Study of axial strain induced torsion of single wall carbon nanotubes by 2D continuum anharmonic anisotropic elastic model

Recent molecular dynamic simulations have found chiral single wall carbon nanotubes (SWNTs) twist during stretching, which is similar to the motion of a screw. Obviously this phenomenon, as a type of curvature-chirality effect, can not be explained by usual isotropic elastic theory of SWNT. More interestingly, with larger axial strains (before buckling), the axial strain induced torsion (a-SIT) shows asymmetric behaviors for axial tensile and compressing strains, which suggests anharmonic elasticity of SWNTs plays an important role in real a-SIT responses. In order to study the a-SIT of chiral SWNTs with actual sizes, and avoid possible deviations of computer simulation results due to the finite-size effect, we propose a 2D analytical continuum model which can be used to describe the the SWNTs of arbitrary chiralities, curvatures, and lengths, with the concerning of anisotropic and anharmonic elasticity of SWNTs. This elastic energy of present model comes from the continuum limit of lattice energy based on Second Generation Reactive Empirical Bond Order potential (REBO-II), a well-established empirical potential for solid carbons. Our model has no adjustable parameters, except for those presented in REBO-II, and all the coefficients in the model can be calculated analytically. Using our method, we obtain a-SIT responses of chiral SWNTs with arbitrary radius, chiralities and lengthes. Our results are in reasonable agreement with recent molecular dynamic simulations. [Liang {\it et. al}, Phys. Rev. Lett, ${\bf 96}$, 165501 (2006).] Our approach can also be used to calculate other curvature-chirality dependent anharmonic mechanic responses of SWNTs.

The amazing mechanical properties of carbon nanotubes (CNTs), such as high elastic modulus, exceptional directional stiffness, and low density, make them idea candidates for the applications of nanoelectromechanical systems (NEMS) devises 1,2,3 . Recent studies have demonstrated the possibilities of using CNT as actuator 4 , nanotweezers 5 , and nanorelay 6,7,8,9 .
Detailed understanding of mechanical behavior, especially structurally-specific mechanical properties of CNT-based NEMS devises is therefore crucial for their potential applications in NEMS.
Unlike isotropic elastic thin shell, due to special geometries of SWNTs, e.g. chiralities, there is coupling between axial strain and torsion strain, which is similar to ordinary helical spring 10 . More interestingly, recent molecular dynamic simulations found asymmetric behaviors of such coupling in chiral SWNTs 11,12 and double walled carbon nanotubes (DWNTs) 13 , namely, asymmetry of a-SIT for tensile and compression strains 11 . Later, Upmanyu et al.'s finite element method simulation also obtained asymmetric a-SIT response 14 . Main property of asymmetric a-SIT is that a-SIT responses for tension or compression are much different at large strain. Torsion angle per unit length increases when strain increases in tension case.
However, with increasing strain under compression, the torsion angle firstly increases, then decreases to zero, and increases again after changing the direction of twist 11,12,13 .
A-SIT implies the coupling between axial vibration modes and torsional ones for chiral SWNTs, which may play an important role in applications of CNT-NEMS oscillators 15,16 .
To understand a-SIT response, there are very few studies: Gartstein, et al. 10 used a twodimensional continuum elastic model, predicted linear a-SIT effect for chiral SWNTs with small strain, i.e., SWNT twists in opposite directions for tension and compression and rotation angle varies linearly with strain. Gartstein et al found a-SIT response is chirality dependent, it reaches the maximum when the chiral angle is π/12. Liang et al.'s molecular simulations 11 extended the study of a-SIT to large strain region (before buckling), obtained asymmetric a-SIT. By comparing a-SIT response with changes of geometry of carbon-carbon bonds, they found asymmetry a-SIT is relevant to microscopic lattice structure of SWNT. All these efforts are valuable in understanding a-SIT. Nevertheless, Gartstein et al's theory was restrict to linear a-SIT response, while the molecular dynamic or finite element method simulations for a series of SWNTs with some special chiral index were time-consuming, a lot of computer resource were needed, which limits their further application to study of properties of actual SWNTs. Also, there is a general question for these simulation work: can the results of the simulations for small systems be extrapolated to SWNTs at equilibrium state with actual sizes?
In our knowledge, there lacks a easily handled theoretical frame capturing basic physics of asymmetric a-SIT which can obtain this response for actual SWNTs at equilibrium states with arbitrary radius and chiralities. To fulfill this task, we propose a quasi-analytical approach based on continuum elastic theory. In our model, the carbon-carbon interactions in SWNT are described by REBO-II potential 17 , which is a classic many-body potential for solid carbon and hydrocarbons. The advantages for REBO-II potential are it has analytical form of carbon-carbon pair potentials with the bond length and bond angle as variables of energy functions, the parameters of REBO-II potential were fitted from a large data sets of experiments and ab initio calculations. REBO-II potential can accurately reproduce elastic properties of diamond and graphite, In Ref. 11, molecular dynamic simulation was also based on REBO-II potential.
The carbon-carbon interaction energy near the equilibrium state without deformations can be obtained analytically by Taylor expansion with inclusion of the most important cubic term, i.e., anharmonic term of bond stretching, Here ij denotes the nearest neighboring atom pairs, θ ijk denotes angle between bonds i − j and i − k. Equilibrium state is denoted by "0". Similar series expansion of quadratic terms for Brenner potential 18 have been reported by Huang et al. 19 .
The non-crossing second and fourth terms in right hand of Eq. 1 were also presented in Lenosky's model 20 . From analytical form of REBO-II potential, the derivatives are, In 2D elastic theory of SWNT, the in-plane deformations of SWNT can be described by 22 with ε 1 ≡ ε 11 , ε 2 ≡ ε 22 , ε 6 ≡ 2ε 12 , are the axial, circumferential, and shear strains, respectively. After deformation, the bond vector from atom i to its three nearest neighboring atoms j, deviates from initial bond vector r 0 ij , A SWNT can be viewed as a cylinder with radius R, its surface can be perfectly embedded by six-member carbon rings 21 . There are three bond curves passing one carbon atoms at the surface of SWNT, in the continuum limit, bond vector can be written as 21 θ(M) is the rotating angle fromê x to tangent vector t, which is related to the chiral angle θ c . 22 After deforming, bond length r ij = | r ij |, and bond angle between bond vectors r ij and r ik are cos θ ijk = u ij · u ik , with unit vector u ij ≡ r ij /r ij . Based on these relations, the 2D continuum limit of elastic energy per unit area of SWNT in Eq. 1, which avoids introducing ill-defined thickness of SWNTs, can be written as, where c ij and c ijk are in-plane elastic constants, i, j, k = 1, 2, 6, they have analytical ex- To study the asymmetric a-SIT, we consider a chiral SWNT with one fixed end, while the other end atoms are allowed to relax both radially and tangentially during deformation. The axial displacement is fixed for each simulation step, ensuring that only axial stress occurs, which is the basic assumption in simulations for a-SIT in SWNTs 11,12,14 .
The free energy per unit area of SWNT under axial stress is, Assumption of equilibrium state leads to the following nonlinear equations They give the relation between torsion angle per nm (in unit of degree) φ = − (180/π) × ε 6 / (R/1nm) and axial strain ε 1 , which is an asymmetric response. There are two critical compressing strains ε * 1 and ε * * 1 , as shown in Fig.1. For axial compression, at ε * 1 , torsion angle reaches its extreme, then SWNT begins to untwist, after totally untwisting at critical strain ε * * 1 , the tube twists again to the opposite direction, i.e., to the direction as the same as that for tension case.
Another interesting result is nonlinear axial stress-strain relation, the axial secant Young's modulus Y s ≡ dσ/ dε 1 of SWNT is a strict monotonically decreasing function, Y s = Y s 0 − t ε 1 , as shown in Fig. 2, thus SWNTs show strain softening under tension, while strain hardening under compression. This phenomenon was also found in recent molecular dynamic simulations 12 .
We find asymmetric a-SIT and nonlinear axial stress-strain of SWNT are tightly related to each other, the nature of which is anharmonicity of atom-atom interaction for SWNTs, such as REBO-II potential in Ref. 11 and present work. This anharmonicity leads to anharmonic bond stretching energy in Eq. 1 and cubic terms in Eq. 3, elastic energy.
Thus, present analysis captures main features of asymmetry a-SIT.
Eq. 3 without cubic terms gives the linear a-SIT response's coefficient with leading term ∼ R −3 characterizing linear a-SIT response, which is in good agreement with Gartstein et al's theoretical results. Therefore present analysis captures the main characters of a-SIT response.
In our continuum elastic theory, we only get symmetric a-SIT without anharmonic terms, however Ref. 14 gave asymmetric a-SIT by finite element simulation based on harmonic elasticity, although much small (∼ 1/1000 of those of molecular simulations). It may be due to the elastic energy we used is the continuum limit of lattice energy, which may lose some subtle microscopic information. Our continuum elastic theory has some advantages, compared to previous simulations, for it is suitable to study SWNTs with actural sizes, and all the elastic constants in the theory are obtained analytically. Obviously, our method can be extended to calculate other anharmonic properties of SWNTs with arbitrary radius and chiralities.
In summary, we emphasize the anharmonicity of inter-atoms interactions and curvaturechirality induced anisotropic elasticity are both important in a-SIT response, and explain the asymmetry a-SIT and nonlinear stress-strain relation all together. We find the unusual asymmetric a-SIT effects is the consequence of the curvature-chirality effect and anharmonic elasticity. We give the analytical expressions of anharmonic elastic energy, as well as curvature-chirality induced anisotropic elasticity based on REBO-II. The calculated results are in reasonable agreement with recent molecular dynamic simulations. Our method can be used to analytically calculate anharmonic properties of SWNTs with arbitrary radius and chirality.