Helicity in Ropes of Chiral Nanotubes: Calculations and Observation

Even though isolated defect-free single-wall carbon nanotubes are straight, bundles of chiral single-wall carbon nanotubes are often helical according to our observations using high-resolution electron microscopy. The driving force for the formation of such helices is the energy gain associated with the optimum orientational alignment of neighboring nanotubes. Our total energy calculations allow us to analyze the torsional and bending stress components in helical nanotube ropes and specify under which conditions straight nanotube bundles gain energy upon forming a helix.

Figure 1

High-resolution transmission electron micrograph of a free-standing rope containing (a) two and (b) seven SWCNTs. The local rope orientation changes along the helix with respect to the electron beam direction, indicated by parallel dotted lines in the insets. Only two dark lines corresponding to the projections of SWCNT walls along the incident electron beams should occur in a two-tube rope for orientations labeled $A_1$ and $A_2$ in (a), which is shown also in Fig. 2a, separated by half the pitch length. Only four such lines are observed in the seven-tube rope in (b) for rope orientations $A_1-A_6$, separated by $\lambda /6$.

Figure 2

Structure of carbon nanotubes and their bundles. (a) A pair of coiled nanotubes as the simplest example of a helical rope. Schematic views of (b) an individual nanotube helix, (c) chiral and achiral nanotubes, with emphasis on the pitch angle $\chi$ associated with the direction of lines of hexagons along the tube, shown by the white solid line, and (d) the optimum entanglement of two chiral nanotubes. The labels in (a) refer to Fig. 1a. The pitch length $\lambda$ and the radius $\rho$ of an individual helix, formed of a nanotube of radius $R$, are shown in (b).

Figure 3

Energies associated with the formation of helical nanotube bundles. (a) Results of Ref. 2 for the interaction energy profile per atom between two parallel (10, 10) carbon nanotubes at equilibrium distance. The equipotential lines are separated by $0.1\mathrm{meV}/\mathrm{atom}$ and the tube orientations ${\phi }_1$ and ${\phi }_2$ are defined in the schematic sketch. (b) Bending (◯) and torsional (□) force constants as a function of the nanotube radius $R$. (c) Coiling energy in helices of constant slope $\mu$ as a function of $\rho$. The data points for a (10, 10) nanotube in (c) are compared to analytical data based on Eq. (1).

Figure 4

Energy considerations for the formation of helical ropes. (a) Coiling energy $E_{\mathrm{coil}}$ per nanotube length $s$ of two identical nanotubes in the case of ideal orientational alignment along the helix, shown by the solid lines. Presence of the intertube interaction $E_i$ stabilizes the helix, as indicated by the dotted lines. The energy reference is a pair of noninteracting (10, 10) nanotubes. (b) Schematic phase diagram, indicating conditions, under which two nanotubes with pitch angles ${\chi }_1$ and ${\chi }_2$ should form a helix. $⟨\chi ⟩=({\chi }_1+{\chi }_2)/2$ is the average pitch angle and $\Delta \chi =|{\chi }_1-{\chi }_2|$ is the pitch angle difference.