Simulation of Young's Modulus of Single-walled Carbon Nanotubes by Molecular Dynamics

5/22/2018 Simulation of Young's Modulus of Single-walled Carbon Nanotubes by Molecular Dynamics

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Physica B 352 (2004) 156163

Simulation of Youngs modulus of single-walled carbon

nanotubes by molecular dynamics

Bao WenXinga,b,, Zhu ChangChuna, Cui WanZhaoa

aSchool of Electronics & Information Engineering, Xian Jiaotong University, Xian 710049, PR ChinabThe Second North-west Institute for Ethnic Minorities, Yinchuan 750021, PR China

Received 15 April 2004; accepted 13 July 2004

Abstract

Based on molecular dynamics (MD) simulation, the Youngs moduli of carbon nanotubes are studied. The inter-

atomic short-range interaction and long-range interaction of carbon nanotubes are represented by a second generation

reactive empirical bond order (REBO) potential and LennardJones (LJ) potential, respectively. The obtained potential

expression is used to calculate the total potential energies of carbon nanotubes. Three types of single-walled carbon

nanotubes (SWCNTs), armchair, zigzag and chiral tubules, are calculated, respectively. The computational results show

that the Youngs moduli of SWCNTs are in the range of 929.8711.5 GPa. From the simulation, the Youngs moduli of

SWCNTs are weakly affected by the tube chirality and tube radius. The numeric results are in good agreement with theexisting experimental results.

r 2004 Elsevier B.V. All rights reserved.

PACS: 61.48.+c; 62.20.Dc; 31.15.Qg

Keywords:Carbon nanotubes; Molecular dynamics; Youngs modulus; Potential function

1. Introduction

Since the first report of carbon nanotubes

(CNTs) by Iijima [1], CNTs have got a lot ofattention because of their unique electronic and

mechanical properties. As a one-dimensional

structure, CNTs can be thought of as one sheet

or multiple sheets of graphene rolled into a

cylinder. They have single or multiple layers ofcarbon atoms in the tube thickness direction,

called single-walled carbon nanotubes (SWCNTs)

or multi-walled carbon nanotubes (MWCNTs),

respectively. According to different chiral angles,

SWCNTs can be classified into zigzag (y=01),

armchair (y=301) and chiral tubule (01oyo

301) [2].

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www.elsevier.com/locate/physb

0921-4526/$ - see front matterr 2004 Elsevier B.V. All rights reserved.

doi:10.1016/j.physb.2004.07.005

Corresponding author. Institute of Vacuum Microelectro-

nics & Microelectromechanical System, School of Electronics &

Information Engineering, Xian Jiaotong University, Xian

710049, PR China. Tel.: +86-29-82668644; fax: +86-29-

82663957.

E-mail address: [email protected] (B. WenXing).
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There are some experimental studies of the

Youngs modulus of CNTs. Liu et al. [3]reported

the Youngs modulus of CNTs is 10.1 TPa with

the diameter increasing from 8 to 40nm bymeasuring resonance frequency of carbon nano-

tubes. Krishnan et al.[4] used TEM to observe the

vibration of an SWCNT at room temperature and

reported Youngs modulus of SWCNTs in the

range from 0.90 to 1.70 TPa, with an average of

1.25 TPa. Tombler et al. [5]used AFM to bend an

SWCNT and reported the Youngs modulus of

SWCNTs around 1.2 TPa. Yu et al. [6]conducted

nanoscale tensile tests of SWCNT ropes pulled by

AFM tips under a scanning electron microscope

and reported the Youngs modulus of SWCNT

ropes ranged from 0.32 to 1.47 TPa (mean

1002GPa). Demczyk et al. [7] reported the

Youngs modulus of MWCNTs range from 0.8

to 0.9 TPa used TEM to bend an individual tubes.

There are also some theoretical studies for

predicting the Youngs modulus of CNTs. Lu [8]

used an empirical force-constant model to deter-

mine several elastic moduli of SWCNTs and

MWCNTs and obtained the Youngs modulus of

about 1 TPa. Li et al.[9] used molecular structural

mechanics method and obtained the Youngs

modulus of 1050750 GPa for MWCNTs. Jin etal.[10]used MD and force-constant approach and

reported the Youngs modulus of SWCNTs to be

about 123677 GPa. Cornwell and Wille [11] used

the MD with the TersoffBrenner potential[12]to

obtain the Youngs modulus of SWCNTs about

0.8 TPa. Zhou et al. [13] used first principles and

reported the Youngs modulus of SWCNTs to be

0.76 TPa. Vodenitcharova et al. [14] used con-

tinuum mechanics model to obtain the Youngs

modulus of SWCNTs about 4.88 TPa.

The objectives of this paper are to compute theYoungs modulus of SWCNTs in detail using MD

simulation. MD simulations are the powerful tools

for investigation of the mechanical properties and

structural formation of carbon nanotubes. The key

to the MD simulation is to choose an exactly

potential energy model. A modified empirical

potential energy model is presented to describe

the bonding and nonbonding interaction. The

effects of carbon nanotubes structure, such as

tube radius and chirality on Youngs modulus of

SWCNTs, are examined. The accuracy and

stability of this method have been verified through

its application to several single layer and multi-

layer graphene sheets with different model sizes.The computational results of the Youngs modulus

of SWCNTs are in good agreement with existing

experimental values. The organization of this

paper is as follows. In the second section, MD

simulation method and Verlet algorithm [15] are

described. The total potential energies and inter-

atomic forces are described in Section 3. The

expression of Youngs moduli of CNTs and

graphite are presented in Section 4. Results and

discussion are given in Section 5. The final section

is the conclusions.

2. Molecular dynamics simulation method

In order to compare theoretical results with

experimental data, the ultimately investigated

model structures should represent stable or meta-

stable minimal energy configurations, which have

to be obtained by finite-temperature structure

optimization. The MD method has to rely on a

mathematical description of the total energy of thesystem as a function of all atomic coordinates.

Focusing on the MD method, the Newtons

equation of motions for all atoms in the structure

has to be solved,

ma md2ri

dt2

Xjai

frij; 1

where a is the acceleration of particle i, ri is the

position of particle i, rij is the distance between

particleiand particlej,fis the total force acting onthe particleiand determined by the gradient of the

total potential energy of the system,f=rEtot. In

order to achieve control of the energy conserva-

tion, it would be desirable to determine the atomic

positions and velocities at each simulation time

step. Standard Verlet algorithm has the obvious

disadvantage that the positions and velocities are

not synchronized. Swope presented a method of

directly getting the positions and velocities at each

simulation time step , and provided an easy way of

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calculating the velocities into Verlet algorithm[16],

rth rt hvt h2at=2; 2

vth vt h ath at =2; 3

where r is the position, v is the velocity, a is the

acceleration at time t and time t+h, respectively,

andh is the time step. As can be seen from Eq. (3),

to calculate the new velocities requires the accel-

erations at time t and time t+h, respectively.

Before going into any MD simulation, the basic

principles of the calculation of the total potential

energies and inter-atomic forces have to be described.

3. Total potential energies and inter-atomic forces

The reliability of MD simulations techniques

depend on the use of appropriate inter-atomic

energies and forces. Inter-atomic energies and forces

were calculated generally using an empirical bond-

order potential that describes the covalent bonding

within both small hydrocarbon molecules and

CNTs. It has been successfully used to study the

structure and mechanical properties of CNTs

[12,1719]. However, the empirical bond-order

potential is short-ranged and does not include

long-range nonbond interactions. For instance,

long-range nonbond interactions can be either

van-der-Waals or Coulomb interactions, or both.

Van-der-Waals interactions normally have a weak

influence on the overall mechanical behavior among

the atomic interactions of the CNTs [20]. A long-

range LJ potential is added to model the van-der-

Waals interactions that exist in the SWCNTs. It

becomes activated only after the short-ranged

REBO potential goes to zero. The total potential

energy of the system may be described by summingbonding energy and nonbonding energy, Etot=E-

bonding+Enonbonding. For CNTs and graphite, the

bonding term can be described precisely by REBO

potential, and the nonbonding term is mainly the

energy of van-der-Waals force.

The van-der-Waals force is most often modeled

using the LJ 612 potential function [21]:

ELJr 4 s

r

6

s

r

12 ; 4

whereris the distance between interacting atoms,and s are the LJ parameters, for graphite and

CNTs, 0:00239 eV and s 0:341 nm [22],

respectively. The cutoff distance of LJ potentialis 2:5s; no interactions are evaluated beyond thisdistance.

Brenner (2002) [23] presented a second-genera-

tion reactive empirical bond order (REBO) poten-

tial energy expression for hydrocarbons. In the

REBO potential, the total potential energy of the

system is given by

EREBO X

i

Xji1

ERrij bijEArij

; 5

where rij

is the distance between pairs of nearest-

nearby atoms iandj, bijis a many-bond empirical

bond-order term between atoms i and j that is

derivable from Huckel electronic structure theory.

ER and EA are the repulsive and attractive pair

terms, respectively,

ERr fcr1Q

rA exp ar; 6

EAr fcrX3n1

Bnebnr; 7

where the parameters Q, A, a; B, b are used to fitthe pairs terms, the values for all the parameters

are given in Ref [23]. The function fc is a cutoff

function, and is given by

fcr

1; roRmin;

1cos prRmin

RmaxRmin

h in o=2; RminoroRmax;

0; r4Rmax

8>:

8

where RmaxRmin defines the distance over the

function goes from one to zero, for graphiteand CNTs, Rmax=0.26 nm and Rmin=0.17 nm,

respectively.

4. Expression of Youngs modulus

Youngs modulus is one of the important

characterizations of mechanical properties of

materials. In classical mechanics the Youngs

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modulus is defined as

Y s

F=A

DL=L0; 9

where s is the axial stress, is the strain, F is theaxial tensile force acted on the object, A is the

cross-sectional area of object, L0 is the initial

length and DL is the elongation under the force F.

The load conditions of CNTs and graphite are

illustrated inFig. 1,respectively.After putting the tensile force on CNTs or

graphite in axial direction and calculating the

variation of the total potential and length, we can

get the Youngs moduli of CNTs or graphite.

5. Results and discussion

In this study, Youngs moduli of graphite and

SWCNTs are simulated using our molecular

dynamics simulation program. The MD simula-

tion program can run on FreeBSD 4.6 and

Windows 2000 platform. The simulation time step

is 1 fs. The Langevin thermostats are applied tosimulate and maintain temperature at 300 K. MD

simulation was performing approximately 5000

steps when the energy attained a constant mini-

mum value. A periodic boundary condition has

been used along the axial direction to mode the

tubes under axial loading.

5.1. Youngs modulus of graphite

The Youngs modulus of graphite is calculated

to verify the feasibility of the present method.

These calculations can provide useful information

for CNTs. In Eq. (9) the crossing-sectional area of

graphite is expressed as A NWdR; Nis numberof graphene layers, W is width of graphite, the

interspacing between graphene layers dR are

assumed to be 0.34 nm.

The computational results of Youngs moduli of

graphite are showed in Tables 1 and 2, respec-

tively. There is a negligible difference of Youngs

modulus in Table 1 between the single layer of

graphite and multi-layers of graphite, and the

results inTable 2are weakly affected by model sizeof graphite. The average of Youngs modulus of

graphite is 1026.176 GPa, it is very close to the

experimental value of 1025.0 GPa [24,25]. Fig. 2

shows the variation of Youngs modulus of

graphite with number of graphene layers.

5.2. Youngs modulus of SWCNTs

Based on this method, the Youngs moduli of

SWCNTs are studied. Eq. (9) is still used for

calculating the Youngs modulus, the cross-sec-tional area A of SWCNTs is 2pRdR; where Rstands for the tube radius and dR stands for the

wall thickness with a value of 0.34 nm[5,8]. Three

types of SWCNTs, armchair, zigzag and chirality,

are considered. By scaling atomic positions along

the axial direction, the variation of the total energy

and variation of tube length are calculated.

The stressstrain curve of armchair (8, 8)

SWCNTs is obtained and given in Fig. 3. It is

seen in the figure that the elastic limit is at the

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Fig. 1. Load conditions of graphite and SWCNTs.



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strain value of 0.08. Beyond the elastic limit, the

stress-strain curve deviate from a straight line.

Computational results of Youngs modulus of

SWCNTs for different tube radius are shown in

Fig. 4. There is a slight difference between the

trends of variation in Youngs modulus of arm-

chair and zigzag SWCNTs inFig. 4.The Youngsmodulus of chiral SWCNTs is slightly lower than

those values of armchair and zigzag. The differ-

ence between maximum and minmum Youngs

modulus values is about 30 GPa. The effect of the

chiral angle on the Youngs modulus of SWCNTs

is shown in Fig. 5. With the variation of chiral

angles, the Youngs modulus value is slightly

fluctuated from 904 to 935 GPa. The effect of tube

radius and tube chirality on Youngs modulus of

SWCNTs is very weak.

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Table 1

The variation of Youngs modulus of Graphite with number of

graphene layers

Number ofgraphene layers

Y (GPa) (presentprediction)

Y (GPa)(experimental

value)

1 1030.530

2 1032.033

3 1033.771 1025.000

4 1022.156

5 1012.390

Average 1026.176

Model size: height is 10.508 nm, width is 1.107 nm.

Table 2The variation of Youngs modulus of graphite with model size

Height (nm) Width (nm) Y (GPa)

16.898 2.336 1012.230

8.378 0.861 1061.025

33.938 1.107 1036.707

16.898 1.107 1033.246

10.508 1.107 1030.530

Fig. 2. Youngs modulus of graphite.

Fig. 3. The curve of variation of axial stresss with the axialstrain (armchair (8,8)).

Fig. 4. Youngs modulus of SWCNTs for different tube radius.



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In order to compare numerical results with

other predictions of Youngs modulus, all the

computational results in this study are given in

Tables 3 and 4, respectively. Table 3 lists thevalues of Youngs modulus of armchair, zigzag,

chiral SWCNTs with different tube radius. Table 4

lists the values of Youngs modulus for different

chiral angers. The present approach gives slightly

lower values of Youngs modulus than those of

reported in [810,26]. The average of Youngs

modulus of SWCNTs is 929.8711.5 GPa and

agrees very well with the existing experiment

values [35]. Comparisons between numerical

results and experimental results and other theore-

tical prediction of Youngs modulus of SWCNTs

listed inTable 5.

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Fig. 5. The effect of the chiral angle on the Youngs modulus ofSWCNTs.

Table 3

Youngs modulus of SWCNTs for different tube radius

(n, n) Number of atoms Radius (nm) Length (nm) Y (GPa)

Armchair

(8,8) 1168 0.542 8.854 934.960

(10,10) 1460 0.678 8.854 935.470

(12,12) 1752 0.814 8.854 935.462

(14,14) 2324 0.949 10.084 935.454

(16,16) 3040 1.085 11.560 939.515

(18,18) 3924 1.220 13.281 934.727

(20,20) 5000 1.356 15.250 935.048

Average 935.80570.618

Zigzag

(14,0) 840 0.548 6.230 939.032

(17,0) 1360 0.665 8.362 938.553

(21,0) 1890 0.822 9.428 936.936

(24,0) 2400 0.939 10.500 934.201

(28,0) 3080 1.096 11.563 932.626

(31,0) 3720 1.213 12.621 932.598

(35,0) 4900 1.370 14.757 933.061

Average 935.28772.887

Chirality

(12,6) 1344 0.525 9.023 927.671

(14,6) 1896 0.696 11.367 921.616

(16,8) 2240 0.828 11.279 928.013

(18,9) 2520 0.932 11.279 927.113

(20,12) 3920 1.096 14.921 904.353

(24,11) 3844 1.213 13.215 910.605

(30,8) 4816 1.358 14.792 908.792

Average 918.309710.392

Average 929.8711.5

The length-to-radius ratio is greater than 10.



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As can be seen inTable 5, the values of Youngs

modulus obtained by different researchers is

different. The reasons and effect of geometric

structure of SWCNTS of elastic moduli can begiven in the following:

(i) Different wall thickness values were used to

calculate the Youngs moduli: in Refs.

[810,26,27] and this present prediction, dR

0:34 nm was used, and obtained the values ofYoungs modulus ranging from 0.9 to

1.26 TPa. But in Ref. [14] dR 0:0617 nmwas used, and the obtained values of Youngs

modulus was 4.88 TPa. Compared with the

CC bond length of CNTs, 0.0617 nm the

magnitude of wall thickness seems too small.

To use dR 0:34 nm calculated the values ofYoungs modulus were closed to experimental

values than Ref. [14] inTable 5.

(ii) Difference in the tube model sizes: in Refs.

[2730], the length-to-radius ratio is smaller than

10. Therefore, the edge effects may have effect

on the results. In this present case and Ref. [10],

the length-to-radius ratio is greater than 10 to

ensure that edge effects can be ignored.

(iii) Different accuracy of theoretical approach: ab

initio, DFT and tight-binding are quantum-

mechanical-based methods, the results ob-

tained are reliable than other methods, but

the computational model is only composed of

several ten to several hundred atoms [2630].

MD methods can be used to model several

thousand to several million atoms and the

results are more close to real CNT. Used

accurate potential model, the simulation

results of MD method are very closer to

experimental data.

(iv) The effect of endcap on Youngs modulus of

SWCNTs: There are two different models,

open-ended SWCNTs [810,26,28,29] andclose-capped SWCNTs[27], respectively. From

Table 5 we see that the Youngs modulus of

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Table 4

Youngs modulus of SWCNTs for different chiral angle

(n,n) Angle (deg) Radius (nm) Y (GPa)

(35,0) 0.0 1.370 933.061

(20,2) 4.715 0.825 918.019

(12,2) 7.589 0.514 920.674

(8,2) 10.893 0.359 913.530

(12,4) 14.920 0.565 919.823

(10,4) 16.100 0.489 912.852

(14,6) 17.000 0.696 921.616

(18,9) 19.106 0.933 927.113

(20,12) 21.780 1.097 904.353(24,18) 25.284 1.430 907.028

(20,20) 30.0 1.356 935.048

Table 5

Comparisons between numerical results and experiment results and other theoretical prediction of Youngs modulus of SWCNTs

Methods Wall thickness value (nm) Y (GPa) Ref.

Armchair Zigzag Chirality

MD 0.34 935.8 935.3 918.3 This present

Empirical force-constant model 0.34 972.0 975.0 973.0 [8]

Molecular structural mechanics model 0.34 1050750 1080720 [9]MD and Force approach 0.34 123677 [10]

Continuum mechanics model 0.0617 4880 [14]

Tight-binding 0.34 12201250 12201260 12401250 [26]

Tight-binding 0.28 950 [29]

Ab initio 0.34 1060 11401180 1030 [27]

Ab initio 1000 [28]

DF-TB 0.34 11002800 [30]

Measuring resonance frequencya 1001000 [3]

TEMa 9001700 [4]

AFMa 1200 [5]

aExperimental methods.



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open-ended SWCNTs in this present prediction

and other prediction is similar to close-capped

SWCNTs[27]. From Ref.[27],a tube with long

structures are expected to avoid the influenceof endcap effects.

(v) The effect of tube radius and tube chirality on

Youngs modulus of SWCNTs: most studies

[810,14,2629]show that the values of Youngs

moduli of SWCNTs are independent of tube

radius and tube chirality, except for Ref. [30].

This is also validated by this present work.

6. Conclusions

Using MD method, we study the Youngs moduli

of graphite and SWCNTs in detail. The Verlet

algorithm, which was improved by Swope in MD

method, was used to obtain atomic positions and

velocities at each simulation time step. Based on the

modified empirical potential function models, the

van-der-Waals forces in SWCNTs are taken into

account. The results show that the average of

Youngs modulus of graphite is 1026.176 GPa,

which is very well closed to the experiment value.

The average of Youngs modulus of SWCNTs is

929.8711.5 GPa, which is slightly lower than theYoungs modulus of graphite. It is also in good

agreement with the existing experimental ones. The

results show that the Youngs moduli of SWCNTs

are independent of the tube radius and tube

chirality. All these illustrate the simulation of

Youngs modulus of SWCNTs with MD and

modified potential energy model is quite workable

and reliable. It is expected that the present model

will be further developed to model other covalent-

bonded materials and analyze the mechanical

properties of nanotube-based composites.

Acknowledgements

The research is supported by the National Natural

Science foundation of China (Grant No. 60036010)

and China Doctoral foundation of National Educa-

tion Bureau (Grant No. 2000069823).

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Simulation of Young's Modulus of Single-walled Carbon Nanotubes by Molecular Dynamics