## Year

2011

## Degree Name

Doctor of Philosophy

## Department

School of Mathematics and Applied Statistics

## Recommended Citation

Tran-Duc, Thien, Mathematical modelling for three problems in nanotechnology, Doctor of Philosophy thesis, School of Mathematics and Applied Statistics, University of Wollongong, 2011. https://ro.uow.edu.au/theses/3496

## Abstract

The non-covalent interaction energy in multimolecular systems is a topic of considerable interest in nanotechnology, since it holds the key to understanding their equilibrium configurations and mechanical properties. For systems involving large numbers of atoms and molecules, the determination of the interaction energy becomes especially complicated and it provides a major challenge for computational chemistry. This thesis develops mathematical models resulting in analytical formulae for the interaction energy in three specific areas of nanotechnology.

Aromatic interactions are generally found to stabilize the global structure of proteins and the double helical structure of DNA, and they also play an important role in recognition processes in biological and non-biological systems. However, the complexity and variety in the structure of aromatic molecules are major obstacles to a proper investigation of their interactions. In this thesis, we investigate a typical case of an aromatic interaction, namely that for a benzene dimer. The major result obtained in this thesis is an analytical expression for the interaction energy which is used to predict all possible equilibrium configurations of the benzene dimer. At sufficiently large distances, the relative orientation of the two benzene molecules is found to be approximated by the arctan of the ratio of the two separation distances in two mutually perpendicular directions.

Aromatic hydrocarbons are emitted into the atmosphere as byproducts of combustion, and constitute one of the most widespread human pollutants known as carcinogens and mutagens. The fact that under normal environmental conditions, aromatic hydrocarbons prefer to stay on the surface of airborne particles, provides a possible solution for the elimination of aromatic hydrocarbons from the environment by exploiting materials with high surface area, such as graphitic structures. In this thesis, we study the adsorption mechanism of aromatic hydrocarbons onto a graphene sheet from a full investigation of the adsorption of a coronene molecule onto graphene. We find that coronene is adsorbed on graphene in three stages, and finally becomes stable on the surface of the graphene sheet at a parallel configuration, which is completely consistent with experimental and theoretical results. Furthermore, computational results show that under normal environmental conditions, the interaction energies between the aromatic molecules and the graphene sheet are much higher than the kinetic energies of their thermal motions, and they are strongly held on the graphene surface.

Carbon nanotubes possess extreme mechanical, thermal, electrical and opto- electronic properties. For applications in nanowires, certain molecules are en- capsulated inside carbon nanotubes to enhance their conductivity. Nanopeapods are carbon nanotubes containing fullerenes and are a typical example for such a nanowire. However, due to the non-covalent interactions, a particular molecule might not always be accepted by a carbon nanotube, even though it might be smaller in size than the diameter of the carbon nanotube. More importantly, the non-covalent interactions also determine the equilibrium configuration of the molecule inside the carbon nanotube by which the conductivity of the nanowire is determined. In this thesis, the encapsulation of benzene and acetylene into carbon nanotubes for applications in nanowires is investigated. Questions concerning the acceptance radii of the carbon nanotube and the equilibrium configurations of the molecules inside carbon nanotubes are addressed.

In summary, the original contribution of this thesis is the development of mathematical models for the three interaction energy problems; namely the geometry and energetics of a benzene dimer; the adsorption of aromatic hydrocarbons onto graphene; and the encapsulation of benzene and acetylene into a carbon nanotube. From these mathematical models, the interaction energies between the molecules and the nanostructures are determined analytically and may be readily evaluated using standard algebraic computer packages, so that the interacting mechanisms and the equilibrium configurations of the systems may be fully investigated.

## FoR codes (2008)

0102 APPLIED MATHEMATICS, 010506 Statistical Mechanics, Physical Combinatorics and Mathematical Aspects of Condensed Matter

**Unless otherwise indicated, the views expressed in this thesis are those of the author and do not necessarily represent the views of the University of Wollongong.**