Computational Materials Science 56 (2012) 131–140

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Computational Materials Science

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Vibrational and dynamic analysis of C60 and C30 fullerenes using FEM

J.H. Lee a,⇑, B.S. Lee b, F.T.K. Au c, J. Zhang c, Y. Zeng c

a Department of Infrastructure Civil Engineering, Chonbuk National University, JeonJu, South Koreab Department of Mathematics, University of Illinois, Urbana-Champaign, IL, USAc Department of Civil Engineering, The University of Hong Kong, Pokfulam Road, Hong Kong, China

a r t i c l e i n f o a b s t r a c t

Article history:Received 19 October 2011Received in revised form 11 January 2012Accepted 14 January 2012

Keywords:VibrationalDynamicC60

C30

Fullerenevan der Waals forceFEM

0927-0256/$ - see front matter � 2012 Elsevier B.V. Adoi:10.1016/j.commatsci.2012.01.019

⇑ Corresponding author. Tel.: +82 63 270 4789.E-mail address: jinhyu@jbnu.ac.kr (J.H. Lee).

Vibrational analysis of C60 and C30 fullerenes was performed using a finite element method (FEM). Thevibrational behaviors of 10ea C60 and C30 fullerenes with 600ea and 300ea for each atomic carbon, respec-tively, were modeled using three-dimensional elastic beams of carbon bonds and point masses. Further-more, dynamic analysis of fullerenes was performed using the nonlinear elastic elements. The naturalfrequencies, strain and kinetic energy were calculated by considering the van der Waals force betweenthe carbon atoms in the hexagonal and pentagonal lattice. The natural frequencies strain and kineticenergy of fullerenes were estimated depending on the geometric fullerene type with boundary condi-tions. The natural frequencies of the 10ea C60 fullerenes increased significantly at higher modes ofvibration. In the dynamic analysis, the change in displacement over time occurs more significantly alongthe z-axis than the x-, y-axis, and the value of a displacement vector sum is somewhat larger in the C30

than in the C60 fullerenes.� 2012 Elsevier B.V. All rights reserved.

1. Introduction

The initial report regarding the existence and characterizationof C60 fullerenes by Kroto et al. [1] is an important landmark forthe chemistry and physics of fullerenes. The importance of this dis-covery was acknowledged with the Nobel Prize in 1996. Saito andOshiyama [2] presented microscopic total energy calculationswhich provide a cohesive property, and presented electronic struc-tures of a new form of solid carbon, the face-centered-cubic C60

crystal. Goodwin [3] has investigated the stability of the ideal C60

molecule and of some isomers using a molecular dynamicsquenching technique. Damay and Leclereq [4] have reported astudy on the molecular geometry of the C60 fullerene from theanalysis of the structure factor. The fundamental concept proposedfor the composition of three-dimensional fullerene structures byKroto et al. is the introduction of a pentagon consisting of fivemembers that are primarily responsible for the curvature. Theyfunction similarly to defects in a graphite structure and lead tononplanarity of the p-electronic structure. However, the strainenergy will only be minimized when the pentagons are as far apartas possible. This isolated pentagon principle has best beenachieved in the C60 fullerene, which consists of 12 regularlyimplanted pentagon and 20 hexagon members and differs mostmarkedly from two dimensional carbon graphite structures. The

ll rights reserved.

C60 fullerene shows an anisotropic electron distribution as a directresult of the 12 pentagonal faces. The pentagons in the C60 fuller-ene are mostly evenly distributed, but are not as far apart aspossible.

A molecular dynamics study of fullerene was applied by Pałuchaet al. [5] who have carried out studies by inspecting the plots of thetranslational and angular velocity autocorrelation functions, theirFourier transforms, and the temperature dependences of the corre-sponding diffusion coefficients. Skrzypek et al. [6] have performedmolecular dynamics (MD) simulations of a fullerene cluster(C60)19, confined between graphite walls. The fullerenes form twomonolayers parallel to the graphite planes. Raczynski et al. [7] haveperformed molecular dynamics (MD) simulations of small fullereneclusters (C60)n (n = 5, 7, 10, 20) and calculated their interaction-in-duced polarizability anisotropy correlation functions and spectra ofthe depolarized light scattering. The calculated correlation func-tions and spectra depend on the number n of fullerenes in the liquidphase of the (C60)n cluster. Dawid et al. [8] have simulated a systemcomposed of endohedral fullerene K+@C60 molecules. Atomicallydetailed MD simulations have allowed them to analyze the dynam-ics of a potassium ion inside a fullerene cage and the motion ofK+@C60 molecules. The librational frequency of a potassium ioninside a fullerene cage has been estimated. The solid–fluid phasetransition has been observed in the system. Piatek et al. [9] havesimulated (MD method) the dynamics of fullerenes (C60) in anextremely small cluster composed of only seven C60 molecules.The interaction is taken to be the full 60-site pairwise additive

132 J.H. Lee et al. / Computational Materials Science 56 (2012) 131–140

Lennard–Jones (LJ) potential, which generates both the transla-tional and anisotropic rotational motions of each molecule. JeongWon et al. [10] investigated the oscillation dynamics of a C60 fuller-ene encapsulated in a single-walled carbon-nanotube-resonator viaclassical molecular dynamics simulations. The C60 fullerene positionsin a single walled carbon-nanotube-resonator could be controlled byvibrating the resonator. The oscillations of the C60 fullerene alongthe tube axis in a vibrating carbon-nanotube-resonator originatedfrom centrifugal forces exerted at the central position of the vibrat-ing carbon-nanotube. Fang and Weng [11] performed moleculardynamics (MD) simulations to investigate the structural featuresand diffusion properties of fullerene-in-water suspensions. Thenumerical results reveal that an organized structure of liquid wateris formed close to the surface of the fullerene molecule, therebychanging the solid/liquid interfacial structure. The simulation re-sults reveal that the diffusion coefficient of the water moleculesvaries as a linear function of the fullerene loading, but is indepen-dent of the fullerene size. Ren et al. [12] studied the configurationsof hydrogen molecules at 0 K within the vacuum of the C60 fullereneand carbon nanocapsules for the combination of the PM3 semi-empirical method for geometry optimization, and ab initio densityfunctional theory (DFT) for energy calculation. The obtainedstructural information including the molecular arrangement andstructural state of the clusters of H2 are mainly determined bytwo types of repulsive energies. Piatek et al. [13] studied thenanosystem composed of only as little as seven endohedralfullerene K+@C60 molecules using classical molecular dynamicssimulation. The interaction is taken to be the full site–site pairwiseadditive Lennard–Jones (LJ) potential plus Coulomb potentialbetween potassium ions.

To [37] reviewed the experimental and calculated elastic prop-erties of single-walled carbon nanotubes (SWCNTs). Peng et al. [28]studied the order of error to approximate SWCNTs as thin shells viaan atomistic-based finite deformation shell theory which neglectsthe shell thickness and Young’s modulus. Zhang [30] reviewedsome basics in the use of continuum mechanics and moleculardynamics to characterize the deformation of single-walled carbonnanotubes (SWCNTs). Recently, Jomehzadeh and Saidi [32], Reddy[33], Marmu and Pradhan [34], Ansari and Ramezannezhad [35]and Narender et al. [36] formulated equations for an analyticalsolution using the nonlocal elasticity theories.

To be able to investigate the vibrational behavior of fullerenes,two groups were considered, including carbon 1ea and 10ea C60,and 1ea and 10ea C30 fullerenes. For the fullerenes, three majorboundary conditions were applied. The first boundary conditionwas the free beam and the second was the cantilever beam andthe third was the fixed carbon fullerenes. To be able to validatethe proposed model, a numerical simulation of resonant frequen-cies of the 10ea C60 and C30 carbon fullerenes was performed.The natural frequencies and mode shapes of the C60 and C30

fullerenes with the boundary conditions were calculated in thisstudy by applying a mass finite element model. An FEM modelingapproach using ANSYS was implemented to achieve this result andto describe the fullerenes. Furthermore, an FEM modeling ap-proach was used for explicit dynamic analysis using LS-DYNA.The C60 and C30 fullerenes were modeled by the 10ea fullereneswith the fixed boundary condition.

Fig. 1. Geometric shape of the C60 and C30 fullerenes with carbon atoms.

2. Geometries of C60 and C30 fullerene structures

The first fullerene to be characterized was the Ih C60 fullerenewhich was originally identified by its four-band IR absorptionspectrum [16]. The C60 fullerene has a truncated-icosahedral form,with a point group symmetry Ih which allows a degeneracy as highas five. The 30 filled pp orbitals host 60p electrons, in a structural

pattern closely resembling that of free particles on the surface of asphere and, in turn, evoke an equal net atomic charge distributionon each carbon [14]. All 60 carbon atoms have equivalent symme-try, but the bonds fall into two sets, namely, hexagon–pentagonand hexagon–hexagon edges. The 60 Hückel molecular orbitalsgive rise to the reducible representation: 2Ag + 3T1g + 4T2g +6Gg + 8Hg + 1Au + 4T1u + 5T2u + 6Gu + 7Hu. Only the Ag and the Hg

modes are Raman active while the T1u modes are solely IR active[15].

Similarly to all closed geodesic structures, C60 (Fig. 1) has 12pentagons that are required to transform a two dimensional net-work of hexagons into a spheroid. The remaining carbon atomsare configured as 20 hexagons to form the molecule’s soccerballshape. Fullerenes follow the net closing formula postulated byLeonhard Euler [18], who states that, for any polygon with n edges,at least one polyhedron can be constructed with 12 pentagons and(n � 20)/2 hexagons. In order to characterize the highly symmetricC60, it is necessary to determine only two independent geometricalparameters, for example, the lengths of the bonds between thesixfold rings and the bonds sharing the fivefold and sixfold rings.Experiments show that the average distances between the nearestneighbors in the structure are 1.40 Å and 1.45 Å, and that carbonatoms are located at 3.51 Å from the center [17]. The high symme-try of a fullerene molecule causes the appearance of simple vibra-tional [19] and electronic spectra [20]. Isolated C60 has anicosahedral symmetry Ih the highest among the point groups.

Heath [21] proposed a model for the clustering sequence inwhich there is a linear chain of carbon atoms in C10, rings in theC10–C20 range, and a fullerene type structure for C30. He claims thatthe successive C2 additions follow until the isolated pentagon ruleis satisfied. Recently, the drift tube ion chromatography experi-ments of laser vaporized carbon clusters show the existence ofpolycyclic rings and the possibility of annealing such a structureto a fullerene [22]. The C30 fullerene (Fig. 1) has 12 pentagonsand five hexagons.

The C60 and C30 fullerenes are interesting materials for techno-logical applications since they present a structure with atomswhich can be used to achieve specific mechanical properties. Forthe C60 and C30 fullerenes, we analyzed only fullerenes consistingof 1ea and 10ea. The studied fullerenes with carbon atoms areshown in Fig. 1. The fullerenes were prepared to volume sizes sothat the C60 adopted the total elements of: 150, total nodes: 60,min: x �3.3496, y �3.4884, z �3.5524, max: x 3.3453, y 3.5, z3.5359 and the C30 adopted the total elements of: 77, total nodes:30, min: x�2.4869, y�2.0366, z�2.0088, max: x 2.4862, y 2.0363, z2.1415.

In this research, the 10ea C60 and C30 fullerenes were connectedin order to analyze the vibration and dynamic behavior, respec-tively. The generated structure of the 10ea C60 fullerenes is shownin Fig. 2. The fullerenes were prepared to volume sizes so that the

Fig. 2. Geometric shape of the 10ea C60 fullerenes with carbon atoms.

Fig. 3. Geometric shape of the 10ea C30 fullerenes with carbon atoms.

J.H. Lee et al. / Computational Materials Science 56 (2012) 131–140 133

adopted total elements were: 1508, total nodes: 600, min: x�3.3496, y �3.4884, z �3.5524, max: x 3.3453, y 3.5, z 67.3306.

The generated structure of the 10ea C30 fullerenes is shown inFig. 3. The fullerenes were prepared to volume sizes so that theadopted total elements were: 750, total nodes: 300, min: x�2.4869, y�2.0366, z�2.0088, max: x 2.4862, y 2.0363, z 39.4942.

3. Simulation of molecular bonds by VDW

The fullerenes were treated as a frame structure in which theirbonds were beam members and their carbon atoms were joints. Toestablish the linkage between the force constants in molecularmechanics and the beam element stiffness in structural mechanics,the energy equivalence concept proposed by Li and Chou [25] wasapplied.

In general, the total potential energy of the force field in MD canbe expressed as the sum of the energies of the bonded andnonbonded interactions [26] as

Utotal ¼ RUr þ RUh þ RUu þ RUx þ RUvdw ð1Þ

where Ur, Uh, Uu, Ux, and Uvdw are the energies for bond stretching,bond angle bending, dihedral angle torsion, out-of-plane torsion,and nonbonded van der Waals interactions, respectively. A sketchrepresenting the above-mentioned interatomic interactions isshown in Fig. 4 [31].

The van der Waals force is an intermolecular force that arisesfrom a fluctuating electromagnetic field resulting in instantaneous(electrical and magnetic) polarizations between atoms/molecules.The van der Waals force has been studied extensively by Israelach-vili et al. [24] and Montgomery et al. [27]. In addition, Yang andQian [29] have developed the non-uniform rational B-spline(NURBS) surface based approach for the van der Waals force com-puting. In a system of two carbon atom spherical particles 1 and 2of radii r1 and r2 (shown in Fig. 4 [31]), with a separation of d, thenon-retarded van der Waals force between the two spheres is asproposed by Chen and Anandarajah [23]:

F ¼ ½�Aðdþ r1 þ r2Þ=3�½�1=2dðr1 þ r2Þ þ 1=4r1r2 þ 1=4d2ðr1

þ r2Þ2 þ r1r2=8r1r2 þ r1r2=8r1r2� ð2Þ

From Eq. (2), we used the concept that three geometric parameters,i.e., separation distance d and sphere radii r1 and r2, are required tocalculate the van der Waals forces for sphere–sphere interaction. Inthis case, we adopt the same separation distance d = 2.442 nm andfix the sphere radii as r1 = r2 = 0.07 nm and the Hamaker constantA = 0.284e�19.

Also, Chen and Anandarajah [23] performed closed-form solu-tions for sphere–half and sphere–sphere systems. Fig. 4 shows a

system of two spheres, 1 and 2, of radii r1 and r2, respectively,placed at a center-to-center separation distance of h. Here, d isthe closest separation between the spheres. The van der Waalsattractive force F(h) is given by

FðhÞ ¼ A½F0ðhÞ þ F1ðhÞ� ð3Þ

where

F0ðhÞ ¼ h=3 �1=d1d3 þ 1=d2d4 þ 2r1r2=d21d2

3 þ 2r1r2=d22d2

4

h ið4Þ

F1ðhÞ ¼ 1=ð30c4h2Þ½4cr1r2ð4r21 þ 4r2

2 � 2h2 þ 9c2Þ þ chX4

i¼1

ðbi=eidiÞ

þ 2c4h2ðc þ 5hÞð1=d1d3 � 1=d2d4Þ þ c5 ln d2d4=d1d3ð Þ

þX4

i¼1

ðgi lnðei=diÞÞ� ð5Þ

From Eq. (5)

d1 ¼ h� r1 � r2;d2 ¼ hþ r1 � r2; d3 ¼ hþ r1 þ r2; d4 ¼ h� r1 þ r2

and ei ¼ di þ c;bi ¼ ð�1Þðiþ1Þe4i ½4ei � 5ðhþ cÞ� �20r1r2e3

i ;

gi ¼ ð�1Þiþ1e3i ½4e2

i �5ð4hþ cÞei þ 20hðhþ cÞ� þ20r1r2e2i ð3h� eiÞ:

Israelachvili [38] reported that for an interatomic van der Waalsforces, Eq. (6) is used for two atoms or small molecules, Eq. (7) isused for a two spheres or macromolecules and C is the coefficientin the atom–atom pair potential:

F ¼ �6C=r7 ð6Þ

F ¼ ð�A=6d2Þ½r1r2=ðr1 þ r2Þ� ð7Þ

Kalamkarov et al. [41] stated that the non-covalent interactions,for example van der Waals forces, can be adequately describedusing the Lennard–Jones potential. The corresponding energy canbe expressed by,

VLJ ¼ 4e½�12ðr=rÞ12 þ ðr=rÞ6� ð8Þ

Here, the terms r (nm) and e (kJ/mol) are defined as theLennard–Jones parameters. They are material specific anddetermine the nature and strength of the interaction. The term rcorresponds to the distance between the interacting particles. TheLennard–Jones force between C–C atoms can also be computedusing the following expression:

FLJ ¼ dVLJ=dr ¼ 4e=r½�12ðr=rÞ12 þ ðr=rÞ6� ð9Þ

where r = 0.34 nm and e = 0.0556 kcal/mol.

Fig. 4. Interatomic interactions in molecular mechanics.

Fig. 5. Finite element model of the 1ea C60 and C30 fullerenes.

134 J.H. Lee et al. / Computational Materials Science 56 (2012) 131–140

Calculating each van der Waals force for Eqs. (3), (6), (7), and (9)as an equivalent value to the variable applied to Eq. (2), modeanalysis about fullerenes is implemented. We adopt the sameseparation distance of h = 2.582 nm. The van der Waals force calcu-lated for Eq. (2) is 812.993(N), for Eq. (3) is �2.882e�08(N), forEq. (6) is �0.045e�75(N), for Eq. (7) is �0.278e�24(N) and forEq. (9) is 4.49e�06(N). The values for Eqs. (3), (6), (7), and (9) areignored as they are too small, and the value for Eq. (2) was used.

Table 1Adjusted dimensions for ANSYS.

Division Units Original dimensions ANSYS dimensions

Length m L 1010LForce N F 1020FMass kg M 1026MYoung’s modulus Pa E EShear modulus Pa G GNatural frequency Hz f 10�8fEnergy J J 1030J

4. Implementation of the finite element model

Based on the modeling concept described above, the finite ele-ment model for fullerenes was implemented using the commercialprogram ANSYS. First, we will briefly summarize the simulatingelement type used in ANSYS for the considered problem. The C–Cbond was simulated as the beam element of type BEAM188, andthe carbon atoms were simulated as the mass element of typeMASS21.

Concentrated masses were used to model carbon atoms as pointelements with up to six degrees of freedom, translations in the no-dal x, y, and z directions and rotations about the nodal x-, y-, and

z-axes. A different mass and rotary inertia could be assigned toeach coordinate direction. The mass element was defined as a sin-gle node, with concentrated mass components in the element coor-dinate directions, or rotary inertias about the element coordinateaxes. The element coordinate system was initially parallel to theglobal Cartesian coordinate system or to the nodal coordinatesystem and rotated to the nodal coordinate rotations during thelarge deflection analysis. If the element required only one mass in-put, it was assumed to act in all appropriate coordinate directions.The mass element had no effect on the static analysis solution

Table 2Input data prepared for the BEAM188 and MASS21 elements.

Division Units ANSYS dimensions Values

C–C bond diameter m d 0.1466Poisson’s ratio m 0.3Density kg/m3 q 2.3E3Young’s modulus Pa E 5.448E12Shear modulus Pa G 0.8701E12Mass of carbon atom kg Mc 2.0

J.H. Lee et al. / Computational Materials Science 56 (2012) 131–140 135

unless acceleration or rotation was present. The 10ea C60 fullerenesare modeled with 0.169771 nm spacing and the 10ea C30 fullerenesare modeled with 0.593614 nm spacing from the nearest two ful-lerenes along the x, y, and z-axes. As an example, the establishedfinite element models of the single C60 and C30 fullerenes areshown in Fig. 5.

Before we input the data of the BEAM188 and MASS21 elementproperties, the dimensions of the parameters stated above werefurther adjusted to avoid possible flow errors during the computa-tion with ANSYS. Thus, the dimensions were adjusted as shown inTable 1 [31]. After adjustment, the numerical parts of the inputdata were prepared for the BEAM188 and MASS21 elements, asshown in Table 2 [31].

After mode analysis using ANSYS, from the 20th mode of eachfullerene we find the node which has the maximum absolute valueof displacement. Then, in order to determine the accurate displace-ments about the x-, y- and z-axes of the found nodes, we performedan explicit dynamic analysis using LS-DYNA.

5. Vibrational and dynamic analysis

5.1. Numerical method for the vibrational analysis

For the most basic problem involving a linear elastic materialwhich obeys Hooke’s Law, the matrix equations were designed inthe form of a dynamic three-dimensional spring mass system.The generalized equation of motion is given as follows

½M�f€ug þ ½C�f _ug þ ½K�fug ¼ ½F� ð10Þ

where [M] is the mass matrix, f€ug is the second time derivative ofthe displacement {u} (i.e., the acceleration), f _ug is the velocity, [C]is a damping matrix, [K] is the stiffness matrix, and [F] is the forcevector.

Vibrational analysis was used for natural frequency and modeshape determination. For vibrational analysis, damping was gener-ally ignored. The equation of motion for an undamped system, ex-pressed in matrix notation, is given in

½M�f€ug þ ½K�fug ¼ f0g ð11Þ

Note that [K], the structure stiffness matrix, may include pre-stress effects. For a linear system, free vibrations will be harmonicwith the following form

fug ¼ fugi cos xit ð12Þ

where {u}i, xi, and t are the vector representing the mode shape ofthe ith natural frequency, the ith natural circular frequency (radiansper unit time), and time, respectively.

Thus, Eq. (11) can be rewritten as

ð�x2i ½M� þ ½K�Þfugi ¼ f0g ð13Þ

This equality is satisfied if either {u}i = {0} or the determinant of([K] �x2[M]) is zero. The first option is trivial and therefore is notof interest. The second option gives the following solution

j½K� �x2½M�j ¼ 0 ð14Þ

This is an eigenvalue problem which may be solved for up to nvalues of x2 and n eigenvectors {u}i that satisfy Eq. (12), where n isthe number of DOFs. The eigenvalue and eigenvector extractiontechniques are used in the Block Lanczos method. Rather than out-putting the natural circular frequencies {x}, the natural frequen-cies (f) are output as

fi ¼ xi=2p ð15Þ

where fi, is the ith natural frequency (cycles per unit time). Normal-ization of each eigenvector {u}i to the mass matrix is performedaccording to

fugTi ½M�fugi ¼ 0 ð16Þ

In the normalization, {u}i is normalized such that its largestcomponent is 1.0 (unity).

The natural frequency of a structure is related to its geometry,mass, and boundary conditions. For the fullerenes considered here,the mass was assumed to be that of each carbon atom,2.0 � 10�26 kg, and the rotational degrees of freedom of the atomwere neglected due to its extremely small diameter.

5.2. Numerical method for the dynamic analysis

For the dynamic analysis progress, we consider the single de-gree of freedom damped system with forces acting on mass mfor time integration. The equations of equilibrium are obtainedfrom d’Alembert’s principle as [39]

fI þ fD þ fS ¼ pðtÞ ð17Þ

Inertia forces fI ¼ m€u; €u ¼ d2u=dt2. . . acceleration

Damping forces fD ¼ c _u; _u ¼ du=dt . . . velocityLinear elasticity fS ¼ ku; u . . . displacementExternal forces pðtÞ

8>>>><>>>>:

where c is the damping coefficient, and k is the linear stiffness. Forcritical damping c = ccr. The equations of motion for linear behaviorlead to a linear ordinary differential equation:

m€uþ c _uþ ku ¼ pðtÞ ð18Þ

However, for the nonlinear case the internal force varies as anonlinear function of the displacement, leading to a nonlinear or-dinary differential equation:

m€uþ c _uþ fSðuÞ ¼ pðtÞ ð19Þ

Analytical solutions of linear ordinary differential equations areavailable, so instead we consider the dynamic response of the lin-ear system subjected to a harmonic loading. It is convenient to de-fine some commonly used terms:

Harmonic loading: pðtÞ ¼ p0 sin �xt. Circular frequency:x ¼

ffiffiffiffiffiffiffiffiffiffik=m

pfor single degree of freedom. Natural frequency:

f = x/2p = 1/T, T = period. Damping ratio: n ¼ c=ccr ¼ c=2mx.Damped vibration frequency. Applied load frequency:xD ¼ x

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� n2

pb ¼ �x=x. The closed form solution is:

Dynamic response of linear undamped system due to harmonicloading:

uðtÞ ¼u0 cos xt þ ð _u0=xÞ sinxt þ ðp0=kÞð1=ð1� b2ÞÞ� ðsin �xt � b sinxtÞ ð20Þ

for the initial conditions: u0 = initial displacement, _u0 = initial veloc-ity, p0/k = static displacement, and 1/(1 � b2) = dynamic magnifica-tion factor. For nonlinear problems, only numerical solutions arepossible. LS-DYNA uses the explicit central difference time integra-tion method [40] to integrate the equations of motion.

136 J.H. Lee et al. / Computational Materials Science 56 (2012) 131–140

6. Numerical results

6.1. Vibrational analysis results

To be able to investigate the vibrational behavior of the C60 andC30 fullerenes (1ea and 10ea) with the boundary conditions (free,cantilever, fixed), the C60 and C30 fullerenes were modeled using3-D beam and mass elements. To be able to validate the proposedmodel a numerical simulation of the resonant frequencies of canti-lever and fixed fullerenes was performed. The numerical results ofthe fullerenes are shown in Table 3 and Figs. 6–8. The deformedshapes of mode 5, 10, 15, and 20 for the 10ea C60 and C30 fullereneswith the fixed boundary condition are illustrated in Figs. 7 and 8.The variations in frequency versus the mode of vibration for the

Table 3Energies for the fixed C60 and C30 fullerenes configuration.

Fullerenes Energy

Strain Kinetic

C60

1ea10ea

C30

1ea10ea

Fig. 6. Variation in the natural frequency of the C60 an

Fig. 7. Deformed shapes in modes 5, 10, 15 and

C60 and C30 fullerenes are compared in Fig. 6. As shown, the fre-quency increased when the number of C–C bonds increased, with-out respect to the boundary conditions.

As can be seen, the frequency did not significantly differ for thefirst six modes of the fullerenes with the free boundary condition.However, a regular pattern of variation for the modes of thefullerenes with the cantilever and fixed boundary condition wasobserved at higher vibrations such that the curves of variationfor the fullerenes increased at all points and proceed periodically.

The results illustrating the frequency for the C60 fullerene dis-play the same trend as those observed for the C30 fullerene inFig. 6. It can be deduced from the figure that the natural frequencyof the C60 fullerene compared with the C30 fullerene increases asthe atom numbers increase and the increase in the natural fre-quency of the C60 fullerene is not close to double that of the C30 ful-lerene. From the comparison of the vibrational modes illustrated inFig. 6 for the two fullerenes with different boundary conditions, itcan be concluded that the modes are mainly the sum of the trans-lation and rotation modes. These two modes are enlarged in thefigures with a greater number of C–C bonds.

As can be observed in Fig. 6, the natural frequencies were in therange of 490–550 GHz for the C60 fullerene and 700–900 GHz forC30 fullerene at mode 5, for the cantilever and fixed 1ea fullerene,respectively. The natural frequencies were in the range of15–30 GHz for the C60 fullerene and 25–50 GHz for C30 fullereneat mode 5, for the cantilever and fixed 10ea fullerenes, respectively.

d C30 fullerenes, with BCs (free, cantilever, fixed).

20 of the 10ea C60 fullerenes with fixed BC.

Fig. 8. Deformed shapes in modes 5, 10, 15 and 20 of the 10ea C30 fullerenes with fixed BC.

J.H. Lee et al. / Computational Materials Science 56 (2012) 131–140 137

Also, the natural frequencies were in the range of 800–1000 GHzfor the C60 fullerene and 1300–1400 GHz for C30 fullerene at mode10, for the cantilever and fixed 1ea fullerene, respectively. Thenatural frequencies were in the range of 30–50 GHz for the C60

fullerenes and 70–100 GHz for C30 fullerenes at mode 10, for thecantilever and fixed 10ea fullerenes, respectively.

In addition, the natural frequencies were in the range of1200–1300 GHz for the C60 fullerene and 1800–2000 GHz for C30

fullerene at mode 15, for the cantilever and fixed 1ea fullerene,respectively. Also, the natural frequencies were in the range of70–90 GHz for the C60 fullerenes and 110–150 GHz for C30 fuller-enes at mode 15, for the cantilever and fixed 10ea fullerenes,respectively. These differences could be due to the boundary con-ditions applied and the number of atoms considered in each model.

In Fig. 1 in the paper of Adhikari and Chowdhury [42], thefundamental frequency of C60 is 8 THz and that of C30 is 17 THz,which is approximately doubled. In this paper, as seen in Fig. 6,the two fullerenes’ frequencies (f) at each mode have decreasedin accordance with the increase of the fullerene mass (M) asf /M�1/2[43–45].

Therefore, the numerical results indicate that the fullereneshave not undergone slight deformation and vibration as the num-ber of fullerenes increased. The deformation shape according toeach modes can be confirmed in the attached video files (C30:Avi 1 ⁄ .avi–Avi 4 ⁄ .avi, C60: Avi 9 ⁄ .avi–Avi 12 ⁄ .avi).

The potential energy including the strain energy of elements is

Epoe ¼ 1=2½fUegTð½Ke� þ ½Se�ÞfUeg� ð21Þ

and the kinetic energy computed only for modal analyses is

Ekie ¼ 1=2½f _UegT ½Me�f _Ueg� ð22Þ

where [Ke]: element stiffness matrix, [Se]: element stress stiffnessmatrix, {Ue}: element DOF vector, f _Ueg: time derivative of elementDOF vector, and [Me]: element mass matrix.

Table 3 show a comparison of the C60 and C30 fullerenes (1eaand 10ea) with the fixed boundary conditions, for strain and ki-netic energies. As can be observed in Table 3, for the fixed 1eaC60 fullerene, the maximum values of the strain energies were inelement no. 45 at mode 15 and the maximum values of the kineticenergies were in element nos. 43 and 73 at mode 20. For the fixed10ea C60 fullerenes, the maximum values of the strain energieswere in element no. 368 at mode 20 and the maximum values ofthe kinetic energies were in element no. 375 at mode 20.

For the fixed 1ea C30 fullerene, the maximum values of thestrain energies were in element no. 10 at mode 20 and the maxi-mum values of the kinetic energies were in element no. 21 at mode20. For the fixed 10ea C30 fullerenes, the maximum values of thestrain energies were in element no. 455 at mode 20 and the max-imum values of the kinetic energies were in element no. 404 atmode 20.

As can be seen, a small difference is noticed between the twomodels of the 1ea C60 and C30 fullerenes and a significant differenceis noticed between the two models of the 10ea C60 and C30 fuller-enes. These difference can be attributed to the geometric differ-ences between the two models. Also, it is evident that the strainenergies of the C30 fullerenes shown in Table 3 were higher thanthose of the C60 fullerenes. Also, the kinetic energies of theC30 fullerenes shown in Table 3 were higher than those of the C60

fullerenes.The modes of deformation illustrated in Figs. 7 and 8 show that

the 10ea fullerenes with the fixed boundary condition do notundergo a similar deformed shape in the same direction as thefullerenes with the same trend pattern in the frequency.

As indicated by the shapes in the figures, the direction of defor-mation varied among the fullerenes at their respective positions.Also, it is evident that the deformation of displacement of higherorder modes shown in Fig. 7 was higher than that of lower ordermodes. Also, the deformation of displacement of higher ordermodes shown in Fig. 8 was higher than that of lower order modes.

In the fifth mode in Fig. 7, the fullerenes shown the deformingwaviness in the vertical direction of the y-axis whereas the fuller-enes for the tenth mode show the deforming waviness in all direc-tions of the x-, y- and z-axes. The fullerenes for the fifteenth modeshow the deforming waviness in the horizontal direction of thex-axis and the fullerenes for the twentieth mode show the deform-ing waviness in two directions of the x- and y-axes, generally.

The other modes of deformation shown in Fig. 8 can beexplained in a similar manner. Therefore, as shown in Fig. 7, thenumerical results indicate that the fullerenes underwent slightdeformation and vibration as the mode order increased. The defor-mation shape according to each modes can be confirmed in the at-tached video files (C30: Avi 5 ⁄ .avi–Avi 8 ⁄ .avi, C60: Avi 13 ⁄ .avi–Avi 16 ⁄ .avi).

6.2. Dynamic analysis results

In addition, the second dynamic analysis results are shown inFigs. 9 and 11 and Table 4 for the C60 fullerenes and

Fig. 9. Displacement vector sums of the C60 fullerene with BCs (free, cantilever, fixed).

Table 4Displacements of a C60 fullerene at chosen nodes 10, 18, 34.

BC C60

Free Cantilever Fixed

Node 10Node 18Node 34

138 J.H. Lee et al. / Computational Materials Science 56 (2012) 131–140

Figs. 10 and 12 and Table 5 for the C30 fullerenes. The analyseswere performed for 20 s.

Each of the node numbers 10, 18 and 34 in Figs. 9 and 11 andTable 4 is the location of the maximum absolute value of displace-ment at the C60 fullerene in the above mode analysis.

Likewise, each of the node numbers 10, 16 and 25 in Figs. 10and 12 and Table 5 is the location of the maximum absolute valueof displacement at the C30 fullerene in the above mode analysis.

Fig. 10. Displacement vector sums of the C30 f

Fig. 11. Deformed shape of the 10e

Fig. 9 shows the largest displacement vector sum (USUM) atnode 34 with the free boundary condition. As can be seen, theUSUM at node 34 has a somewhat larger value with the fixed thanwith the cantilever boundary condition. The USUM at node 10 witha cantilever boundary condition has a larger value than nodes 18and 34 and the C60 fullerene with the fixed boundary condition.Overall, the value of a USUM, which is indicated as USUM in thefigures, appears smaller with the cantilever than with the fixedboundary condition.

Each figure in Table 4 shows the displacements for the x-, y- andz-axes at the chosen nodes (10, 18, 34). The displacements at nodes18 and 34 for the z-axis with a cantilever have a somewhat largewaviness, but the displacements for the x- and y-axes are relativelysmall. The similar phenomenon also appears in the fullerene withthe fixed boundary condition. As can be seen, the fullerenes have alarger waviness in displacement as time passes. It is shown that thedisplacement waviness is especially significant in the z-axis com-pared to other axes.

ullerene with BCs (free, cantilever, fixed).

a C60 fullerenes with fixed BC.

Fig. 12. Deformed shape of the 10ea C30 fullerenes with fixed BC.

Table 5Displacements of a C30 fullerene at chosen nodes 10, 16, 25.

BC C30

Free Cantilever Fixed

Node 10Node 16Node 25

J.H. Lee et al. / Computational Materials Science 56 (2012) 131–140 139

The waviness of the displacements for the x-axis was regular atlower periods of displacement but was irregular at higher periods.The increase in the pattern of waviness of displacement for the ful-lerenes with a cantilever was more regular at lower displacementtimes than that for the fullerenes with a fixed boundary condition.This is closely related to the asymmetric shape of the fullerenes.The deformation shape according to each nodes can be confirmedin the attached video files (Avi 20⁄.avi–Avi 22⁄.avi).

Fig. 10 shows the largest displacement vector sum (USUM) atnode 10 with the free boundary condition. As can be seen, theUSUM at node 10 has a somewhat larger value with a cantileverthan with the fixed boundary condition. The USUM at node 25 witha fixed boundary condition has a larger value than at nodes 16 and10 and the C30 fullerene with a cantilever. Overall, the value of a dis-placement vector sum, which is indicated as USUM in the figures,appears smaller with a cantilever than with a fixed boundarycondition.

Each figure in Table 5 shows the displacements for the x-, y- andz-axes at the chosen nodes (10, 16, 25). The displacements at nodes10 and 25 for the z-direction with a cantilever have a somewhatlarge waviness, but the displacements for the x- and y-directionsare relatively small. The displacements at node 10 for the z-axiswith the fixed boundary condition have somewhat small waviness,but the displacement at node 16 for the y- and z-axes is relativelylarge. The similar phenomenon also appears in the fullerene withthe cantilever boundary condition. As can be seen, the fullereneshave a larger waviness in the displacements for the y-axis withthe fixed boundary condition as time passes. It is shown that thedisplacement waviness is especially large in the x-direction com-pared to other directions.

The waviness of the displacements for the y-axis was regular atlower displacement times but was irregular at higher displacementtimes. The increase in the pattern of waviness of the displacementfor the fullerenes with a cantilever was more regular at lower dis-placement times than that for the fullerenes with the fixed bound-ary condition. This is closely related to the asymmetric shape of thefullerenes. The deformation shape according to each nodes can beconfirmed in the attached video files (Avi 17 ⁄ .avi–Avi 19 ⁄ .avi).

Figs. 11 and 12 show the deformed shapes of the 10ea C60 andC30 fullerenes with the fixed boundary condition. In node 193 ofthe fourth fullerene from the right side in Fig. 11, the node showsthe maximum displacement vector sum. On the other hand, innode 3 of the first fullerene from the right side in Fig. 12, the nodeshows the maximum displacement vector sum. Comprehensively,the deformation of the fullerenes was irregularly generated tothe x-, y- and z-axes.

According to the analysis in Fig. 11, the largest deformation is innode 193 generated in the x-axis and in node 3 generated in they-axis according to the analysis in Fig. 12 among the x-, y- andz-axes. The deformation shape according to each displacementtime can be confirmed in the attached video files (Avi 23 ⁄ .aviand Avi 24 ⁄ .avi).

7. Conclusions

The vibrational and dynamic behaviors of single and multi C60

and C30 fullerenes with boundary conditions were studied usinga finite element method comprised of beam and mass elementsin a three-dimensional coordinate system. In order to explore thevibrational behavior, the influences of van der Walls forces onthe natural frequencies of the boundary conditions were consid-ered. In order to explore the dynamic behavior, nodes selected inmode analysis were interpreted and analyzed via re-performingFEM modeling.

The increase in a number of carbon atoms for fullerenesresulted in a decrease in frequency. The decrease, however, wasnot severe for the first six modes of fullerenes.

A difference is noticed between the two models (C60 and C30

fullerenes) that can be attributed to the geometric differencesbetween the two models.

The energies illustrated in Table 3 show that the fullerenes withthe decreases in a number of atoms and bonds undergo moredeformation than do the fullerenes with the increases for the sameboundary condition.

Therefore, the increase in a number of carbon atoms for fuller-enes can result in a more structural equilibrium. The results implythat the stable frequency of the fullerenes can be due to the bound-ary conditions and geometric configuration at all modes ofvibration.

In the dynamic analysis, the displacements of the 1ea C60

fullerene show the larger waviness along the z-axis and the 1eaC30 fullerene shows the larger waviness along the x- and y-axes.The value of a displacement vector sum appears larger for C30 thanfor the C60 fullerene. This is thought to be related to mass. Refer tothe attached video files.

Acknowledgment

This paper was supported by the research funds of Chonbuk Na-tional University, 2011.

Appendix A. Supplementary data

Supplementary data associated with this article can be found, inthe online version, at doi:10.1016/j.commatsci.2012.01.019.

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