Ideal Polyhedral Model for Boron Nanotubes with Distinct Bond

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J. Phys. Chem. C 2009, 113, 19794–19805

Ideal Polyhedral Model for Boron Nanotubes with Distinct Bond Lengths Richard K. F. Lee,* Barry J. Cox,* and James M. Hill Nanomechanics Group, School of Mathematics and Applied Statistics, UniVersity of Wollongong, Wollongong, NSW 2522, Australia ReceiVed: May 28, 2009; ReVised Manuscript ReceiVed: September 11, 2009

In this Article, we extend both the rolled-up model and the polyhedral geometric model for single-walled boron nanotubes with equal bond lengths to the corresponding models but having distinct bond lengths. The boron nanotubes considered here are assumed to be formed by sp2 hybridization and π-bonds and may have as many as three different bond lengths, and the nanotube lattice is assumed to comprise a triangular pattern. Beginning with the two postulates that all equivalent bond lengths lying on the same helix are equal and that all of the atoms are equidistant from a common axis of symmetry, we derive exact formulas for the geometric parameters such as chiral angles, bond angles, radius, and unit cell length. Results for both the rolled-up model and the prior geometric model are shown to emerge from the new geometric model in the limit of large radius and in the limit of equal bond lengths, respectively. Numerical results for the new polyhedral model are related to existing numerical results for a novel boron nanotube structure, which involves 1/9 of the atoms missing. This novel structure is believed to be more energetically favorable than the conventional model. 1. Introduction Since recent interest in carbon nanotubes was initiated by Iijima’s discovery in 1991,1 other nanotube materials have also become of considerable interest for synthesis. Because of its similarity with carbon and its rich chemistry, boron is a natural choice for constructing nanostructures, such as nanoclusters, nanowires, and nanotubes. Boron nanowires were discovered in 2002,2,3 and single-walled pure boron nanotubes were synthesized in 2004.4 Boron and carbon both have very stable structures formed from sp2 hybridized bonds and π-bonds,5-10 but they have a different lattice configuration. Carbon nanotubes are constructed from a honeycomb hexagonal lattice,1,11-13 while boron nanotubes are formed from a triangular lattice.5,14-22 Previous work on the structure of boron nanotubes is based on the conventional rolled-up model,16-22 which is entirely analogous to the rolled-up model for carbon nanotubes,11-13 and the bond lengths are assumed to be all equal. Recently, Cox and Hill23,24 proposed a new geometric polyhedral model for single-walled carbon nanotubes, in which all bond lengths are assumed to be equal, and which makes very accurate predictions on the geometric parameters of the nanotubes, which are in excellent agreement with certain first-principles calculations.23 However, carbon nanotubes do indeed have different bond lengths and different bond angles,25-31 and these features cannot be ignored. Accordingly, boron nanotubes also might have different bond lengths and different bond angles. The ideal geometric polyhedral model with equal bond lengths for boron is formulated by the authors.32 For the present work, we employ a similar polyhedral model to represent single-walled boron nanotubes but with distinct bond lengths. The polyhedral model with distinct bond lengths for carbon nanotubes has been established.25 The naming convention for boron nanotubes follows Gindulyte et al.,16 which is exactly the same as for carbon nanotubes, and boron nanotubes are termed zigzag when * Corresponding author. E-mail: [email protected] (R.K.F.L.); [email protected] (B.J.C.).

m ) 0 and armchair when m ) n. In all other cases, when 0 < m < n, they are termed chiral nanotubes. For carbon nanotubes, the bond lengths and the bond angles vary with the nanotube curvature,26 and the bonds along the nanotube axis are shorter than the bonds around the nanotube circumference. Accordingly, zigzag and armchair tubes can have two different bond lengths and two different bond angles, while chiral tubes can have three different bond lengths and three different bond angles.25,31 Similarly, the bond lengths for zigzag (σ1, σ2 ) σ3) and armchair (σ1 ) σ2, σ3) boron nanotubes might have two different bond lengths and two different bond angles. The geometric polyhedral model for boron nanotubes with distinct bond lengths is developed in a manner similar to that for the ideal polyhedral model for carbon nanotubes23,24 and for boron nanotubes32 and 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 from a common axis. Typical boron nanotubes obtained from the polyhedral model with distinct bond lengths are shown in Figure 1. The zigzag (5,0) tube has two different bond lengths with σ1 ) 1.68 Å and σ2 ) σ3 ) 1.65 Å. The chiral (5,3) tube has three different bond lengths with σ1 ) 1.67 Å, σ2 ) 1.66 Å, and σ3 ) 1.65 Å. The armchair (5,5) tube has two different bond lengths with σ1 ) σ2 ) 1.66 Å and σ3 ) 1.64 Å. In this figure, the boron atoms are represented by black dots, and the bonds between boron atoms are indicated by black lines. A novel boron nanotube structure is proposed by Tang and Ismail-Beigi8 that is a triangular lattice structure with 1/9 hexagonal holes, as it is believed to be energetically more stable than a pure triangular lattice.33 Following Tang and IsmailBeigi,8 we adopt the definition of hexagonal hole density as the number of hexagonal holes divided by the total number of atoms in the triangular structure. As a first approximation, this novel structure may be considered to be a pure triangular lattice structure with 1/9 hexagonal holes. The transformation of the

10.1021/jp904985r CCC: $40.75  2009 American Chemical Society Published on Web 10/23/2009

Ideal Polyhedral Model for Boron Nanotubes

J. Phys. Chem. C, Vol. 113, No. 46, 2009 19795

Figure 1. Polyhedral model for boron nanotubes for zigzag, chiral, and armchair tubes.

Figure 2. Polyhedral model for boron nanotubes with 1/9 hexagonal holes for zigzag, chiral, and armchair tubes.

chiral number from the novel boron nanotube structure to the pure triangular structure is (3N,3M), where N and M are the chiral vector numbers of the novel boron nanotubes structure. The boron nanotubes with 1/9 missing atoms, which is obtained from the polyhedral model with distinct bond lengths and missing atoms, are as shown in Figure 2. The zigzag (9,0) tube has two different bond lengths σ1 ) 1.68 Å and σ2 ) σ3 ) 1.65 Å. The chiral (9,6) tube has three different bond lengths σ1 ) 1.67 Å, σ2 ) 1.66 Å, and σ3 ) 1.65 Å. The armchair (9,9) tube has two different bond lengths σ1 ) σ2 ) 1.66 Å and σ3 ) 1.64 Å. 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. In sections 2 and 3, we extend the rolled-up model and the polyhedral model respectively for boron nanotubes to the case of three distinct bond lengths. In section 4, we give the asymptotic expansions for the formulas from section 3 including the first two leading order terms. We comment that the present polyhedral model for the triangular structure with distinct bond lengths might also be applied to other nanotube materials, which are based on the triangular structure. Although the triangular lattice structure with 1/9 hexagonal holes is believed to be energetically more stable than a pure triangular lattice, here, however, we propose an ideal or pristine model comprising only a triangular lattice, but we compare our predictions for the radius with Yang et al.33 in section 5. Some concluding remarks are

made in section 6, and various asymptotic expansion details are presented in Appendix A.

2. Rolled-Up Model with Distinct Bond Lengths In the present work, we follow the (n,m) naming scheme employed for boron nanotubes following Gindulyte et al.16 The naming scheme identifies the specific configuration of the boron nanotube originating from the rolled-up model, in which the boron nanotube is conceptualized as a flat plane of sixcoordinated boron atoms, which is then rolled into a right circular cylinder. From Figure 3, three different categories of boron nanotubes may be defined, depending on the values of n and m. When m ) 0, which is equivalent to rolling up the nanotube in the direction of OE, the boron nanotubes are termed zigzag nanotubes. The second type occurs when m ) n, which is equivalent to rolling the nanotube in the direction of OD, and these tubes are termed armchair. If the direction of rolling lies between OD and OE, then the nanotubes are termed chiral. The direction of rolling of the nanotube is represented by the chiral vector Ch as shown in Figure 3. In this figure, the boron nanotube (4,2) is shown. The vector OB is called the conventional translational vector T0, which is normalized by the greatest common divisor d0R of its components so that we have

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J. Phys. Chem. C, Vol. 113, No. 46, 2009

Lee et al.

Ch ) na1 + ma2 T0 ) (n + 2m)a1/d0R - (2n + m)a2/d0R

cos θ02 )

where n and m are non-negative integer chiral vector numbers, a1 and a2 are basis vectors in real space such that |a1| ) σ1 and |a2| ) σ2, where σ1 and σ2 are different bond lengths, and d0R is the greatest common divisor of n + 2m and 2n + m. We note that throughout we use the subscript 0 to designate quantities associated with the conventional rolled-up model. The vector a3 is also a basis vector with length |a3| ) σ3, where σ3 is the third distinct bond length. In Figure 3, the origin O is located at an arbitrary lattice point. The conventional translational vector T0 is perpendicular to the chiral vector Ch when the bond lengths are equal, but in general they are not perpendicular. The sheet is rolled up to form a nanotube where the point A will coincide with the origin O and the point B will coincide with the point B′. Because 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 φ1, φ2, and φ3, which are shown in Figure 3, and are given by

cos φ1 )

σ22 + σ23 - σ21 σ21 + σ23 - σ22 , cos φ2 ) , 2σ2σ3 2σ1σ3 cos φ3 )

σ21 + σ22 - σ23 2σ1σ2

(1)

From Figure 3, the conventional chiral angle θ01 is found to be

cos θ01 )

2nσ21 + mτ 2σ1√n2σ21 + m2σ22 + nmτ

(2)

where τ ) (σ12 + σ22 - σ32). The angles θ02 and θ03 are termed the conjugate chiral angles and are defined as φ3 - θ01 and φ2 + θ01, respectively, and might be expressed as

Figure 3. Boron nanotube constructed from two-dimensional sheet.

2mσ22 + nτ 2σ2√n2σ21 + m2σ22 + nmτ cos θ03 )

,

(2nσ21 + mτ) - (2mσ22 + nτ) 2σ3√n2σ21 + m2σ22 + nmτ

(3)

The conventional chiral angle θ01 is the angle between the chiral vector Ch and the basis vector a1. The conjugate chiral angle θ02 is the angle between the chiral vector Ch and the basis vector a2, and the conjugate chiral angle θ03 is the angle between the chiral vector Ch and the basis vector a3. The conventional chiral angle θ01 and the conjugate chiral angles θ02 and θ03 provide the component angles for the correction term in the asymptotic expansions for the polyhedral model, the details of which are presented in section 4. The conventional radius for the nanotube is obtained from the circumference and is given by the magnitude of the chiral vector |Ch| divided by 2π, thus

r0 ) √n2σ21 + m2σ22 + nmτ/(2π)

(4)

Some special cases have simple exact values for θ01 and r0. For zigzag (m ) 0) nanotubes, we have cos θ01 ) 1 and r0 ) nσ1/(2π). In the case of armchair (m ) n) nanotubes with σ1 ) σ2, then we have cos θ01 ) (1 - [σ3/(2σ1)]2)1/2 and r0 ) n(4σ12 - σ32)1/2/(2π). Boron nanotubes are assumed to be constructed from a repeating unit cell that occurs when the square of the ratios of the bond lengths λ12 ) σ12/σ32 ) p1/p3 and λ22 ) σ22/σ32 ) p2/p3 are rational, so that it is possible to write the ratios σ12:σ22:σ32 as p1:p2:p3, where p1, p2, and p3 are integers with no divisor common to all three values. In all other cases, the nanotubes have no repeating unit cell. The unit cell length for the rolled-up model is not the length of the conventional translational vector |T0|, because in general the conventional translational vector T0 and the chiral vector Ch are not perpendicular. Based on the fact that the bond lengths are



not equal, the new translational vector T, which is perpendicular to the chiral vector Ch, is given by

T ) T1a1/dR - T2a2/dR p3 p3 T1 ) 2 (2mσ22 + nτ), T2 ) 2 (2nσ21 + mτ) σ3 σ3 λ21

σ21

p1 2 σ22 p2 ) 2 ) , λ2 ) 2 ) , dR ) gcd(T1, T2) p3 p3 σ3 σ3

where p1, p2, and p3 are integers, and gcd(x,y) is the greatest common divisor of x and y. The ratios of the bond lengths are necessarily rational so that the values of T1 and T2 are integers, and in the case of equal bond lengths, the values of p1 ) p2 ) p3 ) 1. The conventional translational vector T0 is a special case of the new translational vector T arising from equal bond lengths. The unit cell length for the rolledup model L0 is the length of the new translational vector |T|, which is given by

L0 ) p3√κ(n2σ21 + m2σ22 + nmτ)/(σ23dR)

(5)

where κ ) (σ1 + σ2 + σ3)(σ1 + σ2 - σ3)(σ1 - σ2 + σ3)(-σ1 + σ2 + σ3). Therefore, the unit cell length for the rolled-up model is the length of the new translational vector |T|. The number of atoms in a unit cell N is given by N ) |Ch × T|/ |a1 × a2|, and therefore the number of atoms in the unit cell for the rolled-up model is found to be given by

N ) [2p3(n2σ21 + m2σ22 + nmτ)]/(σ23dR)

(6)

When the bond lengths are equal, the values of the unit cell length and the number of atoms are the same as those given in Lee et al.32 3. Polyhedral Model with Distinct Bond Lengths The polyhedral model with distinct bond lengths for boron nanotubes is similar to the ideal polyhedral model for carbon nanotube23,24 and for boron nanotube.32 For carbon nanotubes, each hexagonal face cannot be coplanar in the rolled-up state because all atoms in the carbon nanotube are equidistant from a common axis; thus elements of the hexagonal lattice are divided into three isosceles triangles and one equilateral triangle.23,24 However, the boron nanotube lattice does not need to be subdivided because the lattice comprises only triangles, each of which is necessarily coplanar. We begin by defining a cylinder, which is traced by helices that correspond to the lattice lines in the direction of a1. Therefore, from Figure 3, it may easily be shown that the number of helices required to trace the entire cylinder is equivalent to the value of m and the boron atoms are positioned on these helices where the helices are shown in Figure 4 with |PQ| ) σ1, |PR| ) σ2, and |PS| ) σ3. From postulate (i), the fundamental parameter for the polyhedral model with distinct bond lengths is the subtend semiangle ψ, which is determined from the following equation: [σ22(n + m)2 - σ32n2] sin2 ψ + [σ32m2 - σ12(n + m)2] sin2 ξ + (σ12n2 - σ22m2) sin2(ξ + ψ) ) 0

(7)

Figure 4. Points lying on three helices and forming a triangle in threedimensional space.

where ξ ) (nψ - π)/m. Equation 7 may have many roots, but a unique solution is determined by imposing the specific requirement that the subtend semiangle ψ must also satisfy the following inequalities:

{

π/(n + m)e ψ e π/n when σ1 g σ2 0e ψ e π/n when σ1 < σ2

Thus, eq 7 is a transcendental equation, which cannot be solved explicitly for arbitrary n, m, σ1, σ2, and σ3, and an accurate numerical value for the required root of eq 7 may be determined after a small number of iterations of Newton’s method, using an initial value of ψ0 ) [π(2nσ21 + mτ)]/[2(n2σ21 + m2σ22 + nmτ)], where τ ) (σ12 + σ22 - σ32). We comment that eq 7 has simple analytical solutions for particular values of n and m, and specifically we may show that ψ ) π/n for zigzag m ) 0 and ψ ) π/(2n) for the special case of armchair (n ) m) and σ1 ) σ2. We see from eq 7 that the subtend semiangle ψ depends on n, m and the three distinct bond lengths σ1, σ2, and σ3, which is in contrast to the ideal polyhedral model for which the subtend semiangle ψ depends only on n and m. The subtend semiangles ψ for the polyhedral model and the ideal polyhedral model are only equal for zigzag tubes (n,0). Because there are three different bond lengths, nanotubes also have two different chiral angles θ1 and θ2, which are ∠QPQ′ and ∠RPR′, respectively. The true chiral angle θ1 is found by considering a triangle comprising the points P, Q, and Q′ (which is the point determined by projecting Q into the xy-plane) as shown in Figure 5. Similarly, the true chiral angle θ2 is found from triangles ∆PRR′ and ∆CPR′ (where R′ is the point determined by projecting R into the xy-plane) as shown in Figure 4. The true chiral angles θ1 and θ2 correspond to the chiral angle θ01 and the conjugate chiral angle θ02 and can be expressed as

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cos θ1 ) 2

Lee et al.

[σ22(n + m)2 - σ23n2] sin2 ψ

respectively. There are three adjacent bond angles φ1, φ2, and φ3 are found the same as the rolled-up model eq 1. There are three next nearest-neighbor bond angles ω1, ω2, and ω3 that are found to be

[σ22(n + m)2 - σ23n2] sin2 ψ + σ23m2 sin2 ξ - σ22m2 sin2(ξ + ψ)

cos2 θ2 )

[σ22(n + m)2 - σ23n2] sin2 ξ σ22[(n + m)2 sin2 ξ - n2 sin2(ξ + ψ)]

(8) The nanotubes considered here have three different bond lengths, and therefore the adjacent bond angles (which are defined as the angle between two bonds where the atoms that are being bonded comprise a single triangle in the nanotube lattice) also have three different values, ∠RPS ) φ1, ∠PSR ) φ2, and ∠PRS ) φ3, as indicated on the left triangle in Figure 6. The next nearest-neighbor bond angles ω1, ω2, and ω3 are the angles between two bonds, which are not adjacent and not opposite; that is, they have one bond separating them. The three opposite bond angles µ1, µ2, and µ3 are the angles between two bonds where the atoms being bonded are collinear in the flat nanotube lattice. Figure 6 indicates the adjacent bond angles, the next nearest-neighbor bond angles, and the opposite bond angles. The adjacent bond angles are found by considering a triangle comprising the points R, S, and Q. The next nearestneighbor bond angles are determined from distinct triangles, where the angle ω1 is found from the triangle ∆PSV, ω2 is found from the triangle ∆PST, and ω3 is found from the triangle ∆RSQ. Similarly, the three opposite bond angles µ1, µ2, and µ3 are found from the triangles ∆RST, ∆VSQ, and ∆PSW,

cos ω1 )

σ22 + σ23 σ21 cos2 θ1 sin2(ψ + 2ξ) 2σ2σ3 2σ σ sin2 ψ 2 3

σ21(2n + m)2 sin2 θ1 2σ2σ3m2 cos ω2 )

σ21 + σ23 σ21 cos2 θ1 sin2(2ψ + ξ) 2σ1σ3 2σ σ sin2 ψ 1 3

σ21(n + 2m)2 sin2 θ1 2σ1σ3m2 cos ω3 )

σ21 + σ22 σ21 cos2 θ1 sin2(ξ - ψ) 2σ1σ2 2σ σ sin2 ψ 1 2

σ21(n - m)2 sin2 θ1 2σ1σ2m2

(9) and three opposite bond angles µ1, µ2, and µ3 are given by

cos µ1 ) 1 cos µ2 ) 1 cos µ3 ) 1 -

( ( (

σ21 cos2 θ1 sin2(2ψ) 2σ21

sin2 ψ

σ21 cos2 θ1 sin2(2ξ) 2σ22

sin2 ψ

+

4m2 sin2 θ1

+

4n2 sin2 θ1

m2 m2

σ21 cos2 θ1 sin2[2(ψ + ξ)] 2σ23

sin2 ψ

)

)

+

4(n + m)2 sin2 θ1 m2

Figure 5. Points forming PQQ′ in three-dimensional space.

)

(10)

The nanotube radius r is the distance from the boron atoms to the axis of the nanotube, which may be found from cos θ1 and the length of |PQ| ) c ) σ1, the length of the boron-boron covalent bond, as shown in Figure 5. Therefore, the nanotube radius r is given by

Figure 6. Indication of adjacent, next nearest-neighbor, and opposite bond angles.


r ) (σ1 cos θ1)/(2 sin ψ)


(11)

As in the rolled-up model, both zigzag (n,0) and the special case of armchair (n ) m) and σ1 ) σ2 have simple exact values for the chiral angle, which are cos θ1 ) 1 and cos θ1 ) (1 - (σ3/2σ1)2)1/2, respectively, and radii given by r ) σ1/[2sin(π/n)] and r ) (4σ21 - σ23)1/2/[4sin(π/2n)], respectively. We comment that the values of the chiral angles are exactly the same as the conventional chiral angles for these special cases. Boron nanotubes might also be considered to be constructed from a repeating unit cell, when the square of the ratios of the bond lengths λ12 and λ22 is rational. The number of atoms in the unit cell N for the polyhedral model is the same as the rolledup model (eq 6). A unit cell length L may be calculated from the number of atoms in a single helix multiplied by the helical vertical spacing coefficient b. The number of atoms in a single helix is found from the number of atoms in a unit cell N divided by m helices. Thus, the unit cell length may be derived from L ) Nb/m and is given by

L ) [2p3(σ21n2 + σ22m2 + nmτ)σ1 sin θ1]/(σ23mdR)

cos θ1 )

(2σ21n + mτ)2

2

+

4σ21(n2σ21 + m2σ22 + nmτ)

π2m2(2σ21n + mτ)2(2σ22m + nτ)2 × [(2σ21n + mτ) - (2σ22m + nτ)]2 64σ21(n2σ21 + m2σ22 + nmτ)5 cos θ2 )

()

(2σ22m + nτ)2

2

1 n4

+O

(14)

+

4σ22(n2σ21 + m2σ22 + nmτ)

π2m2(2σ21n + mτ)2(2σ22m + nτ)2 × [(2σ21n + mτ) - (2σ22m + nτ)]2 64σ22(n2σ21 + m2σ22 + nmτ)5

() 1 n4

+O

where the leading order terms are exactly the corresponding angles θ01 and θ02, which are the conventional expressions of the chiral angle (eq 2) and the conjugate chiral angle (eq 31). The second terms are the first-order corrections to the conventional chiral angle θ01 and the conventional conjugate chiral angles θ02 and θ03, and it may be shown that eq 14 may be expressed as

(12) cos2 θ1 ) cos2 θ01 + (σ21σ22σ23 cos2 θ01 sin2 θ01 cos2 θ02 cos2 θ03)/(r20κ) + O(1/n4)

where p3 is as defined for the rolled-up model.

cos2 θ2 ) cos2 θ02 + (σ21σ22σ23 cos2 θ01 cos2 θ02 sin2 θ02 cos2 θ03)/(r20κ) + O(1/n4)

4. Asymptotic Expansions for Polyhedral Model

(15)

As detailed in Appendix A, the equations of the polyhedral model may be expressed in terms of expansions of n and m in the limit of nf∞ by using the method of asymptotic expansions. The subtend semiangle ψ determined from eq 7 is given by

ψ)

π(2σ21n + mτ) 2(n2σ21

+

m2σ22

+ nmτ)

+ {mπ3(2σ21n + mτ) ×

where κ ) (σ1 + σ2 + σ3)(σ1 + σ2 - σ3)(σ1 - σ2 + σ3) × (-σ1 + σ2 + σ3), θ01, θ02, and θ03 are the conventional chiral angle and the conjugate chiral angles, and r0 is the conventional radius. The next nearest-neighbor bond angles ω1, ω2, and ω3 are expanded by substituting eqs 13 and 141 into eq 9 for cos ω1, cos ω2, and cos ω3 to obtain

(2σ22m + nτ)[(2σ21n + mτ) (2σ22m + nτ)][4σ21n3(σ22 - σ23) + 4σ22m3(σ21 - σ23) +

cos ω1 ) -

nm(m + n)(τ2 + 8σ21σ22)]}/ {96(σ21n2

+

σ22m2

π2(2σ22m + nτ)2[(2σ21n + mτ) - (2σ22m + nτ)]2

+ nmτ) } + O(1/n ) 5

(σ22 + σ23 - σ21) + 2σ2σ3

5

8σ2σ3(n2σ21 + m2σ22 + nmτ)3

(13) cos ω2 ) -

where the O(1/n5) term refers to the maximum order of the magnitude of the next most significant term. The first term of eq 13 gives the leading order behavior for the subtend semiangle ψ, and the second term is a correction term, which takes into account the curvature of the nanotube. It is worth commenting that up to this order, eq 13 is totally in accordance with the special case of zigzag nanotubes m ) 0, where ψ ) π/n. By substituting eq 13 into the expressions for the true chiral angle θ1 and the conjugate chiral angle θ2 given in eq 8 and then by further expansion in terms of 1/n, an expansion for the true chiral angles θ1 and θ2 may be developed, which are given by

1 n4

(σ21 + σ23 - σ22) + 2σ1σ3

π2(2σ21n + mτ)2[(2σ21n + mτ) - (2σ22m + nτ)]2 8σ1σ3(n2σ21 + m2σ22 + nmτ)3 cos ω3 ) -

()

+O

()

+O

1 n4

(σ21 + σ22 - σ23) + 2σ1σ2 π2(2σ22m + nτ)2(2σ21n + mτ)2 8σ1σ2(n2σ21 + m2σ22 + nmτ)3

()

+O

1 n4

(16)

The first terms are exactly the rolled-up model values, and the second terms are the first-order corrections to the conventional values, and it may be shown that eq 16 may be expressed as

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Lee et al.

cos ω1 ) -cos φ1 + (σ2σ3 cos2 θ02 cos2 θ03)/(2r20) + O(1/n4) cos ω2 ) -cos φ2 + (σ1σ3 cos2 θ01 cos2 θ03)/(2r20) + O(1/n4) cos ω3 ) -cos φ3 + (σ1σ2 cos2 θ01 cos2 θ02)/(2r20) + O(1/n4)

(17) Similarly, the expansion equations of the three opposite bond angles µ1, µ2, and µ3 are found by substitution of eqs 13 and 14 into eq 10, from which the expansions for cos µ1, cos µ2, and cos µ3 are given by cos µ1 ) -1 + cos µ2 ) -1 + cos µ3 ) -1 +

π2(2σ12n + mτ)4 8σ12(n2σ12 2

+

m2σ22

+O

+ nmτ)

3

π (2σ22m + nτ)4

+O

8σ22(n2σ12 + m2σ22 + nmτ)3

() () 1 n4 1 n4

π2[(2σ12n + mτ) - (2σ22m + nτ)]4 8σ32(n2σ12

+

m2σ22

+ nmτ)

3

+O

() 1 n4

Figure 7. (4,0) tube of r/σ3 with λ1 ) σ1/σ3 and λ2 ) σ2/σ3.

and these equations may be expressed as

cos µ1 ) -1 + (σ21 cos4 θ01)/(2r20) + O(1/n4) cos µ2 ) -1 + cos µ3 ) -1 +

(σ22 (σ23

cos θ02)/(2r20) cos4 θ03)/(2r20) 4

+ O(1/n ) 4

(18)

+ O(1/n4)

L ) L0 - (L0σ21σ22σ23 cos2 θ01 cos2 θ02 cos2 θ03)/(2r20κ) +

where the first terms are exactly the conventional model values and the second terms are the first-order correction terms. Using the same technique, asymptotic expansions may be developed for the nanotube radius r, which is given by

r)

√n2σ21 + m2σ22 + nmτ + {π[16σ8n4(n + m)2 + 1

2π

16σ82m4(n + m)2 - 16σ83n3m3 + 8σ21n3m(n + m)2(τ3 + σ21(τ + 2σ21)(3τ - 2σ21)) + 8σ22nm3(n + m)2(τ3 + σ22(τ + 2σ22)(3τ - 2σ22)) + n2m2(n + m)2(5τ4 + 8σ21σ22(σ41 + σ42 9σ43)) + 8σ23n2m2(n + m)(nσ61 + mσ62 + 3σ21σ22(nσ22 + mσ21) + 5σ43(nσ21 + mσ22))]}/{192(n2σ21 + m2σ22 + 7/2 3

nmτ) } + O(1/n ) (19)

where the leading order term is exactly the conventional expression (eq 4), and the second term is the first-order correction term. Asymptotic expansion of the unit cell length L yields

L)

√

+ m2σ22 + nmτ) σ23dR mτ)2(2mσ22 + nτ)2

p3 κ(n2σ21

p3π2(2nσ21 + [(2nσ21

+ mτ) -

32σ23dR(n2σ21

+

(2mσ22

m2σ22

×

+ nτ) ]

2 2

+ nmτ) √κ 7/2

(20)

()

+O

the curvature of the structure. The unit cell length may also be written as

1 n3

where we note that the first term is exactly the conventional expression (eq 5). The second term may be viewed as a firstorder correction to the conventional unit cell length arising from

O(1/n3) (21) 5. Results For both carbon and boron nanotubes, zigzag and armchair tubes may have only two different bond lengths and two different bond angles. Otherwise, chiral tubes have three different bond lengths and three different bond angles.25-31 Figure 7 shows that the radius r for a zigzag tube varies linearly with the bond lengths σ1 and σ3 and the radius r is independent of σ2, and the asymptotic expansion (eq 13) shows that the subtend semiangle ψ is independent of the bond lengths for the zigzag type for which the value of the subtend semiangle is ψ ) π/n. For all other cases, the subtend semiangle ψ depends on all of the bond lengths. Because the subtend semiangle ψ is independent of the bond lengths when m ) 0 for zigzag tubes, the chiral angle θ1 is zero, and therefore the equation of the radius r for zigzag tubes is r ) σ1/[2 sin(π/n)], which is a linear relationship with σ1. We now compare our results for zigzag TABLE 1: Comparison of Radii from Conventional Model, Polyhedral Model, and First-Principles Method of Yang et al.33,34 (n,m)

σ1 (Å)

(12,0) (15,0) (18,0) (21,0) (24,0) (27,0) (9,9) (12,12) (15,15) (18,18) (21,21)

1.679 1.679 1.679 1.679 1.679 1.679 1.670 1.670 1.670 1.670 1.670

σ2 (Å)

1.670 1.670 1.670 1.670 1.670

σ3 (Å)

radius r0 (Å)

radius r (Å)

Yang et al.34 (Å)

1.647 1.647 1.647 1.647 1.647

3.207 3.008 4.810 5.612 6.413 7.215 4.162 5.549 6.937 8.324 9.712

3.244 4.038 4.834 5.633 6.332 7.231 4.183 5.565 6.950 8.335 9.721

3.242 4.036 4.836 5.634 6.431 7.229 4.180 5.568 6.944 8.329 9.736



TABLE 2: Different Bond Lengths for First-Principles Method of Yang et al.33,34 hexagonal base radius

outer radius

inner radius

(n,m)

σ1 (Å)

σ2 (Å)

σ3 (Å)

σ1 (Å)

σ2 (Å)

σ3 (Å)

σ1 (Å)

σ2 (Å)

σ3 (Å)

(12,0) (15,0) (18,0) (21,0) (24,0) (27,0) (9,9) (12,12) (15,15) (18,18) (21,21)

1.7262 1.7115 1.7035 1.6975 1.6932 1.6901 1.6882 1.6917 1.6825 1.6812 1.6767

1.6751 1.6753 1.6751 1.6751 1.6789 1.6751 1.6881 1.6906 1.6825 1.6812 1.6767

1.6753 1.6751 1.6752 1.6479 1.6787 1.6751 1.6753 1.6604 1.6673 1.6679 1.6779

1.7209 1.7134 1.6995 1.6976 1.6961 1.6954 1.7101 1.6817 1.6899 1.6879 1.6932

1.7062 1.6998 1.7031 1.6998 1.6915 1.6952 1.7101 1.6817 1.6899 1.6879 1.6932

1.7063 1.6994 1.7032 1.7000 1.6914 1.6953 1.6766 1.6849 1.6806 1.6794 1.6742

1.6806 1.6819 1.6836 1.6820 1.6810 1.6816 1.6926 1.6802 1.6845 1.6869 1.6875

1.7106 1.7068 1.7041 1.7009 1.6941 1.6972 1.6924 1.6802 1.6845 1.6869 1.6875

1.7108 1.7071 1.7040 1.7011 1.6935 1.6970 1.6972 1.6879 1.6891 1.6834 1.6819

and armchair tubes with Yang et al.33 We emphasize that the data shown in the final column of Table 1 arise from Yang et al.33 and Yang34 upon averaging three distinct radii, and the comparison is with a different lattice structure involving both triangles and 1/9 missing atoms. In Table 1, we have arbitrarily adopted values of σ1 for zigzag and σ1, σ2, and σ3 for armchair, such that use of our formulas gives an excellent agreement with the results of Yang et al.33 and Yang.34 We comment that because 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.33,34 In reality, the data of Yang33,34 have a far more complex structure than that alluded to in Table 1, and the model of Yang34 involves three distinct radii; three-quarters of the atoms have a hexagonal-based radius, one-eighth have a puckered outer radius, and one-eighth have a puckered inner radius. The full structure of the numerical data generated by Yang34 is as shown in Table 2. The table shows three categories corresponding to the hexagonal base radius, the outer radius, and the inner radius, and each category can have three distinct bond lengths σ1, σ2, and σ3. For the hexagonal base, both atoms are situated on the hexagonal base, while for the outer and inner radii, one atom is

situated on the hexagonal base, while the other is either puckered out or in, respectively. For carbon nanotubes, the bonds along the nanotube axis are shorter than the bonds around the nanotube circumference, and the two bond lengths are equal for zigzag (σ2 ) σ3) and armchair (σ1 ) σ2).25-31 Similarly, Table 2 also shows that σ2 ) σ3 for zigzag and σ1 ) σ2 for armchair, and the bonds around the nanotube circumference from the hexagonal base or the outer radius are longer than the bonds along the nanotube axis (σ1 > σ2 ) σ3). Similarly, for the armchair tubes, σ3 < σ1 ) σ2. However, for the inner radius of both zigzag and armchair tubes, the situation is reversed (σ1 < σ2 ) σ3 for zigzag and σ3 > σ1 ) σ2 for armchair). From Figure 8, we comment that for a (4,2) chiral tube, the radius r depends on σ1 more than σ2. This can be explained because n > m, and the leading term of the asymptotic expansion for the radius (eq 19) depends on nσ1 and mσ2, and hence there is a greater dependence on the former. The subtend semiangle ψ for armchair type m ) n depends on all three bond lengths, and therefore the radius also depends



19802


Lee et al.

Figure 10. Radius versus csc(π/2n) for special case of armchair (n ) m) and σ1 ) σ2.

on all bond lengths. The value of the subtend semiangle ψ for the special case of armchair (n ) m) and σ1 ) σ2 is ψ ) π/(2n), and the radius is r ) (4σ12 - σ32)1/2/[4 sin(π/2n)]. For the armchair, the bond lengths σ1 and σ2 have the same the nanotube curvature when they are equal, and therefore when one is longer it makes a greater contribution to the radius, and consequently Figure 9 shows a plane of symmetry. For the special case of armchair (n ) m) and σ1 ) σ2, the radius is given by r ) (4σ12 - σ32)1/2/[4 sin(π/2n)], which increases when the bond length σ3 decreases. Figure 10 shows the radii given in Table 1 and calculated from the first-principles method34 plotted against the cosecant of π/2n, csc(π/2n), which as expected is shown to be a straight line with gradient 0.729, indicating that the bond lengths σ1 and σ3 may take any values such that 4σ12 - σ32 ) 8.503. By examining the asymptotic expansions of the equations for the polyhedral model for chiral angles (eq 15), next nearestneighbor bond angles (eq 17), opposite bond angles (eq 18), radius (eq 19), and unit cell length (eq 21), we observe that in every case the analytical expressions give the conventional

TABLE 3: Main Equations for Rolled-Up Model parameter name chiral vector Ch

conventional translational vector T0 new translational vector T

equation

Ch ) na1 + ma2 T0 ) (n + 2m)a1 /d0R - (2n + m)a2 /d0R T ) [p3(2mσ22 + nτ)]a1 /(σ23dR) - [p3(2nσ21 + mτ)]a2 /(σ23dR)

adjacent bond angle φ1

cos φ1 ) (σ22 + σ23 - σ21)/(2σ2σ3)


cos φ2 ) (σ21 + σ23 - σ22)/(2σ1σ3)


cos φ3 ) (σ21 + σ22 - σ23)/(2σ1σ2)

chiral angle θ01

cos θ01 ) (2nσ21 + mτ)/(2σ1√n2σ21 + m2σ22 + nmτ)

conjugate chiral angle θ02

cos θ02 ) (2mσ22 + nτ)/(2σ2√n2σ21 + m2σ22 + nmτ)


cos θ03 ) [(2nσ12 + mτ) - (2mσ22 + nτ)]/[2σ3√n2σ12 + m2σ22 + nmτ]

nanotube radius r0

r0 ) √n2σ21 + m2σ22 + nmτ/(2π)

unit cell length L0

L0 ) p3√κ(n2σ21 + m2σ22 + nmτ)/(σ23dR)

number of atoms in unit cell N

N ) [2p3(n2σ21 + m2σ22 + nmτ)]/(σ23dR)

parameter τ

τ ) σ21 + σ22 - σ23



TABLE 4: Main Equations for Polyhedral Model parameter name subtend semiangle ψ

chiral angle θ1

equation

[σ22(n + m)2 - σ32n2]sin2 ψ + [σ32m2 - σ12(n + m)2]sin2 ξ + (σ12n2 - σ22m2)sin2(ξ + ψ) ) 0

cos2 θ1 )

[σ22(n + m)2 - σ32n2]sin2 ψ [σ22(n + m)2 - σ32n2]sin2 ψ + σ32m2sin2 ξ - σ22m2sin2(ξ + ψ)

cos2 θ2 )


[σ22(n + m)2 - σ32n2]sin2 ξ σ22[(n + m)2sin2 ξ - n2sin2(ξ + ψ)]


cos φ1 )

σ22 + σ32 - σ12 2σ2σ3


cos φ2 )

σ12 + σ32 - σ22 2σ1σ3


cos φ3 )

σ12 + σ22 - σ32 2σ1σ2

next nearest-neighbor bond angle ω1

cos ω1 )

σ12(2n + m)2sin2 θ1 σ22 + σ32 σ12cos2 θ1sin2(ψ + 2ξ) 2 2σ2σ3 2σ σ sin ψ 2σ σ m2 2 3



cos ω2 )

2 3

σ12(n + 2m)2sin2 θ1 σ12 + σ32 σ12cos2 θ1sin2(2ψ + ξ) 2σ1σ3 2σ1σ3sin2 ψ 2σ1σ3m2

cos ω3 )

σ12(n - m)2sin2 θ1 σ12 + σ22 σ12cos2 θ1sin2(ξ - ψ) 2σ1σ2 2σ σ sin2 ψ 2σ σ m2 1 2

opposite bond angle µ1

cos µ1 ) 1 -


cos µ2 ) 1 -


cos µ3 ) 1 -

(

unit cell length L

σ12 cos2 θ1sin2(2ψ) 2σ12

sin2 ψ

(

σ12 cos2 θ1sin2(2ξ) 2σ22

sin2 ψ

σ12 cos2 θ1sin2[2(ψ + ξ)] 2σ32

sin2 ψ

r)

nanotube radius r

number of atoms in unit cell N

(

1 2

N)

L)

4m2sin2 θ1

+

4n2sin2 θ1

+

4(n + m)2sin2 θ1

m2

m2

σ1cos θ1 2sin ψ

2p3(n2σ12 + m2σ22 + nmτ) σ32dR

2p3(n2σ12 + m2σ22 + nmτ)σ1sin θ1 σ32mdR

)

+

m2

) )

19804


formulas as their highest order term, and the second term is a first-order correction to the conventional model. This demonstrates that the polyhedral model converges to the conventional model for large n. The equations for the subtend semiangle (eq 7), the chiral angles (eq 8), the adjacent bond angles (eq 1), the next nearest-neighbor bond angles (eq 9), the opposite bond angles (eq 10), the radius (eq 11), and the unit cell length (eq 12) all coincide with those given by Lee et al.34 when all of the bond lengths are equal. Finally, we comment that the present polyhedral model for the triangular structure with distinct bond lengths might also be applied to other nanotube materials, which are based on the triangular structure.

Lee et al. where ξ ) [ψ - π/n]/h and h ) m/n. The numbers m and n are assumed to be of the same magnitude, so that h is assumed to be of order one and ψ becomes small as n increases, and therefore eq 22 is expanded in terms of ψ and 1/n where we define the series as

ψ0(h) ψ2(h) ψ1(h) + + ... + 3 n n n5 a1(h) a2(h) cos2 θ1 ) a0(h) + 2 + 4 + ... n n ψ)

6. Conclusion Boron has a very stable lattice structure, which is formed from sp2 hybridized bonds and π-bonds. Here, we extend the nanotube structure for single-walled boron nanotubes with distinct bond lengths. The main geometric parameters for the conventional rolled-up model are shown in Table 3. By formulating the polyhedral model that is based on the two fundamental postulates, which are that all of the bond lengths lying on the same helix are equal and all atomic nuclei are equidistant from a common axis, we have derived equations for the key geometric parameters that arise, the subtend angle 2ψ, chiral angles θ1 and θ2, adjacent bond angles φ1, φ2, and φ3, next nearest-neighbor bond angles ω1, ω2, and ω3, opposite bond angles µ1, µ2, and µ3, radius r, number of atoms in unit cell N, and unit cell length L (see Table 4). The subtend semiangle ψ is the fundamental variable on which all of the other parameters depend, and we find that it is determined from the transcendental eq 7 and cannot be written as a simple analytical function of σ1, σ2, σ3, n, and m. The radius r for zigzag tubes has a linear relationship with the bond length σ1, and it is independent of σ2 and σ3. For armchair tubes, the radius r is dependent on all bond lengths. The polyhedral model converges to the conventional model for large n because the leading term of the analytical expressions gives the conventional formulas as the highest order term, while the second-order term may be viewed as a first-order correction to the conventional model. The polyhedral model with distinct bond lengths includes and extends the polyhedral model with equal bond lengths.32 We comment that, as previously noted in the results section, the puckered model proposed by Yang33,34 is far more complicated than the model formulated here, and the geometric modeling of this novel structure presents many challenges for future work. Acknowledgment. The support of the Australian Research Council, both through the Discovery Project Scheme and for providing an Australian Professorial Fellowship for J.M.H. and an Australian Postdoctoral Fellowship for B.J.C., is gratefully acknowledged. We are also very grateful to Dr. Xiaobao Yang34 for providing the raw numerical data from Yang et al.33 Appendix A: Asymptotic Expansions of Exact Formulas Using a series expansion in powers of 1/n, the root of the subtend semiangle ψ (eq 7) is determined, and we then use this as the basis for determining the series expansions for all of the other parameters. First, eq 7 is written in the form

[σ22(1 + h)2 - σ23] sin2 ψ + [σ23h2 - σ21(1 + h)2] sin2 ξ + (σ21 - σ22h2) sin2(ψ + ξ) ) 0 (22)

For the subtend semiangle ψ, we may derive by the method of asymptotic expansions the following expressions:

ψ0(h) )

π(2σ21 + hτ)

(23)

2(σ21 + hτ + σ22h2)

ψ1(h) ) {π3h(2σ21 + hτ)(2σ22h + τ)[(2σ21 + hτ) (2σ22h + τ)][4σ21(σ22 - σ23) + 4σ22h3(σ21 - σ23) + h(1 + h)(τ2 + 8σ21σ22)]}/{96(σ21 + hτ + σ22h2)5}

(24)

which gives ψ in its asymptotic form (eq 13), by substituting for h in eqs 23 and 24. Second, the equation for cos 2θ1 (eq 8-1) is extended by substituting the asymptotic expansion of ψ. Finally, the expansion coefficients of the chiral angle are found in terms of h to be given by

a0(h) ) a1(h) )

(2σ12 + hτ)2 4σ12(σ12 + hτ + h2σ22) π2h2(2σ12 + hτ)2(2σ22h + τ)2[(2σ12 + hτ) - (2σ22h + τ)]2 64σ12(σ12 + hτ + h2σ22)5

Similarly, the asymptotic equation of the conjugate chiral angle cos 2θ2 (eq 8-2) is expressed as

cos2 θ2 )

(2σ22h + τ)2 4σ22(σ12 + hτ + h2σ22)

+

π2(2σ12 + hτ)2(2σ22h + τ)2[(2σ12 + hτ) - (2σ22h + τ)]2 1 + 64σ22(σ12 + hτ + h2σ22)5 n2 O(1/n4)

Now the next nearest-neighbor bond angles ω1, ω2, and ω3 are found from substituting the asymptotic expansions for ψ and cos θ1 into eq 9, giving

Ideal Polyhedral Model for Boron Nanotubes cos ω1 ) -


(σ22 + σ32 - σ12) + 2σ2σ3

From eqs 25 and 26, and the series expansions 13 and 14 for ψ and C ) cos θ1, we may produce an expansion for the unit cell length L, given in eq 20.

()

π2(2σ22h + τ)2[(2σ12 + hτ) - (2σ22h + τ)]2 1 1 +O 4 8σ2σ3(σ12 + hτ + h2σ22)3 n2 n

References and Notes

(σ12 + σ32 - σ22) + cos ω2 ) 2σ1σ3

()

π2(2σ12 + hτ)2[(2σ12 + hτ) - (2σ22h + τ)]2 1 1 +O 4 8σ1σ3(σ12 + hτ + h2σ22)3 n2 n cos ω3 ) -

(σ12 + σ22 - σ32) + 2σ1σ2

()

π2(2σ12 + hτ)2(2σ22h + τ)2 1 1 +O 4 8σ1σ1(σ12 + hτ + h2σ22)3 n2 n

The series expansion of the opposite bond angles µ1, µ2, and µ3 is found by substituting the series expansion for cos 2θ1 and expanding the asymptotic equation for ψ. As a result, opposite bond angles of the series expansion are given by

cos µ1 ) -1 + cos µ2 ) -1 + cos µ3 ) -1 +

π2(2σ21 + hτ)4 8σ21(σ21 2

+ hτ +

1

h2σ22)3 n2 4

π (2σ22h + τ)

1

() ()

+O

1 n4

+O

1 n4

8σ22(σ21 + hτ + h2σ22)3 n2 π2[(2σ21 + hτ) - (2σ22h + 8σ23(σ21 + hτ + h2σ22)3

τ)]4 1 + n2

()

O

1 n4

Substitution of the series expansion for cos θ1 in the formula for the nanotube radius r (eq 11) is given in a series involving powers of ψ by

r ) σ1C/2ψ + σ1Cψ/12 + O(ψ3)

(25)

where C ) cos θ1, and substitution of the expansions for ψ and C yields eq 19. Finally, the unit cell length L given by eq 12 can be expressed by the following expansion:

L)

p3√κ(σ12 + hτ + h2σ22) σ32dR

n-

p3π2(2σ12 + hτ)2(2σ22h + τ)2[(2σ12 + hτ) - (2σ22h + τ)]2 1 + n 32σ2d (σ2 + hτ + h2σ2)7/2√κ 3 R

1

2

O(1/n3)

(26)

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Ideal Polyhedral Model for Boron Nanotubes with Distinct Bond

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