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.

Carbon is a unique and, in many ways, a fundamental element. It can form several different structures, or allotropes. Up to the end of the 1970s, diamond, graphite and graphite-based fibres were the only known forms of carbon assemblies. The discovery in 1985, and subsequent synthesis, of the class of cage-like fullerene molecules, starting with the discovery of the spherically shaped C60 buckyball molecule composed of 20 hexagonal and 12 pentagonal carbon rings, led to the emergence of the third form of condensed carbon. By the beginning of the 1990s, elongated forms of the fullerene molecule, consisting of several layers of graphene sheets, i.e. graphite basal planes, rolled into multi-walled cylindrical structures and closed at both ends with caps containing pentagonal carbon rings, were discovered. Soon afterwards, another form of these cylindrical structures, made from only a single graphene sheet, was also discovered. These two structures have come to be known as carbon nanotubes, and they form the fourth allotrope of carbon.

Three experimental techniques have so far become available for the growth of carbon nanotubes. These are the arc-discharge technique, the laser ablation technique and, recently, the chemical vapour deposition technique. The use of these techniques has led to the worldwide availability of this material, and this has ushered in new, very active and truly revolutionary areas of fundamental and applied research within many diverse and already established fields of science and technology, and also within the new sciences and technologies associated with the new century.

Fluid flow through nanoscopic structures, such as carbon nanotubes, is very different from the corresponding flow through microscopic and macroscopic structures. For example, the flow of fluids through nanomachines is expected to be fundamentally different from the flow through large-scale machines since, for the latter flow, the atomistic degrees of freedom of the fluids can be safely ignored, and the flow in such structures can be characterised by viscosity, density and other bulk properties. Furthermore, for flows through large-scale systems, the no-slip boundary condition is often implemented, according to which the fluid velocity is negligibly small at the fluid–wall boundary.

Reducing the length scales immediately introduces new phenomena, such as diffusion, into the physics of the problem, in addition to the fact that at nanoscopic scales the motion of both the walls and the fluid, and their mutual interaction, must be taken into account. It is interesting to note that the movement of the walls is strongly size-dependent. On the conceptual front, the use of standard classical concepts, such as pressure and viscosity, might also be ambiguous at nanoscopic scales, since, for example, the surface area of a nanostructure, such as a nanotube, may not be amenable to a precise definition.

Introduction

In Chapter 9 we saw how to perform molecular dynamics simulations, although they were not very sophisticated because there was no means of including the effect of the environment. Ways of overcoming this limitation were introduced in the last chapter, in which we discussed methods for calculating the energy and its derivatives for a system within the PBC approximation. As an example, a molecular dynamics simulation for a periodically replicated cubic box of water molecules was performed. Here, we shall start by describing in more detail the type of information that can be computed from molecular dynamics trajectories and also by indicating how the quality of that information can be assessed. Later we shall talk about more advanced molecular dynamics techniques including those that allow simulations to be carried out in various thermodynamic ensembles and those that permit the calculation of free energies.

Analysis of molecular dynamics trajectories

We have seen how to generate trajectories of data for a system, either in vacuum or with an environment, with the molecular dynamics technique. The point of performing a simulation is, of course, that we want to use these data to calculate some interesting quantities, preferably those which can be compared with experimentally measured ones. The aim of this section is to give a brief overview of some of the techniques that can be used to analyse molecular dynamics trajectories and some of the types of quantities that can be calculated.

In Section 9.4 we defined a time series for a property as a sequence of values for the property obtained from successive frames of a molecular dynamics trajectory.

This edition of A Practical Introduction has two major differences from the previous one. The first is a discussion of quantum chemical and hybrid potential methods for calculating the potential energies of molecular systems. Quantum chemical approaches are more costly than molecular mechanical techniques but are, in principle, more ‘exact’ and greatly extend the types of phenomena that can be studied with the other algorithms described in the book. The second difference is the replacement of fortran 90 by Python as the language in which the dynamo module library and the book's computer programs are written. This change was aimed to make the library more accessible and easier to use. As well as these major changes, there have been many minor modifications, some of which I wanted to make myself but many that were inspired by the suggestions of readers of the first edition.

Once again, I would like to acknowledge my collaborators at the Institut de Biologie Structurale in Grenoble and elsewhere for their comments and feedback. Special thanks go to all members, past and present, of the Laboratoire de Dynamique Moléculaire at the IBS, to Konrad Hinsen at the Centre de Biophysique Moléculaire in Orléans and to Troy Wymore at the Pittsburgh Supercomputing Center. I would also like to thank Michelle Carey, Anna Littlewood and the staff of Cambridge University Press for their help during the preparation of this edition, Johny Sebastian from TechBooks for answers to my many LATEX questions, and, of course, my family for their support.

Introduction

In the last chapter we explored various ways of specifying the composition of a molecular system. Many of these representations contained not only information about the number and type of atoms in the system but also the atoms' coordinates, which are an essential element for most molecular simulation studies. Given the nature of the system's atoms and their coordinates the molecular structure of the system is known and it is possible to deduce information about the system's physical properties and its chemistry. The generation of sets of coordinates for particular systems is the major goal of a number of important experimental techniques, including X-ray crystallography and NMR spectroscopy, and there are data banks, such as the PDB and the Cambridge Structural Database (CSD), that act as repositories for the coordinate sets of molecular systems obtained by such methods.

There are several alternative coordinate systems that can be used to define the atom positions. For the most part in this book, Cartesian coordinates are employed. These give the absolute position of an atom in three-dimensional space in terms of its x, y and z coordinates. Other schemes include crystallographic coordinates in which the atom positions are given in a coordinate system that is based upon the crystallographic symmetry of the system and internal coordinates that define the position of an atom by its position relative to a number of other atoms (usually three).

The aim of this chapter is to describe the various ways in which coordinates can be analysed and manipulated. Because numerous analyses can be performed on a set of coordinates, only a sampling of some of the more common ones will be covered here.

The reason that I have written this book is simple. It is the book that I would have liked to have had when I was learning how to carry out simulations of complex molecular systems. There was certainly no lack of information about the theory behind the simulations but this was widely dispersed in the literature and I often discovered it only long after I needed it. Equally frustrating, the programs to which I had access were often poorly documented, sometimes not at all, and so they were difficult to use unless the people who had written them were available and preferably in the office next door! The situation has improved somewhat since then (the 1980s) with the publication of some excellent monographs but these are primarily directed at simple systems, such as liquids or Lennard-Jones fluids, and do not address many of the problems that are specific to larger molecules.

My goal has been to provide a practical introduction to the simulation of molecules using molecular mechanical potentials. After reading the book, readers should have a reasonably complete understanding of how such simulations are performed, how the programs that perform them work and, most importantly, how the example programs presented in the text can be tailored to perform other types of calculation. The book is an introduction aimed at advanced undergraduates, graduate students and confirmed researchers who are newcomers to the field. It does not purport to cover comprehensively the entire range of molecular simulation techniques, a task that would be difficult in 300 or so pages.

Introduction

We saw in Chapter 7 how it was possible to explore relatively small parts of a potential energy surface and in Chapter 8 how to use some of this information to obtain approximate dynamical and thermodynamic information about a system. These methods, though, are local – they consider only a limited portion of the potential energy surface and the dynamics of the system within it. It is possible to go beyond these ‘static’ approximations to study the dynamics of a system directly. Some of these techniques will be introduced in the present chapter.

Molecular dynamics

As we discussed in Chapter 4, complete knowledge of the behaviour of a system can be obtained, in principle, by solving its time-dependent Schrödinger equation (Equation (4.1)), which governs the dynamics of all the particles in the system, both electrons and nuclei. To progress in the solution of this equation we introduced the Born–Oppenheimer approximation, which allows the electronic and the nuclear problems to be treated separately. This separation leads to the concept of a potential energy surface, which is the effective potential that the nuclei experience once the electronic problem has been solved. In principle, it is possible to study the dynamics of the nuclei under the influence of the effective electronic potential using an equivalent equation to Equation (4.1) but for the nuclei only. This can be done for systems consisting of a very small number of particles but proves impractical otherwise.

Fortunately, whereas it is difficult or impossible to study the dynamics of the system quantum mechanically, a classical dynamical study is relatively straightforward and provides much useful information.

Introduction

In the last chapter we dealt with how to manipulate a set of coordinates and how to compare the structures defined by two sets of coordinates. This is useful for distinguishing between two different structures but it gives little indication regarding which structure is the more probable; i.e. which structure is most likely to be found experimentally. To do this, it is necessary to be able to evaluate the intrinsic stability of a structure, which is determined by its potential energy. The differences between the energies of different structures, their relative energies, will then determine which structure is the more stable and, hence, which structure is most likely to be observed.

This chapter starts off by giving an overview of the various strategies that are available for calculating the potential energy of molecular systems and then goes on to describe a specific class of techniques based upon the application of the theory of quantum mechanics to chemical systems.

The Born–Oppenheimer approximation

Quantum mechanics was developed during the first decades of the twentieth century as a result of shortcomings in the existing classical mechanics and, as far as is known, it is adequate to explain all atomic and molecular phenomena. In an oft-quoted, but nevertheless pertinent, remark, P. A. M. Dirac, one of the founders of quantum mechanics, said in 1929:

Introduction

The last two chapters considered two distinct classes of methods for calculating the potential energy of a system. Chapter 4 discussed QC techniques. These are, in principle, the most ‘exact’ methods but they are expensive and so are limited to studying systems with relatively small numbers of atoms. MM approaches were introduced in Chapter 5. These represent the interactions between particles in a simpler way than do QC methods and so are more rapid and, hence, applicable to much larger systems. They have the disadvantage, though, of being unsuitable for treating some processes, notably chemical reactions. Hybrid potentials, which are described in this chapter, seek to overcome some of the limitations of QC and MM methods by putting them together.

Combining QC and MM potentials

Although a hybrid potential is, in principle, any method that employs different potentials to treat a system, either spatially or temporally, the potentials we focus upon in this chapter use a combination of QC and MM techniques. These methods are also known as QC/MM or QM/MM potentials. The first potential of this type was developed in the 1970s by A. Warshel and M. Levitt who were studying the mechanism of the chemical reaction catalyzed by the enyzme lysozyme. Enzymes are proteins that can greatly accelerate the rate of certain chemical reactions. How they achieve this is still a matter of active research but the reaction itself occurs when the substrate species are bound close together in a specific part of the enzyme called the active site.