Mechanics of nested spherical fullerenes inside multi-walled carbon nanotubes

European Journal of Mechanics A/Solids 49 (2015) 283e292

Contents lists available at ScienceDirect

European Journal of Mechanics A/Solids journal homepage: www.elsevier.com/locate/ejmsol

Mechanics of nested spherical fullerenes inside multi-walled carbon nanotubes R. Ansari, F. Sadeghi* Department of Mechanical Engineering, University of Guilan, P.O. Box 3756, Rasht, Iran

a r t i c l e i n f o

a b s t r a c t

Article history: Received 25 January 2014 Accepted 9 August 2014 Available online 23 August 2014

Based on the continuum approximation and Lennard-Jones (LJ) potential, mechanics of nested spherical fullerenes, known as carbon onions, inside multi-walled carbon nanotubes (MWCNTs) is investigated in this study. To this end, direct method is first utilized to determine van der Waals (vdW) interaction force and potential energy between a carbon onion molecule and a semi-infinite MWCNT. According to this method, the interactions between each pair of shells from carbon onion and CNT are summed up over all of the pairs. Thereafter, the suction and acceptance energies for carbon onions entering semi-infinite MWCNTs are evaluated. On the basis of Newton's second law, an analytical expression is then presented to predict the oscillation frequency of a carbon onion molecule inside a MWCNT of finite length. The effect of geometrical parameters on the nature of suction and acceptance energies, vdW interactions and oscillatory characteristics of carbon onion-MWCNT oscillators is thoroughly examined. For a given carbon onion structure, it is found that there exists an optimal value for the number of nanotube shells beyond which the maximum oscillation frequency does not increase considerably. Furthermore, the maximum oscillation frequency decreases as the carbon onion gets larger. © 2014 Elsevier Masson SAS. All rights reserved.

Keywords: Continuum approximation Acceptance and suction energies Oscillation frequency

1. Introduction Nanostructured materials such as carbon nanotubes (CNTs) (Iijima, 1991) and fullerenes (Kroto et al., 1985) have found a wide variety of applications in many nanoelectromechanical systems (NEMS) in recent years (Kim and Lieber, 1999; Postma et al., 2001; Fennimore et al., 2003; Liu et al., 2005). This is mainly due to their exceptional properties such as low weight, high strength and flexibility making them promising candidates for such purposes (Saito et al., 1992; Yeung et al., 2009; Wong et al., 1997). Highfrequency nanoscale oscillators, known as gigahertz (GHz) oscillators, are one of the proposed devices which have captured tremendous attention worldwide. Potential applications of these oscillators might include nanoantennae sensitive to high-frequency electromagnetic signals and optical filters for fiber optic systems (Tuzun et al., 1995; Damnjanovi
c et al., 1999). In the year 2000, Cumings and Zettl (2000) reported an ideal low frictional effect for a sliding inner tube inside multi-walled carbon nanotubes (MWCNTs). They demonstrated that if the inner core pulls out and set free, it spontaneously retracts inside the

* Corresponding author. Tel./fax: þ98 131 6690276. E-mail addresses: [email protected], Fatemeh.sadeghi.1985@ gmail.com (F. Sadeghi). http://dx.doi.org/10.1016/j.euromechsol.2014.08.003 0997-7538/© 2014 Elsevier Masson SAS. All rights reserved.

outer shells due to the van der Waals (vdW) interatomic interaction acting on the core. Subsequently, by using a static mechanical model, Zheng and Jiang (Zheng and Jiang, 2002) exposed that CNTbased oscillators are capable of reaching frequencies up to several GHz. Also, Zheng et al. (2002) incorporated the frictional forces into their model and reported that these forces have almost insignificant effect on the oscillation frequency. Up to now, several types of nano-oscillators have been proposed in the published literature (Hilder and Hill, 2007a; Cox et al., 2008; Alisafaei et al., 2011; Ansari et al., 2013a). A great number of researchers employed molecular dynamics (MD) simulations to predict the behavior of such oscillatory systems (Rivera et al., 2003; Legoas et al., 2004; Kang et al., 2006; Ansari et al., 2013b). On the basis of these simulations, Legoas et al. (2003) confirmed that CNT oscillators can generate frequencies in the GHz range. They also reported that these nano-oscillators are dynamically stable when the interwall spacing is about 0.34 nm. Rivera et al. (Rivera et al., 2005) adopted MD-based models to examine the damping effects of these oscillatory motions. Some other investigations have been also conducted on the issues of energy dissipations and frictional forces through MD simulations (Guo et al., 2003; Zhao et al., 2003; Servantie and Gaspard, 2006). In spite of the fact that MD models may be the most reliable approaches for analysis of nano-oscillators, their main drawback is that they are not computationally efficient, especially in the

284

R. Ansari, F. Sadeghi / European Journal of Mechanics A/Solids 49 (2015) 283e292

case of large number of atoms for simulation. Furthermore, as a result of numerical evaluations of MD-based models, it is impossible to derive analytical expressions for evaluation of vdW interactions while applying these kinds of models. Another approach for modeling vdW interactions between two nanostructures is the continuum approximation of discrete atoms where a constant atomic surface density is used on the surface of each molecule instead of the discrete distribution of carbon atoms. Generally, in order to evaluate vdW interactions between two nanostructures through continuum approach, surface integrals must be performed over both smeared surfaces which leads to a quadruple integral (Ansari et al., 2012). Girifalco and his elaborates (Girifalco et al., 2000; Hodak and Girifalco, 2001; Kniaz et al., 1995) demonstrated successful application of this continuum model to several types of molecular systems. They also asserted that these models are preferable to the discrete model of atoms in some cases. Hilder and Hill (2007b) reported that interactions of CNTeCNT, C60eCNT, C70eCNT and C80eCNT modeled through continuum approach are in reasonable agreement with those modeled through discrete model. Using the continuum model together with elementary mechanical principles, Hill and his colleagues (Cox et al., 2007a) presented an analytical formula for the oscillation frequency of C60 fullereneCNT oscillator. In their model, for the case of long nanotubes, they estimated the vdW interaction force with an impulse-like function acting near the ends of nanotube. This model was then extended to examine the oscillatory behavior of nested spherical fullerenes inside single-walled carbon nanotubes (SWCNTs) (Thamwattana and Hill, 2008). Besides, Ansari and his colleagues employed continuum model to present new semianalytical expressions to determine vdW interactions as well as oscillation frequency of different types of nano-oscillators such as nested CNT (Ansari and Motevalli, 2009, 2011), spherical fullerene-CNT (Ansari et al., 2013c) and ellipsoidal fullerene-CNT (Ansari and Sadeghi, 2012) systems. Their studies revealed some distinctive features of such oscillatory systems for the first time in the literature. Applying the continuum approach, acceptance condition and suction energy of different nanoparticles, which are going to be encapsulated into a nanotube, have been also studied in the open literature (Cox et al., 2007b; Alisafaei and Ansari, 2011; Sadeghi and Ansari, 2012; Ansari and Kazemi, 2012; Ansari et al., 2013d). In the present study, continuum approximation is employed to fully investigate the mechanics of nested spherical fullerenes inside MWCNTs. Nested spherical fullerenes, or the so-called carbon onions, were synthesized in 1992 under electron beam irradiation of CNTs and nanoparticles (Ugarte, 1992). Experimental studies also demonstrated that this shape of carbon onion is more energetically favorable than ellipsoidal and tetrahedral ones (Ugarte, 1992; Banhart and Ajayan, 1996). Besides, Korto and McKay (Korto and McKay, 1988) proposed a plausible structure for carbon onions made up of concentric Goldberg type 1 fullerenes of Ih symmetry. This kind of carbon onion, which is considered in this article, consists of C60@ C240@ C540@C960@ … @CN spherical fullerenes where N denotes the number of carbon atoms. In addition, the intershell spacing of this structure is very close to the interlayer spacing of graphene which is equal to 0.34 nm (Girifalco et al., 2000). In the following sections, first of all, the continuum LennardJones (LJ) model is introduced. Afterward, direct method is used to evaluate vdW interaction force and potential energy between nested spherical fullerenes and semi-infinite MWCNTs. Also, suction and acceptance energies for a carbon onion located coaxially near an open end of a semi-infinite MWCNT are evaluated. On the basis of Newton's second law and neglecting the

frictional forces, an analytical expression is then given to estimate the oscillation frequency of a carbon onion traveling along the axis of a finite MWCNT. Finally, numerical results are presented to get an insight into the effect of geometrical parameters on the distributions of suction and acceptance energies, vdW interactions and oscillatory behavior of carbon onion-MWCNT oscillators. 2. Continuum approximation Here, it is assumed that the vdW interaction between two carbon atoms at a distance r apart is given by the following 6e12 LJ potential function as

A B FðrÞ ¼  6 þ 12 r r

(1)

where A and B specify the attractive and repulsive constants, respectively. Eq. (1) can be also rewritten in the following form

A FðdÞ ¼ 6 a

1 d60 1  2 d12 d6

! (2)

in which d shows the normalized distance between the atoms, a is the carbonecarbon bond length and d0 denotes the normalized equilibrium distance expressed as

d0 ¼

  1 2B 1=6 a A

(3)

The total potential energy between the two nanostructures can be obtained by summing the potential energy between the pair of atoms from each molecule

Ftot ¼

XX   F rij i

(4)

j

where F(rij) is the potential energy function for atoms i and j on each molecule at a distance rij apart. Based on the continuum approximation, which assumes that carbon atoms are uniformly distributed over the surfaces of the molecules with a constant atomic surface density, a double surface integral can be used instead of the double summation in Eq. (4) as

Ftot ¼ h1 h2

Z Z FðrÞd

X X d 1

(5)

2

in which h1 and h2 denote the mean atomic surface density of atoms on each molecule and r represents the distance between two P P typical surface elements d 1 and d 2 . In addition, the corresponding vdW interaction force can be attained through differentiating the total potential energy as follows

FvdW ¼ VFtot

(6)

3. VdW interaction force and potential energy between a carbon onion and a semi-infinite MWCT In this section, direct method is applied as a straight method to evaluate vdW interaction force and potential energy between nested spherical fullerenes and semi-infinite MWCNTs. On the basis of this method, the interaction between each shell of carbon onion molecule with each shell of MWCNT is first determined and then the total interaction is obtained through summing the interactions between each pair of shells.

R. Ansari, F. Sadeghi / European Journal of Mechanics A/Solids 49 (2015) 283e292

3.1. Interaction force Consider a carbon onion entering a semi-infinite MWCNT as depicted in Fig. 1. It is assumed that the origin of the Cartesian coordinate system (x, y, z) is situated at the left end of nanotube, whereas the center of the carbon onion molecule traveling along the axis of nanotube lies at a distance Z from the origin. The number of shells corresponding to carbon onion and MWCNT are denoted by NF and NC, respectively. Also, the radius of jth shell of carbon onion molecule is represented by RFj where j varies from 1 to NF. It must be remarked that the space between each shell of MWCNT is considered to be s ¼ 3.4 Å (Ansari and Motevalli, 2009). Accordingly, the radius of kth shell of MWCNT is defined by RCk ¼ RC þ (k  1)s in which k varies from 1 to NC and RC is the radius of the innermost nanotube. Following (Cox et al., 2007b), potential energy between the jth shell of carbon onion and a typical atom on the kth shell of MWCNT can be stated as

phFj RFj Pjk ðrk Þ ¼ rk

A 1 1   4  2 r þR 4 rk  RFj Fj k !! B 1 1   5 r þ R 10 r  R 10 Fj Fj k k

!

(7)

where hFj is the mean atomic surface density of spherical fullerene qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi of radius RFj and rk ¼ R2Ck þ ðZ  zÞ2 denotes the distance between the tube surface element and the center of the carbon onion molecule. It should be remarked that owing to the symmetry of the problem, only the axial interaction force between the two molecules is considered herein. Based on Fig. 1, the vdW interaction force between the jth shell of carbon onion and a typical atom on the kth shell of MWCNT is of the form FvdW ¼ dPjk(rk)/drk and thus the axial force is obtained as Fz(Z) ¼ ((Zz)/rk)(dPjk(rk)/drk). Accordingly, employing continuum approach, the total axial force between the jth shell of carbon onion and the kth shell of MWCNT is expressed as

Z∞ Z2p Fz;jk ðZÞ ¼ hC RCk 0

0

ðZ  zÞ dPjk ðrk Þ dqdz rk drk

     ¼ 2phC RCk P rk;2  P rk;1

285

Now, rewriting the right-hand side of Eq. (7) in terms of powers of ðrk2  R2Fj Þ as

A 2rk RFj

1

1

!

4   4  rk þ RFj rk  RFj 0 1 2R2Fj 1 B C ¼ 4A@ 3 þ  4 A ; rk2  R2Fj rk2  R2Fj

(9-a)

! 1  10  10  rk þ RFj rk  RFj 0 80R2Fj 336R4Fj 4B B 5 ¼  @ 7 þ  6 þ  8 5 rk2  R2Fj rk2  R2Fj rk2  R2Fj 1 512R6Fj 256R8Fj C þ 9 þ  10 A rk2  R2Fj rk2  R2Fj

B 5rk RFj

1

(9-b)

and substituting the prior equation into Eq. (7), the final form of axial interaction force between the jth shell of carbon onion and the kth shell of MWCNT is stated as follows

Fz;jk ðZÞ ¼ 

10 X



i¼3

Cijk l2jk þ Z 2

(10)

i

is5

where ljk ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R2Ck  R2Fj and

8 jk > C3 ¼ 4AKjk > > > > < jk 0 C7 ¼ 80R2Fj Kjk > > > > > : C jk ¼ 256R8 K 0 Fj jk 10

C4 ¼ 2R2Fj C3

0 C6 ¼ 5Kjk

0 C8 ¼ 336R4Fj Kjk

0 C9 ¼ 512R6Fj Kjk

Kjk ¼ 2p2 RCk R2Fj hFj hC

Kjk ¼

jk

jk

jk

jk

jk 0

4B K 5 jk

(8)

(11)

in which hC denotes the mean surface density of CNT and rk,2 ¼ ∞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi and rk;1 ¼ R2Ck þ Z 2 .

Thus, using direct method, the total interaction force between a nested spherical fullerene and a semi-infinite MWCNT is calculated as

Fztot ð Z Þ ¼

NF X NC X

Fz;jk ð Z Þ

(12)

j¼1 k¼1

3.2. Potential energy Knowing the interaction force between the jth shell of carbon onion and the kth shell of MWCNT, the corresponding potential energy can be achieved through negative integration of interaction force expression with respect to Z. Thus, after performing an extensive manipulation, the analytical form of this potential energy is obtained as

Fig. 1. Geometry of a nested spherical fullerene entering a semi-infinite MWCNT.

9 X   1 Fjk Z ¼ Gjk Gjk 1 tan ujk þ 2i1  i¼1

ujk 1 þ u2jk

i

(13)

286

R. Ansari, F. Sadeghi / European Journal of Mechanics A/Solids 49 (2015) 283e292

in which ujk ¼ Z/ljk and other constants parameters are as follows

(14)

Thus, using Eqs. (13) and (15), the suction energy of system takes the following form jk 2m1 ; m ¼ 3; 4; …; 10; ms5: wheremjk m ¼ Cm =ljk Thus, based on the direct method, the total vdW potential energy between a nested spherical fullerene and a semi-infinite MWCNT is given by

  Ftot Z ¼

NF X NC X

  Fjk Z

(15)

j¼1 k¼1

Ws ¼ p

Z∞ ∞

(17)

The acceptance energy is also defined to examine if the nanoscale object is sucked into the nanotube or not. Herein, the acceptance energy is equal to the total work done by the vdW interaction force for moving the carbon onion from Z ¼ ∞ to Z ¼ Z0 where Z0 denotes the positive root of Fztot ðZÞ ¼ 0. Thus,

ZZ0

Cox and his colleagues (Cox et al., 2007b) first introduced the concept of suction energy and acceptance condition for the systems of atoms and C60 fullerenes inside SWCNTs. These issues are the two main characteristics of nanotube-based systems for medical applications such as drug delivery and so forth. The term suction energy is defined as the value of total energy which is imparted to a nanoscale object in the form of kinetic energy upon entering the nanotube. In fact, the suction energy is a measure of the total increase in the kinetic energy which is experienced by the nanoscale object. The positive value of this energy implies that the inside of nanotube is energetically suitable for the suction phenomena if reachable. For the considered mechanism in this study, the suction energy is equal to the total work performed by the vdW interaction force for moving the carbon onion from Z ¼ ∞ to Z ¼ ∞ and is given by

Fztot ðZÞdZ ¼ Ftot ð∞Þ  Ftot ðZ0 Þ

Wa ¼

(16)

(18)

∞

Therefore, the acceptance energy of system using Eqs. (13) and (15) can be expressed as Table 1 Numerical values of the constants used in the model (Cox et al., 2007a). Attractive constant (eV Å6) Repulsive constant (eV Å12) Mean surface density of CNT (Å2) Mass of a single carbon atom (kg)

A ¼ 17.4 B ¼ 29  103 hC ¼ 0.3812 m0 ¼ 1.993  1026

Table 2 Mean surface density and radius corresponding to different spherical fullerenes (Thamwattana and Hill, 2008). Spherical fullerene

Fztot ðZÞdZ ¼ Ftot ð∞Þ  Ftot ð∞Þ

jk

G1

j¼1 k¼1

4. Suction energy and acceptance condition

Ws ¼

NF X NC X

2

hF (Å ) RF (Å)

C60

C240

C540

C960

0.3789 3.55

0.3767 7.12

0.3898 10.5

0.4011 13.8

R. Ansari, F. Sadeghi / European Journal of Mechanics A/Solids 49 (2015) 283e292

287

Fig. 2. Distribution of suction energy versus interwall spacing corresponding to different carbon onions entering a SWCNT.

Table 3 Values of especial interwall spacing related to suction energy of different carbon onions (NC ¼ 1). NF

1

2

3

4

AÞ ts0 ð tsmax ð AÞ

2.720 3.233

2.703 3.211

2.695 3.201

2.690 3.195

0 NF X NC 9   X X B jk p jk þ tan1 u0jk þ Wa ¼  G2i1  @G1 2 i¼1 j¼1 k¼1

1 u0jk 1þ

u20jk

C i A (19)

in which u0jk ¼ Z0/ljk. Note that as demonstrated by Cox and his colleagues (Cox et al., 2007b), for a nanoscale object to be completely accepted by a nanotube, the sum of acceptance energy and initial kinetic energy must be positive. 5. Oscillation of nested spherical fullerenes inside MWCNTs Consider a carbon onion located on the axis of a finite MWCNT of length 2L. Here, it is assumed that the origin of the coordinate system is located at the center of nanotube. Based on the Newton's

second law and ignoring the frictional effects (Zheng et al., 2002), the equation of motion can be expressed as follows

  € MZðtÞ ¼ Fztot Z

(20)

where M ¼ m0N is the total mass of the carbon onion in which m0 P F 2 denotes the mass of a single carbon atom and N ¼ 60 N j is the j¼1 total number of atoms in the carbon onion molecule. Following (Thamwattana and Hill, 2008), for the case when the radii of carbon onion and nanotube are extremely smaller than the nanotube length, the total axial interaction force can be considered as two equal and opposite Dirac delta functions at the tube ends. Accordingly, Eq. (20) can be rewritten in the following form

€ MZðtÞ ¼ Ws ðdðZ þ LÞ  dðZ  LÞÞ

(21)

in which Ws is the constant pulse strength defined by Eq. (17) and d stands for Dirac delta function. Using the same procedure applied in Cox et al. (2007a); Thamwattana and Hill (2008), it is concluded that the carbon onion molecule travels inside the MWCNT with a qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi constant velocity equal to V ¼ 2Ws =M þ V 20 where V0 is the initial velocity. Accordingly, the oscillation frequency is calculated as

Fig. 3. Distribution of acceptance energy against interwall spacing corresponding to different carbon onions entering a SWCNT.

288

R. Ansari, F. Sadeghi / European Journal of Mechanics A/Solids 49 (2015) 283e292

Table 4 Values of especial interwall spacing related to acceptance energy of different carbon onions (NC ¼ 1).

Table 5 Positive real root of force function corresponding to different carbon onions (Wa ¼ 0, NF ¼ 3, NC ¼ 1).

NF

1

2

3

4

NF

1

2

3

4

AÞ ta0 ð t0 (Å)

2.788 2.9593

2.772 2.9465

2.764 2.9401

2.759 2.9362

AÞ Za0 ð

1.4837

1.8661

2.1685

2.4292

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u NF X NC X 1 u jk tV 2  2p f ¼ G1 0 4L M j¼1

(22)

k¼1

It should be noted that the initial velocity is defined for the case when the carbon onion molecule is not sucked into the nanotube due to the strong repulsive force. 6. Numerical results and discussions Herein, with respect to the formulations given in previous sections, an extensive study on the variations of suction and acceptance energies, vdW interactions and oscillation frequency of carbon onion-MWCNT oscillators is conducted by varying geometrical parameters so that the important features of these parameters are explored. The constant values for numerical evaluations performed herein are outlined in Table 1. In addition, the mean surface density and radius associated with different spherical fullerenes are listed in Table 2. Fig. 2 depicts variation of suction energy with the interwall spacing corresponding to different carbon onion molecules entering a SWCNT. Note that interwall spacing is defined as the difference between the radii of innermost nanotube and the outermost spherical fullerene of carbon onion molecule. According to this figure, for a given interwall spacing, increasing the number of carbon onion shells intensifies the magnitude of suction energy. Furthermore, suction energy almost vanishes when the interwall

spacing approximately reaches 10 Å. Closer inspection to this figure also reveals that there exists a specific interwall spacing beyond which suction energy becomes positive. This specific interwall spacing is symbolized by ts0 and is computed for various carbon onions in Table 3. The results indicate that as the carbon onion molecule gets larger, the value of ts0 gradually decreases. However, this value is nearly equal to 2.7 Å for each configuration. Besides, for a certain interwall spacing, which is marked by check point in Fig. 2, suction energy reaches its greatest value. It should be remarked that in this geometry, the preferred position of carbon onion molecule occurs along the tube axis which leads to generating the highest oscillation frequency (Thamwattana and Hill, 2008). This special interwall spacing is denoted by tsmax and is computed for different carbon onions in Table 3. The results demonstrate that the value of tsmax gradually declines as the number of carbon onion shells increases. Nevertheless, this value is approximately equal to 3.2 Å for each carbon onion structure which is less than the interspacing between two graphene sheets. It should be remarked that the results of tsmax obtained herein are in good agreement with ones previously reported in Thamwattana and Hill (2008). Fig. 3 shows alteration of acceptance energy against the interwall spacing associated with various carbon onions entering a SWCNT. As depicted in this figure, if the interwall spacing is held constant, increasing the number of carbon onion shells causes the magnitude of acceptance energy to increase. Likewise Fig. 2, there is a specific interwall spacing beyond which acceptance energy is positive. This specific interwall spacing is denoted by ta0 and is calculated for different carbon onions in Table 4. As seen, for each configuration, the value of ta0 is a little greater than that of ts0 . Based

Fig. 4. Variation of a) vdW interaction force b) potential energy with the separation distance of carbon onion inside semi-infinite SWCNT associated with different values of interwall spacing (NF ¼ 3).

R. Ansari, F. Sadeghi / European Journal of Mechanics A/Solids 49 (2015) 283e292

289

Fig. 5. Variation of a) vdW interaction force b) potential energy with the separation distance of carbon onion inside semi-infinite CNT associated with different numbers of carbon onion shells and nanotube shells (t ¼ tsmax ).

Fig. 6. Effect of number of nanotube shells on the distribution of suction energy (NF ¼ 3).

on the acceptance condition, it can be concluded that for the interwall spacing less than ta0 the carbon onion molecule cannot be sucked into the nanotube by suction force alone and it necessarily needs an initial velocity to overcome the negative acceptance energy. As mentioned earlier, acceptance energy is only definable when the force function has a positive real root. In this regard, knowing the ultimate value of nanotube radius or interwall spacing at which acceptance energy exists is of importance. This special interwall spacing is symbolized by t0 and is computed for different carbon onions in Table 4. According to the results, the value of this interwall spacing gradually increases with decreasing the number of carbon onion shells. However, this value is almost equal to 2.9 Å for each carbon onion structure. For all special interwall spacing introduced in Figs. 2 and 3, variations of vdW interaction force and potential energy with the separation distance of a carbon onion molecule comprising of three

shells are displayed in Fig. 4 (a) and (b), respectively. As demonstrated in Cox et al. (2007b), the positive root of force function is real only when t  t0. When the acceptance energy is zero, the positive real root of force function corresponding to different carbon onions are symbolized by Za0 and computed in Table 5. As observed, increasing the number of carbon onion shells intensifies the value of Za0 . Also, as the interwall spacing gets larger up to t0,

Table 6 Optimal number of nanotube shells and the associated maximum suction energy corresponding to different carbon onions (NF ¼ 3). NF

1

2

3

4

NC* Ws*max ðeVÞ

5 3.8884

5 10.4874

5 19.2922

4 29.6823

290

R. Ansari, F. Sadeghi / European Journal of Mechanics A/Solids 49 (2015) 283e292

Fig. 7. Variation of interwall spacing related to maximum suction energy with the number of nanotube shells corresponding to different carbon onions.

Table 7 Optimal interwall spacing of different carbon onions related to maximum suction energy. NF

1

2

3

4

AÞ ts*max ð

3.204

3.183

3.173

3.167

the two roots of force graph become closer to each other so that both force roots occur at the origin of reference coordinate when the interwall spacing is equal to t0. Note that for the interwall spacing higher than t0, the force graph does not cross the axis and consequently the carbon onion will be accepted by nanotube in this

geometry. According to Fig. 4 (b), it can be also observed that regardless of the interwall spacing value, all potential curves are zero when the carbon onion is located at the center of nanotube. This conclusion can be easily demonstrated via Eq. (13). Fig. 5 (a) and (b) highlights the effect of both carbon onion shells number and nanotube shells number on the distributions of vdW interaction force and potential energy, respectively. Note that the nanotube is considered to be semi-infinite in length and the interwall spacing is taken in a way so that suction energy reaches its maximum value. In conformity with this figure, enlarging either the number of carbon onion shells or the number of nanotube shells results in increasing the magnitude of vdW interaction force and

Fig. 8. Distribution of maximum oscillation frequency in terms of number of nanotube shells corresponding to various carbon onions (L ¼ 75 Å).

R. Ansari, F. Sadeghi / European Journal of Mechanics A/Solids 49 (2015) 283e292

potential energy. As depicted, the increase in the values of vdW interactions is more pronounced via enlarging the number of carbon onion shells. Graphically illustrated in Fig. 6 is distribution of suction energy versus the interwall spacing corresponding to different numbers of nanotube shells. Herein, the carbon onion molecule is assumed to be comprised of three shells. As can be observed, higher suction energies are obtained for larger numbers of tube shells. In addition, there exists an optimal value for the number of tube shells beyond which suction energy does not increase considerably. Note that the same conclusion can be drawn for other carbon onions. This optimal number of nanotube shells and the associated maximum suction energy are symbolized by NC* and WS*max , respectively and are evaluated for various carbon onion molecules in Table 6. For the case of SWCNT, the value of tsmax related to different carbon onions was calculated in Table 3. Here, in order to extend our investigation, this specific interwall spacing is obtained for a wide range of number of nanotube shells. The distribution of tsmax versus the nanotube shells number related to different carbon onion molecules is depicted in Fig. 7. According to this figure, for a given number of tube shells, smaller carbon onions provide greater value for tsmax . Besides, for each carbon onion, as long as NC < NC* ; increasing the number of nanotube shells leads to the reduction of tsmax , whereas this certain interwall spacing almost remains unchanged when NC > NC* . The optimal value of interwall spacing corresponding to NC* is denoted by ts*max and is computed for different carbon onions in Table 7. The variation of maximum oscillation frequency with the number of nanotube shells corresponding to different carbon onion molecules is depicted in Fig. 8. It should be noted that to obtain the maximum oscillation frequency, the maximum suction energy must be used for each configuration. In accordance with this figure, for a given number of nanotube shells, smaller carbon onions oscillate with higher maximum frequencies inside nanotube. This can be justified with the lower mass of smaller carbon onions. Moreover, as long as NC < NC* ; the maximum frequency increases by increasing the nanotube shells number. However, the maximum frequency does not increase considerably and almost remains unchanged when NC > NC* . The maximum oscillation frequency related * to NC* is denoted by fmax and is calculated for different carbon onion molecules in Table 8. The distribution of normalized binding energy against the half length of nanotube related to different values of interwall spacing is shown in Fig. 9. In this figure, it is assumed that carbon onion molecule is comprised of three shells and also the corresponding optimal number of nanotube shells is considered. The normalized binding energy used in this figure is defined as

Normalized Binding Energy ¼

Binding EnergyðLÞ jminðBinding EnergyÞj

(23)

As shown for all values of interwall spacing, the normalized binding energy reaches a plateau as the length of nanotube increases and all normalized binding energy curves fall on the same trend for a wide range of tube lengths.

291

Fig. 9. Distribution of normalized binding energy against half length of nanotube related to different values of interwall spacing (NF ¼ 3, NC ¼ 5).

Based on the direct method, analytical expressions were first derived to determine vdW interaction force and potential energy between a carbon onion and a semi-infinite MWCNT. Afterward, these expressions were employed to evaluate the suction and acceptance energies of a carbon onion located co-axially near the open end of a semi-infinite MWCNT. On neglecting the frictional forces and estimating vdW interaction force with an impulse-like force, an analytical expression was then proposed to predict the oscillation frequency of system. Finally, a comprehensive study was conducted to examine the effect of geometrical parameters such as number of carbon onion shells and nanotube shells on the distributions of suction and acceptance energies, vdW interactions and maximum oscillation frequency of carbon onion-MWCNT oscillators. Based on the numerical results, the most important findings of this research can be summarized as: ✓ The interwall spacing at which acceptance energy is zero is a little greater than that at which suction energy is zero. ✓ Increasing either the number of carbon onion shells or the number of nanotube shells result in enlarging the value of vdW interaction force and potential energy. ✓ There exists an optimal number for nanotube shells beyond which maximum suction energy and maximum oscillation frequency do not increase considerably. ✓ As the number of nanotube shells increases up to its optimal number, the value of interwall spacing at which suction energy is maximized decreases, while maximum oscillation frequency increases. ✓ Interwall spacing at which suction energy is maximized and maximum oscillation frequency almost remain constant when the number of tube shells exceeds the optimal number. ✓ Normalized binding energy reaches a plateau as the length of nanotube increases.

7. Conclusion The mechanics of carbon onions inside MWCNTs was studied using continuum approximation in conjunction with LJ potential. Table 8 Optimal maximum frequency related to different carbon onions (L ¼ 75 Å). NF

1

2

3

4

* ðGHzÞ fmax

34.0238

24.9887

20.2546

17.1626

References Alisafaei, F., Ansari, R., 2011. Mechanics of concentric carbon nanotubes: interaction force and suction energy. Comp. Mater. Sci. 50, 1406e1413. Alisafaei, F., Ansari, R., Rouhi, H., 2011. Continuum modeling of van der Waals interaction force between carbon nanocones and carbon nanotubes. J. Nanotechnol. Eng. Med. 2, 031002. Ansari, R., Kazemi, E., 2012. Detailed investigation on single water molecule entering carbon nanotubes. Appl. Math. Mech. 33, 1287e1300.

292

R. Ansari, F. Sadeghi / European Journal of Mechanics A/Solids 49 (2015) 283e292

Ansari, R., Motevalli, B., 2009. The effects of geometrical parameters on force distributions and mechanics of carbon nanotubes: a critical study. Commun. Nonlinear Sci. Numer. Simul. 14, 4246e4263. Ansari, R., Motevalli, B., 2011. On new aspects of nested carbon nanotubes as gigahertz oscillators. J. Vib. Acoust. 133, 051003. Ansari, R., Sadeghi, F., 2012. On the oscillation frequency of ellipsoidal fullerenecarbon nanotube oscillators. J. Nanotechnol. Eng. Med. 3, 011001. Ansari, R., Alisafaei, F., Alipour, A., Mahmudinezhad, E., 2012. On the van der Waals interaction of carbon nanocones. J. Phys. Chem. Solids 73, 751e756. Ansari, R., Sadeghi, F., Alipour, A., 2013a. Oscillation of C60 fullerene in carbon nanotube bundles. J. Vib. Acoust. 135, 051009. Ansari, R., Sadeghi, F., Ajori, S., 2013b. Continuum and molecular dynamics study of C60 fullerene-carbon nanotube oscillators. Mech. Res. Commun. 47, 18e23. Ansari, R., Sadeghi, F., Motevalli, B., 2013c. A comprehensive study on the oscillation frequency of spherical fullerenes in carbon nanotubes under different system parameters. Commun. Nonlinear Sci. Numer. Simul. 18, 769e784. Ansari, R., Mahmoudinezhad, E., Alipour, A., Hosseinzadeh, M., 2013d. A comprehensive study on the encapsulation of methane in single-walled carbon nanotubes. J. Comput. Theor. Nanosci. 10, 2209e2215. Banhart, F., Ajayan, P.M., 1996. Carbon onions as nanoscopic pressure cells for diamond formation. Nature 382, 433e435. Cox, B.J., Thamwattana, N., Hill, J.M., 2007. Mechanics of atoms and fullerenes in single-walled carbon nanotubes. II. Oscillatory behaviour. Proc. R. Soc. Lond. Ser. A. 463, 477e494. Cox, B.J., Thamwattana, N., Hill, J.M., 2007. Mechanics of atoms and fullerenes in single-walled carbon nanotubes. I. Acceptance and suction energies. Proc. R. Soc. Lond. Ser. A. 463, 461e477. Cox, B.J., Thamwattana, N., Hill, J.M., 2008. Mechanics of nanotubes oscillating in carbon nanotube bundles. Proc. R. Soc. Lond. Ser. A. 464, 691e710. Cumings, J., Zettl, A., 2000. Low-friction nanoscale linear bearing realized from multiwall carbon nanotubes. Science 289, 602e604. Damnjanovi
c, M., Milosevi
c, I., Vukovi
c, T., Sredanovi
c, R., 1999. Full symmetry, optical activity, and potentials of single-wall and multiwall nanotubes. Phys. Rev. B 60, 2728e2739. Fennimore, A.M., Yuzvinsky, T.D., Han, W.-Q., Fuhrer, M.S., Cumings, J., Zettl, A., 2003. Rotational actuators based on carbon nanotubes. Nature 424, 408e410. Girifalco, L.A., Hodak, M., Lee, R.S., 2000. Carbon nanotubes, buckyballs, ropes, and a universal graphitic potential. Phys. Rev. B 62, 13104e13110. Guo, W., Guo, Y., Gao, H., Zheng, Q., Zhong, W., 2003. Energy dissipation in gigahertz oscillators from multiwalled carbon nanotubes. Phys. Rev. Lett. 91, 125501. Hilder, T.A., Hill, J.M., 2007a. Oscillating carbon nanotori along carbon nanotubes. Phys. Rev. B 75, 125415. Hilder, T.A., Hill, J.M., 2007b. Continuous versus discrete for interacting carbon nanostructures. J. Phys. A: Math. Theor. 40, 3851e3868. Hodak, M., Girifalco, L.A., 2001. Fullerenes inside carbon nanotubes and multiwalled carbon nanotubes: optimum and maximum sizes. Chem. Phys. Lett. 350, 405e411. Iijima, S., 1991. Helical microtubules of graphitic carbon. Nature 354, 56e58.

Kang, J.W., Song, K.O., Hwang, H.J., Jiang, Q., 2006. Nanotube oscillator based on a short single-walled carbon nanotube bundle. Nanotechnology 17, 2250e2258. Kim, P., Lieber, C.M., 1999. Nanotube nanotweezers. Science 286, 2148e2150. Kniaz, K., Girifalco, L.A., Fischer, J.E., 1995. Application of a spherically averaged potential to solid C70 in the disordered phase. J. Phys. Chem. 99, 16804e16806. Korto, H.W., McKay, K., 1988. The formation of quasi-icosahedral spiral shell carbon particles. Nature 331, 328e331. Kroto, H.W., Heath, J.R., Obrien, S.C., Curl, R.F., Smalley, R.E., 1985. C60: Buckminsterfullerene. Nature 318, 162e163. Legoas, S.B., Coluci, V.R., Braga, S.F., Coura, P.Z., Dantas, S.O., Galv~ ao, D.S., 2003. Molecular dynamics simulations of carbon nanotubes as gigahertz oscillators. Phys. Rev. Lett. 90, 055504. Legoas, S.B., Coluci, V.R., Braga, S.F., Coura, P.Z., Dantas, S.O., Galv~ ao, D.S., 2004. Gigahertz nanomechanical oscillators based on carbon nanotubes. Nanotechnol 15, S184eS189. Liu, P., Zhang, Y.W., Lu, C., 2005. Oscillatory behavior of C60-nanotube oscillators: a molecular-dynamics study. J. Appl. Phys. 97, 094313. Postma, H. W. Ch., Teepen, T., Yao, Z., Grifoni, M., Dekker, C., 2001. Carbon nanotube single-electron transistors at room temperature. Science 293, 76e79. Rivera, J.L., McCabe, C., Cummings, P.T., 2003. Oscillatory behavior of double-walled nanotubes under extension: a simple nanoscale damped spring. Nano Lett. 3, 1001e1005. Rivera, J.L., McCabe, C., Cummings, P.T., 2005. The oscillatory damped behaviour of incommensurate double-walled carbon nanotubes. Nanotechnology 16, 186e198. Sadeghi, F., Ansari, R., 2012. Mechanics of ellipsoidal carbon onions inside multiwalled carbon nanotubes. J. Nanotechnol. Eng. Med. 3, 011002. Saito, R., Fujita, M., Dresselhaus, G., Dresselhaus, M.S., 1992. Electronic structure of graphene tubules based on C60. Phys. Rev. B 46, 1804e1811. Servantie, J., Gaspard, P., 2006. Translational dynamics and friction in double-walled carbon nanotubes. Phys. Rev. B 73, 125428. Thamwattana, N., Hill, J.M., 2008. Oscillation of nested fullerenes (carbon onions) in carbon nanotubes. J. Nanopart. Res. 10, 665e677. Tuzun, R.E., Noid, D.W., Sumpter, B.G., 1995. The dynamics of molecular bearings. Nanotechnology 6, 64e74. Ugarte, D., 1992. Curling and closure of graphitic networks under electron-beam irradiation. Nature 359, 707e709. Wong, E.W., Sheehan, P.E., Lieber, C.M., 1997. Nanobeam mechanics: elasticity, strength, and toughness of nanorods and nanotubes. Science 277, 1971e1975. Yeung, C.S., Tian, W.Q., Liu, L.V., Wang, Y.A., 2009. Chemistry of single-walled carbon nanotubes. J. Comput. Theor. Nanosci. 6, 1213e1235. Zhao, Y., Ma, C.-C., Chen, G.H., Jiang, Q., 2003. Energy dissipation mechanisms in carbon nanotube oscillators. Phys. Rev. Lett. 91, 175504. Zheng, Q., Jiang, Q., 2002. Multiwalled carbon nanotubes as gigahertz oscillators. Phys. Rev. Lett. 88, 045503. Zheng, Q., Liu, J.Z., Jiang, Q., 2002. Excess van der Waals interaction energy of a multiwalled carbon nanotube with an extruded core and the induced core oscillation. Phys. Rev. B 65, 245409.