Chiral tube

Fig. 8. Diffraction space according to the "disordered stacking model" (a) achiral (zigzag) tube (b) chiral tube. The parallel circles represent the inner rims of diffuse coronae, generated by streaked reflexions. The oo.l nodes generate sharp circles. In (a) two symmetry related 10.0 type nodes generate one circle. In the chiral case (b) each node generates a separate corona [9].

Several sections of the diffraction space of a chiral SWCNT (40, 5) are reproduced in Fig. 11. In Fig. 11(a) the normal incidence pattern is shown note the 2mm symmetry. The sections = constant exhibit bright circles having radii corresponding to the maxima of the Bessel functions in Eq.(7). The absence of azimuthal dependence of the intensity is consistent with the point group symmetry of diffraction space, which reflects the symmetry of direct space i.e. the infinite chiral tube as well as the corresponding diffraction space exhibit a rotation axis of infinite multiplicity parallel to the tube axis. 

Three examples of particular structures of SWCNTs, depending on the orientation of the hexagons related to the tube axis, (a) armchair-type tube (0 = 30°), (b) zigzag type tube (0 - 0°), and chiral tube (0 < 0 < 30°). Reprint from Carbon, vol. 33, No. 7, Dresselhaus M.S., Dresselhaus G., Saito R., Physics of carbon nanotubes, pages 883-891, Copyright (1995) with permission from Elsevier. 

Let us recall that nanotubes can be considered as graphene sheets rolled up in different ways. If we consider the so-called chiral vectors c = nai + na2, in which a and a2 are the basis vectors of a 2D graphite lattice, depending on the value of the integers n and m, one can define three families of tubes armchair tubes (n = m), zig-zag tubes (n or m = 0), and chiral tubes (n m 0). Band structure calculations have demonstrated that tubes are either metallic compounds, or zero-gap semiconductors, or semiconductors [6,7]. More commonly, they are divided into metallic tubes (when n-m is a multiple of 3) or semiconducting ones. 

The electrical properties of CNTs depend sensitively on the (n,m) indices and, therefore, on the diameter and chirality [4, 9, 12]. According to the m,n structural parameters values, SWCNTs can be either a metal, semiconductor or small-gap semiconductor [1,4,9,12]. When n=m, the CNTs are metallic. If n - m = 3 x integer, the CNTs present an extremely small band gap and at room temperature they have metallic behavior. For other intermediate values of n - m the behavior is that of a semiconductor with a given band gap [4,9]. This extreme sensitivity of electronic properties on structural parameters is one of the most important aspects of nanotubes that make them very unique. Calculahons have predicted that all the armehair tubes are metallie while the zigzag and chiral tubes are either metallie or semieonductor depending on their diameter and chiral angle [6,13],... 

A template-guided synthesis of water-soluble chiral-conducting PAn in the presence of (S)-(—)- and (R)-(+)-2-pyrrolidone-5-carboxylic acid [(S)-PCA and (R)-PCA] has been reported to produce nanotubes.228 The structures prepared have outer diameters of 80-220 nm with an inner tube diameter of 50-130 nm. It was proposed that the tubular structures form as a result of the hydrophobic aniline being templated by the hydrophilic carboxylic acid groups of the PCA in aqueous media during chiral tube formation. The resultant tubes were shown to be optically active, suggesting that the PAn chains possess a preferred helical screw. 

Fig.21 Structure of biotinylated lipid 35 [165] schematic representation of the helical crystallization of streptavidin on a chiral tubular structure, a Formation of a chiral tubule functionalized with biotin b helical crystallization of streptavidin on the preexisting chiral tubes c secondary binding of biotinylated macromolecules on the remaining binding sites of streptavidin. Right TEM image of a helical array of RNA polymerase on a functionalized ceramide tubule (positive surface charge). The diffraction pattern below, with visible peaks to 1/38 A illustrates the crystalline nature of the helices. Photographs reprinted with permission from [166]. Copyright 1998 National Academy of Sciences USA...

Most theoretical studies of inorganic and carbon nanotubes deal with achiral structures, i.e. zigzag and armchair nanotubes. In terms of their integer denominators these are (n,n) and (w,0) nanotubes. There is, however, a plethora of other tubular structures with denominators (n,m) (and n m) that are chiral. Using periodic-boundary conditions, the calculational unit cells of chiral nanotubes contain a lot more atoms than those of the achiral tubes. Thus, calculations of chiral tubes with the generally used techniques are computationally demanding. 

Figure 10.1 Armchair (n,m = 5,5) (top) zigzag (9,0) (middle) and chiral (10,5) single-wall nanotubes. All armchair tubes are metallic, whereas only 1/3 of the chiral tubes have metallic character. (n,m), the roll-up vectors are proportional to the tube diameter. The dangling bonds at the tube ends are saturated by hemispherical fullerene caps.

Merges R, Deichmann M, Wakita T, Okamoto Y (2003) Synthesis of a chiral tube. Angew Chem Int Ed 42(10) 1170-1172... 

Amphiphilic compounds can form chiral tubes and nano-structures. These systems are rather crystalline than liquid crystalline and will not be discussed here. 

Figure 13.9. Top a graphene sheet with the ideal lattice vectors denoted as ai, 02. The thicker lines show the edge profile of a (6,0) (zig-zag), a (4,4) (armchair), and a (4, 2) (chiral) tube. The tubes are formed by matching the end-points of these profiles. The hexagons that form the basic repeat unit of each tube are shaded, and the thicker arrows indicate the repeat vectors along the axis of the tube and perpendicular to it, when the mbe is unfolded. Bottom perspective views of the (8,4) chiral tube, the (7,0) zig-zag tube and the (7, 7) armchair tube along their axes.

There are three types oftubular structures the first corresponds tom = Oor(n, 0), which are referred to as zig-zag tubes the second corresponds to m =nov(n, n), which are referred to as armchair tubes and the third corresponds to m 7 n or n, m), which are referred to as chiral tubes. Since there are several ways to define the same chiral tube with different sets of indices, we will adopt the convention that m < n, which produces a unique identification for every tube. Examples of the three types of tubes and the corresponding vectors along the tube axis and perpendicular to it are shown in Fig. 13.9. The first two types of tubes are quite simple and correspond to regular cylindrical shapes with small basic repeat units. The third type is more elaborate because the hexagons on the surface of the cylinder form a helical structure. This is the reason why the basic repeat units are larger for these tubes. 

If however the interaction of the chiral tube and the achiral iodine chain is dissymmetrical, then an induced Cotton effect should be present here too. We have in fact been able to show that an aqueous iodine-starch solution has a strong Cotton effect around the absorption maximum at 600 nm [49]. 

Figure 20.9 (a) Schematic honeycomb structure of a graphene sheet. Single-walled carbon nanotnbes can be formed by folding the sheet along lattice vectors. The two basis vectors , and U2 are shown. Folding of the (8,8), (8,0), and (10,-2) vectors lead to (b) armchair, (c) zigzag, and (d) chiral tubes. Dai [51]. Reproduced with permission of American Chemical Society. 

Merges and Okamoto have also reported the synthesis of a three-dimensional macrocydic structure [34]. The chiral tube 81 was achieved in 15% yield and the two enantiomers were separated using chiral reverse-phase HPLC. CD spectroscopic measurements were acquired for both enantiomers, confirming the chiral separation as well as the structure assignment. 

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