Radial deformation and stability of single-wall carbon nanotubes under hydrostatic pressure

Abstract

In this paper we have developed a theory of energetics for isolated single-wall carbon nanotubes ͑SWNTs͒ deformed in the radial direction, and applied this theory to investigate their deformation characteristics and stability under hydrostatic pressure. The starting point of the theory is the strain energy of SWNTs predicted by ab initio calculations based on the density functional theory ͑DFT͒, which shows the same behavior as that obtained for the continuum elastic shell model. We extend this result for inflated SWNTs with circular cross section to calculate the deformation energy of a deformed SWNT without performing further DFT calculations. This extension is then complemented by a van der Waals interaction, which is not fully taken into account in the DFT approximations currently in use but becomes important in highly deformed tubes. We find that the minimum pressure, P 1 , for the radial deformation to occur is proportional to the inverse cube of tube diameter, D, in agreement with the recent theoretical predictions as well as the classical theory of buckling. The radial deformation of SWNTs with D Ͻ 2.5 nm is found to be elastic up to very high pressure and they hardly collapse. On the other hand, SWNTs with D Ͼ 2.5 nm show a plastic deformation and collapse if the applied hydrostatic pressure exceeds a critical value, which is about 30-40% higher than P 1 and also varies as D −3 though approximately. These SWNTs with large D collapse when the cross-sectional area is about 60% reduced with respect to the circular one. It is also found that for SWNTs with D Ͼ 7.0 nm, the plastically deformed ͑collapsed͒ state is more stable than the inflated one. This critical value of D is somewhat larger than previously predicted.

Energetics of interlayer binding in graphite: The semiempirical approach revisited

“…In fact, using the atom-atom interactions, DFT ͑r͒ and vdW ͑r͒, for graphite obtained by our method, we have already developed a simple theory for the radial deformation of single-walled carbon nanotubes. 57 These results can also be directly applicable to the calculations of the wall-wall interaction in multiwalled carbon nanotubes and the interaction between nanotubes in bundles.…”

“…͑2͔͒ and the 2 3 4 5 6 7 8 2 3 4 5 6 7 8 2 3 4 5 6 7 8 2 3 4 5 6 corresponding total C-C interactions, ͑r͒ = DFT ͑r͒ + vdW ͑r͒. These results provide a useful means to investigate the energetics of the radial deformation of carbon nanotubes, 57 where DFT ͑r͒ and vdW ͑r͒ are conveniently treated separately. These results for ͑r͒ may also be directly applicable to the calculations of the wall-wall interaction in multiwall carbon nanotubes and the interaction between nanotubes in bundles.…”

“…We also note that the continuum model used in the present approach makes the implementation of our semiempirical method quite simple without sacrificing quantitative accuracy. This model is much more useful in applications of the present results to other graphitic systems, 57 for which the discreteatom model brings about formidable complexities in implementations with little quantitative advantage.…”

“…This point is important in some cases as it is in the application to the radial deformation of carbon nanotubes. 57 In such applications to other graphitic systems, it is also useful to parametrize DFT ͑r͒, and we found that it was well represented by a Morse potential, 24,57…”

“…In this treatment, we employ as before 24 the continuum model conveniently used in such applications. 57 Then, the interaction energy between an atom in one layer at z = 0 and all the atoms in another layer at z = ld ͑l = ±1, ±2, ...͒ is given by…”