Optimal configurations for interacting carbon nanotori

Abstract

The optimal or lowest energy configuration of two carbon nanotori is investigated. With the assumption that the tori are symmetrically situated and parallel with the centres lying on their common axis, the interaction energy between the two tori is evaluated using the continuum approximation and the Lennard-Jones potential function. This mathematical modelling approach provides a fast and accurate computation as compared to other simulation methods. The numerical results obtained determine the most stable configuration corresponding to the minimum energies for five different sized tori, and a relation between the major and minor radii of the tori and the equilibrium position is demonstrated. The interaction energies between two nanotori determined here might be utilised to model many promising devices for future developments in nano and biotechnology.

Appendices

Appendix A: Derivation of energy expression between atom and torus

From (3) the integrand does not depend on $\theta$, and the expression can be reduced to the form

$$\begin{aligned} I_n^\text {(pt)}&=2\pi r\int _{-\pi }^\pi \frac{R+r\cos \phi }{[r^2+R^2+\delta ^2+2r(R\cos \phi -\delta \sin \phi )]^n} \, \text {d}\phi \\&=2\pi r\int _{-\pi }^\pi \frac{R+r\cos \phi }{[a+b\sin (\phi -\phi _0)]^n} \, \text {d}\phi , \end{aligned}$$

where $a=r^2+R^2+\delta ^2, b=-2r(R^2+\delta ^2)^{1/2}$ and $\phi _0=\sin ^{-1}(R/(R^2+\delta ^2)^{1/2})$. Consequently, substituting $\lambda =\phi -\phi _0$ gives

$$\begin{aligned} I_n^\text {(pt)}&=\frac{\pi Rr}{R^2+\delta ^2}\left[ (R^2+\delta ^2-r^2)\int _{\theta _0}^{\theta _0+2\pi }\frac{1}{(a+b\sin \lambda )^n}\,\text {d}\lambda \right. \nonumber \\&\quad \quad \left. +\int _{\theta _0}^{\theta _0+2\pi }\frac{1}{(a+b\sin \lambda )^{n-1}}\,\text {d}\lambda \right] , \end{aligned}$$

(A1)

where $\theta _0$ is a certain angle in $(-\pi ,\pi ]$. Therefore, we primarily focus on integrals of the form

$$\begin{aligned} {\tilde{I}}_m^\text {(pt)}=\int _{\theta _0}^{\theta _0+2\pi }\frac{1}{(a+b\sin \lambda )^m}\,\text {d}\lambda =\int _{-\pi }^{\pi }\frac{1}{(a+b\cos \lambda )^m}\,\text {d}\lambda . \end{aligned}$$

Upon utilising the Legendre polynomials property, $P_{-n}(x)=P_{n-1}(x)$, and the integral representation

$$\begin{aligned} P_n(x)=\frac{1}{\pi }\int _0^\pi [x+(x^2-1)^{1/2}\cos t]^n\,\text {d}t, \end{aligned}$$

(A2)

we obtain

$$\begin{aligned} {\tilde{I}}_m^\text {(pt)}=\frac{2\pi }{(R^2+\delta ^2-r^2)^m} P_{m-1}\left( \frac{R^2+\delta ^2+r^2}{R^2+\delta ^2-r^2}\right) . \end{aligned}$$

Finally, substituting the result back to (A1) gives

$$\begin{aligned} I_n^\text {(pt)}&=\frac{2\pi ^2Rr}{(R^2+\delta ^2)(R^2+\delta ^2-r^2)^{n-1}}\\&\quad \quad \times \left[ P_{n-1}\left( \frac{R^2+\delta ^2+r^2}{R^2+\delta ^2-r^2}\right) +P_{n-2}\left( \frac{R^2+\delta ^2+r^2}{R^2+\delta ^2-r^2}\right) \right] . \end{aligned}$$

Appendix B: Derivation of energy expression between two tori

Now, we consider (4) and by rearranging $\epsilon$, we have

$$\begin{aligned} \epsilon ^2=S^2+s^2+\delta ^2+2s(S^2+\delta ^2)^{1/2}\sin (\omega +\omega _0), \end{aligned}$$

where $\omega _0=\sin ^{-1}(S/(S^2+\delta ^2)^{1/2})$. After substituting the new distance $\epsilon$ into (3), it is observed that the integrand does not depend on $\psi$, so that the energy integral for two tori becomes

$$\begin{aligned} I_n^\text {(tt)}=4\pi ^3Rrs\left( {\tilde{I}}_{n-1}^\text {(tt)}+{\tilde{I}}_{n-2}^\text {(tt)}\right) , \end{aligned}$$

(B1)

where with $\alpha =R^2+S^2+s^2+\delta ^2,\beta =\alpha -r^2,\gamma =\alpha +r^2,\kappa =2s(S^2+\delta ^2)^{1/2}$ and ${\tilde{I}}_m^\text {(tt)}$ for $m=n-1$ and $m = n-2$ are defined as

$$\begin{aligned} {\tilde{I}}_m^\text {(tt)}=\int _{-\pi }^\pi \frac{(S+s\cos \omega )P_m\left( \frac{\gamma +\kappa \sin (\omega +\omega _0)}{\beta +\kappa \sin (\omega +\omega _0)}\right) }{[\alpha +\kappa \sin (\omega +\omega _0)][\beta +\kappa \sin (\omega +\omega _0)]^{n-1}}\,\text {d}\omega . \end{aligned}$$

Making a substitution $\mu =\omega +\omega _0$ and applying the angle addition formula for the cosine function provide

$$\begin{aligned} {\tilde{I}}_m^\text {(tt)}=\frac{S}{2(S^2+\delta ^2)}\left[ (S^2+r^2+\delta ^2-R^2-s^2){\hat{I}}_1^\text {(tt)}+{\hat{I}}_2^\text {(tt)}\right] , \end{aligned}$$

(B2)

where ${\hat{I}}_j^\text {(tt)}$ for $j=1$ and 2 are defined as

$$\begin{aligned} {\hat{I}}_j^\text {(tt)}=\int _{\omega _0-\pi }^{\omega _0+\pi }\frac{P_m\left( \frac{\gamma +\kappa \sin \mu }{\beta +\kappa \sin \mu }\right) }{(\alpha +\kappa \sin \mu )(\beta +\kappa \sin \mu )^{n-j}}\,\text {d}\mu . \end{aligned}$$

(B3)

Upon converting the Legendre polynomial of the first kind into the series representation

$$\begin{aligned} P_n(x)=2^n\sum _{k=0}^n\left( {\begin{array}{c}n\\ k\end{array}}\right) \left( {\begin{array}{c}(n+k-1)/2\\ n\end{array}}\right) x^k, \end{aligned}$$

and using the fact that the integrand is integrated over $2\pi$ and is an even function, the equation above may be rewritten as

$$\begin{aligned} {\hat{I}}_j^\text {(tt)}&=2^{m+1}\sum _{k=0}^m\left( {\begin{array}{c}m\\ k\end{array}}\right) \left( {\begin{array}{c}(m+k-1)/2\\ m\end{array}}\right) \nonumber \\ &\quad \times \int _0^\pi \frac{(\gamma +\kappa \cos \mu )^k}{(\alpha +\kappa \cos \mu )(\beta +\kappa \cos \mu )^{n+k-j}}\,\text {d}\mu . \end{aligned}$$

Adopting the trigonometric property, $\cos \mu =1-2\sin ^2\mu /2$, and making a substitution $t=\sin ^2\mu /2$ give

$$\begin{aligned} {\hat{I}}_j^\text {(tt)}&=2^{m+1}\sum _{k=0}^m\left( {\begin{array}{c}m\\ k\end{array}}\right) \left( {\begin{array}{c}(m+k-1)/2\\ m\end{array}}\right) \nonumber \\ &\quad\times \int _0^1\frac{(\gamma +\kappa -2\kappa t)^kt^{-1/2}(1-t)^{-1/2}}{(\alpha +\kappa -2\kappa t)(\beta +\kappa -2\kappa t)^{n+k-j}}\,\text {d}t. \end{aligned}$$

Subsequently, applying the binomial theorem to the term in the numerator and writing the result in terms of the Appell’s hypergeometric function using the integral representation

$$\begin{aligned} F_1(a;b,b';c;x,y)&=\frac{\varGamma (c)}{\varGamma (a)\varGamma (c-a)} \nonumber \\ &\quad \times \int _0^1t^{a-1}(1-t)^{c-a-1}(1-tx)^{-b}(1-ty)^{-b'}\,\text {d}t, \end{aligned}$$

the equation above becomes

$$\begin{aligned} {\hat{I}}_j^\text {(tt)}&=\sum _{k=0}^m\sum _{l=0}^k 2^{m+2}\left( {\begin{array}{c}m\\ k\end{array}}\right) \left( {\begin{array}{c}k\\ l\end{array}}\right) \nonumber \\ &\quad \times\left( {\begin{array}{c}(m+k-1)/2\\ m\end{array}}\right) \frac{(\gamma +\kappa )^{k-l}(-2\kappa )^l}{(\alpha +\kappa )(\beta +\kappa )^{n+k-j}} \frac{\varGamma (l+1/2)\varGamma (1/2)}{\varGamma (l+1)} \nonumber \\ &\quad \times F_1\left( l+1/2;1,n+k-j;l+1;\frac{2\kappa }{\alpha +\kappa },\frac{2\kappa }{\beta +\kappa }\right) . \end{aligned}$$

Next, we use the series form of the Appell’s function

$$\begin{aligned} F_1(a;b,b';c;x,y)=\sum _{m=0}^\infty \frac{(a)_m(b)_m}{m!(c)_m}F(a+m,b';c+m;y)x^m, \end{aligned}$$

and some properties of the Pochhammer symbol and gamma function, we may deduce

$$\begin{aligned} {\hat{I}}_j^\text {(tt)}&=\pi \sum _{k=0}^m\sum _{l=0}^k\sum _{p=0}^\infty 2^{m+l+1}\left( {\begin{array}{c}m\\ k\end{array}}\right) \left( {\begin{array}{c}k\\ l\end{array}}\right) \left( {\begin{array}{c}2l\\ l\end{array}}\right) \nonumber \\ &\quad \times \left( {\begin{array}{c}(m+k-1)/2\\ m\end{array}}\right) \frac{(\gamma +\kappa )^{k-l}(-\kappa )^l}{(\alpha +\kappa )(\beta +\kappa )^{n+k-j}} \frac{(l+1/2)_p}{(l+1)_p} \nonumber \\ &\quad\times F\left( l+p+1/2,n+k-j;l+p+1;\frac{2\kappa }{\beta +\kappa }\right) \left( \frac{2\kappa }{\alpha +\kappa }\right) ^p, \end{aligned}$$

(B4)

where F(a, b; c; x) is the ordinary hypergeometric function. Finally, substituting the result back into Eqs. (B2) and (B1), respectively, leads to the complete form of the energy expression for two tori interaction.

However, if we further assume that the major radius R is much larger than the minor radius r, the energy expression defined by (B3) can be rewritten as

$$\begin{aligned} {\hat{I}}_j^\text {(tt)}=2\int _0^\pi \frac{1}{(\alpha +\kappa \cos \mu )^{n-j+1}}\,\text {d}\mu . \end{aligned}$$

Similarly, applying the properties of the Legendre polynomials as given in (A2) brings

$$\begin{aligned} {\hat{I}}_j^\text {(tt)}=\frac{2\pi }{(\alpha ^2-\kappa ^2)^{(n-j+1)/2}}P_{n-j}\left( \frac{\alpha }{(\alpha ^2-\kappa ^2)^{1/2}}\right) . \end{aligned}$$

(B5)

Subsequently, following the same procedure as described previously by substituting it back into (B2) and (B1) to obtain the final expression for the interaction energy between two tori.