Year

2018

Degree Name

Doctor of Philosophy

Department

School of Mathematics and Applied Statistics

Abstract

Graphitic nanomaterials have been some of the most widely studied materials of the last decade. The isolation of monolayer graphene in 2004 sparked attention from numerous researchers due to the sheer breadth of its exceptional properties. Thus, it is important that theoretical models keep up with the pace of experimental discovery.

Remarkably, macroscale modelling techniques translate well to nanoscale application. Treating graphitic structures as continuous surfaces allows the application of continuum mechanics and variational calculus amongst others. This thesis involves the formulation and subsequent analysis of three mathematical models relating to the conformation of graphitic structures.

Firstly, we study the interaction energy between sheets of graphene oxide, utilising a continuum model designed for carbon nanostructures. We model graphene oxide as a layered structure consisting of the central graphene sheet and attached functional groups. Varying the oxidation and hydration of these layers is shown to capture the nonstoichiometric nature of graphene oxide. We then model a single-walled carbon nanotube intercalated within the graphene oxide sheets and determine its equilibrium configuration. Molecular dynamics simulations are shown to agree with our analytical results.

Next, we investigate the conformation of the structure consisting of a single-walled carbon nanotube intercalated within a folded graphene sheet. Intercalation provides a potential enhancement of graphitic structures through surface functionalisation. Due to the isotropic nature of the fold, we can consider the problem as curves in two dimensions. The calculus of variations is employed to determine the minimum energy configuration from balancing the bending and adhesion energies. The results are shown to be in excellent agreement with molecular dynamics simulations.

Finally, we examine wrinkle structures formed in a graphene sheet grown on a substrate. Understanding this process has the potential to assist with tuning the properties of the structure or simply eliminating undesirable effects. The calculus of variations is again used to calculate equilibrium wrinkle configurations. We observe several potential configurations of wrinkle depending on the choice of substrate.

Overall, this thesis employs analytical approaches to model graphitic structures, giving rise to explicit solutions that can provide insight into interactions between the nanostructures. The results are shown to agree well with molecular dynamics simulations and existing data in literature. The thesis demonstrates the power of mathematical modelling tools and techniques that can provide reliable predicting capability in the area of nanoscience and nanotechnology.

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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.