## 12/17/10

## Nanotube conformations

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Two classes of nanotube conformations are (i) achiral and (ii) zigzag. Achiral have their structure completely symmetric, and the carbon hexagons do not spiral about the tube axis, but lie in lines exactly parallel to the axis. The above two highly-symmetric conformations are the (n,0) and the (n,n) tubes. The (n, 0) tubes are often called "zigzag" tubes since, if the tube is cut so that the open end has carbon atoms in a regular zigzag pattern. Likewise the (n,n) tubes has the carbon atoms at the cut end are in a pattern with two up, two down, and then two up again, which is reminiscent of the seat and arms of an armchair. The minimum size for the graphene sheet cylinder for the production of nanotubes is believed to be a (4, 4) tube of an approximate diameter of 0.5 nm.

**How many different conformations?**

Consider a maximum tube diameter of 3.5 nm. Since isolated one-wall carbon nanotubes tend to collapse above this diameter, a 3.5 nm diameter (n,n) tube would be a (26,26) tube. So we need to count the tubes with n and m in the range between 4 and 26, inclusive, where n and m can take on any values within that range. To find the number of different ordered pairs of two numbers between 0 and 26, we find the number of permutations of 27 objects taken two at a time. This results in 27! / (27-2)! ordered pairs, where y! is the factorial of y. Hence the value of 27!/25! = 27 x 26 = 702. Since we understand that tubes for which both n and m are less than 4 do not exist, we deduct the number of permutations of 4 objects taken 2 at a time, which is 4!/2! = 12. Thus a possible number of relevant conformations for single-wall carbon nanotubes works out to 690 (=702-12). Now we deduct the number of (0,n) conformations since they are indistinguishable from the (n,0) conformations, that is deduct 23 (=27-4) from 690 giving 667 different conformations of carbon nanotubes smaller than 3.5 nanometers in diameter.

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