Study of axial strain induced torsion of single wall carbon nanotubes by 2D continuum anharmonic anisotropic elastic model
aa r X i v : . [ c ond - m a t . m e s - h a ll ] J u l Study of axial strain induced torsion of single wall carbonnanotubes by D continuum anharmonic anisotropic elastic model
Weihua MU, Ming Li, Wei Wang, and Zhong-can Ou-Yang
1, 4 Institute of Theoretical Physics, The Chinese Academy of Sciences,P.O.Box 2735 Beijing 100190, China Graduate University of Chinese Academy of Sciences, Beijing 100190, China College of Nanoscale Science and Engineering (CNSE),University at Albany, State University of New York, NY 12203, USA Center for Advanced Study, Tsinghua University, Beijing 100084, China bstract 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 atype 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 suggestsanharmonic elasticity of SWNTs plays an important role in real a-SIT responses. In order to studythe a-SIT of chiral SWNTs with actual sizes, and avoid possible deviations of computer simulationresults due to the finite-size effect, we propose a 2D analytical continuum model which can be usedto describe the the SWNTs of arbitrary chiralities, curvatures, and lengthes, with the concerning ofanisotropic and anharmonic elasticity of SWNTs. This elastic energy of present model comes fromthe continuum limit of lattice energy based on Second Generation Reactive Empirical Bond Orderpotential (REBO-II), a well-established empirical potential for solid carbons. Our model has noadjustable parameters, except for those presented in REBO-II, and all the coefficients in the modelcan be calculated analytically. Using our method, we obtain a-SIT responses of chiral SWNTswith arbitrary radius, chiralities and lengthes. Our results are in reasonable agreement with recentmolecular dynamic simulations. [Liang et. al , Phys. Rev. Lett, , 165501 (2006).] Our approachcan also be used to calculate other curvature-chirality dependent anharmonic mechanic responsesof SWNTs. PACS numbers: 62.25.-g, 46.70.Hg . Recent studies havedemonstrated the possibilities of using CNT as actuator , nanotweezers , and nanorelay .Detailed understanding of mechanical behavior, especially structurally-specific mechanicalproperties of CNT-based NEMS devises is therefore crucial for their potential applicationsin 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 helicalspring . More interestingly, recent molecular dynamic simulations found asymmetric behav-iors of such coupling in chiral SWNTs and double walled carbon nanotubes (DWNTs) ,namely, asymmetry of a-SIT for tensile and compression strains . Later, Upmanyu et al. ’sfinite element method simulation also obtained asymmetric a-SIT response . Main propertyof asymmetric a-SIT is that a-SIT responses for tension or compression are much differentat 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, thendecreases to zero, and increases again after changing the direction of twist .A-SIT implies the coupling between axial vibration modes and torsional ones for chiralSWNTs, which may play an important role in applications of CNT-NEMS oscillators .To understand a-SIT response, there are very few studies: Gartstein, et al. used a two-dimensional continuum elastic model, predicted linear a-SIT effect for chiral SWNTs withsmall strain, i.e., SWNT twists in opposite directions for tension and compression and ro-tation angle varies linearly with strain. Gartstein et al found a-SIT response is chiralitydependent, it reaches the maximum when the chiral angle is π/
12. Liang et al. ’s molecularsimulations extended the study of a-SIT to large strain region (before buckling), obtainedasymmetric a-SIT. By comparing a-SIT response with changes of geometry of carbon-carbonbonds, they found asymmetry a-SIT is relevant to microscopic lattice structure of SWNT.Geng et al studied both of torsion induced by axial strain and axial strain induced by tor-sion, and showed nonlinear axial stress-strain relation occurring in the same time. Upmanyu et al ’s finite element method simulation also obtained a-SIT.All these efforts are valuable in understanding a-SIT. Nevertheless, Gartstein et al ’s theorywas restrict to linear a-SIT response, while the molecular dynamic or finite element method3imulations 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 ofproperties of actual SWNTs. Also, there is a general question for these simulation work: canthe results of the simulations for small systems be extrapolated to SWNTs at equilibriumstate with actual sizes?In our knowledge, there lacks a easily handled theoretical frame capturing basic physicsof asymmetric a-SIT which can obtain this response for actual SWNTs at equilibrium stateswith arbitrary radius and chiralities. To fulfill this task, we propose a quasi-analyticalapproach based on continuum elastic theory. In our model, the carbon-carbon interactionsin SWNT are described by REBO-II potential , which is a classic many-body potential forsolid carbon and hydrocarbons. The advantages for REBO-II potential are it has analyticalform of carbon-carbon pair potentials with the bond length and bond angle as variables ofenergy functions, the parameters of REBO-II potential were fitted from a large data sets ofexperiments and ab initio calculations. REBO-II potential can accurately reproduce elasticproperties of diamond and graphite, In Ref. 11, molecular dynamic simulation was also basedon REBO-II potential.The carbon-carbon interaction energy near the equilibrium state without deformationscan be obtained analytically by Taylor expansion with inclusion of the most important cubicterm, i.e., anharmonic term of bond stretching, V = V + 12 X h ij i ) (cid:18) ∂ V∂r ij (cid:19) (cid:0) r ij − r ij (cid:1) + X h ij i X k = i,j (cid:18) ∂ V∂r ij ∂ cos θ ijk (cid:19) (cid:0) r ij − r ij (cid:1) (cid:0) cos θ ijk − (cos θ ijk ) (cid:1) (1)+ 12 X h ij i X k = i,j ∂ V∂ (cos θ ijk ) ! (cid:0) cos θ ijk − (cos θ ijk ) (cid:1) + X h ij i X k,l = i,j (cid:18) ∂ V∂ cos θ ijk ∂ cos θ ijl (cid:19) · (cid:0) cos θ ijk − (cos θ ijk ) (cid:1) (cid:0) cos θ ijl − (cos θ ijl ) (cid:1) + 13! X h ij i (cid:18) ∂ V∂r ij (cid:19) (cid:0) r ij − r ij (cid:1) . Here h ij i 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 terms4or Brenner potential have been reported by Huang et al. .The non-crossing second and fourth terms in right hand of Eq. 1 were also presented inLenosky’s model . From analytical form of REBO-II potential, the derivatives are, (cid:18) ∂ V∂r ij (cid:19) ≈ . · ˚A − , (cid:18) ∂ V∂r ij ∂ cos θ ijk (cid:19) ≈ − . · ˚A − , ∂ V∂ (cos θ ijk ) ! ≈ . , (cid:18) ∂ V∂ cos θ ijk ∂ cos θ ijl (cid:19) ≈ − . , and (cid:18) ∂ V∂r ij (cid:19) ≈ − . · ˚A − . In 2D elastic theory of SWNT, the in-plane deformations of SWNT can be described by ε = ε ε / ε / ε , with ε ≡ ε , ε ≡ ε , ε ≡ ε , are the axial, circumferential, and shear strains,respectively. After deformation, the bond vector from atom i to its three nearest neighboringatoms j , deviates from initial bond vector ~r ij , ~r ij ≈ (1 + ε ) ~r ij . A SWNT can be viewed as a cylinder with radius R , its surface can be perfectly embeddedby six-member carbon rings . There are three bond curves passing one carbon atoms atthe surface of SWNT, in the continuum limit, bond vector can be written as ~r ( M ) = ~r ij = (cid:2) − a κ ( M ) / (cid:3) a ~t ( M )+ (cid:2) a κ ( M ) / a κ s ( M ) / (cid:3) a ~N ( M ) (2)+ (cid:2) κ ( M ) τ ( M ) a / (cid:3) a ~b ( M ) , where a = 1 . A is carbon-carbon bond length without strains, M = 1 , , sp − bonded curves from atom i to atoms j on the surface of SWNT. Vectors ~t, ~N and ~b areunit tangential, normal, and binormal vectors of the bond curves from atom i to j , κ , τ and s are the curvature, torsion, and arc parameter of bond curve, respectively, κ s ≡ dκ/ds . The vectors ~t ( M ) = cos θ ( M )ˆ e x +sin θ ( M ) ~e y , ~b ( M ) = sin θ ( M )ˆ e x − cos θ ( M )ˆ e y , where ˆ e x and5 e y are the unit axial and circumferential vectors at the i -atom’s site on the SWNT surface, θ ( M ) is the rotating angle from ˆ e x to tangent vector ~t , which is related to the chiral angle θ c . 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 2Dcontinuum limit of elastic energy per unit area of SWNT in Eq. 1, which avoids introducingill-defined thickness of SWNTs, can be written as, E elsticity = 12 X ij c ij ε i ε j + X i ≤ j ≤ k c ijk ε i ε j ε k , (3)where c ij and c ijk are in-plane elastic constants, i, j, k = 1 , ,
6, they have analytical ex-pressions, see Appendix. Among them, c , harmonic elastic constant for coupling betweenaxial strain and torsional twist is proportional to ( a /R ) sin(6 θ c ), which clearly shows a-SIT response is curvature and chirality effect, only occurs in chiral SWNT. Obviously lineara-SIT response is distinct at θ c = π/
12 and significant for SWNTs with small diameters,which are in accord with previous theoretical and simulation works. For tubes with largediameters and small strains, anharmonic elastic energy can be ignored along with c and c terms, then the isotropic thin shell model for SWNTs is recovered, and the calculatedin-plane Young’s modulus and Possion’s ratio are similar to the results in Ref. 22.To study the asymmetric a-SIT, we consider a chiral SWNT with one fixed end, while theother end atoms are allowed to relax both radially and tangentially during deformation. Theaxial 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 .The free energy per unit area of SWNT under axial stress is, F = E elsticity − σ ε . (4)Assumption of equilibrium state leads to the following nonlinear equations ∂ F ∂ε i = 0 , (5)They give the relation between torsion angle per nm (in unit of degree) φ = − (180 /π ) × ε / ( R/ ε , which is an asymmetric response. There are two criticalcompressing strains ε ∗ and ε ∗∗ , as shown in Fig.1. For axial compression, at ε ∗ , torsion anglereaches its extreme, then SWNT begins to untwist, after totally untwisting at critical strain6 ∗∗ , the tube twists again to the opposite direction, i.e., to the direction as the same as thatfor tension case.Another interesting result is nonlinear axial stress-strain relation, the axial secant Young’smodulus Y s ≡ dσ/ dε of SWNT is a strict monotonically decreasing function, Y s = Y s − t ε , as shown in Fig. 2, thus SWNTs show strain softening under tension, while strainhardening under compression. This phenomenon was also found in recent molecular dynamicsimulations .We find asymmetric a-SIT and nonlinear axial stress-strain of SWNT are tightly related toeach other, the nature of which is anharmonicity of atom-atom interaction for SWNTs, suchas REBO-II potential in Ref. 11 and present work. This anharmonicity leads to anharmonicbond stretching energy in Eq. 1 and cubic terms in Eq. 3, elastic energy.To illustrate it, we start from a simplified linear elastic energy per unit area of SWNT,˜ F = 12 c ε + c ε ε + 12 c ε − σ ε , (6)After substituting nonlinear stress-strain relation σ = Y s ε − ( t / ε to ˜ F , using equi-librium condition ∂ ˜ F /∂ε = 0 , the torsion angle, which is proportional to ε , is a quadraticfunction of axial strain. φ ( ε ) curve is a parabola with its symmetric axial located at ε < . Thus, present analysis captures main features of asymmetry a-SIT.Eq. 3 without cubic terms gives the linear a-SIT response’s coefficient dφdε | ε = c c − c c c c (7)with leading term ∼ R − characterizing linear a-SIT response, which is in good agreementwith Gartstein et al ’s theoretical results. Therefore present analysis captures the maincharacters 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 harmonicelasticity, although much small ( ∼ / R. H. Baughman, A. A. Zakhidov, and W. A. de Heer, Science, , 787 (2002). H. G. Craighead, Science, , 1532 (2000). S. Sapmaz, Y. M. Blanter, L. Gurevich, and H. S. J. van der Zant, Phys. Rev. B ,235414 (2003). R. H. Baughman et al. , Science , 1340 (1999). Philip P. Kim, and C. M. Lieber, Science , 2148 (1999). J. M. Kinaret, T. Nord, and S. Viefers, Appl. Phys. Lett. , 1287 (2003). S. W. Lee et al , Nano Lett. , 2027 (2004). J. E. Jang et al , Appl. Phys. Lett. , 163114 (2005). J. E. Jang et al. , Appl. Phys. Lett. , 113105 (2008). Y. N. Gartstein, A. A. Zakhidov, and R. H. Baughman, Phys. Rev. B , 115415 (2003). H. Liang, and M. Upmanyu, Phys. Rev. Lett. , 165501 (2006). J. Geng, and T. Chang, Phys. Rev. B , 245428 (2006). H. W. Zhang, L. Wang, J. B. Wang, Z. Q. Zhang, and Y. G. Zheng, Phys. Lett. A, ,3488 (2008). M. Upmanyu, H.L. Wang, Haiyi Liang, and R. Mahajan, J. R. S. interface, , 303 (2008). V. Sazonova, et al. , Nature , 284 (2004). Y. Zhao, C-C. Ma, G. H. Chen, and Q. Jiang, Phys. Rev. Lett. , 175504 (2003). D. W. Brenner, O. A. Shenderova, J. A. Harrison, S. J. Stuart, B. Ni, and S. B. Sinott, J. Phys.Condens. Matter, , 783 (2002). D. W. Brenner, Phys. Rev. B , 9458 (1990). Y. Huang, J. Wu, and K. C. Hwang, , 245413 (2006). T. Lenosky, X. Gonze, and M. Teter, Nature (London), , 333 (1992). O-Y. Zhong-can, Z-B. Su, and C-L. Wang, Phys. Rev. Lett. , 4055 (1997). Z-C. Tu, and Zhong-can. Ou-Yang, Phys. Rev. B , 233407 (2002). Unpublished. T o r s i on ang l e i nduced b y ax i a l s tr a i n ( deg / n m ) Axial Strain (%) (8,0) (8,2) (8,4) (8,6) (8,8) FIG. 1: Torsion angle-axial strain relations for a series of (8 , m ) SWNTs, which shows chiralitydependence of a-SIT response. Only chiral SWNTs have a-SIT response, as shown. I n - p l ane ax i a l s ecan t Y oung ’ s m odu l u s ( e V / n m ) Axial Strain (%) (8,0) (8,2) (8,4) (8,6) (8,8)
FIG. 2: ’ Relation between in-plane axial secant Young’s modulus and axial strain, for a series of(8 , m ) SWNTs. ppendix Elastic constants c , c , . . . , c presented in Eq. 3 can be described by, cc = Mb , where, cc is a column vector with the components cc to cc being the sixteen elasticconstants of SWCNT, i.e., c to c , respectively, M is a 16 × b is a columnvector with components, b = (cid:18) ∂ V∂r ij (cid:19) · a Ω , b = ∂ V∂ (cos θ ijk ) ! · ,b = (cid:18) ∂ V∂r ij ∂ cos θ ijk (cid:19) · a Ω , b = (cid:18) ∂ V∂ cos θ ijk ∂ cos θ ijl (cid:19) · ,b = (cid:18) ∂ V∂r ij (cid:19) · a Ω . Here, a = 1 . A is carbon-carbon bond length without strains, and Ω = 2 . A is thearea occupied by one carbon atom at the surface of SWCNTs.All 34 non-zero elements of matrix M are analytically written as , M , = 916 + (cid:18) − (cid:19) α ,M , = 316 + (cid:18) − (cid:19) α + (cid:18) − (cid:19) α cos(6 θ ) ,M , = 916 + (cid:18) − (cid:19) α + (cid:18) (cid:19) α cos(6 θ ) ,M , = (cid:18) (cid:19) α sin(6 θ ) ,M , = (cid:18) − (cid:19) α sin(6 θ ) ,M , = 316 + (cid:18) − (cid:19) α + (cid:18) − (cid:19) α cos(6 θ ) ,M , = 2716 + (cid:18) − (cid:19) α + (cid:18) − (cid:19) α cos(6 θ ) ,M , = − (cid:18) (cid:19) α + (cid:18) (cid:19) α cos(6 θ ) , , = 2716 + (cid:18) − (cid:19) α cos(6 θ ) ,M , = (cid:18) − (cid:19) α sin(6 θ ) ,M , = (cid:18) (cid:19) α sin(6 θ ) ,M , = 2716 + (cid:18) − (cid:19) α + (cid:18) (cid:19) α cos(6 θ ) ,M , = −
98 + (cid:18) − (cid:19) α + (cid:18) − (cid:19) α cos(6 θ ) ,M , = 98 + (cid:18) − (cid:19) α + (cid:18) − (cid:19) α cos(6 θ ) ,M , = −
98 + (cid:18) − (cid:19) α + (cid:18) (cid:19) α cos(6 θ ) ,M , = (cid:18) − (cid:19) α sin(6 θ ) ,M , = (cid:18) − (cid:19) α sin(6 θ ) ,M , = −
98 + (cid:18) (cid:19) α + (cid:18) − (cid:19) α cos(6 θ ) ,M , = − (cid:18) (cid:19) α + (cid:18) (cid:19) α cos(6 θ ) ,M , = 2732 + (cid:18) − (cid:19) α + (cid:18) − (cid:19) α cos(6 θ ) ,M , = − (cid:18) (cid:19) α cos(6 θ ) ,M , = (cid:18) − (cid:19) α sin(6 θ ) ,M , = (cid:18) − (cid:19) α sin(6 θ ) ,M , = − (cid:18) (cid:19) α + (cid:18) − (cid:19) α cos(6 θ ) ,M , = 564 + (cid:18) (cid:19) cos(6 θ ) ,M , = 364 + (cid:18) − (cid:19) cos(6 θ ) ,M , = (cid:18) (cid:19) sin(6 θ ) ,M , = 364 + (cid:18) (cid:19) cos(6 θ ) , , = (cid:18) − (cid:19) sin(6 θ ) ,M , = 364 + (cid:18) − (cid:19) cos(6 θ ) ,M , = (cid:18) (cid:19) + (cid:18) − (cid:19) cos(6 θ ) ,M , = (cid:18) (cid:19) sin(6 θ ) ,M , = (cid:18) (cid:19) + (cid:18) (cid:19) cos(6 θ ) ,M , = (cid:18) − (cid:19) sin(6 θ ) . Here, α ≡ a /R/R