Twisting graphene nanoribbons into carbon nanotubes

Although carbon nanotubes consist of honeycomb carbon, they have never been fabricated from graphene directly. Here, it is shown by quantum molecular-dynamics simulations and classical continuum-elasticity modeling, that graphene nanoribbons can, indeed, be transformed into carbon nanotubes by means of twisting. The chiralities of the tubes thus fabricated can be not only predicted but also externally controlled. This twisting route is an opportunity for nanofabrication, and is easily generalizable to ribbons made of other planar nanomaterials.

Figure 1

Quantum molecular-dynamics simulation of GNR transformation into a CNT. (a) The simulation of a 12-ZGNR with snapshots at different twist parameters $\tau W$ (for GNR notations, see Appendix A). The red atoms highlight one row of atoms across the ribbon, and their cross-section projections are shown below; the cross denotes the twist axis. The rightmost tube is the final, fully optimized $(9,3)$ CNT. (b) The tubularity parameter $\Omega$ that shows the stages of flat ribbon, buckled ribbon, and CNT. Fluctuations in $\Omega$ arise from nonzero temperature. (c) Potential energy per unit length as a function of twist. The final curve segments in panels (b) and (c) (the green ones) represent the subsequent full structural optimization of the resulting CNT.

Figure 2

Ribbon buckling and tube formation can be demonstrated with a strap of a backpack. Inset: Proposal for an experimental realization, adapting the successful scheme of Refs. 9 and 10.

Figure 3

Tube formation by elasticity theory. (a) Visualizations of twisted elastic ribbons for $\tilde{h}=0.01$ (width $\approx 24$ Å in graphene) and $\tau W=0$, $0.44$, $0.88$, $1.32$, and $1.65$. Cross-section projections are also shown. (b) Tubularity parameter $\Omega$ by elasticity theory ($\tilde{h}=0.01$) and from simulation results for a zigzag (12-ZGNR, $\tilde{h}\approx 0.0104$) and an armchair ribbon (22-AGNR, $\tilde{h}\approx 0.0097$). (c) Buckling (${\tau }_b$) and tube-formation points (${\tau }_t$) as a function of elastic thickness, which results from elasticity theory and from simulations for different ZGNRs. For the rightmost ribbon, 10-ZGNR, the simulation was repeated 10 times, and the vertical bars denote the range where ${\tau }_b$ and ${\tau }_t$ were observed to fluctuate. (d) Buckling and tube-formation points that result from elasticity theory ($\tilde{h}=0.012$) and from simulation results for 10-ZGNR ($\tilde{h}\approx 0.0117$), as a function of pre-strain.

Figure 4

Predicting CNT chiralities. (a) Axial shift $\Delta z$ of the opposite edges in the tube as given by the elasticity theory as a function of ribbon width (lines) for three values of pre-strain (denoted by percentages). Circles are the simulated values for ZGNRs. (b) The CNT chiral angles $\theta$ that correspond to the shifts in panel a. Full lines are the predictions for $\theta$ given by Eq. (1) with $\phi =0^{\circ }$ and with $\Delta z/W$ given by the elasticity theory. (c) Shift $\Delta z$ for 10-, 12-, and 14-ZGNRs (symbols from left to right) and the corresponding elasticity-theory prediction (full curve) with zero pre-strain. Each of these ribbons was simulated 10 times, and the intensity of the color inside the circles is proportional to the number of times the room temperature simulation gave the given $\Delta z$. (d) The CNT chiral angles, augmented by the chiral indices, corresponding to the $\Delta z$ of c, with the same definitions for the symbols. The line is the elasticity-theory prediction by Eq. (1).

Figure 5

Tube formation for armchair graphene nanoribbons. (a) Buckling (${\tau }_b$) and tube-formation points (${\tau }_t$) as a function of elastic thickness given by the elasticity theory (full lines; Eq. (1) with $\phi ={30}^{\circ }$) and by simulations (squares) for AGNRs of varying width [corresponds to Fig. 3c for ZGNRs]. The inset: AGNR prior to tube formation, illustrating the radial bulging of the edges. Pre-strain is zero. (b) Shift $\Delta z$ as given by the elasticity theory (full lines) and by atomic simulations (squares) as a function of ribbon width for three values of pre-strain [corresponds to Fig. 4a for ZGNRs] (c) The CNT chiral angles corresponding to the shifts in (b).

Figure 6

Energy gaps for selected $N$-AGNRs from quasi-static simulations (optimized atomic coordinates for given $\tau W$) representing three different $N$ families. The narrow panel shows the gaps in the resulting CNTs with torque and axial stress removed.

Figure 7

Finite ribbon simulations. Figure shows snapshots from the simulation of $L=38$ nm long 12-ZGNR with clamped ends and with increasing end-to-end twist angle (the first six snapshots; structures are rotated to give the best view angles). The last snapshot is the (partially unzipped) $(9,3)$ CNT after removing the clamps and performing a constraint-free simulation at 300 K.

Figure 8

ZGNRs and AGNRs illustrating the $N$ alternation in the opposite-edge profiles; alternation shows up, for example, in Eq. (2). The unit-cell length is $a_{\mathrm{zz}}\approx 2.46$ Å in ZGNRs and $a_{\mathrm{ac}}\approx 4.26$ Å in AGNRs for zero pre-strain. Ribbon relaxation leads to slightly longer ribbons, more so for narrow ribbons. Width $W_0$ is measured from atomic positions [$W_0\approx (2.13N-1.42)$ Å for ZGNRs and $W_0\approx 1.23(N-1)$ Å for AGNRs]. The angle $\phi$ is the angle between the ribbon axis and the adjacent zigzag direction.

Figure 9

The ratio of CNT circumference to $W_0$. Fitting given for the tube circumference $\pi D\equiv W\approx W_0+2.5$ Å (Poisson effect caused by the pre-strain was removed). The CNT circumference was obtained more accurately from the chiral indices, $W=\sqrt{3}{d}_{\mathrm{bond}}\sqrt{n^2+m^2+mn}$ with $(n,m)$ predicted by Eq. (2).

Figure 10

(a) Torque at the tube-formation point as determined by numerical energy minimization (full line) is compared with the scaling estimate $M_t/K\approx 1.8/\sqrt{\tilde{h}}$ (dashed line). (b) Tube-formation point, ${\tau }_t$, and buckling point, ${\tau }_b$, are compared with the scaling estimates ${\tau }_t\approx 17.2\sqrt{\tilde{h}}$ and ${\tau }_b\approx 6.6\sqrt{\tilde{h}}$, respectively.