Bonding Between Carbon Atoms

Carbon-based materials, clusters, and molecules are unique in many ways. One distinction relates to the many possible configurations of the electronic states of a carbon atom, which is known as the hybridization of atomic orbitals and relates to the bonding of a carbon atom to its nearest neighbors.

Carbon is the sixth element of the periodic table and has the lowest atomic number of any element in column IV of the periodic table. Each carbon atom has six electrons which occupy 1s2, 2s2, and 2p2 atomic orbitals. The 1s2 orbital contains two strongly bound core electrons. Four more weakly bound electrons occupy the 2s22p2 valence orbitals. In the crystalline phase, the valence electrons give rise to 2s, 2px, 2py, and 2pz orbitals which are important in forming covalent bonds in carbon materials. Since the energy difference between the upper 2p energy levels and the lower 2s level in carbon is small compared with the binding energy of the chemical bonds, the electronic wave functions for these four electrons can readily mix with each other, thereby changing the occupation of the 2s and three 2p atomic orbitals so as to enhance the binding energy of the C atom with its neighboring atoms. The general mixing of 2s and 2p atomic orbitals is called hybridization, whereas the mixing of a single 2s electron with one, two, or three 2p electrons is called spn hybridization with n = 1, 2, 3 [1,2].

Thus three possible hybridizations occur in carbon: sp, sp2 and sp3, while other group IV elements such as Si and Ge exhibit primarily sp3 hybridization. Carbon differs from Si and Ge insofar as carbon does not have inner atomic orbitals, except for the spherical 1s orbitals, and the absence of nearby inner orbitals facilitates hybridizations involving only valence s and p orbitals for carbon. The various bonding states are connected with certain structural arrangements, so that sp bonding gives rise to chain structures, sp2 bonding to planar structures and sp3 bonding to tetrahedral structures.

The carbon phase diagram (see Fig. 1) guided the historical synthesis of diamond in 1960 [4], and has continued to inspire interest in new forms of carbon, as they are discovered [3]. Although we have learned much about carbon since that time, much ignorance remains about the possible phases of carbon. While sp2 bonded graphite is the ground state phase of carbon under ambient conditions, at higher temperatures and pressures, sp3 bonded cubic diamond is stable. Other regions of the phase diagram show stability ranges for hexagonal diamond, hexagonal carbynes [5,6,7], and liquid carbon [8]. It is believed that a variety of novel ^-electron carbon bulk phases remain to be discovered and explored.

In addition to the bulk phases featured in the carbon phase diagram, much attention has recently focussed on small carbon clusters [9], since the

Fig. 1. A recent version of the phase diagram of carbon [3]. Solid lines represent equilibrium phase boundaries. A: commercial synthesis of diamond from graphite by catalysis; B: rapid solid phase graphite to diamond synthesis; C: fast transformation of diamond to graphite; D: hexagonal graphite to hexagonal diamond synthesis; E: shock compression graphite to hexagonal diamond synthesis; F: shock compression graphite to cubic-type diamond synthesis; B, F, G: graphite or hexagonal diamond to cubic diamond synthesis; H,I,J: compressed graphite acquires diamond-like properties, but reverts to graphite upon release of pressure

Fig. 1. A recent version of the phase diagram of carbon [3]. Solid lines represent equilibrium phase boundaries. A: commercial synthesis of diamond from graphite by catalysis; B: rapid solid phase graphite to diamond synthesis; C: fast transformation of diamond to graphite; D: hexagonal graphite to hexagonal diamond synthesis; E: shock compression graphite to hexagonal diamond synthesis; F: shock compression graphite to cubic-type diamond synthesis; B, F, G: graphite or hexagonal diamond to cubic diamond synthesis; H,I,J: compressed graphite acquires diamond-like properties, but reverts to graphite upon release of pressure discovery of fullerenes in 1985 by Kroto et al. [10] and of carbon nanotubes in 1991 by Iijima [11]. The physical reason why these nanostructures form is that a graphene layer (defined as a single 2D layer of 3D graphite) of finite size has many edge atoms with dangling bonds,indexdangling bonds and these dangling bonds correspond to high energy states. Therefore the total energy of a small number of carbon atoms (30-100) is reduced by eliminating dangling bonds, even at the expense of increasing the strain energy, thereby promoting the formation of closed cage clusters such as fullerenes and carbon nanotubes.

The rolling of a single graphene layer, which is a hexagonal network of carbon atoms, to form a carbon nanotube is reviewed in this volume in the introductory chapter [12], and in the chapters by Louie [13] and Saito/Kataura [14], where the two indices (n,m) that fully identify each carbon nanotube are specified [9,15]. Since nanotubes can be rolled from a graphene sheet in many ways [9,15], there are many possible orientations of the hexagons on the nanotubes, even though the basic shape of the carbon nanotube wall is a cylinder.

A carbon nanotube is a graphene sheet appropriately rolled into a cylinder of nanometer size diameter [13,14,15]. Therefore we can expect the planar sp2 bonding that is characteristic of graphite to play a significant role in carbon nanotubes. The curvature of the nanotubes admixes a small amount of sp3

bonding so that the force constants (bonding) in the circumferential direction are slightly weaker than along the nanotube axis. Since the single wall carbon nanotube is only one atom thick and has a small number of atoms around its circumference, only a few wave vectors are needed to describe the periodicity of the nanotubes. These constraints lead to quantum confinement of the wavefunctions in the radial and circumferential directions, with plane wave motion occurring only along the nanotube axis corresponding to a large number or closely spaced allowed wave vectors. Thus, although carbon nano-tubes are closely related to a 2D graphene sheet, the tube curvature and the quantum confinement in the circumferential direction lead to a host of properties that are different from those of a graphene sheet. Because of the close relation between carbon nanotubes and graphite, we review briefly the structure and properties of graphite in this chapter. As explained in the chapter by Louie [13], (n, m) carbon nanotubes can be either metallic (n — m = 3q, q = 0,1, 2,...) or semiconducting (n — m = 3q± 1, q = 0,1, 2,...), the individual constituents of multi-wall nanotubes or single-wall nanotube bundles can be metallic or semiconducting [13,15]. These remarkable electronic properties follow from the electronic structure of 2D graphite under the constraints of quantum confinement in the circumferential direction [13].

Actual carbon nanotube samples are usually found in one of two forms: (1) a Multi-Wall Carbon Nanotube (MWNT) consisting of a nested coaxial array of single-wall nanotube constituents [16], separated from one another by approximately 0.34nm, the interlayer distance of graphite (see Sect. 2), and (2) a single wall nanotube rope, which is a nanocrystal consisting of ~10-100 Single-Wall Nanotubes (SWNTs), whose axes are aligned parallel to one another, and are arranged in a triangular lattice with a lattice constant that is approximately equal to dt + ct, where dt is the nanotube diameter and ct is approximately equal to the interlayer lattice constant of graphite.

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