Barry J. Cox

Mechanics of atoms and fullerenes in single-walled carbon nanotubes. I. Acceptance and suction energies

Owing to their unusual properties, carbon nanostructures such as nanotubes and fullerenes have caused many new nanomechanical devices to be proposed. One such application is that of nanoscale oscillators which operate in the gigahertz range, the so-called gigahertz oscillators. Such devices have potential applications as ultrafast optical filters and nano-antennae. While there are difficulties in producing micromechanical oscillators which operate in the gigahertz range, molecular dynamical simulations indicate that nanoscale devices constructed of multi-walled carbon nanotubes or single-walled carbon nanotubes and C60 fullerenes could feasibly operate at these high frequencies. This paper investigates the suction force experienced by either an atom or a C60 fullerene molecule located in the vicinity of an open end of a single-walled carbon nanotube. The atom is modelled as a point mass, the fullerene as an averaged atomic mass distributed over the surface of a sphere. In both cases, the carbon nanotube is modelled as an averaged atomic mass distributed over the surface of an open semi-infinite cylinder. In both cases, the particle being introduced is assumed to be located on the axis of the cylinder. Using the Lennard-Jones potential, the van der Waals interaction force between the atom or C60 fullerene and the carbon nanotube can be obtained analytically. Furthermore, by integrating the force, an explicit analytic expression for the work done by van der Waals forces is determined and used to derive an acceptance condition, that is whether the particle will be completely sucked into the carbon nanotube by virtue of van der Waals interactions alone, and a suction energy which is imparted to the introduced particle in the form of an increased velocity. The results of the acceptance condition and the calculated suction energy are shown to be in good agreement with the published molecular dynamical simulations. In part II of the paper, a new model is proposed to describe the oscillatory motion adopted by atoms and fullerenes that are sucked into carbon nanotubes.

Mechanics of atoms and fullerenes in single-walled carbon nanotubes. II. Oscillatory behaviour

The discovery of carbon nanotubes and C60 fullerenes has created an enormous impact on possible new nanomechanical devices. Owing to their unique mechanical and electronic properties, such as low weight, high strength, flexibility and thermal stability, carbon nanotubes and C60 fullerenes are of considerable interest to researchers from many scientific areas. One aspect that has attracted much attention is the creation of high-frequency nanoscale oscillators, or the so-called gigahertz oscillators, for applications such as ultrafast optical filters and nano-antennae. While there are difficulties for micromechanical oscillators, or resonators, to reach a frequency in the gigahertz range, it is possible for nanomechanical systems to achieve this. This study focuses on C60–single-walled carbon nanotube oscillators, which generate high frequencies owing to the oscillatory motion of the C60 molecule inside the single-walled carbon nanotube. Using the Lennard-Jones potential, the interaction energy of an offset C60 molecule inside a carbon nanotube is determined, so as to predict its position with reference to the cross-section of the carbon nanotube. By considering the interaction force between the C60 fullerene and the carbon nanotube, this paper provides a simple mathematical model, involving two Dirac delta functions, which can be used to capture the essential mechanisms underlying such gigahertz oscillators. As a preliminary to the calculation, the oscillatory behaviour of an isolated atom is examined. The new element of this study is the use of elementary mechanics and applied mathematical modelling in a scientific context previously dominated by molecular dynamical simulation.

Lithiated Porous Aromatic Frameworks with Exceptional Gas Storage Capacity

Kristina Konstas, James W. Taylor, Aaron W. Thornton, Cara M. Doherty, Wei Xian Lim, Timothy J. Bastow, Danielle F. Kennedy, Colin D. Wood, Barry J. Cox, James M. Hill, Anita J. Hill, Matthew R. Hill

Einstein's special relativity beyond the speed of light

We propose here two new transformations between inertial frames that apply for relative velocities greater than the speed of light, and that are complementary to the Lorentz transformation, giving rise to the Einstein special theory of relativity that applies to relative velocities less than the speed of light. The new transformations arise from the same mathematical framework as the Lorentz transformation, displaying singular behaviour when the relative velocity approaches the speed of light and generating the same addition law for velocities, but, most importantly, do not involve the need to introduce imaginary masses or complicated physics to provide well-defined expressions. Making use of the dependence on relative velocity of the Lorentz transformation, the paper provides an elementary derivation of the new transformations between inertial frames for relative velocities v in excess of the speed of light c, and further we suggest two possible criteria from which one might infer one set of transformations as physically more likely than the other. If the energy–momentum equations are to be invariant under the new transformations, then the mass and energy are given, respectively, by the formulae and where denotes the limiting momentum for infinite relative velocity. If, however, the requirement of invariance is removed, then we may propose new mass and energy equations, and an example having finite non-zero mass in the limit of infinite relative velocity is given. In this highly controversial topic, our particular purpose is not to enter into the merits of existing theories, but rather to present a succinct and carefully reasoned account of a new aspect of Einsteins theory of special relativity, which properly allows for faster than light motion.

Mechanics of nanotubes oscillating in carbon nanotube bundles

Carbon nanotubes are nanostructures that promise much in the area of constructing nanoscale devices due to their enhanced mechanical, electrical and thermal properties. In this paper, we examine a gigahertz oscillator that comprises a carbon nanotube oscillating in a uniform concentric ring or bundle of carbon nanotubes. A number of existing results for nanotube oscillators are employed to analyse the design considerations of optimizing such a device, and significant new results are also derived. These include a new analytical expression for the interaction per unit length of two parallel carbon nanotubes involving the Appell hypergeometric functions. This expression is employed to precisely determine the relationship between the bundle radius and the radii of the nanotubes forming the bundle. Furthermore, several pragmatic approximations are also given, including the relationships between the bundle radius and the constituent nanotube radius and the oscillating tube radius and the bundle nanotube radius. We also present a simplified analysis of the force and energy for a nanotube oscillating in a nanotube bundle leading to an expression for the oscillating frequency and the maximum oscillating frequency, including constraints on configurations under which this maximum is possible.

High performance hydrogen storage from Be-BTB metal-organic framework at room temperature.

The metal-organic framework beryllium benzene tribenzoate (Be-BTB) has recently been reported to have one of the highest gravimetric hydrogen uptakes at room temperature. Storage at room temperature is one of the key requirements for the practical viability of hydrogen-powered vehicles. Be-BTB has an exceptional 298 K storage capacity of 2.3 wt % hydrogen. This result is surprising given that the low adsorption enthalpy of 5.5 kJ mol(-1). In this work, a combination of atomistic simulation and continuum modeling reveals that the beryllium rings contribute strongly to the hydrogen interaction with the framework. These simulations are extended with a thermodynamic energy optimization (TEO) model to compare the performance of Be-BTB to a compressed H2 tank and benchmark materials MOF-5 and MOF-177 in a MOF-based fuel cell. Our investigation shows that none of the MOF-filled tanks satisfy the United States Department of Energy (DOE) storage targets within the required operating temperatures and pressures. However, the Be-BTB tank delivers the most energy per volume and mass compared to the other material-based storage tanks. The pore size and the framework mass are shown to be contributing factors responsible for the superior room temperature hydrogen adsorption of Be-BTB.

The geometric structure of single-walled nanotubes

In this paper, we survey a number of existing geometric structures which have been proposed by the authors as possible models for various nanotubes. Atoms assemble into molecules following the laws of quantum mechanics, and in general computational approaches to predicting the molecular structure can be arduous and involve considerable computing time. Fortunately, nature favours minimum energy structures which tend to be either very symmetric or very unsymmetric, and which therefore can be analyzed from a geometrical perspective. The conventional rolled-up model of nanotubes completely ignores any effects due to curvature and the present authors have proposed a number of exact geometric models. Here we review a number of these recent developments relating to the geometry of nanotubes, including both the traditional rolled-up models and some exact polyhedral constructions. We review a number of formulae for four materials, carbon, silicon, boron and boron nitride, and we also include results for the case when the bond lengths may take on distinct values.

Orientation of spheroidal fullerenes inside carbon nanotubes with potential applications as memory devices in nano-computing

A spheroid is an ellipsoid for which two of the axes are equal, and here the interaction between spheroidal fullerenes and carbon nanotubes is modeled using the Lennard–Jones potential and the continuum approximation. The resulting surface integrals are evaluated analytically for a number of configurations, including lying and standing as well as spheroids with an arbitrary tilt angle, and centered on the nanotube axis. Analytical expressions for off-axis spheroids in all three orientations are also given, and the findings are shown to agree well with previously published work. However, the major contribution of this work is the derivation of new exact analytical formulae to calculate the van der Waals interaction energy for these configurations, and in particular the results for the tilting and off-axis configurations which are far more general than those which have appeared in the literature previously. From these exact expressions, five primary regimes are identified: lying on-axis, tilting on-axis, standing on-axis, standing off-axis and finally lying off-axis. Also identified in this study is a precisely prescribed radius, for the transition between regimes four and five, for which two equally energetically favorable orientations exist and for which these two configurations are separated by a known energy barrier. The notion arises that such configurations may be exploited for nano-scaled memory devices used in nano-computing.

Mechanics of fullerenes oscillating in carbon nanotube bundles

In this paper, we examine the mechanics of a nano-scaled gigahertz oscillator comprising a fullerene that is moving within the center of a bundle of carbon nanotubes. Although numerical results specifically for a C60 fullerene are presented, the method is equally valid for any fullerene which can be modeled as a spherical molecule. A general definition of a nanotube bundle is employed which can comprise any number of parallel carbon nanotubes encircling the oscillating fullerene. Results are presented which prescribe the dimension of the bundle for any nanotube radius and the optimal configurations which give rise to the maximum suction energy for the fullerene. Prior results for fullerene single-walled nanotube oscillators are employed, and new results are also derived. These include a calculation of optimum nanotube bundle size to be employed for a C60-nanotube bundle oscillator, as well as new analytical expressions for the force and energy for a semi-infinite nanotube and a fullerene not located on the axis of the cylinder.

An exact polyhedral model for boron nanotubes

An exact idealized polyhedral model is formulated to describe the geometry of single-walled boron nanotubes. The boron nanotubes considered here are assumed to be formed by sp2 hybridization and adopt a flat equilateral triangle pattern. Beginning from the two fundamental postulates that all bond lengths are equal and all atoms are equidistant from a common cylindrical axis, we derive exact formulae for the geometric parameters of the nanotube radius, bond angle and unit cell length, and we present asymptotic expansions for these quantities to the first two orders of magnitude. Good agreement is demonstrated for the predictions of the polyhedral model, compared with the results obtained from first-principles simulations. The polyhedral model allows the possible identification of an inner radius, so that the notion of nanotube wall thickness can be introduced. Finally, we examine the geometric structure of some ultra-small boron nanotubes.