Carbon-nanotube geometries: Analytical and numerical results

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)