Restricted three body problems at the nanoscale

In this paper, we investigate some of the classical restricted three body problems at the nanoscale, such as the circular planar restricted problem for three C60 fullerenes, and a carbon atom and two C60 fullerenes. We model the van der Waals forces between the fullerenes by the Lennard-Jones potential. In particular, the pairwise potential energies between the carbon atoms on the fullerenes are approximated by the continuous approach, so that the total molecular energy between two fullerenes can be determined analytically. Since we assume that such interactions between the molecules occur at sufficiently large distance, the classical three body problems analysis is legitimate to determine the collective angular velocity of the two and three C60 fullerenes at the nanoscale. We find that the maximum angular frequency of the two and three fullerenes systems reach the terahertz range and we determine the stationary points and the points which have maximum velocity for the carbon atom for the carbon atom and the two fullerenes system

Some novel plane trajectories for carbon atoms and fullerenes captured by two fixed parallel carbon nanotubes

The movement of atoms and molecules at the nanoscale constitutes a fundamental problem in physics, especially following the motion of atoms in many-body systems condensing together to form molecular structures. A number of simplified nanoscale dynamical problems have been analyzed and here we investigate the classical orbiting problem around two centers of attraction at the nanoscale. An example of such a system would be a carbon atom or a fullerene orbiting in a plane which is perpendicular to two fixed parallel carbon nanotubes. We model the van der Waals forces between the molecules by the Lennard-Jones potential. In particular, the total pairwise potential energies between carbon atoms on the fullerene and the carbon nanotubes are approximated by the continuous approach, so that the total molecular energy can be determined analytically. Since we assume that such interactions occur at a sufficiently large distance, the classical two center problem analysis is legitimate to determine various novel trajectories of the atom and fullerene numerically. It is clear that the oscillatory period of the atom for some bounded trajectories reaches terahertz frequencies. We comment that the continuous approach adopted here has the merit of a very fast computational time and can be extended to more complicated structures, in contrast to quantum mechanical calculations and molecular dynamics simulations.Y. Chan and J.M. Hil

Packing configurations for methane storage in carbon nanotubes

In this paper we investigate methane packing in single-walled carbon nanotubes. We employ classical applied mathematical modelling using the basic principles of mechanics to exploit the Lennard- Jones potential function and the continuous approximation, which assumes that intermolecular interactions can be approximated by average atomic surface densities.We consider both zigzag and spiral configurations formed by packing methane molecules into (9, 5), (8, 8) and (10, 10) carbon nanotubes, and we derive analytical expressions for the interaction potential energy of these configurations. Our findings indicate that for the zigzag configuration for a (9, 5) tube, the potential energy of the system is minimized when the methane molecules simply form a linear chain along the tube axis, but genuine zigzag patterns are found as the tube size increases such as for the (8, 8) and (10, 10) tubes. For the spiral configuration, the potential energy of the system is minimized when the angular spacing is approximately equal to π for the (9, 5) and (8, 8) tubes, and π/2 for the (10, 10) tube. Overall, our results are in good agreement with molecular dynamics simulations in the literature and show that the most energetically efficient packing configuration of the three tubes studied, occurs for a (10, 10) tube with a zigzag packing, while a (10, 10) tube with a spiral packing configuration has the largest free-cavity volume for methane adsorption at higher temperatures.O. O. Adisa, B. J. Cox, and J. M. Hil

Interaction of double-stranded DNA inside single-walled carbon nanotubes

Deoxyribonucleic acid (DNA) is the genetic material for all living organisms, and as a nanostructure offers the means to create novel nanoscale devices. In this paper, we investigate the interaction of deoxyribonucleic acid inside single-walled carbon nanotubes. Using classical applied mathematical modeling, we derive explicit analytical expressions for the encapsulation of DNA inside single-walled carbon nanotubes. We adopt the 6–12 Lennard–Jones potential function together with the continuous approach to determine the preferred minimum energy position of the dsDNA molecule inside a single-walled carbon nanotube, so as to predict its location with reference to the cross-section of the carbon nanotube. An analytical expression is obtained in terms of hypergeometric functions which provides a computationally rapid procedure to determine critical numerical values. We observe that the double-strand DNA can be encapsulated inside a single-walled carbon nanotube with a radius larger than 12.30 Å, and we show that the optimal single-walled carbon nanotube to enclose a double-stranded DNA has radius 12.8 Å.Mansoor H. Alshehri; Barry J. Cox; James M. Hil

Modeling on ion rejection using membranes comprising ultra-small radii carbon nanotubes

In this paper, we investigate the complete ion rejection using carbon nanotube membranes comprising ultra-small radii nanotubes. Three acceptance radii for a water molecule, a sodium ion and a chloride ion are determined assuming the continuous approximation. Given the acceptance radii, we may confine the scope of the nanotube radius so that only water molecules can pass through but the heavier sodium and chloride ions are repulsed from the nanotube ends. We assume that the collective motion of water molecules inside a sufficiently long nanotube is driven by atomic vibrations so that classical phonon theory might be used to deduce the average water transit time inside the nanotube for ion rejection. We predict that for carbon nanotube membranes comprising nanotubes of radii lying between 3.4 and 3.9 Å, only water molecules will pass through, and sodium and chloride ions will not, which together using phonon theory, we deduce that the smaller the nanotube radius, the lower the average water transit time and the higher the efficiency of the membrane for ion rejection purposes. The present theoretical approach has the merit of rapid computational times and indicates those nanotube radii where future experimental work might be focussed. © 2012 EDP Sciences, SIF, Springer-Verlag Berlin Heidelberg.Y. Chan and J.M. Hil