16 Allotropy of Carbon Compounds

5.3.1 Diamond and Graphite

5.3.2 Fullerene

5.3.2 Carbon nanotubes

5.3.2 Graphene

An assemble of carbon atoms can exists in different distinct forms differing in the crystal structure with significantly varied physical properties and this property is known as allotropy. These

Figure 5.17: Different allotropic forms of an assemble of carbon atoms. (a) graphite, (b) diamond, (c) lonsdaleite, (d) single-walled carbon nanotube, (e) multi-walled carbon nanotube, (f) fullerene-C60, (g) fullerene-C76, (h) carbon nanohorns, (i) onion-like carbon, (j) graphene, (k) carbon nanoribbons.

are molecules and structures which are made up entirely of carbon atoms and such allotropes of carbon are shown in Figure 5.17. One such molecule, Fullerene discovered in 1985, has generated excitement over the last two decades. Another example of carbon nanotubes which are tubes made out of networked carbon atoms, with a typical diameter of a few nanometers and lengths of several hundreds of microns.

Interest on carbon nanotubes zoomed out after its rediscovery in 1991. Carbon nanotubes are interesting in many ways. The most interesting aspcet of carbon nanotubes is the existence of both metallic and semiconducting nanotubes depending on the diameter and how the tube is rolled up. The other two most popular allotropes are graphite and diamond. Graphite is a layered material made up of weakly coupled planar sheets of carbon atoms arranged in a hexagonal lattice structure. The most popular form is the graphene, which was proposed by Wallace in 1947 as a good starting point for studying graphite is to study the single sheet of graphite. Before 2004, graphene was not existed. In 2004, it was demosnstrated in an experimental breakthrough it is possible to obtain atomically thin, large two-dimensional crystals of carbon by a fairly simple technique. In all these allotropic forms, one aspect is common i.e. carbon atoms are arranged in different topology by covalent bond with different hybridization sp3, sp2 and sp or combination of these hybrid orbitals. It can be shown that the covalent bonding of graphitic carbon lead to a simple set of geometrical rules that relate the shape of a carbon nanostructure with the topology of carbon rings within it.

Based on dimensionalities, carbon nanomaterials can be divided into four groups: (i) zero-dimensional (0D) such as fullerene, carbon quantum dots, and nanodiamonds, (ii) one-dimensional (1D) such as carbon nanotubes and carbon nanohorns, and (iii) two-dimensional (2D) such as graphene, carbon (graphene) nanoribbons, and few-layered graphenes. and (iv) three dimensional(3D) carbon allotropes such as graphite and diamond. Carbon nanomaterials can also be classified based on covalent bonding (sp1, sp2 and sp3) between carbon atoms in a respective carbon. In terms of orbital hybridization, carbon is found to display sp3, sp2, and sp1 configurations, allowing a great variety of crystalline and disordered structures. In addition to these allotropes, there are several theoretically predicted carbon based nanomaterials, such as body-centered tetragonal carbon, T-carbon, M-carbon and nanorope.

5.3.1 Diamond and Graphite

Diamond, the hardest known natural mineral, has tetrahedrally coordinated carbon atoms bonded each other with sp3 hybridization that form an extended three-dimensional network with a crystal structure of the face-centered cubic Bravais lattice. Most natural diamonds are formed at high temperature and pressure at depths of 150 to 200 kilometers in the Earth’s mantle. Carbon-containing minerals provide the carbon source, and the growth occurs over periods from 1 billion to 3.3 billion years. Theoretical phase diagram of carbon is shown in Figure 5.18 which shows how it is possible to obtain one phase to another phase under different conditions.. Most notable properties of carbon are its extreme hardness, high thermal conductivity (25 W·cm−1·K−1) and wide bandgap of 5.47 eV. Diamond converts to graphite above 1700 °C in vacuum and above 700 °C in air.

The chemical bonds in diamonds are weaker than those in graphite. In diamonds, sp3 hybridized bonds form an inflexible three-dimensional lattice, whereas in graphite, the atoms are tightly bonded into sheets, which can slide easily over one another due to weak van der Waals bond between sheets, making the overall structure weaker.

Figure 5.19 Lattice structure of diamond and graphite..

Figure 5.18 Theoretical phase diagram of carbon, showing the state of matter at different temperatures and pressures. The hatched regions indicate conditions under which one phase is metastable, so that two phases can coexist.

Graphite is another crystalline allotrope of carbon. As discussed before graphite is the most stable form of carbon under standard conditions. In contrast to diamond, graphite has three-coordinated sp2 carbon atoms arranged in a honeycomb lattice with layered structure, in which layers of carbon atoms are held together by weak van der Waals forces. Different physical properties of graphite and diamond are manifested due to the different layout of carbon atoms in the lattice of diamond and graphite. Graphite is opaque, metallic and earthy looking whereas diamond is transparent, insulator and brilliant looking. The individual layers are called graphene. In each layer, the carbon atoms are arranged in a honeycomb lattice with separation of 0.142 nm, and the distance between planes is 0.335 nm. There are two forms of graphite, alpha (hexagonal) and beta (rhombohedral). They have very similar physical properties, except for that the graphene layers stack slightly differently in these two forms of graphite. The alpha form can be transformed to the beta form by mechanical treatment, similarly the beta form can be transformed to the alpha form by heating above 1300 °C. The crystal structures of diamond and graphite are shown in Figure 5.19.

5.3.2 Fullerene

Fullerenes, a new carbon allotrope, was discovered in 1985. In fullerenes, the closed-cage structure is established by the presence of five-membered rings; the arrangement of carbon atoms is not planar but rather slightly pyramidalized as a result of a “pseudo” sp3 bonding component present in the essentially sp2 carbons. Among the known forms of fullerenes, C60 is the most symmetrical and stable fullerene molecule consisting of 20 hexagonal and 12 disjoint pentagonal faces where a carbon atom is located at each corner of the individual polygons which are arranged into a highly symmetric truncated icosahedrons. There are two bond lengths for C60 molecule. The 6:6 ring bonds, between two hexagons, can be considered double bonds and are shorter than the 6:5 bonds between a hexagon and a pentagon and their average bond length is 1.4 angstroms. In addition to C60, other fullerenes have been reported, such as C70, C76, C82, and C84. Fullerenes can be obtained by a low pressure method in which an electric discharge is passed across a gap between two carbon electrodes in a helium atmosphere. Other popular method is based on laser vaporization of carbon or arc vaporization of graphite in an inert atmosphere. Fullerenes and their derivatives are extremely important due to their fascinating optical properties, superconductivity, and ferromagnetic behavior with high Curie temperatures. Fullerenes can be functionalized with a different organic and inorganic molecules to extend their range of potential applications.

Figure 5.20 Graph of (a) C20 fullerene, (b) C26 fullerene, (c) C60 fullerene and (d) C70 fullerene.

Mathematically, the structure of a fullerene is a trivalent convex polyhedron with pentagonal and hexagonal faces. According to graph theory, the term fullerene refers to any set of graph with 3-regular, planar graph with all faces of size five or six. It follows from Euler’s polyhedron formula, V − E + F = 2, where V, E, F are the numbers of vertices, edges, and faces, respectively and there are exactly 12 pentagons in a fullerene and V/2 − 10 hexagons. The smallest fullerene is the dodecahedral C20. The number of fullerenes C2n grows with increasing n = 12, 13, 14, …, except n=11, roughly proportional to n9 . Accordingly, there are 1812 non-isomorphic fullerenes C60. Out of which only one form of C60, has no pair of adjacent pentagons. To further illustrate, there are 214,127,713 non-isomorphic fullerenes C200, of which 15,655,672 have no adjacent pentagons. The graphical representation of different fullerene according to Euler’s polyhedran formula is shown in Figure 5.20.

5.3.3 Carbon nanotubes

Figure 5.21 Single wall carbon nanotube (SWCNT) and multiwall carbon nanotube (MWCNT).

Another allotrope of carbon discovered in 1991, known as multi-walled carbon nanotube (MWCNT), in a carbon soot prepared by an arc-discharge method, subsequently single-walled carbon nanotube (SWCNT) was discovered in 1993. A MWCNT is made of concentric cylinders with an interlayer spacing of 0.34 nm and a typical diameter ranging from ~2 to ~30 nm and the concentric cylinders are held together by van der Waals interactions, shown in Figure 5.21. The length of SWCNT can be as high as hundreds of microns. The aspect ratio i.e., length-to-diameter ratio, can exceeds 10000 and thus, carbon nanotubes is the most anisotropic materials. Most SWNTs have a diameter of close to 1 nanometer, and can be of hundreds of micron length. The structure of a SWNT can be thought of as wrapping a one-atom-thick layer of graphite called graphene into a seamless cylinder. A pair of integers (n,m) defines how the graphene sheet is wrapped leading to SWCNT. The integers n and m denote the number of unit vectors along two directions in the honeycomb crystal lattice of graphene.There are three possibilities. (i) m = 0, the nanotubes are called zigzag nanotubes, (ii)   n = m, the nanotubes are called armchair nanotubes and (ii) otherwise, they are called chiral (Figure 5.22). The diameter of an ideal nanotube can be calculated from its (n,m) indices as follows

Figure 5.22 The (n,m) nanotube naming scheme is based on vector (Ch) in an infinite graphene sheet that describes how to “roll up” the graphene sheet to obtain the three different types (armchair, zigzag and chiral) nanotube. T denotes the tube axis, and a1 and a2 are the unit vectors of graphene in real space.

In carbon nanotubes, the carbon atoms are held together by sp2 bonds. Carbon nanotubes are the strongest and stiffest materials yet discovered and existing in nature with a Young’s modulus of ~1.2 TPa and tensile strength of ~100 GPa, which is about a hundred times higher than steel. The electrical characteristics of carbon nanotubes are strongly governed by the diameter and chirality of the nanotubes.

Armchair nanotubes (i.e., n = m) are expected to exhibit metallic behavior; they have a finite density of states at the Fermi level. However, they may become semiconducting if their diameter drops below a threshold value because the energy gaps in the semiconducting carbon nanotubes scale with 1/d , where d is the nanotube diameter. Chiral nanotubes, for which m ≠ n and (m – n) is a multiple of three are semiconducting. The band structure of such nanotubes features a very small band gap with a zero density of states inside the gap due to the modest degree of sp2-to-sp3 hybridization induced by the non-flat nature of the hexagons on the nanotube walls.

Bothe SWCNT and MWCNT show remarkable electrical, transport, and optical properties. In particular, they have been investigated as functional components of energy and gas storing devices, building blocks of nanoelectronic, spintronic, and nanophotonic devices and medical tools for drug-delivery carriers for cancer treatment, gene delivery systems, and in photo-thermal therapy. In addition to these practical applications, carbon nanotubes are often used as model systems for the study of various quantum phenomena that occur in quasi-1D solids, including single-electron charging, ballistic transport, weak localization, and quantum interference.

5.3.4 Graphene

As mentioned before the most stable state of elemental carbon at ambient pressure and temperature is graphite which consists of individual graphene layers, each composed of interlinked hexagonal carbon rings tightly bonded to each other, however stacked loosely by van der Waals interaction into a three-dimensional material. Within a single graphene layer, each carbon atom is tightly bonded to three neighbors with sp2 bonding in a plane. As discussed before, the 2s, 2px and 2py orbitals are superimposed to form three new linear combinations which result three lobes of charge reaching outward from the carbon atom at 120 degree angles to each other, all within the x-y plane. The remaining 2pz orbital, which points perpendicular to the x-y plane, overlaps with 2pz orbitals on neighboring atoms to form an extended sheet-like bonding state that covers the upper and lower

Figure 5.23 Graphene is a crystalline allotrope of carbon with carbon atoms densely packed in a regular atomic-scale hexagonal pattern. Each atom has four bonds, one σ bond with each of its three neighbors and one π-bond that is oriented out of plane.

surfaces of the graphene sheet. To have π bonds with maximum strength, the 2pz orbitals of neighboring atoms should point in the same direction, so the 2D carbon sheet will be most stable when it is perfectly flat (Figure 5.23). The ability of carbon to form these highly stable effectively two-dimensional structures is fundamental to possibility to have different carbon based topological structures which can be realized by distorting this two-dimensional graphene sheet in the third dimension to form a very rich family of structures. The energy cost to perform these distortions is relatively small. Graphene is a single atomic layer, so it can be bent without changing the in-plane bond lengths significantly. To achieve stable distortion from two to three dimension by bending graphene sheet into interesting structures, any dangling bonds at the edges of such a distorted sheet must be eliminated. This can be achieved by wraping the carbon structure around onto itself so that it forms a closed sheet with no edges. Examples of such structures are carbon nanotube and fullerene. Another fundamental requirement in a closed graphene-like structure is that every carbon atom have three bonds to neighboring atoms, and that the entire structure fold back on itself without any dangling bonds.