Superelasticity of Carbon Nanocoils from Atomistic Quantum Simulations
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A structural model of carbon nanocoils (CNCs) on the basis of carbon nanotubes (CNTs) was proposed. The Young’s moduli and spring constants of CNCs were computed and compared with those of CNTs. Upon elongation and compression, CNCs exhibit superelastic properties that are manifested by the nearly invariant average bond lengths and the large maximum elastic strain limit. Analysis of bond angle distributions shows that the three-dimensional spiral structures of CNCs mainly account for their unique superelasticity.
KeywordsNanocoil Nanotube Superelasticity Young’s modulus
There is a large class of novel nanostructures with helical geometries including boron carbide , SiC  and ZnO [3, 4] nanosprings, carbon  and ZnO  nanohelices, and carbon nanocoils [7, 8]. Among them, carbon nanocoil (CNC) (also known as coiled carbon nanotube) has attracted particular attention due to its structural correlation with carbon nanotubes (CNTs). Intuitively, CNCs may inherit some of the fundamental properties of carbon nanotubes but exhibit other unique mechanical, electronic, and magnetic properties associated with their coiled geometries and the intrinsic distribution of five-membered and seven-membered rings.
In early 1990s, Dunlap  and Ihara et al. [10, 11, 12] proposed several structural models for coiled carbon nanotubes and discussed the relationships between the geometric parameters (diameter, pitch length, rotational symmetry) and the energetic, elastic, and electronic properties. Molecular dynamics simulations and tight-binding calculations have demonstrated the structural stability of CNCs; they have higher cohesive energy (~7.4 eV/atom) than that of C60 (7.29 eV/atom) [10, 13]. Electronic properties of CNCs including band structures and density of states were investigated using a tight-binding model [11, 14], and it was predicted that some carbon nanocoils could be semi-metals, in contrast to the conventionally semiconducting and metallic behavior known for the straight carbon nanotubes.
Since Zhang et al. first fabricated carbon nanocoils (700 nm in pitch and ~20 nm in tubular diameter) via catalytic decomposition of acetylene in 1994 , there have been large experimental efforts in synthesizing CNCs of high quality. Production of CNCs by chemical vapor deposition (CVD) [15, 16, 17, 18, 19], laser evaporation of the fullerene/Ni particle mixture , and opposed flow flame combustion method  has been reported. Pan and coworkers realized diameter control of CNCs via tuning the particle size of the nanoscale catalysts . In addition to the conventionally synthesized multi-walled CNCs with tubular diameters of 15–100 nm [7, 16, 17, 18, 19], evidence of ultrathin single-walled carbon nanocoils (with both tubular diameter and pitch length down to 1 nm) was found in the products of carbon nanotubes from catalytic decomposition of hydrocarbon molecules by Biró’s STM experiments .
With their unique three-dimensional (3D) helical structures, the CNCs are expected to exhibit spring-like behavior in their mechanical properties. In an experiment by Chen et al. , multi-walled CNCs with outer tubular diameter of ~126 nm have been elastically elongated to a maximum strain of ~42%. A spring constant of 0.12 N/m in the low strain region was obtained. According to the structural parameters of nanocoil given by Chen et al.  (tubular diameter of 120 nm, coil radius of 420 nm, and pitch of 2,000 nm), Fonseca et al.  computed the CNC’s Young’s modulus within the framework of the Kirchhoff rod model and obtained a value of 6.88 GPa. Using finite element analysis at the continuum level, Sanada et al. also predicted a similar result (about 4.5 GPa) for carbon nanocoil with tubular radius of 240 nm, coil radius of 325 nm, and coil pitch of 1,080 nm . However, the experimentally measured Young’s modulus values are much higher than these theoretical predictions. Volodin et al.  reported a Young’s modulus ~0.7 TPa for CNCs with coil diameter >170 nm from AFM measurement. Using a manipulator-equipped SEM, Pan et al. determined the Young’s modulus of CNCs to be up to 0.1 TPa for coil diameter ranging from 144 to 830 nm . The large discrepancy between experiment and theory has been attributed to the usage of mechanical parameters of bulk materials in the continuum mechanics simulations .
Despite the above efforts, our theoretical knowledge of the CNCs is still limited. In particular, there have been no atomistic simulations of the mechanical properties of the CNCs. In this paper, we proposed a new way of constructing structural models of carbon nanocoils and computed the Young’s moduli and spring constants for a series of ultrathin CNCs. Most interestingly, we observed an unusual superelasticity in these CNCs owing to their 3D spiral geometries.
Structural Model and Computational Methods
Upon relaxation, the nanotube segment is bent around the defect site in order to release the strain energy induced by the pentagons and heptagons. The pentagon (heptagon) pair locates in the convex (concave) part of the segment (see Fig. 1b), passing through a bisector after we adjust the number of carbon atoms on the two ends to make the segment symmetric. Depending on how these basic structural segments are connected, either a nanocoil or a nanotori [9, 29, 30] is formed. As shown in Fig. 1c, two segments are connected with a certain rotating angle to make the combined structure spiral and to form a seamless hexagonal carbon network.
The structures and energetics of these CNCs were described by a nonorthogonal tight-binding (TB) model developed by our group previously . This TB total energy model is based on the extended Hückel approximation and employed an exponential distance-dependent function for the hopping integral overlap. The TB parameters were especially developed for hydrocarbon molecules and nanostructures. The experimental or ab initio data on the geometry structures, binding energies, on-site charge transfer, and vibrational frequencies of a variety of hydrocarbon molecules have been well reproduced. In addition, a few test calculations on the carbon fullerenes and nanotubes also showed satisfactory agreement between TB and DFT results.
Within 1D periodic boundary condition, the lattice parameter (pitch) of each nanocoil was carefully adjusted to minimize the total energy. Starting from the equilibrium 1D lattice, the CNCs were either compressed or elongated by gradually varying the lattice parameter to investigate the mechanical properties of these nanocoils. At any given lattice parameter, the atomic coordinates of CNCs were fully relaxed without any symmetry constraint. To validate the results from TB calculations, we performed all-electron density functional theory (DFT) calculations on the smaller (5, 5) CNC. In the DFT calculations, we adopted generalized gradient approximation (GGA) with the PW91 parameterization  and the double-numerical plus d polarization (DND) basis set as implemented in the DMol3 package .
Results and Discussion
Young’s Modulus and Spring Constant
where U is the elastic potential energy of the system (total energies differences of different lengths),Open image in new windowandOpen image in new windoware the 1D displacement and strain under elongation/compression, respectively, L is the length of 1D unit cell and the L0 is its equilibrium value, and V0 is the effective volume of the 1D structural unit in its equilibrium configuration. For a carbon nanocoil, V0 = S × L0, where S is the area of cross section of the nanocoil from the top view (see Fig. 2). Similarly, for a single-walled carbon nanotube, V0 = 2πr × L0 × Δd, where Δd = 3.4 Å is the shell thickness of tube wall and r is the tube radius [35, 36].
Young’s modulus (E) of different armchair carbon nanotubes from DFT (EDFT) and TB (ETB) calculations
Young’s modulus (E) and spring constant (k) of carbon nanocoils (CNCs) from TB and DFT (values in brackets) calculations
Although the computed Young’s modulus for nanocoil varies with the tubular diameter and coil diameter (see Table 2), there seems no clear diameter-dependent trend, in agreement with the experimental observations [27, 28]. For carbon nanocoils of diameters between 144 and 830 nm, Hayashida et al.  found that the Young’s modulus changes irregularly from 0.04 TPa to 0.10 TPa. Volodin’s measurement of Young’s modulus also revealed no apparent dependence on the coil diameter .
The spring constants of the CNCs were also computed using Eq. (1), and the results are listed in Table 2. For the (5, 5), (6, 6), and (7, 7) CNCs, the spring constants are around 15–19 N/m, whereas the (8, 8) CNC possesses a very large spring constant of 44.36 N/m. Previous experiment by Chen et al.  obtained a k = 0.12 N/m for a mesoscale CNC (tubular diameter of 120 nm, coil radius of 420 nm, and pitch of 120 nm). The discrepancy between the present theoretical values and the measured data might be understood by the different length scales of the systems (nanometers in our model systems versus hundreds of nanometers in experimental CNCs).
For macroscopic materials, the superelastic (or pseudoelastic) effect in the shape memory alloys results in a variety of useful industrial and medical applications . In the nanostructured materials, similar superelastic phenomena were recently revealed in nanocoils and microcoils. Gao et al. reported superelasticity in ZnO nanohelices (~560 nm in coil diameter) with an experimental maximum elongation of 69.8% measured by AFM and a theoretical maximum elongation of 72% calculated by classical elasticity theory . A Si4N3 microcoil with coil diameter of 160 μm also exhibited good recovery ability under repeated load, corresponding to the superelasticity . In particular, even when stretched to a nearly straight shape for several cycles, the Si4N3 microcoil recovered its original state without damage after the load was released. As for the coiled carbon structures, Motojima et al. revealed that carbon microcoils could be extended and contracted by 3–15 times  and 5–10 times  with regard to the original coil length. Meanwhile, carbon nanocoils also demonstrated superior elasticity with a maximum relative elongation of ~42% .
With increasing tubular diameter, the variation of average bond length in the nanocoil is less sensitive to elongation strain (see Fig. 3), implying that the nanocoil can undertake higher strain. On the contrary, the elastic limit of compression for a CNC reduces with increasing tubular diameter. For example, the maximum compressive strain is 35% for (5, 5) CNC, 25% for (6, 6) CNC, and 20% for (7, 7) CNC. It is interesting to note that the carbon nanocoils can undertake higher elongation strain (up to ~60%) than compressive one (up to 20–35%).
We have constructed a series of carbon nanocoils by periodically introducing pentagons and heptagons in the segments of carbon nanotubes to make them coiled. The computed Young’s moduli of carbon nanocoils (3–6 GPa) are much lower than those of carbon nanotubes (~1 TPa). Under large elongation/compressive strains, the average bond lengths of CNCs almost remain invariant, while the elastic energy is stored via bond angle redistributions, corresponding to the superelastic behavior. Compared to the carbon nanotubes with same chirality, nanocoils show much smaller Young’s moduli and unusual superelasticity, which might lead to some future nanotechnology applications.
This work is supported by NCET Program provided by the Ministry of Education of China (NCET-060281), and the Scientific Research Foundation for the Returned Overseas Chinese Scholars.
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