Received 29 January 2016; accepted 23 February 2016; published 26 February 2016
Dedicated to geo-carbon expert, Prof. Gustaf Arrhenius, of Scripps Institution of Oceanography, CA
Carbon  is a wonder element of Nature. One of its forms, graphite   , crystallizes in hexagonal and rhombohedral forms. It is a solid stack of layers of graphene   with the inter-planar spacing of 0.34 nm in both forms   , held by weak van der Waals (vdW) bonds  . This short article provides for the first time a new and simple interpretation of the length and nature of this bond.
2. Atomic Structure of Graphene
Graphene is a two-dimensional network of regular hexagons of identical carbon atoms, with equal inter-atomic spacing of d(CC)res = 0.142 nm  -   -  . It is usually represented like a wire mesh in the ball and stick model  . Its electrical conductivity as an atomically thin material makes it very useful in many ways, see e.g.,  . Its structure at the atomic level was worked out by the author  - , in which all the details can be found. The Figure 4 in  shows the alternately charged cations and anions, which are the ionic resonance forms (as for the H2 molecule  ) of the adjacent C atoms bound by covalent bonds, responsible for electrical conduction in graphene. This is shown in Figure 1(a) here. The sum of the radii of the cation R(+) and anion R(−) is equal to the covalent bond length, d(CC),   . The atomic and ionic radii in Figure 1(a) are related by the Golden ratio, φ = (1+51/2)/2 = 1.618 as explained in   and in the legend for Figure 1. The covalent radius, RCres = d(CC)res/2 = (φ − 1/2)aB = 0.71 nm, where aB is the Bohr radius obtained from the first ionization potential   .
3. Atomic Structure of Graphite
In graphite   , the graphene layers are connected by weak vdW bonds. The vdW bonds exist in a variety of molecules  and nano-structures  . These bonds are, in general, longer than the covalent bonds. Pauling  describes vdW bonds as arising out of the attractive and repulsive forces, and defines the vdW radius as half the corresponding bond length. In the case of carbon, the reported value of vdW radius is around 0.17 nm    . The unit cell of graphite  has the lattice constants, a = 0.246 nm and c = 0.671 nm. The inter-planar spacing in (both hexagonal and rhombohedral) graphite is c/2 = 0.336 nm, which is the length of the vdW bond. Half of this is the van der Waals radius, RvdW = 0.168 nm.
Figure 1. (a) Graphene  . Bond length d(CC)= 2RCres = 0.142 nm. The alternate red and blue circles are cations and anions (responsible for conduction) of radii, RCres(+) = d(CC)/φ2 and RCres(−) = d(CC)/φ and d(CC) = RCres(+) + RCres(−), where φ is the Golden ratio (see  for details); (b) Graphite. Atoms C(1) and C(2) are in adjacent graphene layers connected by the van der Waals bond as shown by the dotted line.
4. The van der Waals Bond in Graphite
In Figure 1(b) are shown two carbon atoms, C(1) and C(2) in two adjacent graphene layers (1) and (2) connected by the vdW bond of length d(C..C)vdW = 2RvdW = c/2 = 0.336 nm. On examining this distance, c/2, it was astonishing to find that it is actually the sum of the lattice constant, a, which is the width, 31/2d(CC)res, of the graphene hexagon (0.246 nm, see Figure 1(a)) and the anionic radius, RC(−) = 0.088 nm of carbon,
where, d(CC)res = 0.142 nm, a = 0.142(31/2) = 0.246 nm and RC(−) = 0.142/φ = 0.088 nm. It can be seen that all the distances are related to the Golden ratio, φ    .
This shows, for the first time, that the vdW bond in graphite is a long range coulombic bond between the positively charged carbon cation, C(1)(+) on graphene layer (1) and the negatively charged carbon anion , C(2)(−) in layer (2), separated by the distance, a + RC(−).
Graphene, a hexagonal network of identical carbon atoms, has become the wonder material of attention for its many useful properties    . Its structure at the atomic level was established by the author  -  based on the discovery  that atomic and/or ionic radii are additive in chemical bonds in small as well as large molecules. For a power point review talk, see  . The electrical conduction in graphene was attributed  to the alternate positively and negatively charged carbon atoms in the ionic resonance forms (as for H2 molecule  ). As described in    , the exact radii R(+) and R(−) of these ionic forms could be obtained as the Golden sections of the interatomic distance in graphene, d(CC) = 0.142 nm. Graphite is a stack of graphene layers (crystallizing in hexagonal and rhombohedral forms depending on the conditions  ) bound by vdW bonds    . Although the inter-planar distance, C(1)..C(2) = 0.336 nm (see Figure 1(b)) held by the vdW forces is known   , a simple exact interpretation of this bond in graphite had been lacking. In this paper, it is shown for the first time that the known vdW bond length is the sum of the width of graphene hexagon (=a, the unit cell parameter of graphite) and the anionic radius, R(−). This means that the vdW bond is of coulombic nature and is the distance between the C(1)(+) cation and C(2)(−) anion as shown in Figure 1(b). This work accounts for the first time why graphite is a conductor and finds its use as an electrode. For the coulomb drag in graphene layers at longer distances of 50 nm and 100 nm, and for scaling laws for van der Waals interactions in nanostructured materials, see   respectively.
The author is grateful to the Editor and reviewers for their useful comments and to SCIRP for partial financial assistance towards APC.
*A preliminary report of this work as preprint is at: http://www.vixra.org/abs/1601.0273.