Relative stability and local curvature analysis in carbon nanotori

We introduce a concise formalism to characterize nanometer-sized tori based on carbon nanotubes and to determine their stability by combining ab initio density functional calculations with a continuum elasticity theory approach that requires only shape information. We find that the high strain energy in nanotori containing only hexagonal rings is significantly reduced in nanotori containing also other polygons. Our approach allows to determine local curvature and link it to local strain energy, which is correlated with local stability and chemical reactivity.

Figure 1

Atomic structure of (a) a polyhex nanotorus with only hexagons and (b) a polygonal nanotorus containing nonhexagonal rings. (c) A model of a perfect torus with major radius $R$ and minor radius $r$. $\theta$ is the longitudinal angle and $\varphi$ is the azimuthal angle. The yellow shading in (a) and (b) indicates specific polygons in the nanotori.

Figure 2

(a) Schematic of a striped nanotube (left), which is bent to a nanotorus (right). (b) Contour plot representing the in-plane strain energy $\Delta E_s(R,r)$ according to Eq. (9).

Figure 3

Definition of the principal radius $R_2$ of a nanotorus in (a) its interior and (b) its exterior part as a function of the zenith angle $\theta$. (c) Contour plot displaying the out-of-plane curvature energy $\Delta E_c(R,r)$ of nanotori according to Eq. (11).

Figure 4

Strain energy $\Delta E$ in different polygonal nanotori. (a) Energy differences per atom $\Delta E_{\mathrm{tot}}^{\mathrm{DFT}}$ based on DFT and $\Delta E_{\mathrm{tot}}^{\mathrm{Keating}}$ based on the Keating potential are presented next to curvature energies $\Delta E_c^{\mathrm{Keating}}$ based on Eq. (2) for Keating-optimized geometries, described in more detail in the text. (b) Comparison between the analytical expression in Eq. (11) for the curvature energy $\Delta E_c^A$ based on the elastic description of a parametric torus and the curvature energy $\Delta E_c^{\mathrm{Keating}}$ obtained using Eq. (2) for Keating-optimized discrete torus geometries.

Figure 5

Local Gaussian curvature $G$(left panels) and local curvature energy $\Delta E_c/A$ (right panels) across the surface of selected nanotori. Representative examples shown are (a) the flattened ${\mathrm{C}}_{320}$ nanotorus with 320 atoms and (b) the elongated ${\mathrm{C}}_{280}$ nanotorus with 280 atoms. $G$ and $\Delta E_c/A$ are interpolated from their values at the atomic sites.

Figure 6

(a) A schematic model of a polygonal nanotorus with $n$-fold symmetry $(n=6$ is shown here), consisting of $n$ nanotube segments that are connected by $n$ nanotube elbow joints. The nanotube segments are characterized by the width $W$, length $L$, and height $H$. (b)–(d) Curvature energies per atom based on Eq. (2) for Keating-optimized nanotori of different shapes. $\Delta E_c$ is presented for nanotori with (b) $n,W,H=\mathrm{const}.$, but nanotube length $L$ changing, (c) $n,L,W=\mathrm{const}.$, but nanotube height $H$ changing, and (d) $L,W,H=\mathrm{const}.$, but rotational symmetry number $n$ changing. The insets show nanotorus models with different values of $L,H$ and $n$. The lines in (b) and (c) are analytical extrapolations discussed in the text. Two sets of data points in (d), connected by lines to guide the eye, correspond to two structural families of nanotori, described in the text.

Figure 7

(a) Structural models, (b) local Gaussian curvature $G$, and (c) local curvature energy $\Delta E_c/A$ across the surface of torus isomers with different lengths of the nanotube segments $L$, defined in Fig. 6. The nonhexagonal rings in (a) are shaded. The values of $G$ and $\Delta E_c/A$ have been interpolated from their values at the atomic sites.

Figure 8

(a) Structural models, (b) local Gaussian curvature $G$, and (c) local curvature energy $\Delta E_c/A$ across the surface of torus isomers with different heights $H$, defined in Fig. 6. The nonhexagonal rings in (a) are shaded. The values of $G$ and $\Delta E_c/A$ have been interpolated from their values at the atomic sites.

Figure 9

[(a) and (d)] Structural models, [(b) and (e)] local Gaussian curvature $G$, and [(c) and (f)] local curvature energy $\Delta E_c/A$ across the surface of torus isomers with different rotational symmetry numbers $n$, defined in Fig. 6. Structures in [(a) and (c)] and [(d) and (f)] represent two distinct torus families. The nonhexagonal rings in (a) and (d) are shaded. The values of $G$ and $\Delta E_c/A$ have been interpolated from their values at the atomic sites.

Figure 10

(a) A schematic model of a polygonal nanotorus with an $n$-fold $(n=6$ in this case) symmetry. (b) Distribution of the optimal rotational symmetry numbers $n_{\mathrm{opt}}$ in the 43 families of polygonal nanotori covered in this study.