Introduction

The electromagnetic field has been described in two different ways. Following the nineteenth century approach (pre quantum mechanics), a field is introduced having appropriate transformation properties. The price one pays is that not every field represents a physically allowed state: such fields must be annihilated by appropriate equations.

Introduction

Applications of group theory in physics start with two very important principles. These are Galileo's principle of relativity (of observers) and Einstein's principle of equivalence (of states).

Nanostructures form the building blocks in nanoscale science and nanoscale technology. These structures operate at highly reduced energy, length and time-scales. They are ultra small for direct observation and measurement, yet may be too large to be described completely by quantum-mechanical computational techniques. Therefore, predictive computer-based numerical modelling and simulations to study their energetics and dynamics, and nanoscale processes, have come to play an increasingly significant role in the conceptual design, synthesis, manipulation, optimisation and testing of functional nanoscale components, nanostructured materials, composed of nanosized grains, and other structures dominated by nanointerfaces. The importance of these computational approaches, from the perspective of nanoscience and nanotechnology, rests on the fact that they can provide essentially exact data pertinent to nanoscale model systems which, as a result of the reduced energy, length and time-scales involved, may be very difficult to obtain otherwise.

Many key questions in nanoscience are related to the properties of the constituent nanostructures. It is known that the stability of the different phases is altered in the nanometre regime, which is influenced by both kinetic and thermodynamic factors. Therefore, we need to investigate the mechanics and thermodynamics of phase transformations in nanostructures. Many mechanical, thermal and electronic properties of nanoscale building blocks vitally depend on the size, shape and the precise geometrical arrangement of all the atoms within the block.

Various types of continuum-based elasticity theories have been extensively employed to model the nanomechanics of free-standing SWCNTs, MWCNTs and nanotubes that are embedded in elastic media, such as a polymeric matrix. The results from these modelling studies have been compared with the results obtained from the atomistic-based studies where the discrete nature of the nanotube structure has been explicitly taken into account. Remarkably, as we shall see later on, close agreements have been obtained between these results, indicating that the laws of continuum-based elasticity theories can still be relevant in modelling structures and systems in the nanoscale domains. The continuum-based theories that have been used include the nonlinear thin-shell theories, the theories of curved plates, the theories of vibrating rods and the theories of bending beams. In this chapter we shall present the essential tenets of all these theories, and provide enough details so that the current research materials can be followed and future problems can be formulated with their aid.

Basic concepts from continuum elasticity theory

Hooke's laws in isotropic elastic materials

Let us first consider some of the essential topics from the theory of elasticity that are routinely employed in studies concerned with the mechanical properties of solid structures.

Computational modelling of the adsorption and flow of various types of gas in nanotubes forms a very active area of research. The adsorption can take place inside SWCNTs and, in the case of a bundle of SWCNTs, it can also take place at three additional bundle sites. These sites are the interstitial channels between the SWCNTs, i.e. the interior space between the SWCNTs in the bundle, the outer surfaces of the nanotubes composing the rope, and the ridges (or groove) channels, i.e. the wedge-shaped spaces that run along the outer surface of the rope where two SWCNTs meet. Computational modelling studies that we will consider show that while H2, He and Ne particles can adsorb in the interstitial channels, other types of atom are too large to fit into such tiny spaces. As a result, the clarification of the adsorption sites, to determine where a given gas atom can be accommodated, is a focal point of research. The insight obtained from this research has important ramifications for the application of nanotechnology to gas-storage devices, molecular sieves and filtration membranes. We will first consider the all-important case of modelling hydrogen storage in nanotubes.

Atomic and molecular hydrogen in nanotubes

Modelling H2 storage in carbon nanotubes occupies a very prominent position in the ongoing research in this field, and several, rather detailed, numerical simulations have been devoted to the study of various aspects of this problem.

Computational investigation of the mechanical properties of carbon nanotubes is one of the most active research fields in the physics of nanotubes due to the importance of these properties in the practical applications of nanotubes in nanotechnology devices. Experimental and theoretical/computational modelling studies indicate that SWCNTs and MWCNTs enjoy extraordinary mechanical properties. The computational modelling in this field has employed some of the highly sophisticated atomistic and continuum-elasticity models, showing that, for instance, carbon nanotubes have high tensile strengths, large bending flexibilities and high aspect ratios. These are properties that make nanotubes an ideal material for superstrong nanofibres. Defect-free nanotubes have no exposed edges in the direction parallel to the axis of the nanotube, in contrast to graphene sheets, and as a result they can resist fracture or crack-formation in the direction perpendicular to the externally applied strain.

A rather extensive part of the research concentrates on the computation of the elastic constants, Poisson's ratios and Young's moduli of SWCNTs, MWCNTs and their respective bundles (ropes), aiming to show the dependence of these properties on the diameter and chirality of the nanotubes. A very interesting aspect of the computational modelling of the mechanical properties of nanotubes that has clearly emerged from the research in this area of nanotube physics is the relevance of the well-established continuum-based theories of curved plates, thin shells, beams and vibrating rods, to model and interpret the response of nanotubes to external influences, such as large strains, or the flow of fluids inside nanotubes.

The appearance of powerful high-performance computational facilities has led to the emergence of a new approach to fundamental research, namely computational modelling and computer-based numerical simulations, with applications in practically all areas of basic and applied sciences, from physical sciences to biological sciences, from medical sciences to economic and social sciences. The applications of computational simulations over the past two decades have been so phenomenal that a new academic discipline, called computational science, with its own research centres, laboratories and academia, has appeared on the educational and industrial scenes in almost all the developed, and many developing, countries. Computational science complements the two traditional strands of research, namely analytical theory-building, and laboratory-based experimentation, and is referred to as the third approach to research. Numerical simulations present the scientists with many unforeseen scenarios, providing a backdrop to test the physical theories employed to model the energetics and dynamics of a system, the approximations and the initial conditions. Furthermore, they offer clues to the experimentalists as to what type of phenomena to expect and to look for. It is no exaggeration to suggest that we are now experiencing a monumental numerical revolution in physical, biological and social sciences, and their associated fields and technologies.

Carbon nanotubes will form the essential components in all sorts of functional devices, from nanoscale transistors to nanofluidic devices and functionalised array medical nanosensors, to name but a few. Besides their mechanical and electronic properties, it is also very important to know their thermal properties and thermal performances. In contrast to the mechanical, electronic and storage properties of nanotubes, for which a rather significant number of modelling and experimental studies have been carried out, the investigations into the thermal properties of nanotubes have been rather scant. Measurements of the specific heat and thermal conductivity of microscopic structures, such as mats covered with compressed ropes of carbon composed of hundreds of nanotubes, have been made, providing valuable information on the ensemble-average thermal properties of these bulk-phase materials, rather than on individual nanotubes.

The measurement of the thermal conductivity of nanotubes, like the measurement of their other properties, is subject to a degree of uncertainty owing to the impurities that are likely to be present in the composition of synthesised nanotubes. For example, the MWCNTs grown by the chemical vapour deposition (CVD) technique at temperatures as low as T ∼ 600 K are not perfect, and this is borne out by the fact that their thermal and electrical conductivities are two orders of magnitude lower than those of perfect crystalline graphite at room temperature.

Information concerning the mechanical properties of nanostructures, such as their elastic constants, fracture strength, stress distribution maps, etc., can be obtained directly from the underlying interatomic potential energies. Here, we consider the necessary theoretical tools for the atomistic-based computation of these properties, and show in detail the steps that could be followed to derive the analytical expressions that can be used in computer-based simulations. We assume that the energetics of an N-atom system, as described by an interatomic potential energy function of whatever variety, are known. However, we illustrate our derivations on the basis of central two-body potentials. Derivation of these properties, based on more complex many-body potentials, can follow similar techniques to those discussed here for the two-body potentials. A concise derivation of the pertinent expressions for the atomic-level stress tensor and elastic constants is very desirable, and this task has been performed in the work of Nishioka et al., to which we shall refer.

Atomic-level stress tensor

The notion of obtaining the elastic constants and atomic-level stresses of a crystal in terms of interatomic forces was extensively studied by Max Born in his classic treatise with Huang using the method of small homogeneous deformations. Let us, as before, denote the stress by the rank-two tensor σαβ(i), where the index i is now introduced to designate a particular atom in an N-atom system, and α, β = 1, 2, 3 refer to the x1-,x2- and x3-Cartesian components.

Carbon is the first element in Group IV of the Periodic Table, with the properties listed in Table 2.1. The isolated carbon atom has an electronic configuration 1s22s22p2, composed of two electrons in the 1s orbital, the filled K shell, and the remaining four electrons distributed according to two electrons in the filled 2s orbital of the L shell and two electrons in the two half-filled 2p orbitals of the same shell. In the ground state of the carbon atom, the s orbital is spherically symmetric and the p orbital is in the shape of a dumbbell which is symmetrical about its axis. While the s orbital is non-directional, the p orbital has directional properties. The ionisation energies of the electrons in a carbon atom are very different, and they are listed in Table 2.2.

Carbon atoms bond together by sharing electron pairs that form covalent bonds. The two electrons in the very stable K shell are not involved in any bonding that takes place. The bonding can lead to various known carbon allotropes, i.e. diamond, graphite, various types of fullerene and several kinds of nanotube. Since carbon bonding occurs as a result of the overlap of atomic orbitals, one might think that a carbon atom can form only two bonds with other atoms since it has only two 2p electrons available as valence electrons.