The geometric structure of single-walled nanotubes

Abstract

In this paper, we survey a number of existing geometric structures which have been proposed by the authors as possible models for various nanotubes. Atoms assemble into molecules following the laws of quantum mechanics, and in general computational approaches to predicting the molecular structure can be arduous and involve considerable computing time. Fortunately, nature favours minimum energy structures which tend to be either very symmetric or very unsymmetric, and which therefore can be analyzed from a geometrical perspective. The conventional rolled-up model of nanotubes completely ignores any effects due to curvature and the present authors have proposed a number of exact geometric models. Here we review a number of these recent developments relating to the geometry of nanotubes, including both the traditional rolled-up models and some exact polyhedral constructions. We review a number of formulae for four materials, carbon, silicon, boron and boron nitride, and we also include results for the case when the bond lengths may take on distinct values.

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Richard K. F. Lee

Richard K. F. Lee completed a Bachelor of Science in Physics and Mathematics with Distinction in 2006 from the University of Wollongong, Australia. In 2008, he enrolled in a Master of Science in Mathematics by Research in the area of Mathematical Modelling in Nanotechnology, and in the same year converted this enrolment to a PhD. His research interests include the mathematical modelling of the mechanics and geometric structures of nanotubes and fullerenes.

Barry J. Cox

Barry Cox obtained his PhD at the University of Wollongong in 2007. Following this he began as a postdoctoral researcher in the Nanomechanics Group at the University of Wollongong and in 2009 was awarded an Australian Postdoctoral (APD) Fellowship from the Australian Research Council. He is currently a senior lecturer at the University of Wollongong and has authored and co-authored 30 research articles in applied mathematical modeling of nanoscale materials, devices and phenomena. His current research focus is on producing mathematical and mechanical models for nano-engineering and nanomedicine.

James M. Hill

Hill completed his PhD at the University of Queensland in 1972 followed by a Postdoctoral appointment in Theoretical Mechanics at the University of Nottingham. He has had a research focused career at the University of Wollongong. His areas of Applied Mathematics include finite elasticity; heat transfer, nonlinear diffusion, granular materials and presently works in nanotechnology. He works with scientists, engineers and industry, and is committed to research with practical outcomes. He received the ANZIAM Medal for 2008 in recognition of his personal research and sustained contributions to the discipline.

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1 Introduction

Nanotechnology has brought a revolutionary period, namely the age of nano-materials, for which the physical size of the smallest devices has changed from the micrometre to the nanometre scale. The age of nanotechnology has its origins in the 20th century, and it is defined as the science of components or devices which are nanometres in scale. The nanometre is one of the International System of Units (SI Unit) which is one billionth, or 10⁻⁹, of a meter. Many materials at the nanometre scale display exceptional physical characteristics such as their mechanical and electronic properties,¹ and these properties are quite different compared to those at the micro-scale. In the 21st century, nanotechnology is expected to be even more important in many areas such as lab-on-chip devices and targeted drug and gene delivery.²

Recent interest in carbon nanotubes was initiated by Iijima's discovery in 1991,³ which also triggered interest in other nanotube materials such as B,^4–6 Si,^7–9 BN,^10–17 WS₂,¹⁸ Bi₂S₃,^19,20 ZnS,^21,22 ZnO,^23–28 GaN,²⁹ AlN,³⁰ InP,³¹ SiO₂,³² TiO₂^33–35 and Eu₂O₃.³⁶ The traditional conceptualisation of nanotube structure is a rolled-up model that is developed from the work of Dresselhaus et al.^37–39 for carbon nanotubes, Gindulyte et al.^40,41 for boron nanotubes and Rubio et al.⁴² and Saxena et al.⁴³ for boron nitride nanotubes.

In the traditional rolled-up models,^37–43 nanotubes are conceptualized by beginning with a two dimensional sheet which is then rolled into a right circular cylinder. The curvature of the structure is not considered in these traditional rolled-up models and therefore values for the radius differ appreciably from those predicted by molecular dynamics simulations, especially for the smaller radii nanotubes where curvature is significant. Polyhedral models for single-walled nanotubes are proposed that inherently include the curvature of the nanotube structure, for carbon nanotubes,^44,45 silicon nanotubes,⁴⁶ boron nanotubes⁴⁷ and boron nitride nanotubes⁴⁸ which give predictions for the geometric parameters of the tube that are in excellent agreement with first-principles calculations.^44–48 The traditional rolled-up models^37–43 and the idealized polyhedral models for carbon nanotubes^44,45 and for silicon nanotubes⁴⁶ all assume that all bond lengths and bond angles are equal, while for boron nanotubes⁴⁷ and boron nitride nanotubes⁴⁸ it is assumed that only all the bond lengths are equal, but the bond angles are allowed to vary.

The nanotube radius depends on the bond lengths and the chiral vector numbers (n, m), so that when the bond lengths are not equal, the nanotube radius may vary greatly from the predictions of the traditional rolled-up models and the ideal geometric polyhedral models. From molecular dynamics simulations results for carbon nanotubes^49–54 and silicon nanotubes,^55,56 the bond lengths of nanotubes may vary depending on the direction of the bond. These features should not be ignored and therefore the traditional rolled-up models and the idealized polyhedral models are extended to include distinct bond lengths for carbon nanotubes,⁵⁷ silicon nanotubes⁵⁶ and boron nanotubes⁵⁸ and these models are shown to be in excellent agreement with first-principles calculations.

The carbon nanotubes considered here are assumed to be formed by sp² hybridization³ and the nanotube structure is assumed to comprise hexagonal lattices as shown in Fig. 1. Silicon prefers sp³ bond formation^55,60,61 and it is predicted that the four-coordinated atoms form a skewed rhomboid lattice^55,60,61 as shown in Fig. 2. The boron nanotubes considered here are assumed to be formed by sp² hybridization and π-bonds^62–67 The lattice structures for boron nanotubes that have been proposed include flat equilateral triangles,^{40,41,62,71,77–81} puckered equilateral triangles^{63–65,71,72,81–83} and a novel lattice pattern⁶⁶ that is a triangular lattice structure with 1/9 hexagonal holes, a structure which is believed to be energetically more stable than a pure triangular lattice.⁷³ However, analysis of the raw data of Yang^73,74 reveals a far more complex structure and the model of Yang⁷⁴ involves three distinct radii; three-quarters of the atoms have an hexagonal based radius, one-eighth have a puckered outer radius and one-eighth have a puckered inner radius. However, although we acknowledge that the polymorphism of boron nanotubes is an important consideration with a number of possible structures which may be more stable than a homogeneous triangular lattice,⁶⁶ here for simplicity we consider the lattice for boron nanotubes to comprise only equilateral triangles with vertices which are all equidistant from a common axis. As a result, the nanotube lattice assumed here comprises a flat triangular pattern^62–67 as shown in Fig. 3. Boron nitride nanotubes have a complex electronic structure such that boron atoms are formed by sp² hybridization and nitrogen atoms are formed by the unhybridized s and p orbitals.⁵⁹ Accordingly, the boron nitride nanotube structure is similar to the carbon nanotube but it has two distinct radii.⁵⁹Fig. 1, 2 and 3 show the general polyhedral model and the atoms are represented by black dots and the bonds between atoms are indicated by black lines. Fig. 4 is the ideal polyhedral model for boron nitride nanotubes and the boron atoms and nitride atoms are represented by solid dots and hollow dots, respectively, and the black lines indicate the bonds. In Fig. 1 and 4 the gray lines represent the conceptual division of each hexagon into four triangles.

Fig. 1 General polyhedral model for carbon nanotubes for zigzag, chiral and armchair tubes.

Fig. 2 General polyhedral model for silicon nanotubes for prismatic, chiral and antiprismatic tubes.

Fig. 3 General polyhedral model for boron nanotubes for zigzag, chiral and armchair tubes.

Fig. 4 Ideal polyhedral model for boron nitride nanotubes for zigzag, chiral and armchair tubes.

In the follow sections, we present the traditional rolled-up model, the rolled-up model with distinct bond lengths, the ideal polyhedral model and the general polyhedral model for carbon nanotubes (section 2), silicon nanotubes (section 3) and boron nanotubes (section 4). The traditional rolled-up model and the ideal polyhedral model for boron nitride nanotubes are presented in section 5. In section 6, the nanotube radius is discussed and some concluding remarks are made. Throughout this paper we use the subscripts re, pe, rd and pd to designate respectively quantities associated with rolled-up model with equal bond lengths, polyhedral model with equal bond lengths, rolled-up model with distinct bond lengths and polyhedral model with distinct bond lengths.

2 Carbon nanotubes

The traditional rolled-up model for carbon nanotubes assumes that all bond lengths and all bond angles are equal and the nanotube can be envisaged as a two dimensional sheet which is rolled into a seamless right circular cylinder.^37–39 Carbon nanotubes are categorized into three types, namely zigzag, armchair and chiral, based on the values of the chiral vector C_h = na₁ + ma₂ where n and m are non-negative integer chiral vector numbers, a₁ and a₂ are basis vectors and the naming of the nanotubes is indicated by the chiral vector numbers (n, m). Carbon nanotubes are termed zigzag when m = 0 and armchair when m = n. In all other cases, when 0 < m < n they are termed chiral nanotubes.

The traditional chiral angle is the angle between the chiral vector C_h and the basis vector a₁. The chiral angle for the traditional rolled-up model is given by

which has a simple exact value for certain special cases. For zigzag type (m = 0), the chiral angle θ_re is zero, while in the case of armchair tubes (m = n), the chiral angle θ_re is π/6. We remind the reader that throughout this paper we use the subscript re to denote the rolled-up model with equal bond lengths. The nanotube radius r_re for the traditional rolled-up model is obtained from the magnitude of the chiral vector |C_h| divided by 2π and we find that where σ is the bond length.

Experimental results^49–54 show that the bond lengths of the nanotubes vary depending on the bond direction because the bond lengths and the bond angles may vary with nanotube curvature.⁴⁹ The bonds paralleled to the nanotube axis, are found to be shorter than the bonds around the nanotube circumference.^49–53 Therefore, from symmetric considerations zigzag and armchair tubes can have two distinct bond lengths and bond angles, while chiral tubes can have three different bond lengths and bond angles.^49–54,57

The general rolled-up model must accommodate three different bond lengths⁵⁷ and thus requires three basis vectors a₁, a₂ and a₃, as shown in Fig. 5. Employing the same notion as the traditional rolled-up model, the two dimensional sheet with distinct bond lengths is rolled along the chiral vector which is still given by C_h = na₁ + ma₂, but the lengths and direction of the basis vectors may be different from the traditional rolled-up model.

Fig. 5 Carbon nanotube constructed from two dimensional sheet with distinct bond lengths.

Since there are three different bond lengths, the nanotubes may have three different bond angles in the plane but the rolled-up model with distinct bond lengths and distinct bond angles could restructure into many different hexagonal patterns such that the sum of the bond angles is less than 2π.^50–52,54 The three different bond angles all asymptote to a sum of 2π in the limit as n becomes large^50–52 and therefore for the rolled-up model we assume that the three bond angles are equal ϕ_re = 2π/3.

In Fig. 5, the origin O is located at an arbitrary lattice point. The chiral vector C_h goes from O to A. The sheet is rolled so that the point A coincides with the origin O and the point B coincides with the point B′. When m = 0, which is equivalent to rolling up the nanotube in the direction of OE, the carbon nanotubes is called zigzag. When m = n, which is equivalent to rolling the nanotube in the direction of OD, the resulting to be termed armchair. In all other cases, when 0 < m < n they are termed chiral.

The chiral angle for the general rolled-up models bases on the different bond lengths and given by

where Δ = (a₁² + a₂² − a₃²) and a₁, a₂ and a₃ are the length of the basis vectors. The lengths of the basis vectors are , and . We note that throughout we use the subscript rd for the rolled-up model with distinct bond lengths. For the special case (m = 0), the chiral angle θ_rd is zero.

The nanotube radius for the general rolled-up model is given by

and when the bond lengths are equal, the equations for the chiral angle (3) and the nanotube radius (4) for the general rolled-up model are exactly the same as eqn (1) and (2) for the traditional rolled-up model, and therefore the traditional rolled-up model may be viewed as a special case of the general rolled-up model.⁵⁷

The conventional rolled-up model^37–39 and the general rolled-up model,⁵⁷ all assume that a flat sheet of graphene is rolled into a seamless right circular cylinder. However, a real nanotube has a curvature⁴⁹ and recently, Cox and Hill^44,45 propose a new ideal geometric polyhedral model of single-walled carbon nanotubes, in which all bond lengths are assumed to be equal but the curvature is properly accommodated. This model makes very accurate predictions on the geometric parameters of the nanotube which are in excellent agreement with certain first-principles calculations.^68–70

The ideal polyhedral model is based on three fundamental postulates:^44,45 (i) all bond lengths are equal σ; (ii) all the adjacent bond angles are equal ϕ_pe; and (iii) all atomic nuclei are equidistant from a common axis r_pe. We note that throughout we use the subscript pe for the polyhedral model with equal bond lengths. For carbon nanotubes, each component of the hexagonal lattice is divided into three isosceles triangles and one equilateral triangle.^44,45 The equilateral triangular pattern is shown in Fig. 1 by gray lines and the dots denote the atomic location on each of the isosceles triangles making up a pyramid.

A fundamental parameter for the ideal polyhedral model is the subtend semi-angle ψ_pe which is the semi-angle subtended at the nanotube axis in the xy-plane of one edge of a equilateral triangle from the lattice. This angle is a root of the following transcendental equation:

(n² − m²) sin²(ξ_pe + ψ_pe) − n(n + 2m) sin²ξ_pe + m(2n + m) sin²ψ_pe = 0 (5)

where ξ_pe = (nψ_pe − π)/m. The subtend semi-angle ψ_pe is determined as a root of (5), which may have many roots. For a physically meaningful structure to result, the subtend semi-angle ψ_pe also satisfies the following inequalities:

π/(n + m) ≤ ψ_pe ≤ π/n (6)

While no explicit equation is know for ψ_pe, an accurate numerical value for the root of the transcendental eqn (5) may be determined after a small number of iterations of Newton's method, using the initial estimate

ψ_0pe = π(2n + m)/[2(n² + nm + m²)] (7)

The chiral angle θ_pe for the ideal polyhedral model, by which we mean ΔQPQ′, may be derived from the two triangles, ΔPQQ′ and ΔPCQ′ as shown in Fig. 6. The point Q′ defines as the point determined by projecting Q into the xy-plane. The length of PQ is the length of the basis vector a₁ and the lengths of CP and CQ′ are the radii r_pe. The chiral angle is given by

Fig. 6 Points lying on some helices in three dimensional space for (a) CNTs/BNTs/BNNTs and for (b) SiNTs.

For certain special cases, the chiral angle has two simple exact values. The chiral angle θ_pe is zero for zigzag tubes (m = 0), and takes the value π/6 for armchair tubes (m = n).

The adjacent bond angle ϕ_pe is defined as the angle between two bonds when the atoms that are being bonded comprise a single lattice. We need to introduce a new parameter to provide an equation for the adjacent bond angle, since it includes a new parameter which is the angle of incline ζ_pe of the pyramidal components of the surfaces. The incline angle ζ_pe is given by

while the adjacent bond angle ϕ_pe is given by

cos ϕ_pe = (– 2 + k²)/(4 + k²) (10)

where k is related to the perpendicular height of the constituent pyramids and given by the positive root of k²ρ + kχ + ν = 0, where the coefficients are defined by

The nanotube radius for the ideal polyhedral model is given by

r_pe = [σ sin(ϕ_pe/2) cos θ_pe]/sin ψ_pe (12)

Using the asymptotic expansions method, the chiral angle θ_pe for the ideal polyhedral model may be written in terms of the traditional chiral angle θ_re and the traditional nanotube radius r_re and is given by

where the O(1/n⁴) term refers to the maximum order of the magnitude of the next most significant term. Similarly, the ideal polyhedral nanotube radius r_pe may also be written as

By examining the asymptotic expansions of the equations for the ideal polyhedral model for the chiral angle (8) and the nanotube radius (12) we observe from eqn (13) and (14) that the leading order term of the analytical expressions gives the traditional formulae for rolled-up model (1) and (2) as their leading order term and the next term may be viewed as a first-order correction to the traditional model. This demonstrates that the ideal polyhedral model converges to the traditional rolled-up model for large n.

Table 1 compares the traditional rolled-up model, the ideal polyhedral model and the simulation results Cabria et al.,⁶⁸ and shows that the results of the ideal polyhedral model are in good agreement with those from the simulation results. The traditional rolled-up model is only close to the simulation results for large values of the nanotube radius. However, the ideal polyhedral model provides much closer results to these from simulation than the traditional rolled-up model.

Table 1 Comparison of carbon nanotube radii from traditional rolled-up model, ideal polyhedral model and ab initio calculations of Cabria et al.⁶⁸ using σ = 1.44 Å

(n, m)	r re/Å	r pe/Å	r ^*/Å
(4,0)	1.5878	1.7125	1.71
(3,2)	1.7303	1.8109	1.80
(4,1)	1.8191	1.9178	1.91
(5,0)	1.9848	2.0840	2.06
(3,3)	2.0626	2.1258	2.12
(4,2)	2.1005	2.1724	2.17
(5,1)	2.2102	2.2934	2.28
(6,0)	2.3817	2.4641	2.45
(4,3)	2.4146	2.4703	2.46

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The ideal polyhedral model, which assumes all bond lengths are equal, may be extended to having distinct bond lengths to give the general polyhedral model. The general polyhedral model is similar to the ideal polyhedral model in that it is also based on certain postulates. For carbon nanotubes there are two general polyhedral models, (Model I) and (Model II). Model I is the polyhedral model with distinct bond lengths and distinct bond angles. Model II is the polyhedral model with distinct bond lengths but equal bond angles. In other words, the difference between the two carbon models is the values of the bond angles. Model I assumes three prescribed distinct bond angles but Model II assumes that all the bond angles are the same. In Model I, the bond angles cannot be determined from the bond lengths, and therefore the bond lengths and bond angles must be prescribed. Without prescribed bond angles the general polyhedral model for Model I is not fully determined, and therefore to model nanotubes with the bond angles are not known we have developed the general polyhedral Model II.

The general polyhedral model for carbon nanotubes (Model I)⁵⁷ is based on two fundamental postulates: (i) corresponding bonds lying on the same helix are equal in length, either σ₁, σ₂ or σ₃ which are assumed generally to be distinct; and (ii) all atomic nuclei are equidistant r_pd from a common axis. For Model II,⁵⁷ there are three fundamental postulates: (i) corresponding bonds lying on the same helix are equal in length to either σ₁, σ₂ or σ₃ which are assumed generally to be distinct; (ii) all adjacent bond angles are equal to ϕ_pd; and (iii) all atomic nuclei are equidistant r_pd from a common axis. We note that throughout we use the subscript pd for the polyhedral model with distinct bond lengths. We use the superscripts (I) and (II) for models I and II, respectively.

The leading parameter for both models is the subtend semi-angle ψ_pd which is determined as a root of the following transcendental equations:

[a₂²(n + m)² − a₃²n²] sin²ψ_pd + [a₃²m² − a₁²(n + m)²] sin²ξ_pd + (a₁²n² − a₂²m²) sin²(ξ_pd + ψ_pd) = 0 (15)

where ξ_pd = (nψ_pd − π)/m. The lengths of the basis vectors arewhere ϕ^(I)_1pd, ϕ^(I)_2pd and ϕ^(I)_3pd are distinct bond angles for Model I. Model II assumes that the bond angles are all equal to ϕ^(II)_pd, and therefore ϕ^(I)_1pd = ϕ^(I)_2pd = ϕ^(I)_3pd = ϕ^(II)_pd. An initial value of the root of the transcendental eqn (15) for Model I is given by

ψ_0pd = [π(2na²₁ + mΔ)]/[2(n²a²₁ + m²a²₂ + nmΔ)] (17)

where Δ = (a₁² + a₂² − a₃²). We are not able to use Newton's method for Model II due to the values of a₁, a₂ and a₃ being unknown.

The chiral angle for both Models I and II is given by

while the nanotube radius for the general polyhedral model is given by

r_pd = a₁ cos θ_pd/(2 sin ψ_pd) (19)

where a₁ can be calculated for Model I because the bond angles values are known but for Model II, the subtend semi-angle ψ^(II)_pd and the chiral angle θ^(II)_pd cannot be determined.

The subtend semi-angle ψ^(II)_pd and the bond angle ϕ^(II)_pd for Model II both appear in the transcendental eqn (15), and therefore the angles cannot be determined and we need further constraints. For the ideal polyhedral model, we need to determine the incline angle parameter ζ_pe by considering the pyramidal components of the surfaces and through the properties of the pyramid and the nanotube radius, we obtain the three unknowns ψ^(II)_pd, ϕ^(II)_pd and ζ^(II)_pd in two equations which are given by

where (X, Y, Z) are the Cartesian coordinates of the vertex of the pyramid and are given by

and cos β = (a₁² + a₃² − a₂²)/(2a₁a₃).

For eqn (15) and (20), it is not easy to determine simple strict inequalities on the variables. However, we have found that accurate numerical values can be determined using MAPLE and restricting the angles to the following ranges:

0 ≤ ψ^(II)_pd ≤ π/n, 5π/9 ≤ ϕ^(II)_pd ≤ 2π/3, − π/4 ≤ ζ^(II)_pd ≤ π/4 (21)

which we emphasize are empirically determined approximate ranges. The incline angle ζ^(I)_pd for Model I may be found from either of (20)₁ or (20)₂ by a numerical method. Equations for the chiral angle (18) and the nanotube radius (19) for the general polyhedral model are the same as eqn (8) and (12) for the ideal polyhedral model, with the same bond lengths and the same bond angles, so that the ideal polyhedral model is a special case of the general polyhedral model.

Using the method of asymptotic expansions, the chiral angle θ_pd is given by

Similarly, the nanotube radius may be rewritten as

By an examination of these asymptotic expansions for the general polyhedral model for the chiral angle θ_pd(22) and for the nanotube radius r_pd(23), we observe that the leading order terms coincide exactly with the general rolled-up model eqn (3) and (4) and we view the second terms as first-order corrections to the general rolled-up model.

In Table 2, the values of the various bond lengths and bond angles for carbon nanotubes are provided from Table 1 of Jiang et al.⁵¹ and compared with the general polyhedral model for Model I. The values of the carbon bond lengths for Table 3 are provided from Table 1 of Budyka et al.,⁴⁹ while the bond angle values are not given. Accordingly, we use the bond angle arising from the general polyhedral model with distinct bond lengths and the same bond angle and the general polyhedral model for carbon nanotubes for Model II, in order to compare with Budyka et al.⁴⁹Tables 2 and 3 show that the results of the general polyhedral model to be in excellent agreement with the first-principle simulations.

Table 2 Comparison of carbon nanotube radii for general polyhedral model with distinct bond lengths and distinct bond angles (Model I) and interatomic potential method of Jiang et al.⁵¹

(n, m)	σ 1/Å	σ 2/Å	σ 3/Å	ϕ ^(I)1pd (°)	ϕ ^(I)2pd (°)	ϕ ^(I)2pd (°)	r ^(I) pd/Å	θ ^(I) pd (°)	r rd/Å	r ^*/Å	θ ^* (°)
(5,0)	1.4669	1.4669	1.4542	118.99	118.99	112.59	2.0761	0.000	2.0219	2.0761	0.000
(7,0)	1.4586	1.4586	1.4528	119.40	119.40	116.28	2.8553	0.000	2.8146	2.8552	0.000
(10,0)	1.4544	1.4544	1.4518	119.69	119.69	118.20	4.0385	0.000	4.0093	4.0386	0.000
(20,0)	1.4516	1.4516	1.4510	119.92	119.92	119.55	8.0178	0.000	8.0031	8.0180	0.000
(30,0)	1.4511	1.4511	1.4508	119.96	119.96	119.80	12.0103	0.000	12.0005	12.0150	0.000
(4,2)	1.4603	1.4675	1.4555	116.92	120.59	114.19	2.1678	19.650	2.1364	2.1678	18.702
(6,2)	1.4564	1.4592	1.4531	118.61	120.07	116.72	2.9284	14.074	2.8983	2.9284	13.673
(9,3)	1.4532	1.4544	1.4518	119.36	120.02	118.56	4.3557	13.972	4.3351	4.3556	13.794
(16,6)	1.4514	1.4518	1.4510	119.79	120.01	119.57	7.8929	15.320	7.8823	7.8930	15.262
(25,9)	1.4510	1.4511	1.4508	119.91	120.00	119.82	12.2120	14.810	12.2050	12.2125	14.786
(4,4)	1.4548	1.4604	1.4548	117.31	120.45	117.31	2.8036	30.476	2.7856	2.8036	29.833
(5,5)	1.4533	1.4568	1.4533	118.27	120.26	118.27	3.4897	30.299	3.4751	3.4899	29.887
(6,6)	1.4525	1.4549	1.4525	118.80	120.17	118.80	4.1782	30.204	4.1657	4.1782	29.919
(12,12)	1.4511	1.4517	1.4511	119.70	120.04	119.70	8.3229	30.050	8.3165	8.3230	29.979
(18,18)	1.4509	1.4511	1.4509	119.87	120.02	119.87	12.4752	30.023	12.4707	12.4750	29.991

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Table 3 Comparison of carbon nanotube radii for general polyhedral model with distinct bond lengths and the same bond angle (Model II) and the various methods of Budyka et al.⁴⁹

(n, m)	σ 1/Å	σ 2/Å	σ 3/Å	ϕ ^(II)pd (°)	θ ^(II)pd (°)	r ^(II)pd/Å	r rd/Å	r ^*/Å	Method used in^49
(4,4)	1.424	1.429	1.424	118.285	29.942	2.773	2.726	2.759	PM3
(4,4)	1.437	1.442	1.437	118.285	29.943	2.798	2.751	2.786	PBEPBE/3-21C*
(4,4)	1.435	1.439	1.435	118.285	29.954	2.793	2.746	2.783	PBEPBE/6-31C*
(4,4)	1.432	1.435	1.432	118.285	29.965	2.786	2.739	2.773	PBEPBE/6-311C*
(4,4)	1.432	1.439	1.432	118.285	29.919	2.791	2.744	2.773	B3LYP/6-31G
(4,4)	1.429	1.435	1.429	118.285	29.931	2.784	2.737	2.773	B3LYP/3-21G*
(5,5)	1.423	1.426	1.423	118.906	29.965	3.439	3.402	3.427	PM3
(5,5)	1.420	1.425	1.420	118.906	29.942	3.435	3.398	3.421	PM5
(5,5)	1.435	1.438	1.435	118.906	29.965	3.468	3.431	3.381	PBEPBE/3-21G*
(5,5)	1.428	1.431	1.428	118.906	29.965	3.451	3.414	3.422	B3LYP/3-21G*
(6,6)	1.422	1.424	1.422	119.241	29.977	4.109	4.078	4.099	PM3
(8,8)	1.421	1.423	1.421	119.574	29.977	5.456	5.433	5.448	PM3
(9,9)	1.420	1.423	1.420	119.664	29.965	6.131	6.111	6.123	PM3
(10,10)	1.420	1.422	1.420	119.728	29.977	6.805	6.786	6.799	PM3
(10,10)	1.433	1.434	1.433	119.728	29.988	6.864	6.845	6.859	PBEPBE/3-21G*
(12,12)	1.420	1.422	1.420	119.811	29.977	8.159	8.144	8.153	PM3
(15,15)	1.420	1.422	1.420	119.879	29.977	10.192	10.180	10.184	PM3

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3 Silicon nanotubes

The nanotube lattice for silicon nanotubes is assumed to comprise only skewed rhombi. The traditional rolled-up model for silicon nanotubes is similar to that for carbon nanotubes in that it assumes that all bond lengths and all bond angles are equal. An extension of the traditional rolled-up model might be described as the general rolled-up model which assumes two distinct bond lengths and that all bond angles are equal. The chiral vector C_h for both models is C_h = na₁ + ma₂ where the two basis vectors are a₁ and a₂. The lengths of the basis vectors for the traditional rolled-up model are a₁ = a₂ = σ where σ is bond length. However, the general rolled-up model may have different bond lengths and a₁ = σ₁ and a₂ = σ₂ where σ₁ and σ₂ are distinct. Fig. 7 shows the general rolled-up model initially drawn on a two dimensional sheet. The terminology for silicon nanotubes is different to that of carbon nanotubes because of the different lattice structure.^46,56 When m = 0, which is equivalent to rolling up the nanotube in the direction of OE, we term the nanotubes prismatic because the tube comprises regular n-gon prisms. When m = n, we term the tube antiprismatic which occurs when the sheet is rolled following the direction OD. In all other cases 0 < m < n, the nanotube is termed as chiral.

Fig. 7 Silicon nanotube constructed from rolling-up a two dimensional sheet with distinct bond lengths.

The main equations for the traditional rolled-up model are given by

while for the general rolled-up model we have

For m = 0, the chiral angles θ_re and θ_rd are zero. In the case m = n, the chiral angle θ_re for the traditional rolled-up model is π/4 and the chiral angle θ_rd for the general model is which we observe depends upon the bond lengths only. The adjacent bond angle is π/2 for the traditional rolled-up model ϕ_re and the general rolled-up model ϕ_rd. Eqn (25) show that when the bond lengths are equal (σ₁ = σ₂), these equations coincide with eqn (24), and therefore the traditional rolled-up model is a special case of the general rolled-up model.

The ideal polyhedral model is based on three fundamental postulates⁴⁶ that: (i) all bond lengths are equal to σ; (ii) all the adjacent bond angles are equal to ϕ_pe; and (iii) all atomic nuclei are equidistant from a common axis r_pe. In this event, the subtend semi-angle is determined as a root of the following transcendental equation:

n tan ξ_pe + m tan ψ_pe = 0 (26)

where ξ_pe = (nψ_pe − π)/m. For a physically meaningful structure, the subtend semi-angle ψ_pe is similar to the ideal polyhedral model for carbon nanotubes and the subtend semi-angle satisfies the inequalities (6) and an accurate numerical value for the required root of (26) may be determined after a small number of iterations using Newton's method, with an initial value of ψ_0pe = nπ/(n² + m²).

The chiral angle can be shown to be given by

which is zero for prismatic tubes (m = 0), and in this special case m = n the chiral angle θ_pe tends to π/4 as n increases. The silicon adjacent bond angle ϕ_pe may be shown to be given by

cos ϕ_pe = (m² sin²ψ_pe)/(n² cos²ψ_pe + m²),

so that the adjacent bond angle ϕ_pe for the prismatic tube has the constant value π/2. The value of the adjacent bond angle ϕ_pe for chiral and antiprismatic tubes asymptotes to π/2 in the limit as n becomes large, since the surface of the nanotube approaches a flat plane in this limit. The nanotube radius can be shown to be given by

r_pe = (σ cos θ_pe)/(2 sin ψ_pe) (28)

The chiral angle θ_pe and the nanotube radius r_pe may be written in terms of the traditional chiral angle θ_re and the traditional nanotube radius r_re by using the method of asymptotic expansions to yield

Eqn (29) and (30) show that the leading order terms of the analytical expressions give the traditional formulae (24) as their highest order terms and the second terms provide first-order corrections to the traditional model.

Table 4 compares the nanotube radii for the traditional rolled-up model, the ideal polyhedral model and the simulation results, and show that the results of the ideal polyhedral model are in good agreement with the simulation results of Li et al.,⁶¹ who examine a number of prismatic silicon nanotubes using an empirical full-potential linear-muffin-tin-orbital molecular dynamics method. Their silicon nanorings and nanotubes are prismatic silicon nanotubes with atoms at the surface, and their nanotubes are similar to the prismatic nanotubes described here.

Table 4 Comparison of silicon nanotube radii from traditional rolled-up model, ideal polyhedral model and molecular dynamics method of Li et al.⁶¹ using σ = 2.305 Å

(n, m)	r re/Å	r pe/Å	r ^*/Å
(6,0)	2.201	2.305	2.35
(8,0)	2.935	3.012	3.00
(10,0)	3.669	3.730	3.75

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For the general polyhedral model, we employ the following three fundamental postulates:⁵⁶ (i) corresponding bonds lying on the same helix are equal in length to either σ₁ or σ₂ which are assumed generally to be distinct; (ii) all adjacent bond angles are equal to ϕ_pd; and (iii) all atomic nuclei are equidistant r_pd from a common axis. It may be shown that the subtend semi-angle ψ_pd for the general polyhedral model is determined as a root of the following transcendental equation:

4nm(σ²₂ sin²ψ_pd − σ²₁ sin²ξ_pd) + (σ²₁n² − σ²₂m²)[sin²(ξ_pd + ψ_pd) − sin²(ξ_pd − ψ_pd)] = 0,

where ξ_pd = (nψ_pd − π)/m and the initial values of the root of the transcendental equation for Newton's method is ψ_0pd = σ²₁nπ/(σ²₁n² + σ²₂m²). The chiral angle θ_pd can be shown to be given byand the equation for the nanotube radius r_pd is the same as eqn (19) for the general polyhedral model for the carbon nanotubes, but the value of a₁ is different to carbon nanotubes, where for silicon nanotubes a₁ = σ₁.

The adjacent bond angle ϕ_pd is defined as the angle between two bonds where the atoms that are being bonded form part of the same rhombus in the nanotube lattice while the opposite bond angle is the angle between two bonds where the atoms that are being bonded do not form part of the same rhombus in the nanotube lattice. The adjacent bond angle ϕ_pd is given by

cos ϕ_pd = [(λ² + 1)m² sin²ψ_pd − λ²m² cos²θ_pd sin²(ξ_pd + ψ_pd) − λ²(n + m)² sin²θ_pd sin²ψ_pd]/(2λm² sin²ψ_pd) (32)

and the opposite bond angles ω_1pd and ω_2pd can be shown to be given bywhere λ is the ratio of the bond lengths λ = σ₁/σ₂. The adjacent bond angle ϕ_pd and two opposite bond angles ω_1pd and ω_2pd are shown in Fig. 6 (b).

Table 5 shows a comparison of the silicon nanotube radius for the general polyhedral model and the general rolled-up model as compared with the molecular dynamics simulations at 100 K. We have relaxed the idealised silicon nanotube structures with all the bond lengths set equal to 2.35 Å using the LAMMPS software (see Lee et al.⁴⁶ and Plimpton⁷⁵) and we model the silicon pairwise interactions using a Stillinger–Weber potential.⁷⁶ The σ₁*, σ₂* and r* are the molecular dynamics simulations results. The numerical data of the structures considered here shows them to be at least metastable in these simulations, the structures obtained from the molecular dynamics simulations for (4, 0), (4, 1) and (4, 2) are shown in Fig. 8, 9 and 10, respectively.

Fig. 8 Molecular dynamics result for SiNT (4, 0) at 100 K.

Fig. 9 Molecular dynamics result for SiNT (4, 1) at 100 K.

Fig. 10 Molecular dynamics result for SiNT (4, 2) at 100 K.

Table 5 Results of molecular dynamics simulations for stable silicon nanotubes at 100 K

(n, m)	σ 1*/Å	σ 2*/Å	r*/Å	r pd/Å	r rd/Å
(4,0)	2.462 ± 0.031	2.573 ± 0.040	1.74 ± 0.08	1.74	1.57
(4,1)	2.431 ± 0.032	2.588 ± 0.017	1.75 ± 0.07	1.78	1.60
(4,2)	2.479 ± 0.023	2.611 ± 0.040	1.88 ± 0.03	1.96	1.78
(5,0)	2.418 ± 0.051	2.574 ± 0.038	2.12 ± 0.01	2.06	1.92
(5,1)	2.413 ± 0.012	2.571 ± 0.021	2.07 ± 0.05	2.10	1.96
(6,0)	2.429 ± 0.066	2.580 ± 0.087	2.42 ± 0.07	2.43	2.32
(6,1)	2.449 ± 0.055	2.564 ± 0.076	2.47 ± 0.45	2.49	2.37
(7,0)	2.466 ± 0.098	2.571 ± 0.091	2.84 ± 0.69	2.84	2.75
(7,1)	2.447 ± 0.037	2.602 ± 0.028	2.83 ± 0.43	2.85	2.76

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Table 6 shows the bond angles for the polyhedral model with distinct bond lengths compared with these obtained from the molecular dynamics simulations at 100 K. The bond angles ϕ*, ω₁* and ω₂* are the average of the bond angles measured from two rings of the nanotube and the error is two standard deviations. The bond angles ϕ_pd, ω_1pd and ω_2pd are calculated from the polyhedral model using the values σ₁* and σ₂* from Table 5. The results of the adjacent bond angle ϕ_pd and the opposite bond angle ω_1pd of the polyhedral model and the molecular dynamics simulations are in excellent agreement. The calculated adjacent bond angle ϕ* has a large error, and therefore we have grouped the raw data into two groups as either above or below π/2 which are termed the adjacent bond angles ϕ_a* and ϕ_b*. The low value of the error obtained for the adjacent bond angles ϕ_a* and ϕ_b* suggests that the adjacent bond angle may have two distinct values. The opposite bond angle ω_2pd for prismatic tubes does not match well with the molecular dynamics simulations results because the tubes possess an off-centred structure which alternates along the nanotube length as shown in Fig. 8. The structures of the molecular dynamics simulations results for (4,1) and (4,2), which are shown in Fig. 9 and 10, are similar to those predicted by the general polyhedral model. However, Table 5 shows that the results of the general polyhedral model are in excellent agreement with the molecular dynamics simulations.

Table 6 Results of bond angles from molecular dynamics simulations for stable silicon nanotubes at 100 K

(n, m)	ϕ* (°)	ϕ a* (°)	ϕ b* (°)	ω 1* (°)	ω 2* (°)	ϕ pd (°)	ω 1pd (°)	ω 2pd (°)
(4,0)	90.13 ± 15.69	82.57 ± 1.72	97.68 ± 1.34	90.00 ± 5.21	158.08 ± 2.40	90.00	90.00	180.00
(4,1)	88.49 ± 10.82	83.27 ± 0.87	93.71 ± 0.99	98.63 ± 0.56	174.79 ± 1.34	86.98	100.27	170.59
(4,2)	85.58 ± 12.83	79.39 ± 0.97	91.78 ± 1.01	118.03 ± 1.31	162.86 ± 3.04	84.34	119.02	157.60
(5,0)	89.99 ± 12.08	84.70 ± 5.30	95.29 ± 5.54	107.99 ± 2.15	162.30 ± 2.04	90.00	108.00	180.00
(5,1)	89.40 ± 8.37	85.37 ± 1.18	93.44 ± 1.29	112.12 ± 1.08	178.13 ± 1.20	88.79	113.19	175.61
(6,0)	89.98 ± 13.58	83.94 ± 5.69	96.02 ± 5.90	119.94 ± 4.90	160.33 ± 4.69	90.00	120.00	180.00
(6,1)	89.46 ± 14.81	82.49 ± 4.85	96.43 ± 3.41	112.43 ± 11.90	158.94 ± 8.96	89.46	122.87	177.73
(7,0)	89.93 ± 18.04	81.59 ± 5.65	98.26 ± 6.67	128.39 ± 10.76	152.97 ± 8.32	90.00	128.57	180.00
(7,1)	89.52 ± 12.95	83.68 ± 5.42	95.36 ± 5.03	130.32 ± 8.49	160.01 ± 7.15	89.70	130.41	178.57

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4 Boron nanotubes

The traditional rolled-up model^40,41 assumes that the lattice pattern of a boron nanotube comprises flat equilateral triangles and that all bond lengths and all bond angles are equal. The terminology for boron nanotubes is precisely the same as that used for carbon nanotubes with zigzag tubes where m = 0, armchair where m = n and chiral where 0 < m < n. The chiral angle θ_re coincides with that for carbon nanotubes (1) but the radius r_re is different and is given by

In generalising this model, Lee et al.⁴⁷ have proposed a rolled-up model with distinct bond lengths. Since there are three different bond lengths, the nanotubes will have three different adjacent bond angles, which are defined as the angle between two of the bonds where the atoms that are being bonded comprise a single triangle in the nanotube lattice. There are three adjacent bond angles ϕ_1rd, ϕ_2rd and ϕ_3rd, which are shown in Fig. 11, and are given by

and in the figure, the boron nanotube (4,2) is shown. The chiral angle θ_rd and the radius r_rd for the general rolled-up model also coincide with the general rolled-up model for carbon nanotubes (3) and (4) but the value of Δ and the lengths of the basis vectors are different. For a boron nanotube, Δ = (σ₁² + σ₂² − σ₃²) and the lengths of the basis vectors are a₁ = σ₁, a₂ = σ₂ and a₃ = σ₃.

Fig. 11 Boron nanotube constructed from two dimensional sheet with distinct bond lengths.

The ideal polyhedral model is based on the two fundamental postulates⁴⁷ that: (i) all bond lengths are equal σ; and (ii) all atomic nuclei are equidistant from a common axis r_pe. The number of fundamental postulates for boron nanotubes are fewer than is the case for carbon nanotubes and silicon nanotubes because the adjacent bond angle ϕ_pe = π/3 for the boron nanotubes may be derived immediately from postulate (i) trivially, and therefore an independent adjacent bond angle is not postulated.⁴⁷

The basis vectors and the pattern for boron nanotubes are the same as for the ideal polyhedral model for carbon nanotubes with each hexagon divided into six equilateral triangles,^44,45 and therefore only equilateral triangles need be considered for boron nanotubes.⁴⁷Fig. 1 shows that the equilateral triangles pattern by gray lines and without pyramids are the same as in Fig. 3. The length of the basis vectors for boron nanotubes is a₁ = a₂ = σ. As a result, the equations for the subtend semi-angle ψ_pe, an initial value for a small number of iterations of Newton's method, the chiral angle θ_pe and the radius r_pe are the same as the ideal polyhedral model for carbon nanotubes, eqn (5), (7), (8) and (12), respectively. Since the adjacent bond angle ϕ_pe for boron nanotubes is π/3 therefore the radius r_pe might be simplified to r_pe = (σ cos θ_pe)/(2 sin ψ_pe). Table 7 shows that the results of the ideal polyhedral model are in good agreement with local density simulation results.

Table 7 Comparison of boron nanotube radii from traditional rolled-up model, ideal polyhedral model and local density approximation method of Cabria et al.^71,72

(n, m)	σ/Å	r re/Å	r pe/Å	r*/Å
(12,0)	1.72	3.285	3.323	3.35
(15,0)	1.71	4.082	4.112	4.12

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The general polyhedral model for boron nanotubes⁵⁸ is also based on two fundamental postulates that: (i) corresponding bonds lying on the same helix are equal in length, either σ₁, σ₂ or σ₃ which are assumed generally to be distinct; and (ii) all atomic nuclei are equidistant r_pd from a common axis. Similarly, the equations for the subtend semi-angle ψ_pd, an initial value for a small number of iterations of Newton's method, the chiral angle θ_pd and the radius r_pd are the same as the general polyhedral model for carbon nanotubes, eqn (15), (17), (18) and (19), respectively. The lengths of the basis vectors for boron nanotubes are a₁ = σ₁, a₂ = σ₂ and a₃ = σ₃.

Fig. 12 shows the general polyhedral model with 1/9 hexagonal holes. In this Figure the missing boron atoms are represented by gray dots and the image bonds between boron atoms and missing atoms are indicated by gray dashed lines. Table 8 compares the general rolled-up model, the general polyhedral model for zigzag and armchair tubes with Yang et al.⁷³ We emphasize that the data shown in the final column of Table 8 arises from Yang et al.⁷³ and Yang⁷⁴ upon averaging three distinct radii and the comparison is with a different lattice structure involving both triangles and 1/9 missing atoms. In Table 8 we have arbitrarily adopted values of σ₁ for zigzag and σ₁, σ₂ and σ₃ for armchair, such that use of our formulae gives an excellent agreement with the results of Yang et al.⁷³ and Yang.⁷⁴ We comment that since σ₂ and σ₃ do not appear in the formula for the radius for zigzag, these can adopt any values. We see that on adopting this strategy for both zigzag and armchair nanotubes, the polyhedral model is in excellent agreement with these studies.^73,74

Fig. 12 Polyhedral model for boron nanotubes with 1/9 hexagonal holes for zigzag, chiral and armchair tubes.

Table 8 Comparison of boron nanotube radii from general rolled-up model, general polyhedral model and first-principles method of Yang et al.^73,74

(n, m)	σ 1/Å	σ 2/Å	σ 3/Å	r rd/Å	r pd/Å	r ^*/Å
(12,0)	1.679	—	—	3.207	3.244	3.242
(15,0)	1.679	—	—	3.008	4.038	4.036
(18,0)	1.679	—	—	4.810	4.834	4.836
(21,0)	1.679	—	—	5.612	5.633	5.634
(24,0)	1.679	—	—	6.413	6.332	6.431
(27,0)	1.679	—	—	7.215	7.231	7.229
(9,9)	1.670	1.670	1.647	4.162	4.183	4.180
(12,12)	1.670	1.670	1.647	5.549	5.565	5.568
(15,15)	1.670	1.670	1.647	6.937	6.950	6.944
(18,18)	1.670	1.670	1.647	8.324	8.335	8.329
(21,21)	1.670	1.670	1.647	9.712	9.721	9.736

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5 Boron nitride nanotubes

Boron nitride nanotubes have a similar structure to that of carbon nanotubes but they have two radii such that all boron atoms have one radius and all nitrogen atoms have another radius.⁴³ As is the case for carbon, the traditional rolled-up model for boron nitride has both atomic spaces located on a flat two dimensional sheet and therefore all the equations of the traditional rolled-up model are the same as those for carbon nanotubes,⁴² including the chiral angle (1) and the radius (2).

Cox and Hill⁴⁸ propose an ideal polyhedral model for single-walled boron nitride nanotubes that includes the two radii structure. Their ideal polyhedral model has five postulates: (i) all bonds are of equal length σ; (ii) all bond angles ϕ_B for the boron atoms are equal; (iii) all boron atoms lie at an equal distance r_B from the nanotube axis; (iv) all nitrogen atoms lie at an equal distance r_N from the nanotube axis; and (v) there exits a fixed ratio of pyramidal height τ between the boron species, compared to the corresponding height in a symmetric single species nanotube. The boron atoms have a flatter structure of bonds which are located closer to the nanotube axis as compared to the nitrogen atoms. If all the bond angles are equal for the nitrogen atoms then the model would be forced to be exactly that for carbon nanotubes when all the bond angles are equal.

The subtend semi-angle ψ_pe, the chiral angle θ_pe and the incline angle ζ_pe for boron nitride nanotubes are the same as those for carbon nanotubes, namely eqn (5), (8) and (9). The adjacent bond angles and the radii are different to carbon nanotubes but the coefficients of the constituent pyramids are the same as those for eqn (11). The adjacent bond angle for boron atoms is

cos ϕ_B = (−2 + τ²k²)/(4 + τ²k²) (35)

The two radii for boron nitride nanotubes are given by

and the approximate formula for the adjacent bond angles in degrees are given bywhere ϕ_B is the adjacent bond angle for boron atoms and ϕ_1N, ϕ_2N and ϕ_3N are the three distinct adjacent bond angles for nitrogen atoms. The first term in each formula is the traditional angle of 2π/3. The second term in each equation is the correction term which takes into account the curvature of the cylinder, while the final terms are τ dependent corrections for the three distinct nitrogen bond angles.

The value for the constant of proportionality τ may be determined by the boron angles,

where ϕ_pe is the bond angle expected in an equivalent carbon nanotube (10). The τ might also be found from
where

In Table 9, the bond angles ϕ_B* and ϕ_N* in are found by density functional theory within the generalized-gradient approximation, implemented via the Vienna ab initio Simulation Package (VASP).⁸⁴ The value of τ is calculated from (38). A comparison of the bond angles shows that the average of nitrogen bond angles are a close match with the results of Barnard et al.⁸⁴

Table 9 Comparison of boron nitride nanotube angles from polyhedral model and density functional theory results from Barnard et al.⁸⁴

(n, m)	ϕ B* (°)	ϕ N* (°)	ϕ (°)	τ
(6,6)	119.75	118.48 ± 0.82	119.24	0.57 ± 0.01
(9,9)	119.89	119.32 ± 0.38	119.66	0.57 ± 0.03
(12,12)	119.94	119.62 ± 0.21	119.81	0.56 ± 0.04

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(n, m)	ϕ 1N (°)	ϕ 2N (°)	ϕ 3N (°)	[small phi, Greek, macron]N (°)
(6,6)	119.75	117.86	117.86	118.49 ± 0.89
(9,9)	119.89	119.04	119.04	119.32 ± 0.40
(12,12)	119.94	119.45	119.45	119.61 ± 0.23

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6 Results and conclusions

For the general rolled-up model the equations for the chiral angle, (3) and (25)₁ and for radius (4) and (25)₂ are the same as the traditional rolled-up model defined by (1), (2) and (24), when the bond lengths are equal so that the traditional rolled-up model is a special case of the general rolled-up model.^56–58 The equations for the general polyhedral model (18), (19) and (31) are also the same as the equations for the ideal polyhedral model (8), (12), (27) and (28) when the bond lengths are equal, and therefore the general polyhedral model includes the ideal polyhedral model.^56–58 The traditional rolled-model is a special case of the general rolled-up model and similarly the ideal polyhedral model is a special case of the general polyhedral model.

From the asymptotic expansions (13), (14), (29), (30) and (37) for the ideal polyhedral model,^44,46–48 the first terms are exactly the traditional rolled-up model eqn (1), (2) and (24) and the second term may be viewed as a first-order correction to the traditional model. Similarly, the first terms of the asymptotic expansions (22), (23), (29) and (30) for the general polyhedral model^56–58 are exactly the general rolled-up model eqn (3), (4) and (25) and the second term is a first-order correction to the traditional model. This demonstrates that the polyhedral models converge to the rolled-up models for large n. As a result, the polyhedral models include the rolled-up and ideal models or corresponding general cases, and therefore the general polyhedral model incorporates both rolled-up models and the ideal polyhedral model.

Tables 1, 4 and 7 show that the results of the ideal polyhedral model are in good agreement with those arising from simulation results. For the smaller radii nanotubes, there is a large difference in the radius predicted by the traditional rolled-up model and that arising from the molecular dynamics simulations. This phenomenon is due to the increasing curvature of the tube surface as n becomes small, so that the nanotubes with the smallest values of n have the largest curvature. Tables 2, 3, 5 and 8 show comparisons of the radius from computational studies (molecular dynamics simulations or ab initio) with the results of the general polyhedral model and the general rolled-up model. In almost all cases, the results for the general polyhedral model and the computational studies are in excellent agreement.

In summary, the conventional rolled-up model of nanotubes is an approximate model and completely ignores any effects due to curvature. Accordingly, this opens up the possibility of providing exact geometric constructions which correctly take into account the curvature of the nanotube, but are in complete accord with traditional models for large radii nanotubes. The present paper summarizes the work of the authors which addresses exact nanotube construction based upon certain geometric principles. Polyhedral models are proposed for the four materials, carbon, silicon, boron and boron nitride. In general, the proposed new geometric models are in excellent agreement with existing data from purely numerical approaches and the models can be readily extended to incorporate distinct bond lengths.

Table 10 List of symbols

Symbols	Meaning	Equation where first used
β	Parameter cos β = (a1^2 + a3^2 − a2^2)/(2a1a3)	(20)
Δ	Parameter Δ = (a1^2 + a2^2 − a3^2)	(3)
ϕ pe	Bond angle for the ideal polyhedral model	(10)
ϕ re	Bond angle for the traditional rolled-up model for CNT, SiNT and BNNT
ϕ ^(I)1pd, ϕ^(I)2pd, ϕ^(I)2pd	Distinct bond angles for the CNT general polyhedral Model I	(16)
ϕ ^(II)pd	Bond angle for the CNT general polyhedral Model II	(21)
ϕ 1rd, ϕ2rd, ϕ3rd	Distinct bond angles for the rolled-up model for BNT	(34)
ϕ B	Bond angles for the ideal polyhedral model for boron atoms of BNNT	(35)
ϕ 1N, ϕ2N, ϕ3N	Distinct bond angles for the ideal polyhedral model for nitride atoms of BNNT	(37)
λ	The ratio of the bond lengths λ = σ1/σ2	(32)
θ re	Chiral angle for the traditional rolled-up model	(1)
θ rd	Chiral angle for the general rolled-up model	(3)
θ pe	Chiral angle for the ideal polyhedral model	(8)
θ pd	Chiral angle for the general polyhedral model	(18)
ρ, χ, ν	Parameter	(11)
σ	Bond length	(2)
σ 1, σ2, σ3	Distinct bond lengths	(16)
τ	A fixed ratio of pyramidal height	(35)
ω 1pd, ω2pd	Distinct opposite bond angles for SiNT	(33)
ξ	Parameter (nψ − π)/m	(5)
ψ pe	Subtend semi-angle for the ideal polyhedral model	(5)
ψ pd	Subtend semi-angle for the general polyhedral model	(15)
ψ 0pe	Initial value of Newton's method for the ideal polyhedral model	(7)
ψ 0pd	Initial value of Newton's method for the general polyhedral model	(17)
ζ pe	Incline angle for the ideal polyhedral model	(9)
ζ pd	Incline angle for the general polyhedral model	(20)
a 1 , a2, a3	Basis vectors	(3)
C h	Chiral vector
k	Parameter k^2ρ + kχ + ν = 0 and k is the nondimensional pyramidal height	(10)
n, m	Chiral vector numbers	(1)
r re	Nanotube radius for the traditional rolled-up model	(2)
r rd	Nanotube radius for the general rolled-up model	(4)
r pe	Nanotube radius for the ideal polyhedral model	(12)
r pd	Nanotube radius for the general polyhedral model	(19)
r B	Nanotube radius for boron atoms of the ideal polyhedral model	(36)
r N	Nanotube radius for nitride atoms of the ideal polyhedral model	(36)
X, Y, Z	Cartesian coordinates for the atomic location at the top of a pyramid	(20)

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Acknowledgements

The support of the Australian Research Council, both through the Discovery Project Scheme and for providing an Australian Professorial Fellowship for JMH and an Australian Post-doctoral Fellowship for BJC is gratefully acknowledged. The authors are also grateful to Dr Xiaobao Yang⁷⁴ for providing the raw numerical data from Yang et al.⁷³

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