Richard K. F. Lee

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.

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.

Metallofullerenes in Composite Carbon Nanotubes as a Nanocomputing Memory Device

Here, we investigate a hybrid carbon nanostructure, which comprises two single-open host nanotubes of the same radius and joined by another single-open nanotube, which is centrally located between the host nanotubes but has a smaller radius. A metallofullerene is then enclosed inside the structure to represent a bit information and is originally located inside one of the host nanotubes. The geometric parameters, such as the radii of nanotubes and fullerene radius are purposely chosen so that the metallofullerene cannot enter the central nanotube without additional energy. By applying an external electrical field, the metallofullerene can overcome the energy barrier and pass from one end to the other end to form a two-state fullerene shuttle memory device. The key geometric parameters are provided for a range of fullerenes, including C60, C 80, and C100, noting that we assume most metallofullerenes take the form M@C60, M@C80, and M@C100, where M denotes a metal atom or ion located noncovalently inside the fullerene Cn.

An idealized polyhedral model and geometric structure for silicon nanotubes

In this paper, we introduce an idealized model of silicon nanotubes comprising an exact polyhedral geometric structure for single-walled silicon nanotubes. The silicon nanotubes considered here are assumed to be formed by sp(3) hybridization and thus the nanotube lattice is assumed to comprise only squares or skew rhombi. Beginning with the three postulates that all bond lengths are equal, all adjacent bond angles are equal, and all atoms are equidistant from a common axis of symmetry, we derive exact formulae for the geometric parameters such as radii, bond angles and unit cell length. We present asymptotic expansions for these quantities to the first two orders of magnitude. Because of the faceted nature of the polyhedral model we may determine a perceived inner radius for the nanotube, from which an expression for the wall thickness emerges. We also describe the geometric properties of some ultra-small silicon nanotubes. Finally, the values of the diameters for the polyhedral model are compared with results obtained from molecular dynamics simulations and some limited numerical calculations are undertaken to confirm the meta-stability of the proposed structures.

General Rolled-Up and Polyhedral Models for Carbon Nanotubes

In many computational studies of carbon nanotubes, the minimum energy configuration frequently settles on a structure for which the bond lengths are distinct. Here, we extend both the rolled-up and the polyhedral models for SWCNTs to produce general models incorporating either distinct bond lengths and the same bond angle, or distinct bond lengths and distinct bond angles. The CNTs considered here are assumed to be formed by sp2 hybridization but with different bond lengths so that the nanotube structure is assumed to comprise irregular hexagonal patterns. The polyhedral model with distinct bond lengths and distinct bond angles is based on the two postulates that all bonds lying on the same helix are equal in length and that all atoms are equidistant from a common axis of symmetry. The polyhedral model with distinct bond lengths and the same bond angle has the additional postulate that all the adjacent bond angles are equal. We derive exact formulae for the geometric parameters and we present asymptotic expansions for the polyhedral model with distinct bond lengths and distinct bond angles to the first two orders of magnitude. Good agreement is demonstrated for the predictions of the polyhedral model compared with the results obtained from other computational studies.

Design Parameters for a Two-State Nanomemory Device

In this study, we investigate the internal mechanics for a two-state memory device,comprising a charged metallofullerene which is located inside a closed carbon nanotube.Assuming the Lennard-Jones interaction energy and the continuum approximation, the metallofullerenehas two symmetrically placed equal minimum energy positions. On one side theencapsulated metallofullerene represents the zero information state and by applying an externalelectrical field, the metallofullerene can be made to overcome the energy barrier of thenanotube, and pass from one end of the tube to the other, where the metallofullerene thenrepresents the one information state.

General formulae for interacting spherical nanoparticles and fullerenes

Most nanodevices under investigation adopt a computational approach such as molecular dynamics simulations, which gives a numerical value for the potential energy as calculated from the interaction of every atom on one molecule with every atom on a second molecule. Although the simulation only involves short range atom–atom interactions and ignores those interactions at longer distances, the simulation still involves significant computational time. In this paper, we determine analytical formulae for four types of Lennard–Jones interactions: (i) a solid spherical nanoparticle with an atom, (ii) two distinct radii hollow spherical fullerenes, (iii) a solid spherical nanoparticle with a hollow spherical fullerene and (iv) two distinct radii solid spherical nanoparticles. The interaction energy using the 6–12 Lennard–Jones potential for these four situations are determined using the continuum approximation, which assumes that a discrete atomic structure can be replaced by either an average atomic surface density or an average atomic volume density. Using these formulae the computational time for a simulation might be dramatically reduced for those molecular interactions involving spherical nanoparticles or fullerenes. Such formulae might be exploited in hybrid analytical-computational numerical schemes, as well as in metallofullerenes and certain assumed spherical models of molecules such as methane and ammonia. As an illustration of the formulae presented here we determine both the most stable and the maximum radii of a solid spherical nanoparticle inside a fullerene, modelling the centre of a carbon onion or metallofullerenes. We also determine new cut-off formulae for interacting spherical nanoparticles and fullerenes which might be useful in computational schemes.

POLYHEDRAL MODELS AND GEOMETRIC STRUCTURES FOR NANOTUBES

In this thesis, some new polyhedral models for nanotubes are examined. The conventional rolled-up model for carbon nanotubes assumes that a flat sheet of graphene is rolled into a seamless right circular cylinder and therefore in terms of the geometric parameters, the curvature inherent in the structure of nanotubes is not taken into account. The conventional rolled-up model of nanotubes completely ignores any effects due to curvature while the existing ideal polyhedral models for single-walled carbon nanotubes and boron nitride nanotubes, which are both hexagonal structures, are known to give predictions for the geometric parameters of the tube which are in excellent agreement with computational studies (molecular dynamics simulations and ab initio calculations). In this thesis the notion of an ideal polyhedral model is extended to silicon and boron nanotubes, which adopt respectively squares or skew rhombi and flat equilateral triangles as their structure. The silicon nanotubes considered here are assumed to be formed by sp hybridization and thus the nanotube lattice is assumed to comprise only squares or skew rhombi. The boron nanotubes considered here are assumed to be formed by complex bonding type and therefore the nanotube lattice is assumed to comprise a triangular pattern. From molecular dynamics simulation results for carbon nanotubes and silicon nanotubes, the bond lengths are known to vary depending upon the bond direction. Often this aspect can not be ignored and therefore in this thesis both the conventional and the ideal polyhedral models are extended to include distinct bond lengths, and specifically for carbon, silicon and boron nanotubes. These general models are shown to be in excellent agreement with computational studies. We first present the standard geometric parameters for the conventional nanotube model. Noting again that the curvature inherent in this model is completely ignored, for the ideal polyhedral model for silicon nanotubes we begin with three

Geometric model of silicon nanotubes

In this paper, we extend both the rolled-up and the polyhedral models for single-walled silicon nanotubes with equal bond lengths to models having distinct bond lengths. The silicon nanotubes considered here are assumed to be formed by sp3 hybridization with different bond lengths so that the nanotube lattice is assumed to comprise only skew rhombi. Beginning with the three postulates that (i) all bonds lying on the same helix are equal, (ii) all adjacent bond angles are equal, and (iii) all atoms are equidistant from a common axis of symmetry, we derive exact formulae for the polyhedral geometric parameters such as chiral angles, adjacent bond angles and radius. Finally, some molecular dynamics simulations are undertaken for comparison with the geometric model. These simulations start with equal bond lengths and then stabilize in such a way that two distinct bond lengths emerge.