# A Review of Geometry, Construction and Modelling for Carbon Nanotori
## 3 Lennard-Jones Potential and Continuum Approximation  
The main interaction energy between two non-bonded atoms originates from the van der Waals interaction energy which is modelled by the Lennard-Jones potential function [37], namely  
\[E=4\epsilon\left[-\left(\frac{\sigma}{\rho}\right)^{6}+\left(\frac{\sigma}{\rho} \right)^{12}\right]=-\frac{A}{\rho^{6}}+\frac{B}{\rho^{12}},\]  
where \(\rho\) is the distance between the two particular atoms, \(\epsilon\) and \(\sigma\) are the potential well depth and the van der Waals radius, respectively, and \(A=4\epsilon\sigma^{6}\) and \(B=4\epsilon\sigma^{12}\) denote the attractive and repulsive Lennard-Jones constants, respectively.  
For two non-bonded molecules, such as two carbon nanostructures, the total interaction energy can be evaluated by summing the interaction energies between each atom pair of the two molecules, so that we have  
\[E=\sum_{i=1}^{N}\sum_{j=1}^{M}\left(-\frac{A_{ij}}{\rho_{ij}^{6}}+\frac{B_{ij} }{\rho_{ij}^{12}}\right), \tag{1}\]  
where the two molecules are assumed to consist of \(N\) and \(M\) atoms, and the subscripts represent the types of each atomic interaction. For two non-identical atomic interactions, the appropriate Lennard-Jones constants are conventionally evaluated using the mixing rules [38].  
Figure 4 shows the interaction energy between two carbon atoms for any separation distance obtained from the Lennard-Jones potential. Although the energy is a scalar quantity, its sign represents the type of the interaction, where the plus sign denotes a repulsive energy while the minus sign signifies an attractive interaction. We observe that when the distance between the two atoms is small, the repulsive part dominates the attractive one, and when this distance reaches a certain value, the attractive component overtakes the repulsive interaction.  
Figure 3: Carbon nanotori constructed from joining \((5,0)\) and \((4,4)\) nanotubes.  
The continuum approximation [39] assumes that all atoms are uniformly distributed over their own structure, and the interaction energy between two molecules as defined by Equation (1) can be approximated by  
\[E=\eta_{1}\eta_{2}(-AI_{3}+BI_{6}), \tag{2}\]  
where \(\eta_{i}\) for \(i=1,2\) are the average atomic densities for molecule \(i\) and the integrals \(I_{n}\) for \(n=3\) and \(6\) are defined by  
\[I_{n}=\int_{S_{2}}\int_{S_{1}}\rho^{-2n}\,\mathrm{d}S_{1}\mathrm{d}S_{2}, \tag{3}\]  
where \(\rho\) denotes the distance between atoms for a typical atomic pair and \(\mathrm{d}S_{i}\) for \(i=1,2\) denotes the surface elements for each molecule \(i\).  
Mathematical and Theoretical Framework for Interaction Energy of Nanotorus with Other Nanostructures  
In this section, the energy expression as defined in (3) may be evaluated for a number of different molecular configurations and the resulting formulae are summarised in the following sections, and when evaluated, can be substituted into Equation (2) to obtain the total interaction energy. The major advantage of the approach described here is that the analytical expressions so obtained can be readily evaluated using standard algebraic packages, such as MATLAB and MAPLE, and accordingly numerous molecular configurations can be investigated for a low computational effort.