Article Contents
Article Contents

Carbon-nanotube geometries: Analytical and numerical results

• We investigate carbon-nanotubes under the perspective ofgeometry optimization. Nanotube geometries are assumed to correspondto atomic configurations whichlocally minimize Tersoff-type interactionenergies. In the specific cases of so-called zigzag and armchairtopologies, candidate optimal configurations are analytically identifiedand their local minimality is numerically checked. Inparticular, these optimal configurations do not correspond neither tothe classical Rolled-up model [5] nor to themore recent polyhedral model [3]. Eventually, theelastic response of the structure under uniaxial testing is numericallyinvestigated and the validity of the Cauchy-Born rule is confirmed.

Mathematics Subject Classification: Primary: 82D25.

 Citation:

• Figure 1.  Rolling-up of nanotubes from a graphene sheet

Figure 2.  Notation for bonds and bond angles

Figure 3.  Zigzag nanotube

Figure 4.  The construction of the function $\beta_z$

Figure 5.  The angle $\beta_z$ as a function of the angle $\alpha$ (above) and a zoom (below) with the points $(\alpha^{\rm ru}_z,\beta_z(\alpha^{\rm ru}_z))$ and $(\alpha^{\rm ch}_z,\beta_z(\alpha^{\rm ch}_z))$ for $\ell=10$

Figure 6.  The angle $\beta_a$ as a function of the angle $\alpha$ (above) and a zoom (below) with the points $(\alpha^{\rm ru}_a,\beta_a(\alpha^{\rm ru}_a))$ and $(\alpha^{\rm ch}_a,\beta_a(\alpha^{\rm ch}_a))$ for $\ell=10$

Figure 7.  The energy-per-particle $\widehat E_i$ in the zigzag (above) and in the armchair (below) family, as a function of the angle $\alpha$ for $\ell=10$, together with a zoom about the minimum

Figure 8.  Comparison between energies of the optimal configurations and energies of their perturbations in the cases Z1, Z2, Z3 (left, from the top) and A1, A2, A3 (right, from the top). The marker corresponds to the optimal configuration $\mathcal{F}_i^*$ and value $\alpha$ represents the mean of all $\alpha$-angles in the configuration

Figure 9.  Optimality of the configuration $(F^*_L,L)\in \mathscr{F}_z$ (bottom point) for all given $L$ in a neighborhood of $L^*$

Figure 10.  Elastic response of the nanotube Z1 under uniaxial small (left) and large displacements (right). The function $L \mapsto E(F_L^*,L)$ (bottom) corresponds to the lower envelope of the random evaluations (top)

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