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Copyright © 2013 American Scientific PublishersAll rights reservedPrinted in the United States of America

Journal ofComputational and Theoretical Nanoscience

Vol. 10, 1921–1927, 2013

Application of Molecular Dynamics in MechanicalCharacterization of Carbon Nanocones

Mir Masoud Seyyed Fakhrabadi∗, Behnam Dadashzadeh,Vahid Norouzifard, and Akbar Allahverdizadeh

Karaj Branch, Islamic Azad University, Karaj, Iran

The paper presents the molecular dynamics simulation of the elastic, failure, buckling and vibra-tional behaviors of the carbon nanocones (CNCs) with different geometries. First, the elastic moduliof the CNCs are presented for different geometries and the effects of various loading rates andtemperature variations on the rupture points of a CNC are investigated. Second, the buckling phe-nomena of the CNCs with different geometries are studied and the effects of mentioned conditionson the critical buckling stresses of a CNC are discussed. Third, the first natural frequencies of dif-ferent CNCs are presented and the effects of temperature variations on the natural frequencies arestudied.

Keywords: Carbon Nanocones, Molecular Dynamics, Mechanical Behaviors.

1. INTRODUCTION

Investigation of different properties of the nanomaterialsand nano structures using various modeling techniquesincluding continuum, discrete and quantum approachesopened new horizons of possible applications of thenanomaterials and nano systems in novel and hightechnologies.1–8 In addition to modeling and simulationmethods, the field of nanotechnology was deepened usingnano fabrication techniques in order to bring the simula-tion results in reality.9–10 Nanotechnology in general andnano engineering in particular possess diverse aspects. Forexample, nanomaterials, nano electronics, nano mechanicscan be considered the various facets of nano engineering.Each field has its own experts and scientists and they makeattempt to study one or some aspects of nano engineeringand nano systems.11–13

The working conditions of the nanomaterials in differentenvironments and conditions require knowing their prop-erties well to conduct the desirable duties effectively.14–16

Mechanical behaviors of nanomaterials and nanostructuresare the underlying factors in designing and fabricatingnanosystems. Different mechanical characteristics of thenanomaterials in general and carbon-based nanomaterialsin particular have been studied by many researchers allover the world.Yang and Wei applied molecular dynamics to study

the effects of radius and defect on the oscillatory

∗Author to whom correspondence should be addressed.

behaviors of C60-nanotube oscillators.17 They employedthe second-generation empirical bond-order and modifiedLennard-Jones potentials to model the interaction betweencarbon atoms of the carbon nanotubes (CNTs) and C60.They presented the results for the CNTs with differentdimensions. In another study, Kang et al. investigatedthe effects of intertube gap on frequency-controlled CNToscillators.18 They modeled a nano-positioning systemincluding two outer tubes and one inter tube via moleculardynamics and studied the effects of the gap between twoouter tubes on the vibrational frequencies of the intertube.Han and Elliott used molecular dynamics to study the

elastic behaviors of the polymer/CNT composites.19 Theymodeled a CNT with polymer molecules around it andapplied a tensile force to investigate the elastic properties.The effects of various volume fractions were consideredin their paper.Predicting the elastic properties of the double-walled

CNTs was performed by Zhang and Shen using moleculardynamics simulation.20 They applied appropriate potentialsto model the bond between carbon atoms and presentedthe results for the CNTs with different characteristics. Inanother research, Wang et al. applied molecular dynam-ics to investigate twisting of the CNT.21 The ultimate twistangle per unit length and the deformation energy were esti-mated for the CNTs with different geometries. It was recog-nized that the thick CNTs were harder to be twisted whilethe thin CNTs exhibited higher ultimate twisting ratio.Yao et al. presented the mechanical properties of the

CNTs using molecular dynamics.22 They employed the

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Application of Molecular Dynamics in Mechanical Characterization of Carbon Nanocones Fakhrabadi et al.

mentioned technique to compute the elastic modulus ofthe CNT and its tensile strength. They obtained the elasticmodulus of the CNT 3.62 TPa which is not in agreementwith the literature. From other application of moleculardynamics, we may mention the work performed by Negiet al.23 They employed molecular dynamics to investigatea double-walled CNT motor subjected to a sinusoidallyvarying electric field where the inner tube behaves as ashaft and the outer one as a sleeve.Fakhrabadi et al. applied molecular mechanics approach

to study the vibrational properties of the CNTs.24 Theyused this method to investigate the effects of geometryand boundary condition variations on the natural frequen-cies of the CNTs. They also applied a neural network inorder to predict the vibrational properties of the unmod-eled CNTs. Davoodi et al. applied molecular dynamicsto investigate the Melting Transitions of the CNTs. Theresults revealed that the melting temperatures of the sin-gle wall CNTs increased with the increment in the size oftheir radii, but this dependence was not the same for thevarious chiral angles of the CNTs.25

In another research, Khan et al. studied the dynam-ical response of the nonlinear vibration of the singlewall CNTs.26 They showed that for various values ofthe parameters a chaotic behavior was observed. Apply-ing the asymptotic methods of Bogoliubov, Krylov andMitropolsky, non-resonant and resonant cases were investi-gated. Computational studies of the cases through poincaresections, time series, phase plots and poincare map provedthe chaotic behavior or aging effect for the certain valuesof parameters of the CNTs.The above researches are exemplary ones reviewed.

Despite hundreds of papers regarding to the field ofcarbon-based nanomaterials, which cannot be reviewed inthis paper due to their immensity, the CNCs has not beenresearched in details.Naess et al. studied the morphologies of the CNCs with

different apex angles experimentally.27 They investigatedthe structure of these nanostructures via transmission elec-tron microscopy (TEM), synchrotron X-ray and electrondiffraction. Moreover, in another research, the nucleation,thermal stability and nanomechanics of the CNCs usingmolecular dynamics were revealed by Tsai et al.28 Theresults were compared with the values corresponding tothe CNTs and the outcomes showed that, unlike CNTs, theproperties of the CNCs strictly depended on their geome-tries such as the apex angles.Applications of the CNCs in hydrogen and neon adsorp-

tions were studied by Yu et al. and Majidi et al.,respectively.29–30 The former was conducted experimentallywhile the latter was performed by molecular dynamics.In reviewing the literature about mechanical investiga-

tion of the CNCs, we encounter the researches conductedby Wei et al. They applied molecular dynamics simulationto analyze the mechanical behaviors of the CNCs. In one

Fig. 1. Schematic representation of a CNC.

of their papers, the mentioned method was used to studythe elastic moluli of a CNC with a certain apex angle anddifferent lengths.31 In another paper, they studied the com-pressive buckling loads of the CNC.32 In both articles, theyused Brenner potential as well as Lennard-Jones poten-tial between carbon atoms. As mentioned, the results werereported only for apex angle 19.2�.Although the mechanical behaviors of the nanomateri-

als can be classified into various categories, in this paper,the elastic, failure, buckling and vibrational characteris-tics of the CNCs are investigated using molecular dynam-ics. The results are reported for various lengths and apexangles. According to Figure 1, the CNCs are canoni-cal nanostructures composing of carbon atoms in threedimensions. Their two parameters including the lengthsand apex angles can be considered their characteristic fac-tors. Although the lengths of the CNCs are arbitrary, theirapex angles are not and only possess certain values whichare approximately: 123.6�, 86.6�, 60.0�, 38.9� and 19.2�.Moreover, the effects of different loading rates and temper-ature variations on the mentioned behaviors of the CNCsare studied in detail.

2. ATOMISTIC SIMULATION TECHNIQUES

In this section, the concepts of the applied atomistic sim-ulation method are presented. The simulations are per-formed using molecular dynamics. In this technique, theshort range interaction force between atoms was esti-mated via second-generation reactive empirical bond-order(REBO) potential of Brenner as below:32

EREBOij = VR�rij �− �b̄ij �VA�rij � (1)

where, VR�rij � and VA�rij � are respectively repulsive andattractive interactions between carbon atoms (i.e., depend-ing only on rij � and b̄ij is the reactive empirical bond order

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between atoms. The latter represents the multibody cou-pling resulting from the interaction between atoms i, j ,and their environments.33

For Brenner’s potential, we can write:

VR�r�=D�e�

S−1e−

√2s��r−R�e��fc�r� (2)

VA�r�=D�e�

S−1e−

√2/S��r−R�e��fc�r� (3)

where, D�e� = 6�000 eV, S = 1�22, � = 21 nm−1, R�e� =0�1390 nm. In addition, fc is a cut-off function in theform of

fc�r�

=

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

1 r <0�17 nm

12

{1+cos

×[��r−0�17 nm�

0�03 nm

]}0�17 nm< r<0�2 nm

0 r >0�2 nm

(4)

On the other hand, the multibody coupling factor b̄ij canbe formulated as below:

b̄ij =12�bij +bji� (5-a)

bij =[1+ ∑

k� �=ij�

G�ijk�fc�rik�

]−0�5

(5-b)

where, ijk = cos−1�r2ij + r2i − r2ik�/2rij rik represents theangle between carbon bonds i–j and i–k. Furthermore, thefunction G can be defined as:

G��= a0

[1+ c20

d20

− c20d20 + �1+ cos�2

](6)

where, a0 = 0�00020813 c0 = 330d0 = 3�5. The expres-sions of VR, VA and b̄ij for the second generation potentialcan be found in Ref. [34].For the long range interaction forces, the van der Waals

potential is applied.

EvdWij =

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

0 rij <r∗s

c3k�rij−rk�3+c2k�rij−rk�

2 r∗s ≤rij <r∗m

ELJ �rij � r∗m≤rij <r∗b

(7)

where, cnk are cubic spline coefficients, r∗s = 0�2 nm, r∗m =

0�32 nm, r∗b = 1 nm. Furthermore, ELJ is defined as theLennard–Jones 12–6 potential which is calculated from thefollowing relation.

ELJ = 4�[(

�ij

r

)12

−(�ij

r

)6](8)

Therefore, the potential sum becomes

E =∑i

∑j>i

�EREBOij +EvdW

ij � (9)

3. RESULTS AND DISCUSSION

In this section, the results of the mechanical behaviorsof the CNCs are presented and discussed. As mentionedbefore, the results are reported for different geometries andsome boundary conditions. The presented results cover theelastic and failure properties of the nanostructures in ten-sion and buckling as well as the vibrational behaviors ofthe CNCs. Furthermore, the effects of different tempera-tures and various loading rates on the mentioned behaviorsare investigated. Figure 2 depicts the simulation process.In the left figure, the CNC is constrained from the lowerbase and a tensile load (strain rate) is applied on the upperside. This condition corresponds with the tensile loading.In the right figure, the lower base is constrained and theupper side is applied a compressive force (strain rate). Thisloading case relates to the buckling phenomena.

3.1. Tensile Loading

This section presents the elastic and failur behaviors ofthe CNCs under tensile loading. Figure 3 shows the ten-sile stress–strain diagram of a CNC with length 63 A andapex angle 19.2� versus steps. The strain rate is considered0.05%/ps and the temperature is considered 300 K.By applying � = E�, that � is the stress, E is the

elastic modulus and � is the strain, one can obtain theelastic modulus of the mentioned CNC 746.76 GPa.It is worth noting that because of oscillations in thestress values in Figure 3, averaging on different steps isconsidered.It is worth noting that the elastic moduli of the CNCs

depend on the geometries and do not have same values.The variations of the elastic moduli of the CNCs versuslength for different apex angles are shown in Figure 4.As shown in this figure, the elastic moduli decrease withincreasing lengths and apex angles. The gradients of vari-ations are sharper in the smaller apex angles and largerlengths rather than the larger apex angles and smallerlengths.At the beginning of the simulation, the equilibrium

structure of the CNC is obtained and after it, the appliedtensile load makes the nanostructure stretch (Figs. 5(a)–(d)). In the strain 0.08, as shown in Figure 5(e), the CNCcannot tolerate the force anymore and ruptures.The rupture point depends on the working conditions

such as loading rate and temperature. Table I presents theeffects of various strain rates on the ultimate strength. Asshown in this table, with increment in the strain rate, therupture stress increases.Figure 6 shows the effects of temperature variations on

the rupture stress of the mentioned CNC. As shown in thefigure, in general, increment of the temperature weakensthe nanostructure and leads to the lower rupture stresses.For the higher temperatures, the nanostructure weakensmore than lower temperatures.

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(a)

(b)

Fig. 2. Schematic representation of the tesile and compressive loadingof a CNC.

Fig. 3. Stress–strain diagram of a CNC with length 63 A and apex angle19.2� under a tensile force.

Fig. 4. Elastic moduli of the CNCs with different geometries.

Fig. 5. Schematic representation of the CNC behavior under a tensileforce.

Table I. Variations of the rupture stress versus loading rate for the casestudy CNC.

Strain rate (%/ps) 0.05 0.1 0.15 0.2

Stress (GPa) 47.05 48.77 49.05 51.09

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Fig. 6. Effects of temperature variations on the rupture stress of thecase study CNC.

3.2. Buckling of the CNCs

In this section, the results of buckling stresses of the CNCsare presented in detail. Figure 7 shows the stress–straindiagram of the case study CNC in the buckling mode. Thestrain rate is considered −0�05%/ps and the temperatureis considered 300 K. As shown in this figure, after findingthe equilibrium point, the compressive stress in applied onthe CNC. It can bear the force up to a maximum valueand then it buckles.The front and side views of the buckling process of the

CNC are illustrated in Figures 8–9. In this case, the lowerbase of the CNC is constrained and the upper one is onlyallowed to displace in the vertical direction.It should be noted that the buckling stresses of the

CNCs depend on their geometries. This fact is depictedin Figure 10. As shown in the figure, the absolute valesof the buckling stresses increase with the increment in theapex angle and decrease with the increment in the length.Similar to the tensile loading case, the gradients of varia-tions are sharper in the smaller apex angles rather than thelarger ones. Moreover, with length increment, the resultsapproach to each other.

Fig. 7. Stress versus strain diagram of the case study CNC buckling.

Fig. 8. Front view of the buckling process.

Fig. 9. Side view of the buckling process.

Fig. 10. Critical stresses versus length for buckling of different CNCs.

Table II. Effects of loading rates on the buckling stresses of the CNC.

Rate (%/ps) 0.05 0.1 0.15 0.2

Stress (GPa) −38.76 −38.86 −41.66 −43.55

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Fig. 11. Effects of temperature variations on the buckling stresses ofthe case study CNC.

As mentioned before, the working and loading condi-tions can affect the outcomes considerably. Table II pro-vides the results of the loading rate effects (strain rates) onthe buckling stresses of the case study CNC. As presentedin the table, the absolute values of the buckling stressesbecome larger with increment of the loading rates.Furthermore, temperature variation effects on the buck-

ling stresses of the CNC are shown in Figure 11.According to the figure, the increment in the tempera-ture can reduce the absolute vales of the buckling stressesdrastically.

3.3. Vibrational Natural Frequencies

In this section, the vibrational natural frequencies are pre-sented for three distinct boundary conditions:(a) clamped from the smaller base-free,(b) clamped from the larger base-free and(c) doubly clamped.

The values corresponding with the first natural frequen-cies of the CNCs with different geometries and bound-ary conditions are reported in Figures 12(a)–(c). The firstcase corresponds with the clamped from the smaller base-free boundary conditions. The second one relates to theclamped from the larger base-free boundary conditions andthe last one associates with the doubly clamped bound-ary conditions. The figures reveal that the tighter bound-ary conditions result in the higher values. In other words,the CNCs with doubly clamped boundary conditions havethe highest natural frequencies and the clamped from thesmaller base-free boundary conditions lead to the small-est ones. It is clear that the mentioned comparisons areapplied for the CNCs with same geometries.The main points in the diagrams are that with length and

apex angle increments, the natural frequencies decrease.Moreover, for the longer CNCs, the values approach eachother.The effects of temperature variations on the natural fre-

quencies of the case study CNC in the first boundary

(a)

(b)

(c)

Fig. 12. Natural frequencies of the different CNCs.

Fig. 13. Temperature effects on the natural frequencies of the casestudy CNC.

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condition is illustrated in Figure 13. It is shown that thetemperature increment reduces the natural frequency of theCNC. It seems that this consequence can be generalizedfor all CNCs and boundary conditions.

4. CONCLUSION

The paper applied molecular dynamics to study themechanical behaviors of the CNCs including their elas-tic, failure, buckling and vibrational properties and theeffects of loading rates and temperature variations on them.The results offered that the elastic moduli of the CNCsdepended on their geometries and, unlike CNTs, are notsame for all cases. Then, the effects of the loading ratesand temperature variations on the rupture and bucklingstresses of a case study CNC were investigated. The buck-ling stresses of the CNCs with different lengths and apexangles were investigated in detail. Also, the moleculardynamics approach was utilized to obtain the first natu-ral frequencies of the CNCs and the temperature variationeffects on them. As a whole, the temperature increment, ingeneral, weakened the nanostructure and resulted in lowerstresses and vibrational natural frequencies.

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Received: 20 May 2012. Accepted: 12 June 2012.

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