since the first investigation of their chemistry (Haufler et al. 1990). Even before the isolation of bulk quantities of C60 (Kratschmer et al. 1990), a large electron affinity was demonstrated for this molecule in gas phase experiments (Curl & Smalley 1988).

This trend has continued with the development of the physics, chemistry and materials science of the fullerenes. In this paper some of the experiments that have thrown light on the ability of the fullerenes to accept electrons are summarized and qualitative explanations for their extraordinary electron affinity are discussed.