Nanotube chiral

The number of hexagons, N, per unit cell of a chiral nanotube is specified by the integers (n, m) and is given by... 

In general, the number of phonon branches for a carbon nanotube is very large, since every nanotube has 6N vibrational degrees of freedom. The symmetry types of the phonon branches for a general chiral nanotube are obtained using a standard group theoretical analysis [194]... 

Key Words—Single-wall, multi-wall, vibrational inodes, chiral nanotubes, electronic bands, tubule arrays. 

Fig. 1. The 2D graphene sheet is shown along with the vector which specifies the chiral nanotube. The chiral vector OA or Cf, = nOf + tnoi defined on the honeycomb lattice by unit vectors a, and 02 and the chiral angle 6 is defined with respect to the zigzag axis. Along the zigzag axis 6 = 0°. Also shown are the lattice vector OB = T of the ID tubule unit cell, and the rotation angle 4/ and the translation r which constitute the basic symmetry operation R = (i/ r). The diagram is constructed for n,m) = (4,2).

Fig. 3. The 2D graphene sheet is shown along with the vector which specifies the chiral nanotube. The pairs of integers ( , ) in the figure specify chiral vectors Cy, (see Table I) for carbon nanotubes, including zigzag, armchair, and chiral tubules. Below each pair of integers (n,m) is listed the number of distinct caps that can be joined continuously to the cylindrical carbon tubule denoted by (n,wi)[6]. The circled dots denote metallic tubules and the small dots are for semiconducting tubules.

Fig. 8. LDF valence band structure of [9,2] chiral nanotube. The Fermi level lies at midgap at -3.3 eV. Dimensionless wavenumber coordinate k ranges from 0 to t.

For the case = 1, the symmetry group of a chiral nanotube specified by (n,m) is a cyclic group of order N given by... 

Equations (13-15) completely determine the character table of the symmetry group Q for a chiral nanotube. 

Similarly, it can be shown that the nanotube modes at the T-point obtained from the zone-folding eqn by setting Ai = 1), where 0 < ri < N/2, transform according to the , irreducible representation of the symmetry group e. Thus, the vibrational modes at the T-point of a chiral nanotube can be decomposed according to the following eqn... 

Fig. 10.7 Chirality vector and folding scheme for semiconducting and metallic nanotube (a). Zig-zag, armchair, and chiral nanotubes by rolling-up of the graphite lattice (b) (Reprinted from Terrones 2003. With permission from Annual Reviews)... 

There are three general types of CNT structure (Figure 12.10). The zigzag nanotubes correspond to ( ,0) or (0,m) and have a chiral angle of 0°. The carbon-carbon position is parallel to the tube axis. Armchair nanotubes have (n,n) with a chiral angle of 30°. The carbon-carbon positions are perpendicular to the tube axis. Chiral nanotubes have general (n,m) values and a chiral angle of between 0° and 30°, and as the name implies, they are chiral. 

In that nomenclature system, the center of a hexagon is chosen as the origin (0,0) and then it is superimposed with the center m,n) of another hexagon to form the nanotube. There are three types of carbon nanotubes. If the graphene sheet is rolled in the direction of the axis, it will produce either an armchair nanotube m = ) or a zig-zag nanotube m = 0). On the other hand, if the graphene sheet is rolled in any other m,n) direction it will produce a chiral nanotube and the chirality will depend on whether the sheet is rolled upwards or backwards. 

Nanotubes fall into three groups, depending on the chiral angle, 6, between the < 2110 > direction of the hexagons and the tube axis (Figure 17.9). If 0 = 0, a zigzag nanotube results. If 0 = 30°, the nanotube is called an armchair. Chiral nanotubes are those for which 0 < 9 < 30°. These develop twists. Nanotubes can have metallic conduction others are semiconductors or insulators. 

While functionalized chiral nanotubes and chiral fullerenes may provide mo-lecularly well-defined materials in the future, their specialized chemistry is not covered in this overview. 

It is also possible to prepare chiral PANI by in situ polymerisation with CSA, and in this case the reaction can afford chiral nanotubes [63]. The optically active materials contain nanotubes with 80 to 200 nm outer diameter and an internal diameter of between 20 and 40 nm, as revealed through microscopy images. A self-assembly process was proposed in which anilinium cations and CSA anions form micelles which act as templates for the growing polymer chains. Nanotubes are also formed when (R)- or (S)-2-pyrrolidone-... 

The SWNT systems chosen in the present studies include 3 armchair nanotubes and 3 zigzag nanotubes with diameters ranging from 4 A to 12 A, and 1 chiral nanotube with a diameter of 8.28 A. The nanotubes were carefully chosen to address the fundamental issues of curvature and chirality and the effect of each on the adsorption capacity. First, to understand the curvature effect on hydrogen uptake, we selected nanotubes with diameters varying from about 4 A to 12 A. Next, to investigate the effect of nanotube chirality, we intentionally chose the nanotubes of different chiral architectures with similar diameters. Finally, to study the capacity of a given nanotube, we included three different H2 loadings at 0.4 wt. %, 3.0 wt. % and 6.5 wt. %, respectively, in our MD simulations. 

Fig. 2 Schematic representation of the folding of a graphene sheet into (a) zigzag, (b) armchair and (c) chiral nanotubes. 

The indices (n, 0) or (0, m), d = 0°, correspond to the zigzag tube (so-called because of the AA/ shape around the circumference, perpendicular to the tube axis). The indices (n, m) with n = m (6 = 30°) corresponds to the armchair tube (with a / / shape around the circumference, perpendicular to the tube axis). If one of the two indices n or m is zero, the tube is nonchiral it is superimposable on its mirror image. A general chiral nanotube (nonsuperimposable on its mirror image) occurs for all other arbitrary angles. Common nanotubes are the armchair (5, 5) and the zigzag (9, 0). 

Single-walled carbon nanotubes comprise rolled sheets of sp graphene carbon, which form well-defined cylinders with diameters in the range 1 to 2 run. The diameter depends on the synthesis conditions as does the orientation of the six-membered carbon rings with respect to the nanotube axis. Achiral zigzag, achiral armchair, or chiral nanotubes can be obtained as illustrated in Figure 16. ... 

Figure 4 (A) Atomically resolved scanning tunneling microscopy (STM) of a SWNT in the surface of a rope revealing chiral-twist. (Reprinted with permission from Ref 15. 1998 Macmillan Magazines Ltd (www.nature.com).) (B) STM images of SWNTs produced by arc-discharge method (a) chiral nanotube with angle 7°, (b) zigzag nanotube, and (c) armchair nanotube. The tube axis is shown with dashed arrows. (Reprinted with permission Ref. 16. 1998 Macmillan Magazines Ltd)...

Fig. 8.7. Models of (a) armchair, (b) zigzag, and (c) chiral nanotubes. Reproduced from ref [17], with permission.

If the angle of turning the graphene layer before rolling up is between 0° and 30°, chiral nanotubes are obtained. They are characterized by a Une in parallel with the unity vector Oj that spirals up around the tube. Consequently two enantiomeric forms exist for these species. 

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