Buckling of Double-walled Carbon Nanotubes

Transcription

1 Buckling o Double-walled Carbon anotubes Y. H. Teo Engineering Science Programme ational University o Singapore Kent idge Singapore 960 Abstract This paper is concerned with the buckling o double-walled carbon nanotubes (DCTs) under aial load. The DCTs are modelled as two cylindrical shells one shell nested in the other and inkler springs are introduced to connect them in order to simulate the van der aals orces between the two nanotubes. By using the Donnell thin shell theory we derive the governing equations or the buckling o the aorementioned double-shell model under aial compression. The equations are solved together with the boundary conditions and the critical buckling strains or cylindrical shells with dierent diameters and lengths. The critical strains are compared with the results obtained rom molecular dynamic simulations and were ound to be not so good in agreement. Better shell models have to be ound or predicting more accurate results. Introduction Carbon nanotubes (CTs) were discovered in the year o 99 Since their discovery they have attracted the attention o researchers in many ields o science and technology. Eperimental and theoretical studies have also intensiied in recent years as numerous studies have shown that carbon nanotubes possess superior mechanical properties e.g. high stinessto-weight and strength-to-weight ratios Young s modulus value as large as ~ TPa (ishio et al. 005) and hold substantial promise as superstrong ibres and composite. Such mechanical strength is suitable or the nanoscale systems such as probes or scanning probe microscopy (SPM) nanotweezers or nanomanipulations and nano-oscillators or mass detection. Presently there eist several techniques to analyse the compressive ailure o singlewalled carbon nanotubes (SCTs) and multiwalled carbon nanotubes (MCTs) such as eperimental approach which involves methods such as the transmission electron microscopy (TEM) and atomic orce microscopy (AFM) molecular mechanics (MM) molecular

2 dynamics (MD) and continuum mechanics (CM) techniques. Unortunately eperiments on CTs are etremely diicult and epensive to conduct whereas MD simulations are currently limited to very small length and time scales and cannot deal with the large-sized atomic system due to the limitations o current computing power. As or CM technique the computational eort is much less vigorous the results obtained is comparable computed with MD simulations (Yakobson et al. 996). In this paper a simple double-cylindrical shell model based on Donnell s shell theory is presented or the analysis o DCTs buckling under aial load. In this model the inner and outer nanotubes are treated as thin cylindrical shells and the van der aals interations are taken into account by connecting the shells together using lateral (inkler) springs. The buckling equations are derived based on this shell model and solved together with the boundary conditions or the critical buckling strains. The mathematical sotware MATAB was used to compute the buckling results and compared with the results obtained rom MD simulations to eamine the validity o the proposed shell model or the aial buckling analysis on CTs. Elastic Cylindrical Shell Model Based on Donnell s thin shell theory (Donnell 976) we obtain the ollowing equations or the equilibrium o membrane orces in the cylindrical shell: 0 and 0 () where and are the aial circumerential and torsional membrane orces respectively. In order or Eq. () to be true there must eist a unction A() such that A and A (3) Similarly or Eq. () to be true there must also eist a unction B() such that B and B (56) ow we introduce a stress unction φ() into both unctions A() and B() such that we have

3 ϕ ( ) ( A and B( ( ϕ (78) The stress unction is introduced into unctions A() and B() so that the membrane orces and can be represented by ϕ y ( ( ϕ and ( ϕ (9 0 ) The membrane strains are given by ( ) ( ) ε v ε v and γ ( ( v) ϕ ( 3 ) where E is the Young s modulus h is the thickness o the nanotube and v is the Poisson s ratio. By substituting the membrane orces o Eqs. (9) and (0) into Eqs. () and (3) the membrane strains can be rewritten as ε ϕ y ( ϕ( v ( ) ( ) and ϕ ϕ ε v (a 3a) y According to the Donnell thin shell theory the compatibility condition o a thin shell yields ε ε γ ( 0 (5) where () is the radial displacement o the middle surace o the shell along the normal direction and is the radius o the nanotube. By substituting Eqs. (a) (3a) and () into Eq. (5) we obtain ϕ (6) The governing equation or single-walled carbon nanotube (SCT) or aial compression is given by D P (7)

4 3 where D is the leural stiness o the shell and P denotes the normal pressure ( υ ) as shown in Figure. ow we substitute Eq. (0) into Eq. (7) and it becomes ( ) D ϕ P (8) By using Eq. (7) to eliminate the stress unction φ() in Eq. (8) we obtain a single equation in terms o () i.e. 8 D 0 P (9) Van der aals Force between adjacent nanotubes ennard-jones model can be used to describe the van der aals orce between any two carbon atoms. According to Girialco (99) the van der aals orce eerted on any atom on a tube can be estimated by adding up all orces between the atom and all atoms on the other tube. Figure shows the double-cylindrical shell model or a double-walled carbon nanotube under aial compression. Figure : A double-shell model or a DCT under aial compression

5 The data given by Girialco can be used to estimate the van der aals orce eerted on any point o the tube. At any point between the inner and outer tube the pointwise pressure can be assumed to be a unction o the normal distance between the inner and outer tubes at that point. Furthermore due to the act that the interaction orces between the tubes are equal and opposite the pressure p and p eerted on the corresponding points on the inner tube and outer tube respectively should be related by p ( p ( (0) where is the radii o the inner tube whereas is the radii o the outer tube. The pressure caused by the van der aals orces at any point () on the inner tube could be assumed to be a unction o the distance between the outer tube at that point denoted by δ() namely ( G[ δ ( ] p () where G(δ) is a nonlinear unction o the intertube spacing δ as given by u (000 00). Ater buckling the pressure caused by the van der aals orces at any point ()on the inner tube is assumed to be linearly proportional to the jump o the buckling delection between the inner and outer tubes and is shown as p ( c[ w ( w ( ] () ote that the equilibrium distance between a carbon atom and a lat monolayer is around 0.3nm (Girialco and ad 956) the van der aals orce between the inner and outer tubes us zero i the interlayer spacing is 0.3nm. in this case any increase (or decrease) in the interlayer spacing at a point will cause an attractive (or repulsive) van der aals interaction at that point and then c as deined below should be a positive number. According to the data given in Saito (00) one can ind J / m c d 9 / m 3 (3) 0 where d. 0 m. By using Eqs. (0) and () one can show that the pressure eerted on the inner and outer tubes o the carbon nanotube is such as p ( c[ w ( w ( ] and p ( c [ w ( w ( ] ()

6 Critical buckling condition ow we are able to study the elastic buckling o a double-walled carbon nanotube under aial load. By substituting Eqs. () and () into Eq. (9) we obtain the governing equation o buckling or each o the walls i.e. ( ) 8 c D (5) ( ) 8 c D (6) Buckling on the hand leads to a periodic low-amplitude rippling o the shell wall. et us assume the buckling modes are as ollows: sin sin n m π (7) sin sin n m π (8) where and are real constants m and n are respectively the wave numbers in the aial and circumerential directions. The epressions indicate that the carbon nanotubes have the buckling modes with sinusoidal wave pattern both in the aial and circumerential directions. By substituting Eqs. (7) and (8) into Eqs. (5) and (6) respectively we get ( ) c D ω ω (9) ( ) c D ω ω (30) where n mπ n mπ and mπ ω.

7 Equations (9) and (30) may be written in a matri orm i.e. D c ω ω G c D c c ω ω 0 (3) The critical buckling strain is obtained by solving the characteristic equation det G 0 or the lowest value o aial strain ε o the double-walled carbon nanotube with respect to the wave numbers m and n. umerical results and discussion DCT with dierent parameters such as inner tube radii outer tube radii and the length o the nanotube are used in the computation o the critical strain ε /(). Mechanical properties such as Young s modulus E Pa and Poisson s ratio υ 0.9 are ied at constant values throughout the computation with the thickness h m. The computations o the critical strain ε are done with cylindrical shell (CS) model with the mode wave number m n as well as molecular dynamic (MD) simulations being perormed and the results are presented in Table. Table : Comparison o critical strain between CS model and MD model Case Inner tube diameter (0-0 m) Outer tube diameter (0-0 m) ength (0-0 m) Critical strain (MD) Critical strain (CS) Dierence (%)

8 It can be seen rom the results in Table the dierence between molecular dynamic simulations and cylindrical shell model ranges rom approimately 8% to 5%. This shows that results computed by using the cylindrical shell model with small length-to-diameter ratios are not that comparable to the MD simulation results. Hence a more reined shell theory say the non-local shell or beam theory which allows or the small length scale eect should be eplored or better prediction o the critical buckling strains. eerences Donnell. H. (976) Beam Plates and Shells McGraw-Hill ew Your. Girialco. A. and ad. A. (956) Energy o cohesion compressibility and the potential energy unctions o graphite system J. Chem Phys. 5 pp Girialco. A. (99) Interaction potential or C 60 molecules J. Phys. Chem. 95 pp isio M. Akita S. and akayama Y. (005) Buckling test under aial compression or multiwall carbon nanotubes Jap. J. Appl. Phys. pp u C. Q. (000b) Eective bending stiness o carbon nanotubes Phys. ev. B 6 pp u C. Q. (000a) Eect o Van der aals orces on aial buckling o a double-walled carbon nanotube J. Appl. Phys. 87 pp u C. Q. (00) Aially compressed buckling o a double-walled carbon nanotube embedded in an elastic medium J. Mech. Phys. Solids. 9 pp Saito. Matsuo. Kimura T. Dresselhaus G. and Dresselhaus M. S. 00 Anomalous potential barrier o double-wall carbon nanotube Chem. Phys. ett. 38 pp Yakobson B. I. Brabee C. J. and Bernhole J. (996) anomechanics o carbon tubes: instability beyond linear response Phys. ev. ett. 76 pp. 5-5.