Computational modeling of the transverse-isotropic elastic properties of single-walled carbon nanotubes

Computational Materials Science 49 (2010) 544–551

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Computational modeling of the transverse-isotropic elastic properties of single-walled carbon nanotubes A. Montazeri a, M. Sadeghi a, R. Naghdabadi a,b,*, H. Rafii-Tabar c,d a

Institute for Nano Science and Technology, Sharif University of Technology, Tehran, Iran Department of Mechanical Engineering, Sharif University of Technology, P.O. Box 11155-9567, Tehran, Iran c Department of Medical Physics and Biomedical Engineering, and Research Centre for Medical Nanotechnology and Tissue Engineering, Shahid Beheshti University of Medical Sciences, Evin, Tehran, Iran d Computational Physical Sciences Research Laboratory, Department of Nano-Science, Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395-5531, Tehran, Iran b

a r t i c l e

i n f o

Article history: Received 22 February 2010 Received in revised form 29 April 2010 Accepted 25 May 2010 Available online 17 June 2010 Keywords: Single-walled carbon nanotubes Transverse-isotropic material Molecular dynamics simulation Continuum-based elasticity theory

a b s t r a c t Various experimental and theoretical investigations have been carried out to determine the elastic properties of nanotubes in the axial direction. Their behavior in transverse directions, however, has not been well studied. In this paper, a combination of molecular dynamics (MD) and continuum-based elasticity model is used to predict the transverse-isotropic elastic properties of single-walled carbon nanotubes (SWCNTs). From this modeling study, five independent elastic constants of an SWCNT in transverse directions are obtained by analyzing its deformations under four different loading conditions, namely, axial tension, torsion, uniform and non-uniform radial pressure. To find the elastic constants in the transverse directions, the strain energy due to radial pressure is calculated from the MD simulation. Then, a continuum-based model is implemented to find the relation between the strain energy and maximum pressure under these two loading conditions. Based on the energy equivalence between the MD simulation and the continuum-based model, the transverse-isotropic elastic constants of SWCNTs are computed. The effectiveness of this approach is demonstrated by comparing the results with previous experimental and computational works. Ó 2010 Elsevier B.V. All rights reserved.

1. Introduction Carbon nanotubes (CNTs) are quasi-one-dimensional nanostructures that have attracted considerable attention in the fields of mechanics, electronics, physics and chemistry. It has been theoretically and experimentally confirmed that CNTs possess exceptionally high stiffness and strength. Their stiffness and strength are in the order of 1 TPa and 200 GPa, respectively [1]. Studying the mechanical behavior of CNTs is important considering the broad field of applications predicated for them such as in nanosized strain sensors and actuators, nanofluidic components, and as reinforcement agents in nanocomposites. Various experimental and theoretical studies have been performed to determine the mechanical properties of single-walled carbon nanotubes. It is to be noted that many simulation techniques such as molecular dynamics (MD) and molecular mechanics methods, based on prescribed empirical potentials [2–4], tight-binding-based approaches [5], first principles quantum mechanical methods [6], and struc-

* Corresponding author at: Department of Mechanical Engineering, Sharif University of Technology, P.O. Box 11155-9567, Tehran, Iran. Tel.: +98 21 6616 5546; fax: +98 6600 0021. E-mail address: [email protected] (R. Naghdabadi). 0927-0256/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.commatsci.2010.05.047

tural mechanics models [7,8] have been used to predict the isotropic elastic properties, such as Young’s modulus, major Poisson’s ratio and shear modulus of SWCNTs [9]. The in-plane stiffness of graphite has been determined experimentally to be 1.06 TPa, whereas its stiffness in the direction perpendicular to the layers is only 0.036 TPa [10]. Similarly, the transverse Young’s modulus of a CNT is expected to be much smaller than its axial one. While the multi-walled carbon nanotubes have an axial Young’s modulus of 2 TPa [11], they show reduced stiffness in their transverse directions [12]. SWCNTs are even softer than multi-walled nanotubes in transverse directions [13]. TEM observations show that when two nanotubes are brought close to each other, the contact area is flattened due to van der Waals forces between them [14]. Lordi and Yao [15] utilized HRTEM in tandem with MD simulations to study the mechanical force–displacement relationship of SWCNTs. Their results revealed that large radial compressions could be induced by a small force on nanotubes. Relatively few studies have been devoted to analyze the elastic behavior of these nanostructures in transverse directions. Lu [10] used an empirical force-constant model and Lennard–Jones potential to investigate the effect of radius, helicity and number of walls on the transverse-elastic properties of carbon nanotubes and nanoropes. It has been proposed that CNTs can be modeled as

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transverse-isotropic materials [16–20]. Shen and Li [19,20] found closed-form solutions for five independent elastic constants of a carbon nanotube as a transverse-isotropic material. In their work, an energy approach based on molecular mechanics was used to evaluate the local and global deformations of SWCNTs and MWCNTs. This was carried out under four loading conditions, namely axial tension, torsion, in-plane biaxial tension, and in-plane pure shear, and yielded closed-form expressions for the longitudinal Young’s modulus, major Poisson’s ratio, longitudinal shear modulus, plane strain bulk modulus, and in-plane shear modulus. In addition, Jin and Yuan [21] used MD simulations to study the transverse-elastic properties of SWCNTs. Using energy and force approaches, they calculated the transverse elastic constants of SWCNTs. Moreover, Wang et al. [22] determined the five elastic constants via a thin-shell model together with MD simulations and showed that these transverse-elastic properties significantly depend on the size and chirality of the CNT. To properly use CNTs in nanoelectromechanical and nanoelectronic systems, knowledge of their transverse deformability is as important as that of their longitudinal properties [23]. It has been shown that the radial deformation of CNTs may strongly affect their electrical properties [24,25]. Also, the elastic properties of a CNT in the transverse directions significantly affect the mechanical integrity of nanowire templates, hydrogen containers, nanogears, and nanofluidic devices using CNTs as structural components [26]. Furthermore, the transverse elasticity of CNTs plays an important role in the interfacial stresses and failure behavior of CNTreinforced nanocomposites. It is to be noted that because of their high strength and stiffness, as well as high aspect ratio and low density, carbon nanotubes are considered as ideal reinforcement elements in the new generation composites. The proper use of CNTs as reinforcement agents in composites with anisotropic properties and subject to loading conditions in different directions requires data on mechanical properties of CNTs in all directions, especially the transverse ones, and not just in the axial directions. Our motivation has been to derive the mechanical properties of nanotubes in transverse directions so that when these tubes are used as reinforcement elements in nanocomposites under various types of applied loads, the appropriate directional properties of CNTs can be included in the computations. The objective of this paper is to formulate a transverse-isotropic elastic model of SWCNTs that combines methods from continuum elasticity theory and molecular dynamics simulation. This model is employed to predict the transverse-elastic properties of SWCNTs. To achieve this end, MD simulations are used to model the mechanical behavior of SWCNTs under axial, torsional and radial loadings. Also, continuum-based models using the linear elasticity theory are employed to model the mechanical behavior of SWCNTs under these loading conditions. The methodology developed herein combines a unit cell continuum model with MD simulations to determine the transverse-isotropic elastic constants of SWCNTs. The predicted elastic constants are compared with the available published data.

Adopting the coordinate system shown in Fig. 1 for an SWCNT, the strain–stress relations can be written in the matrix from:

8 exx 9 > > > > > > eyy > > > > > > > > > > < =

2

S11 6S 6 12 6 6 S13 ezz ¼6 > > 6 eyz > > 6 0 > > > > 6 > > ezx > > 4 0 > > > > : ; exy 0

S12 S11

S13 S13

0 0

0 0

0 0

S13 0

S33

0

0

0

0

S44 =2

0

0

0

0

0

S44 =2

0

0

0

0

0

S11  S12

ð1Þ

rxy

Regarding the Poisson’s ratio, Young’s modulus and shear modulus in the longitudinal direction and the Poisson’s ratio and Young’s modulus in the transverse plane; namely m, E, G, m0 , and E0 , respectively, the following relations for the engineering elastic constants of a transverse-isotropic solid are obtained:

S11 ¼

1 E0

S12 ¼ 

m0 E0

S13 ¼ 

m E

S33 ¼

1 E

S44 ¼

1 G

ð2Þ

These are the five elastic constants to be determined. To compute these constants, MD simulations were performed under four different loading conditions, namely; axial tension, torsion, uniform and non-uniform radial pressure as schematically shown in Fig. 2. As was previously mentioned, Shen and Li [19,20] developed an energy approach in the framework of molecular mechanics to derive a set of closed-form expressions for local and global deformations of CNTs. To obtain the transverse properties, they used in-plane biaxial tension and in-plane pure shear tests. Here, a new method is proposed based on the combination of MD simulation and continuum-based elasticity theory to determine the SWCNT transverse elastic constants. The model is based on the Cauchy–Born rule, which relates the deformation behavior of a continuum to the deformation behavior of the crystal lattice of a material [9]. Each point in the continuum is enveloped within a representative cell in which the deformation is uniform, and the strain-energy density in the continuum level is computed by summing the energies of all the interatomic bonds contained within that cell, after the deformation is applied. The methodology involves applying a uniform and non-uniform radial pressure to an SWCNT and then computing the potential energy due to interatomic interactions as a function of the imposed pressure. A direct transformation to continuum properties is then made by assuming that the potential energy density of discrete atomic interactions is equal to the strain-energy density of the continuous substance occupying the volume of the SWCNT. In this way, the relations between the imposed pressure and the strain energy of the equivalent solid are obtained. These relations yield the Poisson’s ratio

2. Modeling 2.1. Analysis methodology As discussed before, SWCNTs exhibit transverse-isotropic properties in the plane normal to their longitudinal axis. Transverseisotropic materials possess a unique axis about which the material’s elastic properties are independent of direction [27]. With this type of material symmetry, the number of independent elastic constants in the elasticity tensor is reduced to 5 from a total of 21 independent constants in the case of a fully anisotropic solid.

38 rxx 9 > > > > > 7> > ryy > > > > 7> > > > 7> 7< rzz = 7 7> r > 7> > yz > > > 7> > > > 5> > > rzx > > : ;

Fig. 1. Nanotube coordinate system.

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P

Mt

Mt P

(a)

(b) P0

P0 P=P0 sin2θ

Fig. 3. Symmetric arrangement of C–C covalence bonds about the axial direction for armchair and zigzag CNTs ((a) and (b)) and asymmetric arrangement for chiral ones (c).

(c)

(d)

Fig. 2. A schematic illustration of the four loading conditions of SWCNTs: (a) axial tension, (b) torsion, (c) uniform radial pressure (end view), and (d) non-uniform radial pressure (end view).

and Young’s modulus in the transverse plane; namely m0 , E0 , respectively. It is to be pointed out that few models have been developed for investigating the elastic properties of transverse-isotropic SWCNTs [16–22]. In most of these models, the atomic scale deformations of nanotubes in transverse directions have not been considered [16– 20]. Among these models, only the works of Jin and Yuan [21] and Wang et al. [22] were devoted to the analysis of deformation of CNTs at atomic scales using molecular dynamics simulations. But, it is to be noted that in [21,22], only the circumferential direction of the CNT was considered as the transverse direction. Hence, the calculated elastic constants could not be used when nanotubes are subjected to a general loading condition. In the present model, the transverse-isotropic elastic constants of SWCNTs were computed based on the elasticity definition of the transverse-isotropic material, and not in the circumferential direction only as defined in the previous works [21,22]. As discussed before, materials with a unique longitudinal axis of symmetry are called transverse-isotropic materials. As illustrated in Fig. 3, carbon nanotubes are divided to three types based on their atomic structures, i.e. armchair, zigzag and chiral nanotubes. As can be seen in the figure, only armchair and zigzag nanotubes are symmetric about the axial direction and thus they exhibit transverse-isotropic behavior in the plane normal to the longitudinal axis. Hence, the strain–stress relations of these types of carbon nanotubes are defined by five independent elastic constants. Meanwhile, in the case of chiral nanotubes, the C–C covalent bonds are not symmetric about the longitudinal axis. For these CNTs, there exist no material symmetry about the axial direction and their mechanical properties are fully anisotropic. Therefore, the chiral nanotubes cannot be modeled by transverse-isotropic models.

2.2. Molecular dynamics simulation Here, according to many previous studies of the mechanical properties of CNTs [28–33], the second-generation Brenner potential [34] is used to describe the short-range sp2 covalent bonding between carbon atoms in an SWCNT. This potential provides a significantly better description of binding energies, lattice constants as well as chemical and mechanical properties of solid state carbon molecules compared with its previous generation [35]. The Brenner potential is given by [34]:

Eb ¼

XX R  V A ðr Þ ½V ðrij Þ  b ij ij i

ð3Þ

jð>iÞ

where the functions VR(rij) and VA(rij) denote the repulsive and attractive pairwise interactions, respectively, and rij is the distance  between pairs of nearest neighbouring atoms i and j. The term b ij represents the reactive empirical bond order depending on local bonding environment. In the second-generation potential, the forms

V R ðrÞ ¼ fc ðrÞð1 þ Q=rÞA expðarÞ X Bn expðbn rÞ V A ðrÞ ¼ fc ðrÞ

ð4Þ ð5Þ

n¼1;3

ij is lengthy and are used for the pairwise terms. The expression for b therefore is not reproduced here. Detailed explanation about the terms of the second-generation Brenner potential including the necessary parameters of the pairwise terms are listed in Ref. [34]. The velocity Verlet algorithm [36] was used to integrate the equations of motion with a time step of dt = 1 fs. This time step guarantees good conservation of energy. Also, it provides a good balance between accuracy and computational costs. The Nose– Hoover thermostat [37] was used to maintain the simulation temperature at T = 300 K. The use of this thermostat leads to less fluctuations in the temperature [32]. The important issue is that the initial configuration of the SWCNT in the MD simulation may not correspond to an equilibrium state of the system. Thus, before applying the external loads to produce the desired deformations,

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a relaxation process was implemented in which the SWCNT was initially relaxed at T = 300 K by minimizing the total potential energy of the entire nanotube [30]. Finally, with appropriate boundary conditions, different mechanical loadings, including axial tension, torsion, uniform and non-uniform radial pressure were utilized to obtain the five independent transverse-isotropic elastic constants. 2.3. Continuum model The main goal of this work is to present a new combined MD/ continuum model to calculate the Poisson’s ratio and Young’s modulus of SWCNTs in the transverse plane; namely m0 , E0 , respectively. As was previously mentioned, it is necessary to calculate the strain-energy density due to application of the uniform and nonuniform radial pressure on an SWCNT from the equivalent continuum model. In this section, the relations between the imposed pressure and the strain-energy density in both uniform and nonuniform radial pressure loading conditions are formulated based on the linear elasticity theory. 2.3.1. Uniform radial pressure Consider the equivalent continuum model of an SWCNT as a solid cylinder with radius R subject to a uniform external radial pressure P0 (Fig. 2c). In cylindrical coordinate system wherein the zaxis corresponds to the axial direction of the SWCNT, the stress components can be written as follows. The normal stresses in r and h coordinates are:

rrr ¼ rhh ¼ P0

rzz ¼ mðrrr þ rhh Þ ¼ 

S13 2S13 ðrrr þ rhh Þ ¼ P0 S33 S33

ð7Þ

In addition, in the case of a uniform radial pressure, all shear stress terms vanish.

rrz ¼ rhz ¼ rrh ¼ 0

ð8Þ

The strain-energy density of a solid is calculated as:

1 Sab ra rb 2

a; b ¼ 1; 2; . . . ; 6

ð9Þ

where Sab’s are the compliance coefficients given in (1) and ra ’s are given in contracted notations as:

r1 ¼ rrr ; r2 ¼ rhh ; r3 ¼ rzz ; r4 ¼ rrz ; r5 ¼ rhz ; r6 ¼ rrh e1 ¼ err ; e2 ¼ ehh ; e3 ¼ ezz ; e4 ¼ erz ; e5 ¼ ehz ; e6 ¼ erh ð10Þ Substituting the non-vanishing stress terms from (6) to (7) into (9), the strain-energy density can be determined for a transverse-isotropic material under uniform radial pressure as:

u¼

Ujuniform press: ¼ pR2 L

1 1 S11 r2rr þ S12 rrr rhh þ S13 rrr rzz þ S11 r2hh þ S13 rhh rzz 2 " # 2 1 S213 2 2 P þ S33 rzz ¼ S11 þ S12  2 2 S33 0

ð11Þ

Thus, the total strain energy (U) of a SWCNT due to the uniform radial pressure P0 is given by:

" Ujuniform press: ¼ pR2 L S11 þ S12  2

# S213 2 P S33 0

ð12Þ

where R and L are the radius and the length of the carbon nanotube respectively. Using (2), this strain energy can be rewritten as:

#

1 m0 ðm=EÞ2 2 P0 0  0 2 1=E E E

ð13Þ

2.3.2. Non-uniform radial pressure In a similar manner, the strain energy due to a non-uniform radial pressure (P = P0 sin2 h) is calculated here (Fig. 2d). Toward this goal, Michell solution which is the most general solution of the elasticity equations in polar coordinates is used [38]. According to the loading conditions in this problem, the Airy stress function is chosen in the form:

/ ¼ ½C 1 r 2 þ C 2 r 4  cos 2h þ C 3 r 2 ;

ð14Þ

where C1, C2, and C3 are constants to be determined. The stress components can be obtained by substituting the Airy stress function into the stress equations in the cylindrical coordinates [38]

1 @/ 1 @ 2 / þ ¼ ½2C 1  cos 2h þ 2C 3 ; r @r r 2 @h2 @2/ rhh ¼ 2 ¼ ½2C 1 þ 12C 2 r2  cos 2h þ 2C 3 ; @r 1 @/ 1 @ 2 / ¼ ½2C 1 þ 6C 2 r 2  sin 2h: rrh ¼ 2  r @h r @r@h

rrr ¼

ð15Þ ð16Þ ð17Þ

Applying the boundary conditions:

rrrðr¼aÞ ¼ P0 sin2 h ¼

P0 ðcos 2h  1Þ and 2

rrhðr¼aÞ ¼ 0

ð18Þ

the constants in the Airy stress function (14) are determined as:

ð6Þ

Also, due to the large length of the SWCNT, the axial strain is zero and this qualifies for a plane strain assumption [38]. Thus, the normal stress in the z-direction is:

u¼

"

C1 ¼ 

P0 ; 4

C2 ¼

P0 ; 12a2

and C 3 ¼

P0 4

ð19Þ

Substituting these in (15)–(17), the stress components are obtained as:

P0 2 ðcos 2h  1Þ ¼ P0 sin h ¼ r1 2     r 2 P rhh ¼ 0 2  1 cos 2h  1 ¼ r2 a 2   P 0  r 2 rrh ¼  1 sin 2h ¼ r6 2 a    r 2 S rzz ¼ mðrrr þ rhh Þ ¼  13 P0 cos 2h  1 ¼ r3 a S33

rrr ¼

ð20Þ

Other stress components (rrz = r4 = 0 and rhz = r5 = 0) vanish. From (9), the strain-energy density in case of a non-uniform radial pressure for a transverse-isotropic material like an SWCNT is obtained in the form:

u¼

1 1 S11 r2rr þ S12 rrr rhh þ S13 rrr rzz þ S11 r2hh þ S13 rhh rzz 2 2 1 þ S33 r2zz þ ðS11  S12 Þr2rh 2

ð21Þ

Substituting the stress components from (20) into (21) and integrating over the volume, the total strain energy of the SWCNT is found as:

Z 2p Z R u dV ¼ L ur dr dh 0 0 " # pR2 L S213 2 P ¼ 9S11 þ 5S12  14 24 S33 0

Ujnonuniform press: ¼

Z

ð22Þ

This can be rewritten in terms of the elastic moduli constants from (2) as:

Ujnonuniform press: ¼

pR2 L 24

" 9

# 1 m0 ðm=EÞ2 2 P0  5  14 1=E E0 E0

ð23Þ

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3. Results and discussion Based on the aforementioned molecular dynamics simulation and the continuum elasticity model, the five independent elastic moduli of a (10,10) armchair SWCNT consisting of 1600 atoms at a length of 97.3 A under four prescribed loading conditions were determined. First, based on the MD simulation, the longitudinal Young’s modulus, shear modulus and the major Poisson’s ratio were calculated and verified with the existing results. In the next step, a combination of the MD simulation and the continuum model was used to obtain the Young’s modulus and Poisson’s ratio in the transverse plane of the SWCNT. 3.1. Longitudinal Young’s modulus and major Poisson’s ratio To obtain the longitudinal Young’s modulus (E) and major Poisson’s ratio (m) of the SWCNT, the axial tension loading was used as shown in Fig. 2a. Initially, to equilibrate the system at the specified temperature, all atoms except those on the fixed ends were allowed to move freely for about 2 ps. The tension was then imposed by prescribing an incremental axial displacement of 0.0486 A to the atoms at one end of the SWCNT while the atoms at the other end were fixed. Following this, both ends of the nanotube were fixed again and the remaining atoms were free to evolve in time until the new equilibrium configuration was reached [30]. The axial force needed to impose such a tension could be calculated from the MD simulation. Repeating these processes for 25 steps, gave the axial force vs. displacement curve for the total strain of 1.25%  (Fig. 4). The slope of this curve Pd was used to calculate the longitudinal Young’s modulus (E) as follows:

E¼

P L ¼ 825 GPa dA

ð24Þ

where L and A denote the length and cross-sectional area of the SWCNT respectively. To calculate the cross-sectional area, the interlayer separation distance of graphite, which is 0.34 nm was used as the effective wall thickness of the SWCNT. In addition, the major Poisson’s ratio (m) was determined to be:

m¼

DR=R0

eL

¼ 0:276

of the tube thickness. Good agreement with existing numerical results for the elastic properties of various SWCNTs with tube diameters from 0.5 to 2 nm validates the present analysis. 3.2. Longitudinal shear modulus To find the shear modulus of the SWCNT, torsional loading was used as shown in Fig. 2b. After equilibration for 2 ps, this loading was imposed on the nanotube through a rotation. During the process, keeping the four bottom rings of atoms fixed in position, the torsion of SWCNT was achieved by rotating the four top rings of atoms about the tube axis at a constant increment of 0.25 [31]. After each loading, the entire nanotube was fully relaxed for 2 ps while the ends of the SWCNT were restrained. Then, using the MD simulation, the total torque (Mt) needed to apply such a rotation could be determined by summing the product of the tangential force on each rotated atom and the respective radial distance of the atom from the tube axis. Finally, the longitudinal shear stress (G) of the nanotube can be obtained from:

G¼

M t Leff U J

ð26Þ

The relation between Mt and U can be found by a linear regression of the MD results, as can be seen in Fig. 5. Also, J, the polar moment of inertia of SWCNTs, is defined as:

J¼

4

4 ! t t  R Rþ 2 2 2

p

ð27Þ

where R and t are the radius and the wall thickness of the nanotube respectively. Substituting the proper values in (26), the longitudinal shear modulus of the (10,10) SWCNT was obtained to be 352.1 GPa. Using the molecular dynamics simulation based on the Brenner second-generation potential, Yu et al. [42] calculated the shear modulus of a (10,10) SWCNT to be 370 GPa. Gupta et al. [43] also used the MD method with Tersoff–Brenner potential [44] to study the elastic constants of SWCNTs. They found the shear modulus for a (10,10) carbon nanotube to be 344.3 GPa. It can be seen that our result are in good agreement with the available data.

ð25Þ

where eL is the longitudinal strain equal to 0.0125, R0 is the initial radius of the SWCNT in the equilibrium state after relaxation, and DR is the change of radius in the tensile loading obtained from the molecular dynamics simulation. Cornwell and Wille [39] obtained the longitudinal Young’s modulus of (10,10) SWCNTs to be 0.8 TPa via an MD simulation based on the Tersoff–Brenner potential. Using a combination of the Brenner second-generation potential and van der Waals interactions between the carbon atoms, Agrawal et al. [3] computed the Young’s modulus of armchair SWCNTs and reported the value of 0.76 TPa. In another MD simulation, using the Brenner secondgeneration potential, Mylvaganam and Zhang [40] showed that this elastic constant was determined to be 0.7 TPa for armchair nanotubes. Using a similar method, WenXing et al. [41] showed that the Young’s moduli of SWCNTs were in the range of 929.8 ± 11.5 GPa. Employing an empirical force field model based on harmonic potentials, Lu [10] obtained the Young’s modulus and major Poisson’s ratio of (10,10) SWCNT to be 0.972 TPa and 0.278, respectively. Regarding the major Poisson’s ratio, different methods have been used. Hernandez et al. [5] adopted a tightbinding method to obtain average values of Poisson’s ratio of SWCNTs with different chiralities and found the value of 0.262. Based on a link between molecular and solid mechanics, Natsuki et al. [2] calculated the major Poisson’s ratio of SWCNTs. They found this parameter to be 0.27 and showed it to be independent

3.3. Transverse Young’s modulus and Poisson’s ratio In Section 2, the linear elasticity solution of an SWCNT under a uniform and non-uniform radial pressure was presented. As can be seen from (13) and (23), the total strain energy stored in the equivalent solid during deformation is linearly proportional to the square of the maximum pressure in both cases. These linear functions are obtained using the molecular dynamics simulation in this section. Substituting E and m from (24) and (25) into (13) and (23), and implementing the values from MD, the Poisson’s ratio (m0 ) and Young’s modulus (E0 ) of the SWCNT in the transverse plane were determined. To simulate the effects of a uniform radial pressure (P0) on the radial deformation of the SWCNT in the MD simulation, radially directed external forces with the total magnitude of P0A were imposed on all CNT atoms, where A is the surface area of the SWCNT. The total pressure of 1 GPa was exerted on the (10,10) SWCNT in 10 steps. The strain energy was recorded as a function of the imposed pressure. The process of imposing the non-uniform radial pressure (P = P0 sin2 h) on the SWCNT was the same. The only difference was that the magnitude of the pressure varies for CNT atoms according to their positions with respect to the horizontal axis. It was noted that in this case, the circular cross-section of the nanotube changed to an elliptic one as depicted in Fig. 6. The linear relationships between the strain energy and the square of the maximum pressure are displayed in Fig. 7 for both cases. This

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18.000 16.000

Axial Force (nN)

14.000 12.000

y = 140.4x + 0.7 R2 = 1.0

10.000 8.000 6.000 4.000 2.000 0.000 0.0000000

0.0200000

0.0400000

0.0600000

0.0800000

0.1000000

0.1200000

Imposed Displacement (nm) Fig. 4. MD results of axial force vs. imposed displacement in the axial tension test for (10,10) SWCNT.

3.000

Torque (nN nm)

2.500 2.000

1.500

y = 28.46x + 0.18 R2 = 1.00

1.000 0.500 0.000 0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

Imposed rotation (rad) Fig. 5. MD results of torque vs. imposed rotation in the uniaxial torsion test for (10,10) SWCNT.

makes a good connection with the continuum-based modeling of the SWCNT under corresponding loadings (Eqs. (13) and (23)). As can be seen from the figure, the slopes of the strain energy vs. the square of maximum pressure are 1.93 and 0.91 for uniform and non-uniform cases, respectively. Substituting these values in (13) and (23) yields: 2  2 3 nN 0 0:276=825 2 1 m nm5 ðnmkÞ 7 pð0:678 nmÞ2 ð9:73 nmÞ6 4 0  0 2 5 ¼ 1:93 nN 1=825 nN 2 E E ðnmÞ

2

pð0:678 nmÞ2 ð9:73 nmÞ 6 1 49

24

E0

5

m0 E0

  14

nN 0:276=825 ðnmÞ 2 nN 1=825 ðnmÞ 2

2 3

ð28Þ 5

nm 7 5 ¼ 0:91 nN

ð29Þ

Solving (28) and (29) simultaneously, the transverse-elastic properties of the (10,10) SWCNT are found as: 0

E ¼ 4:54

nN 2

ðnmÞ

¼ 4:54 GPa;

0

m ¼ 0:389

ð30Þ

Using these values, the in-plane shear modulus (G0 ) of the SWCNT is calculated as follows:

G0 ¼

E0 ¼ 1:63 GPa 2ð1 þ m0 Þ

ð31Þ

There exist limited data regarding the transverse-isotropic elastic properties of SWCNTs in the literature. Yu et al. [45] conducted nanoindentation tests on MWCNTs using a tapping-mode atomic force microscope to investigate the radial deformations of these nanotubes. They found that the effective transverse modulus ranges from 0.4 to 3 GPa. Also, Xiao et al. [46] studied the deformation of singleand multi-walled carbon nanotubes under radial pressure using an analytical molecular structural mechanics model. They have pointed out that the elastic constants of large SWCNTs in the transverse (radial) direction are less than 0.0001 times those in the circumferential and the axial directions. In the most relevant work, Shen and Li [20] theoretically studied the five independent elastic moduli of singleand multi-walled carbon nanotubes under four loading conditions. They calculated the effective in-plane shear modulus ðG0 Þ of a (10,10) SWCNT as 1.669 GPa. This parameter was defined as [20]:

G0 ¼

G0  t ðR=2Þ

ð32Þ

where t and R are wall thickness and radius of the nanotube respectively and G0 is the in-plane shear modulus. Substituting the proper values for (10,10) SWCNT gave us:

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Fig. 6. The elliptical cross-section of the (10,10) SWCNT in the non-uniform radial pressure test shown by MD simulation: (a) end view and (b) side view.

(a)

2

Strain energy (nN nm)

1.8 1.6 1.4

y = 1.93x + 0.04 R2 = 1.00

1.2 1 0.8 0.6 0.4 0.2 0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.8

0.9

1

Square of radial pressure (GPa2)

(b)

1

Strain energy (nN nm)

0.9 0.8 0.7 0.6 0.5

y = 0.91x + 0.02 R2 = 1.00

0.4 0.3 0.2 0.1 0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Square of maximum pressure (GPa2) Fig. 7. MD results of total strain energy vs. the square of maximum pressure in (a) uniform and (b) non-uniform radial pressure test for (10,10) SWCNT.

G0 ¼

1:63  0:34 ¼ 1:635 GPa ð0:678=2Þ

ð33Þ

This is in good agreement with the result obtained in Ref [20]. It is noted that the closed-from expressions of the five independent moduli obtained by Shen and Li, illustrate that the in-plane shear modulus of carbon nanotubes explicitly includes the factor 1/R3. So, it is anticipated that as the tube diameter increases, the in-plane shear modulus (G0 ) abruptly decreases. To check this point, the new model presented here for the transverse-elastic properties of SWCNTs was applied to a (15,15) carbon nanotube. Repeating all

the MD simulations for this nanotube gave the value of 0.525 GPa for the effective in-plane shear modulus, in good agreement with the results of Ref. [20] for a (15,15) SWCNT which was found to be 0.496 GPa. We note that, this value is much less than the corresponding value for (10,10) SWCNT (1.635 GPa). As we have discussed above, in addition to the armchair SWCNTs, transverse-isotropic models can also be used to obtain the mechanical properties of zigzag nanotubes. To ascertain the role of SWCNT’s diameter and chirality on the transverse-isotropic elastic constants, we also selected zigzag SWCNTs having the same length (97.3 A) as the armchair (10,10) for different diameters. To

551

A. Montazeri et al. / Computational Materials Science 49 (2010) 544–551 Table 1 The variation of elastic constants of zigzag SWCNTs having different diameters. Type of SWCNT

Diameter (nm)

E (GPa)

G (GPa)

E0 (GPa)

m0

G0 (GPa)

G0 (GPa)

(10,0) (14,0) (16,0)

0.783 1.096 1.253

832.1 855.2 887.6

271.8 330.3 369.6

5.17 5.68 6.04

0.392 0.362 0.328

1.86 2.08 2.27

3.23 2.58 2.46

this end, all of the MD simulations including tensile, torsional, uniform and non-uniform radial pressure were repeated for three different zigzag SWCNTs, i.e. (10,0), (14,0) and (16,0) nanotubes. The results are listed in Table 1 and show that the elastic constants of zigzag nanotubes in the axial directions are greater than the corresponding values for armchair nanotubes having the same diameter. This result is in agreement with the available results [7,42,47,48]. Furthermore, increasing the diameter of zigzag nanotubes causes an increase in the axial elastic constants. This fact has also been shown by [7,42,47,48] and is a further confirmation of our model. Also, as can be seen in Table 1, in case of zigzag SWCNTs, there is a noticeable decrease in the effective in-plane shear modulus with increasing diameter, similar to the armchair nanotubes. The results indicate that the transverse-isotropic elastic model presented here can be successfully used to predict the mechanical properties of armchair and zigzag SWCNTs in the axial and transverse directions. 4. Conclusions In this paper, a combination of molecular dynamics simulation and continuum-based elasticity theory was used to evaluate the five independent elastic constants of single-walled carbon nanotubes. To this end, MD analyses were carried out to investigate the axial, torsional, and radial deformation characteristics of SWCNTs. Also, continuum plane strain models based on linear elastic theory were developed for SWCNTs under both uniform and non-uniform radial pressures. In such loading conditions, good agreements were found between MD and continuum studies regarding the energy–pressure relation. A linear relationship was obtained between the strain energy and square of the maximum external pressure in both MD simulation and continuum solution of SWCNTs under radial deformations from which the transverse elastic constants were derived. The results were in good agreement with existing experimental and numerical ones. The values obtained in the transverse direction, i.e. the Young’s modulus in the transverse plane (E0 ) and the in-plane shear modulus (G0 ) are much lower than the corresponding longitudinal values. The transverseelastic properties of SWCNTs obtained by this method play an essential role in the studies of the mechanical integrity of nanowire templates, nanogears and nanofluidic components based on CNTs. In addition, transverse direction elasticity of CNTs is important especially in the study of the interfacial stresses and failure behavior of nanocomposites where the embedded tubes are subjected to large deformations in the transverse direction under the applied loads exerted on the composite structure. Acknowledgment H. R-T would like to acknowledge the support of Iran National Science Foundation for a Research Chair in nanotechnology.

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