Much research has been devoted to the study of the equilibrium structure of carbon nanotubes. Currently, some information is still begin disputed, but I have compiled recent data on the most basic and neccessary aspects of single-walled carbon nanotubes (SWNT).
Simply put, carbon nanotubes exist as a macro-molecule of carbon, analagous to a sheet of graphite (the pure, brittle form of cabon in your pencil lead) rolled into a cylinder. Graphite looks like a sheet of chicken wire, a tessellation of hexagonal rings of carbon. Sheets of graphite in your pencil lay stacked on top on one another, but they slide past each other and can be separated easily, which is how it is used for writing. However, when coiled, the carbon arrangement becomes very strong. In fact, nanotubes have been known to be up to one hundred times as strong as steel and almost two millimeters long! [17] These nanotubes have a hemispherical "cap" at each end of the cylinder. They are light, flexible, thermally stabile, and are chemically inert. They have the ability to be either metallic or semi-conducting depending on the "twist" of the tube.
Nanotubes form different types, which can be described by
the chiral vector (n, m), where
n and m are integers of the vector
equation R = na_{1} + ma_{2} .
The chiral vector is determined by the diagram at the left.
Imagine that the nanotube is unraveled into a planar sheet. Draw two lines (the blue lines)
along the tube axis where the separation takes place. In other words, if you cut along the
two blue lines and then match their ends together in a cylinder, you get the nanotube that
you started with. Now, find any point on one of the blue lines that intersects one of the
carbon atoms (point A). Next, draw the Armchair line (the thin yellow line), which travels
across each hexagon, separating them into two equal halves. Now that you have the armchair
line drawn, find a point along the other tube axis that intersects a carbon atom nearest
to the Armchair line (point B). Now connect A and B with our chiral vector, R (red arrow).
The wrapping angle ; (not shown) is formed between R and the Armchair line.
If R lies along the Armchair line (=0°), then it is called an
"Armchair" nanotube. If =30°, then the tube is of the "zigzag" type.
Otherwise, if 0°<<30° then it is a "chiral" tube. The vector
a_{1}
The values of n and m determine the chirality, or "twist" of the nanotube. The chirality in turn affects the conductance of the nanotube, it's density, it's lattice structure, and other properties. A SWNT is considered metallic if the value n - m is divisible by three. Otherwise, the nanotube is semiconducting. Consequently, when tubes are formed with random values of n and m, we would expect that two-thirds of nanotubes would be semi-conducting, while the other third would be metallic, which happens to be the case. [23]
Given the chiral vector (n,m), the diameter of a carbon nanotube can be determined using the relationship
d = (n^{2} + m^{2} + nm)^{1/2} 0.0783 nm
The average diameter of a SWNT is 1.2 nm. [1]
However, nanotubes can vary in size, and they aren't always perfectly
cylindrical.
The larger nanotubes, such as a (20, 20) tube, tend to bend under their own weight.
[12] The diagram at right shows the average bond length
and carbon separation values for the hexagonal lattice. The carbon bond length of 1.42 Å was measured
by Spires and Brown in 1996 [1] and later confirmed by Wilder
et al. in 1998. [23] The C-C tight bonding overlap energy is in the
order of
2.5 eV. Wilder et al. estimated it to be between 2.6 eV - 2.8 eV [23]
while at the same time, Odom et al. estimated it to be 2.45 eV
[24]
A (10, 10) Armchair tube was found to have C_{5V} symmetry [2]
which has the following character table:
E | 2C_{5} | 2C_{5}^{2} | 5_{v} | |||
A_{1} | 1 | 1 | 1 | 1 | z | x^{2} + y^{2}, z^{2} |
A_{2} | 1 | 1 | 1 | -1 | R_{z} | |
E_{1} | 2 | 2 cos 72° | 2 cos 144° | 0 | (x, y)(R_{x}, R_{y}) | (xz, yz) |
E_{2} | 2 | 2 cos 144° | 2 cos 72° | 0 | (x^{2} - y^{2}, xy) | |
Further Information: P. W Atkins, M. S. Child, and C. S. G. Phillips, Tables for group theory. Oxford University Press (1970). |
In 1996, Thess et al. measured the properties of "ropes" of carbon nanotubes. [2] As shown in the diagram at right, ropes are bundles of tubes packed together in an orderly manner. They found that the individual SWNTs packed into a close-packed triangular lattice with a lattice constant of about 17 Å. This was later confirmed by Gao, Cagin, and Goddard in 1997. [3] In addition, they concluded that the density, lattice parameter, and interlayer spacing of the ropes was dependent on the chirality of the tubes in the mat. (10, 10) Armchair tubes had a lattice parameter of 16.78 Å and had a density of 1.33 g/cm^{3}. Zigzag tubes of the chirality (17, 0) had a lattice parameter of 16.52 Å and a density of 1.34 g/cm^{3}. Mats made of (12, 6) chiral SWNT's had a lattice parameter of 16.52 Å and a density of 1.40 g/cm^{3}. The space between the tubes was also dependent on chirality. Armchair tubes had a spacing of 3.38 Å, zizzag tubes had a spacing of 3.41 Å, and (2n, n) chiral tubes had a interlayer spacing value of 3.39 Å. Compare these values to the spacing between the layers of graphite sheets, and the spacing between the variant walls of a MWNT, both about 3.4 Å. [13]
As a good estimate, the lattice parameter in CNT ropes (bundled nanotubes) is d + 0.34 nm, where d is the tube diameter given above.