Interatomic Potential for CC Non Covalent Systems

The non-covalent interactions between carbon atoms are required in many of the simulation studies in computational nanoscience and nanotechnology. These can be modeled according to various types of potential [10]. The Lennard-Jones and Kihara potentials can be employed to describe the van der Waals intermolecular interactions between carbon clusters, such as C60 molecules, and between the basal planes in a graphite lattice. Other useful potentials are the exp-6 potential [121], which also describes the C60-C60 interactions, and the Ruoff-Hickman potential [122], which models the C60-graphite interactions.

3.1. The Lennard-Jones and Kihara Potentials: The total interaction potential between the carbon atoms in two C60 molecules, or between those in two graphite basal planes, could be represented by the Lennard-Jones potential [123]

0iLJ(ri!J) = 4eZ¡Zj>¡ [(o/ryIJ)12 - (o/rij1)6], (54)

where I and J denote the two molecules (planes), rij is the distance between the atom i in molecule (plane) i and atom j in molecule (plane) J. The parameters of this potential, (£=0.24127 x 10-2 <?v , o=0.34 $ [nm]), were taken from a study of graphite [124]. The Kihara potential is similar to the Lennard-Jones except for the fact that a third parameter d, is added to correspond to the hard-core diameter, i.e.

®iLJ(r¡jIJ) = 4£Z¡Zj>¡ [{(o-d)/(rijIJ-d)}12-[((o-d)/(rijIJ-d)}6] for r>d ,

3.2. The exp-6 Potential: This is another potential that describes the interaction between the carbon atoms in two C60 molecules

Two sets of values of the parameters are provided, and these are listed in Table 6. These parameters have been obtained from the gas phase data of a large number of organic compounds, without any adjustment.

Table 6. Parameters of the exp-6 potential for C.

A(kcal/mol)

B [kcal/molx(fa [nm])6]

a (fa [nm])-1

Set one

42000

3.58 x 108

35.8

Set two

83630

5.68 x 108

36.0

The measured value of the C60 solid lattice constant is a = 1.404 fa [nm] at T = 11° K. The calculated value using the set one was a = 1.301 fa [nm] and using the set two was a = 1.403fa [nm]. The experimentally estimated heat of sublimation is equal to - 45kcal/mol (extrapolated from the measured value of - 40.1 ± 1.3 kcal/mol at T = 707° K). The computed value using the set one was - 41.5 kcal/mol and using the set two was - 58.7 kcal/mol. We see that whereas the set two produces a lattice constant nearer the experimental value, the thermal properties are better described by using the set one.

3.3. The Ruoff-Hickman Potential: This potential, based on the model adopted by Girifalco [125], describes the interaction of a C60 molecule with a graphite substrate by approximating these two systems as continuum surfaces on which the carbon atoms are 'smeared out' with a uniform density. The sums over the pair interactions are then replaced by integrals that can be evaluated analytically. The C60 is modeled as a hollow sphere having a radius b = 0.355fa [nm], and the C-C pair interaction takes on a Lennard-Jones form fai(rij) = cur'12 - c6r 6,

with c6 = 1.997x10-5 [ev.(Q [nm])6] and cn = 3.4812 x10-8 [ev.(Q [nm])12] [125]. The interaction potential between the hollow C60 and a single carbon atom of a graphite substrate, located at a distance z > b from the center of the sphere, is then evaluated as m = Mz) - m, (57)

where

Mz) = cy[2(n-2)].[N/(bz)W/(z - b)n-2 - 1/(z + b)n-2], (58)

where N is the number of atoms on the sphere (N = 60 in this case) and n = 12, 6. The total interaction energy between the C60 and the graphite plane is then obtained by integrating <p(z) over all the atoms in the plane, giving

where

En(R) = {cJ[4(n-2)(n-3)]}.(N2/b3).[1/(R-bf~3 - 1/(R+b)n-3], (60) and R is the vertical distance of the center of the sphere from the plane.

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