The most stable structure [25
] for a single C SI in a (9,0)@(18,0) DWCNT is an inter-tube bridge. Figure shows the energy variations that accompany the successive transformations which enable this defect to migrate between neighboring sites. The results were obtained by the elastic band method [29
], an approach that typically provides [30
] barriers in satisfactory agreement with experiments. The method simulates the so-called minimum-energy pathway of a process with a sequence of intermediate configurations termed images. The curve of Figure includes three local-energy minima, related to structures that play a role in the diffusion of SI through sequential switches of its bonds with the inner and outer shells. The effective activation energy for migration is about 1.6 eV.
Energy variation during consecutive migration steps of a C SI in a (9,0)@(18,0) carbon nanotube. The arrows show the rate-limiting step that determines a diffusion barrier of about 1.6 eV.
Given that SI defects tend to agglomerate on graphene and SWCNT [4
], it is natural to probe whether similar trends appear for defects in DWCNTs. Indeed, pairs of SIs in a (9,0)@(18,0) DWCNT have significant binding energy against dissociation to individual defects. The most stable pair structure is shown in Figure . It has a binding energy of 5.4 eV, and it resembles the hillock SI defect found on graphene and SWCNTs [4
]. A large binding energy and a SI diffusion barrier of 1.6 eV suggest that the stable hillock is formed under annealing at moderate temperature above room conditions.
Pairs of carbon interstitials in a (9,0)@(18,0) double-wall carbon nanotube. (a) Inter-tube bridge (arrows point to C interstitials), (b) hillock on the inner (9,0) tube (the atoms and bonds of the outer shell are shown in a wire frame).
Compared to the configuration of Figure , the energy of a hillock structures on the outer (18,0) shell is higher by 1.35 eV, while the formation of the double inter-tube bridge of Figure increases the energy by more than 1.9 eV. Clustering changes thus the character of the defect favoring the elimination of inter-shell links. As shown in Figure , the energy variation during sliding of the inner shell is smooth in the absence of inter-tube bridges, without the cusps that are characteristic [25
] to inter-tube shift in the case of individual SI's. The presence of the hillock reduces the amplitude of the variation (also called corrugation) from 0.77 eV of the pristine case to about 0.53 eV for the case of the hillock.
Energy variation during inter-tube sliding of a (9,0)@(18,0) carbon nanotube with no defects (squares-solid line), with a hillock SI pair (diamonds-dashed line) or a di-vacancy (triangles-dotted line) on the inner (9,0) shell.
As for SIs, clustering is energetically favorable also for vacancies. The lowest-energy structure is the di-vacancy on the inner (9,0) shell. The energy of a di-vacancy on the outer (18,0) tube is 0.8 eV higher, while the vacancy-induced inter-tube bridge formation increases the energy by more than 5.5 eV. The results of Figure show that the presence of the di-vacancy affects the response of the (9,0)@(18,0) DWCNT under inter-tube sliding in a similar way as SI pairs. In particular, the formation of the di-vacancy does not introduce any hysteretic effects, but limits corrugation by about 0.24 eV.
We now turn our attention to the stability of point defects in the arm-chair (6,6)@(11,11) DWCNT. Figure depicts several SI configurations, in particular, two structures of C adatoms on the outer and inner shells, and two geometries with inter-tube SI bridges. The most stable configuration is the one depicted in Figure . If we set the energy of this structure equal to zero, then the energies of the geometries of Figure are higher by 1.15, 0.64, and 0.60 eV, respectively.
Carbon SI in a (6,6)@(11,11) carbon nanotube: (a, d) C adatoms on the outer and inner shells, respectively, (b, c) inter-tube bridges.
When the inner shell is pulled with respect to the outer tube, the SI bridge is moving in the direction of sliding and the energy initially increases, as shown in the corresponding diagram of Figure . At a certain displacement, however, one of the bonds between the SI and the inner tube switches to a neighboring site and another cycle of stretching commences. The end result is that the SI stays roughly at the same spot during repeated cycles, despite the relative movement of the DWCNT tubes. When sliding materializes in the opposite direction, a different stretching sequence is traced, shown as open squares in Figure . This difference in paths gives rise to hysteresis during inter-tube sliding, highlighted by an arrow in Figure . Compared to the SI-related energy variation, the corresponding changes for a pristine (6,6)@(11,11) DWCNT or one with a single vacancy on the inner tube are negligible.
Figure 5 Energy variation during inter-tube sliding of a (6,6)@(11,11) carbon nanotube with a C SI (filled and open squares for sliding in opposite directions). The lines with almost vanishing values (filled circles) at the bottom are for the pristine case of (more ...)
Figure shows the variation of energy when the inner shell is rotated in small angles. As in the case of sliding, the first stretching phase is suddenly interrupted through an abrupt transformation to a neighboring inter-tube configuration. Unlike sliding, however, repeated stretching cycles during rotation force the SI to move along with the inner shell, while its bonds to the outer tube switch sequentially to neighboring sites. The abrupt changes depicted in Figure can give rise to hysteresis when rotation direction is reversed. This effect is shown as open squares in the figure.
Figure 6 Energy variation during inter-tube rotation of a (6,6)@(11,11) carbon nanotube with a C SI (filled and open squares for rotation in opposite angles). The lines with almost vanishing values (filled circles) at the bottom are for the pristine case of no (more ...)