Modelling and Mechanics of Carbon-based Nanostructured Materials

By Duangkamon Baowan, Barry J Cox, Tamsyn A Hilder and 2 more

Modelling and Mechanics of Carbon-based Nanostructured Materials sets out the principles of applied mathematical modeling in the topical area of nanotechnology. It is purposely designed to be self-contained, giving readers all the necessary modeling principles required for working with nanostructures.

The unique physical properties observed at the nanoscale are often counterintuitive, sometimes astounding researchers and thus driving numerous investigations into their special properties and potential applications. Typically, existing research has been conducted through experimental studies and molecular dynamics simulations. This book goes beyond that to provide new avenues for study and review.

• Explores how modeling and mechanical principles are applied to better understand the behavior of carbon nanomaterials
• Clearly explains important models, such as the Lennard-Jones potential, in a carbon nanomaterials context
• Includes worked examples and exercises to help readers reinforce what they have read

• Language: English
• Publisher: Elsevier Science
• Release date: Feb 12, 2017
• ISBN: 9780128124642

Duangkamon Baowan

Duangkamon Baowan is Associate Professor of Applied Mathematics at Mahilodi University, Thailand, having previously worked at the University of Wollongong, Australia. Her research is focused on the mechanics of nanoscaled materials

Book preview

Modelling and Mechanics of Carbon-based Nanostructured Materials

First Edition

Duangkamon Baowan

Barry J. Cox

Tamsyn A. Hilder

James M. Hill

Ngamta Thamwattana

Table of Contents

Cover image

Title page

Copyright

Preface

Chapter 1: Geometry and Mechanics of Carbon Nanostructures

Abstract

1.1 Background

1.2 Carbon Nanostructures

1.3 Interaction Between Molecular Structures

1.4 Book Overview

Exercises

Chapter 2: Mathematical Preliminaries

Abstract

2.1 Introduction

2.2 Dirac Delta Function: δ(x)

2.3 Heaviside Function: H(x)

2.4 Gamma Function: Γ(z)

2.5 Beta Function: B(x, y)

2.6 Hypergeometric Function: F(a,b;c;z)

2.7 Appell’s Hypergeometric Function: F1(a;b,b’;c;x,y)

2.8 Associated Legendre Functions: Pνμ(z) and Qνμ(z)

2.9 Chebyshev Polynomials: Tn(x) and Un(x)

2.10 Elliptic Integrals: F(ϕ, k) and E(ϕ, k)

Exercises

Chapter 3: Evaluation of Lennard-Jones Potential Fields

Abstract

3.1 Introduction

3.2 Interaction of Linear Objects

3.3 Interaction of a Spherical Surface

3.4 Interaction of a Cylindrical Surface

Chapter 4: Nested Carbon Nanostructures

Abstract

4.1 Introduction

4.2 Atom@Fullerene—Endohedral Fullerene

4.3 Fullerene@Fullerene—Carbon Onion

4.4 Fullerene@Carbon Nanotube

4.5 Carbon Onion@Carbon Nanotube

4.6 Carbon Nanotube@Carbon Nanotube—Double-Walled Carbon Nanotube

4.7 Nanotube Bundles

4.8 Carbon Nanotube@Nanotube Bundle

4.9 Fullerene@Nanotube Bundle

Exercises

Chapter 5: Acceptance Condition and Suction Energy

Abstract

5.1 Introduction

5.2 C60 Fullerene Inside a Carbon Nanotube

5.3 Double-Walled Carbon Nanotubes

5.4 Nanotube Bundle

Exercises

Chapter 6: Nano-oscillators

Abstract

6.1 Introduction

6.2 Oscillation of a Fullerene C60 Inside a Single-Walled Carbon Nanotube

6.3 Oscillation of Double-Walled Carbon Nanotubes

An alternative approach

6.4 Oscillation of Nanotubes in Bundles

Exercises

Chapter 7: Mechanics of More Complicated Structures: Nanopeapods and Spheroidal Fullerenes

Abstract

7.1 Introduction

7.2 Nanopeapods

7.3 Spheroidal Fullerenes

Exercises

Chapter 8: Nanotubes as Drug Delivery Vehicles

Abstract

8.1 Introduction

8.2 Underlying Mathematics

8.3 Encapsulation of Cisplatin Into a Carbon Nanotube

8.4 Alternative Nanotube Materials

Exercises

Chapter 9: New Formulae for the Geometric Parameters of Carbon Nanotubes

Abstract

9.1 Introduction

9.2 Conventional ‘Rolled-Up’ Model

9.3 New ‘Polyhedral’ Model

9.4 Details of the Polyhedral Model

9.5 Results

9.6 Conclusion

Exercises

Chapter 10: Two Discrete Approaches for Joining Carbon Nanostructures

Abstract

10.1 Introduction

10.2 Nanotori

10.3 Joining Carbon Nanotubes and Flat Graphene Sheets

10.4 Nanobuds

Exercises

Chapter 11: Continuous Approach for Joining Carbon Nanostructures

Abstract

11.1 Introduction

11.2 Calculus of Variations

11.3 Joining Carbon Nanotubes and Flat Graphene Sheets

11.4 Nanobuds

11.5 Nanopeanuts

Exercises

Hints and Solutions

Chapter 1

Chapter 2

Chapter 4

Chapter 5

Chapter 6

Chapter 7

Chapter 8

Chapter 9

Chapter 10

Chapter 11

Bibliography

Index

Copyright

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Preface

D. Baowan; B.J. Cox; T.A. Hilder; J.M. Hill; N. Thamwattana

This book is designed to support third- and fourth-year undergraduate courses in applied mathematical modelling in the currently topical area of nanotechnology. It is purposely designed as a self-contained text, which demonstrates to the student the process of utilising elementary geometry and mechanics combined with some special function theory to formulate simple applied mathematical models in a theoretical physics context, which have been traditionally analysed through computational procedures such as molecular dynamics simulations.

The essential modelling content of the book is the use of the continuum assumption for atomic surface densities and the replacement of double summations with double surface integrals involving atomic potential functions Φ(ρ) = −Aρ−n. For certain surfaces, such integrals can often be evaluated in terms of well-known analytical functions, but generally these integrations are highly nontrivial. For example, even for the electrostatic or Newtonian gravitational potential Φ(ρ) = −Aρ−1, Sir James Jeans states in page 33 of ‘The Mathematical Theory of Electricity and Magnetism’ that ‘An attempt to perform the integration, in even a few simple cases, will speedily convince the student that the form is not one which lends itself to rapid progress’. The mathematical perspective of this book is that many of the integrals can be identified from integral representations of the various hypergeometric functions, such as

valid for Re(γ) > Re(β) > 0, and Γ denotes the usual gamma function; there are similar results for the Appell hypergeometric functions.

Such formulae are important from two points of view. Firstly, such an identification subsequently leads to the evaluation of the integral in terms of better-known special functions, such as Legendre polynomials, and secondly, through algebraic packages such as MAPLE and MATLAB, the hypergeometric functions can be readily evaluated numerically. Throughout the book such packages are frequently adopted for the numerical evaluation of integrals. In situations where a large numerical landscape is required, these approaches are far more computationally effective than molecular dynamics simulations. We further comment that many of the integrals arising from the hypergeometric function or the Appell hypergeometric function can also be evaluated equally well in terms of elliptic functions. These details are not included in the book; the interested reader should consult the original publications cited in the bibliography.

The first six chapters are intentionally introductory in nature, focussed on the key ideas, and restricted to material considered to be accessible by students in their third year of the traditional Australian undergraduate mathematics degree. The final five chapters deal with the same ideas introduced earlier, but the situations and geometry are more complicated than earlier examples. Many of the exercises and examples throughout the book emanate from the research work of the Nanomechanics Group at the University of Wollongong. In nanotechnology and nanobiotechnology at the moment, there is not a lack of information; rather, there is an abundance of information but very little insight into underlying physical or biological mechanisms. In order to move forward there is a need to condense this information in the form of simple models which properly encapsulate the known behaviour. We believe that at least for some situations, this book demonstrates to the student how simple geometry, mechanics, and mathematical analysis might be exploited to obtain physically meaningful models.

The authors are especially grateful to Professor Quanshui Zheng from the Department of Engineering Mechanics at Tsinghua University, Beijing, whose helpful advice has materially assisted in the research work underlying this text.

Chapter 1

Geometry and Mechanics of Carbon Nanostructures

Abstract

Chapter 1 develops the essential elementary ideas necessary to exploit the simple geometry, mechanics, and special function theory, to formulate applied mathematical models in nanotechnology. The first chapter provides an introduction to the physical background including the bonding, the discoveries, the geometric structures, and the practical uses of the various carbon nanostructures; including nanotubes, fullerenes, cones, nanopeapods, and nano-onions. The idea of mathematically modelling such structures through their atomic surface densities coupled with the van der Waals and Leonard Jones potentials is then introduced.

Keywords

Carbon nanostructures; Geometry; Mechanics; Interaction between molecular structures

1.1 Background

The advent of nanoscience and nanotechnology has generated considerable advances in many industries such as composite materials, electronics, and medicine. The prefix ‘nano’ means small and derives from the Greek word for dwarf, namely nanos. In a scientific context, ‘nano’ means one billionth, and a nanometer (nm) means one billionth of a meter or one millionth of a millimeter. One angstrom (Å) is ten nanometers; for convenience angstrom units are used throughout the book. In this book, the term ‘nanoscience’ refers to the study of the structures and properties of materials on the nanometer scale (nanomaterials), and the term nanotechnology refers to the synthesis, control, engineering, and manipulation of the nanomaterials. By nanometer scale, we mean a length scale at the level of several atoms and molecules.

The unique physical properties observed at the nanoscale are often counter-intuitive, sometimes surprising researchers and thus driving numerous investigations into their special properties and their potential applications. Typically, existing research has been conducted through experimental studies and molecular dynamics simulations. However, mathematical modelling often facilitates device development and provides a quicker route to applications of the technology. As stated by Ferrari (2005) ‘Novel mathematical models are needed, in order to secure the full import of nanotechnology into oncology’. Mathematical models are not only important for nanomedicine but in all areas of nanotechnology, where it is important to fully comprehend often subtle or complex phenomena and perhaps save on time-consuming experimental studies. Applied mathematics is an important tool which may provide insight, assess feasibility and deliver overall guidelines for subsequent experimental and molecular dynamics studies. This book provides an introduction to the use of mathematics and mechanics in nanoscience and nanotechnology. We refer the reader to Dresselhaus et al. (1996), Saito et al. (1998), and Harris (2002) for more physically based accounts of the various carbon structures.

Until recently only two types of all-carbon crystalline structures were known, namely the naturally occurring allotropes diamond and graphite. The breakthrough discovery of carbon nanotubes and fullerenes have revolutionised carbon science from experiments on clusters formed by laser vaporisation of graphite. The discovery of fullerenes and nanotubes has generated considerable research into their properties and potential applications, and many more carbon nanostructures have also been discovered. For example, the nanopeapod, which is a carbon nanotube with many fullerenes encapsulated within its interior, is one such novel nanostructure. Other carbon nanostructures which have received attention are nanotori, nano-onions, and nanobundles, which are illustrated in Fig. 1.1. Briefly, a nanotorus or fullerene ‘crop circle’ is a circular formation of a carbon nanotube which joins together seamlessly; a nano-onion, like the name suggests, is a collection of concentric fullerenes of varying radii encapsulated within each other; and a nanobundle is a stable circular arrangement of several nanotubes. Our focus is the use of geometry and mechanics for modelling; in contrast to quantum mechanical, electrical, or optical applications.

Fig. 1.1 Various carbon nanostructures. (A) C 60 fullerene, (B) carbon nanotube, (C) nanopeapod, (D) nanotori, (E) nano-onion, and (F) nanobundle.

As a consequence of their unique mechanical and electronic properties, carbon nanostructures are being investigated for their use in numerous applications such as:

• Nanoreinforced composites: producing more durable, lightweight materials

• Nanocapsules: used for drug delivery, toxin storage, and magnetic resonance imaging scans

• Shock absorbers, or man-made molecular springs

• Nanoscale instruments: nanoprobes for scanning force microscopy, nanobearings with super low friction, molecular oscillators with frequencies as high as several gigahertz, and nanosensors

• Nanostraws: to penetrate living cells and inject molecules without damaging the cellular structure

• Computer memory: to further miniaturise the computer industry

• Superconductors: increasing the current carrying capacity

and there are many more potential applications of nanotechnology other than those listed here.

The vast amount of potentially important applications of carbon nanostructures and their unique physical, mechanical, and electronic properties has stimulated many investigations. Modelling provides a critical basis for providing a predictive capacity and for understanding and designing nanostructures and their applications. This book focuses on the existing mathematics and mechanics, which has proved useful for the modelling of nanostructures. The following section provides an introduction to the topic and discusses various carbon nanostructures. In particular, the section outlines: graphene, including details on carbon–carbon bonding; carbon nanotubes and the rolled up model which is used to describe their structure; and fullerenes and cones. In addition, Section 1.2 includes a discussion on Euler’s theorem and how it may be applied to these carbon nanostructures. Following this, in Section 1.3 the basic equations and the theory which are used to determine the interaction between molecular structures is outlined. Finally, in Section 1.4 the organisation of the book is outlined, and relevant exercises are given at the end of the chapter.

1.2 Carbon Nanostructures

1.2.1 Graphene and C-C Bonding

As a result of the chemical structure of carbon, a carbon atom can bond with itself and other atoms to form an endlessly varied combination of chains and rings. As a result of this ability to form several distinct types of valence bonds, carbon can assume many distinct structural forms. For example, a planar sheet forms graphene or a tetrahedral structure forms diamond. Carbon is the sixth element in the periodic table, and a free carbon atom has six electrons which occupy 1s², 2s², and 2p² atomic orbitals, which are illustrated in Fig. 1.2. In other words, carbon has two electrons which occupy the 1s orbital which is the orbital closest to the nucleus, two in the 2s orbital, and the remaining two electrons occupy two separate 2p orbitals, since the p orbitals have the same energy and the electrons would prefer to be separate. The 2s and 2p orbitals have similar energy levels. In practice, it is not possible to know exactly where the electrons are, since they appear smeared into orbitals. Despite this, the orbital notation is commonly used.

Fig. 1.2 The electron structure of a single carbon atom. Two electrons are in the 1 s orbital, two in the 2 s orbital, and two in the 2 p .

When bonding with other atoms, the electronic structure of a single carbon atom may be hybridised to adapt to various structural arrangements. The energy difference between the upper 2p orbital and the lower 2s orbital is small as compared to the binding of chemical bonds. Therefore, these electrons can readily mix with each other to enhance the binding energy of the carbon atom with neighbouring atoms, and this process is called hybridisation. For example, in order to form covalent bonds in graphene (nanotubes and fullerenes), one of the 2s electrons is promoted to a 2p orbital, and the orbitals are then hybridised to form what are called sp² bonds. In sp² bonding, each carbon atom forms three bonds with three other neighbouring atoms. Similarly in diamond, the bonds are hybridised to form sp³ bonds so that each carbon atom forms four bonds to give a tetrahedral bonding structure. Generally, the bonds which are in-plane and form strong covalent bonds are referred to as σ bonds, and the bonds which are perpendicular to the plane and form weak van der Waals bonds are referred to as π bonds.

The most commonly occurring form of carbon is graphite, which is formed from many layers of graphene sheets. Graphene is a planar sheet consisting of a tessellation of hexagonal rings of carbon atoms, or a honeycomb lattice as shown in Fig. 1.3, all with the hybridised sp² bonds. Graphite, originally found useful for marking sheep in the 16th century, is best known for its use in pencils, where it is commonly referred to as ‘lead’. In graphite, the graphene layers are stacked on top of each other so that for example in a pencil, the sheets slide past each other and onto the page. The graphene layers slide past each other because of the relatively weak van der Waals forces which exist between layers, not as a result of the strong covalent bonds which exist between each atom in the planar sheet.

Fig. 1.3 Planar sheet consisting of a tessellation of hexagonal carbon rings (graphene).

The sp² carbon–carbon bond in the basal plane of graphite is the strongest of all chemical bonds, but the interplanar forces (or van der Waals forces) are relatively weak. In fact, the sp² bonding in graphene is stronger than the sp³ bonding in diamond, as demonstrated by the carbon–carbon bond lengths (1.42 Å and 1.54 Å, respectively); and the shorter the bond length, the stronger the bond. For a more detailed account of the molecular bonding of carbon we refer the reader to Companion (1979).

Immediately after the discovery of carbon nanotubes and fullerenes, there was less interest in graphene. However, graphene sheets have recently shown considerable promise for applications in electronics. For example, it was found that graphene exhibits a much higher thermal conductivity than carbon nanotubes, which is vital in electronic applications to dissipate heat. Since graphene is basically an unrolled carbon nanotube, many of the applications investigated for nanotubes may also apply to graphene. As mentioned in Section 1.1, there are now many forms of carbon structures and two of these, the carbon nanotube and fullerene, are examined in some detail in Sections 1.2.2 and 1.2.3, respectively. Interestingly, fullerenes and nanotubes exhibit structures remarkably similar to those found in nature. For example, C60 and other symmetrical fullerenes have a similar structure to icosahedral viruses, and carbon nanotubes have a similar structure to certain bacteriophages (bacteriolytic viruses).

1.2.2 Carbon Nanotubes and the Rolled-Up Model

The discovery by Iijima (1991) that carbon nanotubes could be grown without a catalyst has generated considerable research into numerous potential applications, ranging from prospective devices in biology to electronics. His paper outlined the experimental identification of multiwalled carbon nanotubes and had a significant impact on subsequent scientific research.

This is often considered to be the first occurrence of carbon nanotubes, but in fact they were observed much earlier. The first evidence, using transmission electron microscopy, of the tubular nature of some nanosized carbon filaments appeared in 1952 by Radushkevich and Lukyanovich in the Russian Journal of Physical Chemistry. Figures from this paper clearly show carbon filaments with a continuous inner cavity and tubes that formed as a result. The nanotubes are now thought to be multiwalled carbon nanotubes with 15–20 layers. However, this paper received very little attention from the world scientific community, perhaps because of the Cold War and the general difficulties of obtaining Russian scientific documents. However, certainly Radushkevich and Lukyanovich should be credited with the discovery that carbon filaments could be hollow and have a nanometer-size diameter or, in other words, for the actual discovery of carbon nanotubes. Of course, Iijima was the first to fully appreciate the nature and importance of these structures.

Laboratory methods to synthesise single-walled carbon nanotubes were discovered in 1993 by Bethune and colleagues at the IBM Almaden Research Center in San Jose, California, and also independently by Sumio Iijima at the NEC Laboratories in Japan. An image found in a paper by Oberlin as early as 1976 shows a nanotube resembling a single-walled carbon nanotube, although this is not explicitly claimed by the authors.

Carbon fibres are the macroscopic analogue of carbon nanotubes. Such fibres have been known for some time, and they were first used by Thomas A. Edison as a filament for an early model of the electric light bulb. Since then, carbon fibres have been used in numerous applications, such as for sporting equipment, boat hulls, and aeronautics. Carbon nanotubes have the potential to vastly improve technologies which already employ carbon fibres, in addition to providing an avenue for many new and exciting applications. Carbon nanotubes have many fascinating and unique mechanical and electronic properties, including but not limited to: high strength and flexibility, low density, completely reversible deformation and their capacity to be either metallic or semiconducting depending upon their geometric structure.

Carbon nanotubes may be thought of as one or several graphene sheets rolled up into a seamless hollow cylinder, forming either a single-wall (having only one atomic layer) or a multiwall carbon nanotube (having two or more walls). Experimental studies confirm that a carbon nanotube consists of a hexagonal lattice sheet, just like graphene, which is rolled into a seamless cylinder. Fig. 1.4 illustrates single-walled and multiwalled carbon nanotubes.

Fig. 1.4 Illustration of (A) single-wall and (B) multiwall carbon nanotubes.

Single-walled carbon nanotubes frequently occur in tightly packed bundles so as to minimise their energy. Laboratory samples of single-walled carbon nanotubes tend to be more uniform than multiwalled carbon nanotubes, and they have a smaller range in diameters and fewer obvious defects. In other words, single-walled carbon nanotubes have more perfect structures than multiwalled carbon nanotubes. Several nanotube bundles can be bound together to form a nanorope. Generally, the diameter of single-walled carbon nanotubes is approximately 7–100 Å, with the majority of observed single-wall nanotubes having a diameter less than 20 Å. The aspect ratio (length/diameter) of nanotubes is extremely large, and it can be as high as 10⁴–10⁵, and therefore nanotubes are often considered to be one-dimensional structures. Multiwalled carbon nanotubes are concentric with a separation distance which is typically of the order of 3.4 Å, which is close to the interlayer separation distance of graphite.

The structure of a carbon nanotube is described by its chiral vector C, which is obtained by unrolling the carbon nanotube onto a planar sheet and connecting two crystallographically equivalent sites. The chiral vector is defined by C = na + mb or alternatively written simply as (n, m), where n and m are integers, and a and b are the basis vectors for one hexagonal unit cell on a graphene sheet. The magnitude of both a and b is 2.46 Å, or the width of one hexagonal unit, as shown in Fig. 1.5. Note that due to the hexagonal symmetry of the honeycomb lattice, it is only necessary to consider 0 ≤ m ≤ n; by convention this is assumed to be the case. In the rolled-up model of carbon nanotubes, the chiral vector C is always perpendicular to the nanotube axis, and its magnitude is equal to the nanotube’s circumference. The indices (n, m) are commonly referred to as the nanotube’s chirality and are sometimes referred to as the helicity.

Fig. 1.5 Conventional rolled up model for carbon nanotubes.

Two special carbon nanotubes are zigzag (n, 0), and armchair (n, n); these two types are normally referred to as achiral. The general carbon nanotube is referred to as chiral (n, m). See ) is visible along this line. Interestingly, carbon nanotubes can be either metallic or semiconducting, and this property is completely dependent upon their geometric structure, namely their diameter and chirality. When (n − m) is a multiple of 3 the tubes are metallic; otherwise, they are semiconducting. For example, an armchair (n, n) tube is always metallic, while a zigzag (n, 0) tube is metallic only if n is a multiple of 3.

A carbon nanotube is also defined by the chiral angle θ0 shown in Fig. 1.5, which is the angle the chiral vector C makes with the zigzag line, and which is defined by

We note that θ0 = 0 for zigzag tubes, while θ0 = π/6 for armchair tubes and generally 0 < θ0 < π/6 for chiral tubes. The radius of a carbon nanotube, written in terms of (n, m), is given by

   (1.1)

where σ is the length of one carbon–carbon bond, which is approximately 1.42 Å for a carbon nanotube. For example, using Eq. (1.1) the radius of a (10, 10) carbon nanotube is 6.78 Å. In fact, the length of the carbon–carbon bond σ in a nanotube is believed to be 1.44 Å, and as such the radius of a (10, 10) nanotube would be 6.875 Å. Despite this discrepancy the length of the carbon–carbon bond for graphene (1.42 Å) is typically assumed for carbon nanotubes.

Worked Example 1.1

Prove the equation for the carbon nanotube radius (Eq. 1.1) for the rolled up model.

Solution

On noting that the magnitude of the chiral vector C is equal to the circumference of the nanotube |C| = 2πr, we obtain

where the magnitude of a . We note that the vectors a and b are not orthogonal and their inner products yield

There are many other formulae and vectors which exist to describe a carbon nanotube, and these are derived from the chiral vector C. One important definition is the length of the unit cell. The translational vector T of a unit cell is parallel to the nanotube axis; therefore it is perpendicular to the chiral vector C, as illustrated in Fig. 1.5, and it is given by

where t1 and t2 are integers, which are defined by

and dR is the highest common divisor of (2n + m, 2m + n), or more specifically

where d is the highest common divisor of (n, m). For example, for a (5, 5) armchair tube dR = 3d = 15, while for a (9, 0) zigzag tube dR = d = 9 and for a (7, 4) chiral tube dR = 3d = 3.

The above relation for d and dR comes from one of Euclid’s Laws (7th Book, Proposition 2). The law is commonly referred to as the Euclidean algorithm or the greatest common divisor, gcd(a, b). The Euclidean algorithm is one of the oldest algorithms known, since it appeared in Euclid’s Elements in 200 BC. Euclid originally formulated the problem geometrically as the greatest common measure for two line lengths.

The length of T is thus given by

   (1.2)

We note that the length of T is greatly reduced when (n, m) has a common divisor or when (n − m) is a multiple of 3d. The number of hexagons per unit cell N; therefore the number of carbon atoms nC per unit cell are given by

   (1.3)

Worked Example 1.2

Prove the formulae given in Eq. (1.3).

Solution

Note that the number of hexagons per unit cell is equal to the area of the unit cell divided by the area of one hexagonal unit.

Fig. 1.6 illustrates the cross product of the vectors a and b, where the cross product is perpendicular to both vectors and its magnitude is equal to the area between them. The cross product of any two vectors may be determined from the determinant det(a b.

Fig. 1.6 The cross product between two vectors a and b .

Therefore the area of the unit cell can be determined by finding the magnitude of the cross product of the vectors C and T, thus

. Note that the vectors C and T are always perpendicular to one another so that the angle between them is always 90°, i.e. π/2.

Similarly, the area of one hexagonal unit is the magnitude of the cross product of the unit vectors a and b, thus

Note that the angle between the vectors a and b is always 60° or π/3 and that |a ×a| = |b ×b| = 0. The number of hexagons per unit cell is thus given by

It then follows that the number of carbon atoms per unit cell is nC = 2N.

As an example, Fig. 1.7 illustrates a unit cell for a (5, 5) carbon nanotube. Further details for this nanotube are given in Table 1.1, where the structural parameters of a selected number of carbon nanotubes are given. For more details on the structural parameters of carbon nanotubes, we refer the reader to Dresselhaus et al. (1995) and Saito et al. (1998).

Fig. 1.7 Unit cell for a (5, 5) armchair nanotube.

Table 1.1

General Structural Parameters for a Selection of Carbon Nanotubes

Generally, carbon nanotubes are assumed to be formed from perfect hexagonal rings of carbon. However, in reality, there are often defects occurring along the nanotube wall. Nanotubes with diameters greater than 20 Å tend to show many more structural defects and are inclined to become unstable due to atmospheric pressure. On the other hand, the very small diameter nanotubes, although they tend to be more perfect with fewer structural defects tend to be intrinsically unstable giving rise to a definite window of diameters which are seen to occur in practice. The Stone–Wales transformation, which is illustrated in Fig. 1.8 and discussed in more detail in Section 1.2.5, is one such