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Torsional buckling of carbon nanopeapods
Carbon 45 (2007) 952–957 www.elsevier.com/locate/carbon
Torsional buckling of carbon nanopeapods A.N. Sohi a, R. Naghdabadi a
a,b,*
Department of Mechanical Engineering, Sharif University of Technology, Box 11365-3567, Tehran, Iran b Institute for Nano Science and Technology, Sharif University of Technology, Tehran, Iran Received 11 May 2006; accepted 28 December 2006 Available online 5 January 2007
Abstract Torsional buckling of carbon nanopeapods (carbon nanotubes filled with fullerenes) is studied using a continuum-based multi-layered shell model. The model takes into account non-bonded van der Waals interactions between nested fullerenes and the innermost layer of host nanotube. For nanopeapods with linearly arranged nested fullerenes, equivalent pressure distribution is proposed to model these interactions. Deriving explicit equations governing the torsional stability, it is concluded that the critical torsional load of a carbon nanopeapod is less than that of a carbon nanotube under otherwise identical geometric and mechanical conditions. Performing numerical calculations, it is also shown that increasing the number of layers of the host carbon nanotube decreases the weakening effect of encapsulated fullerenes on torsional stability of the nanopeapod. Ó 2007 Elsevier Ltd. All rights reserved.
1. Introduction The discoveries of ‘‘buckminsterfullerene’’ or simply ‘‘fullerene’’ in the mid 1980s [1] and subsequently carbon nanotubes (CNTs) in the early 1990s [2,3] have opened up opportunities for different applications of these nanostructures. Such discoveries raised a great interest in studying other nanostructures of carbon, ultimately leading to the discovery of ‘‘carbon nanopeapod’’ or simply ‘‘nanopeapod’’ in 1998 [4]. This new class of self-assembled hybrid structures is in essence the well known CNT filled with ordered arrangements of fullerenes. Based on the characteristics of host CNT and nested fullerenes, different ordered arrangements such as linear chain and zigzag have been predicted theoretically [5,6] and observed experimentally [7]. Due to the physical properties of nanopeapods, different potential applications have been proposed for them [8,9]. Some of these applications entail mechanical loading which necessitates the evaluation of the elastic properties and sta-
*
Corresponding author. Address: Department of Mechanical Engineering, Sharif University of Technology, Box 11365-3567, Tehran, Iran. Tel.: +98 21 66005716; fax: +98 21 66000021. E-mail address: [email protected] (R. Naghdabadi). 0008-6223/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.carbon.2006.12.027
bility of nanopeapod under the applied load. Hence, the elucidation of the elastic behaviors of nanopeapods is one of the critical steps towards their practical usage [10]. Treating such properties involves modeling non-bonded van der Waals (vdW) interactions between the guest fullerenes and the adjacent layer of the host CNTs. These interactions are responsible for self-assemblage of fullerenes along the host CNT and provide critical effects on mechanical behavior of the host CNT. In this analysis, the nanopeapod is characterized by a hybrid structure of encapsulated fullerenes inside an elastic cylindrical shell. The non-bonded vdW interactions between the linearly arranged nested fullerenes and the innermost layer of the host CNT are modeled by equivalent axisymmetric harmonic pressure distribution. Employing a continuum-based multi-layered shell model with proper account of the proposed vdW pressure distribution, the instability of the nanopeapod under torsional loading is investigated. 2. Modeling van der Waals interactions Different packing arrangements have been proposed for encapsulated fullerenes inside nanopeapods [5–7]. These
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953
2.2. van der Waals interactions between nested fullerenes and the innermost layer of the host nanotube
Fig. 1. Schematic of double-walled carbon nanopeapod under torsional load T; y and z denote circumferential and axial coordinates, respectively.
arrangements are of principal importance as we seek to model the corresponding vdW interactions. The linear chain arrangement imposes an axisymmetric harmonic vdW pressure upon the innermost layer of the host CNT. In Fig. 1, schematic of a carbon nanopeapod with linear chain arrangement of nested fullerenes is shown. 2.1. van der Waals interactions between adjacent layers of the host nanotube In a multi-walled carbon nanotube (MWNT), if the vdW pressure exerted on layer i by the adjacent layer j is denoted by pij and the pressure exerted on layer j by the ith layer is denoted by pji, we can write pij Ri ¼ pji Rj
ð1Þ
where Ri and Rj express the mean radius of the ith and jth layers, respectively. Considering the infinitesimal buckling, interlayer vdW interactions are modeled by equivalent pressure distributions which are linear functions of the difference of deflections of adjacent layers [11]. Thus, the total inward vdW pressures on different layers are stated as
The non-bonded vdW interactions between nested fullerenes and the innermost layer of the host nanotube can be described using empirical potentials such as Lennard–Jones (LJ) [13]. The total radial force exerted by nested fullerenes on a specific atom of the CNT wall can be calculated by summing up the radial components of all of the vdW interacting forces between the selected atom and the atoms of neighboring nested fullerenes. In Fig. 2, the vdW radial force exerted by nested C60 fullerenes on carbon atoms of a (10, 10) host CNT wall, ˚ [4], is with average C60 center to center distance of 10 A shown. The horizontal axis represents the axial coordinate of the selected atom of the CNT wall, while the vertical axis shows the magnitude of the radial force. Positive and negative values of the radial force account for repulsive and attractive vdW interactions, respectively. In this figure, the diamonds stand for numerical data while the continuous line shows a typical Fourier series fitting. This Fourier series is in the form of X 2npz 2npz F vdW ðzÞ ¼ a0 þ an cos þ bn sin ð3Þ Df Df n In Eq. (3), Df represents the equilibrium distance between adjacent fullerenes inside the nanopeapod [Fig. 1] which is assumed to remain constant during the torsional loading of the nanopeapod. Also, (z) denotes the axial coordinate of the cylindrical shell and its origin is located at the center of the first nested fullerene of consideration. As can be seen, the periodic distribution of radial force can be precisely approximated by the Fourier series fitting. Any increase in number of terms of the Fourier series would further enhance the precision of the approximation. The Fourier series fitting of Fig. 2 has 17 terms (n = 8 in Eq. (3)).
p1 ¼ p12 ¼ c½w2 w1 p2 ¼ p23 þ p21 ¼ c½w3 w2 ðR1 =R2 Þðw2 w1 Þ .. .
ð2Þ
pN ¼ pN ðN 1Þ ¼ cðRN 1 =RN Þ½wN 1 wN where wi denotes the additional displacement of the ith layer due to the buckling along the inward normal direction. Subscripts 1, 2, . . . , N denote the quantities associated with the first innermost layer (1), second innermost layer (2), and so on up to the outermost layer (N) and c is the vdW characteristic coefficient. It should be noted that the initial vdW forces between adjacent layers are set to zero because the pre-buckling interlayer spacings are equal or very close to the equilibrium spacing between graphene layers. In addition, the van der Waals interactions between non-adjacent structures are neglected since these interactions are not significant when compared with those between adjacent structures [12].
Fig. 2. Radial force on carbon atoms of the host CNT in C60@(10, 10) nanopeapod.
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The relationship between the vdW radial force, FvdW(z) in Eq. (3), and its corresponding vdW pressure distribution over the innermost layer of the host CNT is expressed as pffiffiffi !1 3 3 2 d pvdW ðzÞ ¼ F vdW ðzÞ ð4Þ 4 cc where dcc represents the carbon–carbon bond length whose ˚ . The vdW pressure distribution of Eq. magnitude is 1.42 A (4) causes a circumferential membrane force in the innermost layer of the host CNT which, according to equilibrium conditions, can be stated as N 0y1 ¼ R1 pvdW ðzÞ
ð5Þ
In this analysis, the vdW interactions between nested fullerenes and the layers of the host CNT other than the innermost one are neglected. Hence, in the absence of any external pressure, the circumferential membrane forces in other layers of the host CNT are almost zero. In addition, since it is assumed that the average C60 center to center dis˚ and does not change with slight deformations tance is 10 A of the host CNT, the vdW pressure distribution over the innermost layer of the host CNT (Eq. (4)) remains valid during the torsional loading. 3. Torsional buckling analysis In recent years, some continuum-based shell models have been successfully applied to predict the behavior of carbon nanostructures. Using a linear approximation for the vdW interactions between adjacent layers (Eq. (2)), Ru et al. have systemically studied the axial instability and vibration behavior of CNTs [11,14]. Later on, this approach has been incorporated in studying other cases such as vibration of a multi-layered graphene sheet [15] and torsional and bending instability of CNTs [16–18]. Motivated by the idea of continuum-based modeling of CNT, a nanopeapod can be regarded as a hybrid structure of nested fullerenes inside an elastic cylindrical shell, with the vdW interactions between them. Since the nested fullerenes are bonded by weak vdW interactions, they willingly move to more stable positions along the host CNT when the nanopeapod undergoes structural instability. Thus, the total effect of these encapsulated fullerenes is summarized in a harmonic pressure distribution over the innermost layer of the host CNT (Eq. (4)). It has been shown that the friction between adjacent layers in MWNTs as well as between nested fullerenes and the innermost layer of the host CNT in nanopeapods are negligible [19,20]. Hence, in the absence of any tangential force, Donnell’s equilibrium equation governs the buckling behavior of the host CNT [11,16]. Considering the host CNT as a cylindrical shell with the mean radius R, thickness h, and Young’s modulus E, Donnell’s equation is stated as
Eh o4 w Dr8 w þ 2 4 R oz o2 w o2 w o2 w þ N 0y 2 þ r4 pðz; yÞ ¼ r4 N 0z 2 þ 2N 0zy oz oz oy oy
ð6Þ
where p(z, y) denotes the total inward vdW pressure due to the buckling (Eq. (2)), D is the effective bending stiffness of the elastic shell and N 0z , N 0y and N 0zy denote the axial, circumferential and shearing membrane forces due to the pre-buckling loading conditions, respectively. When torsional load is applied to the shell, the resultant shearing membrane force becomes of primary importance and the axial membrane force vanishes [17]. Consequently, Eq. (6) reduces to 2 2 Eh o4 w 8 4 0 o w 0o w Dr w þ 2 4 ¼ r 2N zy þ N y 2 þ r4 pðz; yÞ ozoy oy R oz ð7Þ If Ti (i = 1, 2, . . . , N) denotes the contribution of the ith layer to the total torque (T), then the total torque applied to the nanopeapod is expressed as T ¼ T1 þ T2 þ þ TN
ð8Þ
Assuming the shearing membrane forces in different layers of the host CNT are the same, we would have N 0zy1 ¼
T1 T2 TN ¼ N 0zy2 ¼ ¼ ¼ N 0zyN ¼ 2pR21 2pR22 2pR2N
ð9Þ
From Eqs. (8) and (9) we obtain a geometry-dependent relationship between the torque in each layer (Ti) and the total applied torque (T) as follows: R2 T i ¼ PN i
2 i¼1 Ri
ð10Þ
T
Similar to the case of a CNT [16,17], one expects that the torsional instability in a nanopeapod with simply supported end conditions would lead to a periodic, low-amplitude rippling of the layers of the host CNT which can be expressed as wi ¼ Ai sinðaz bi yÞ where mp ; a¼ L
bi ¼
n Ri
ð11Þ
ði ¼ 1; 2; . . . ; N Þ
ð12Þ
Ai is a constant defining the amplitude of the radial deflection of the ith layer and L is the length of the nanopeapod [Fig. 1]. Also, m and n are two positive integers representing the axial half wavenumber and circumferential wavenumber, respectively. Substituting Eqs. (10)–(12), along with the pressure distributions of Eq. (2) into Eq. (7), we obtain a set of coupled linear equations of unknown radial deflections wi (i = 1, 2, . . . , N) which can be stated in the following matrix form:
A.N. Sohi, R. Naghdabadi / Carbon 45 (2007) 952–957
2
F 11
6 6 F 21 6 6 6 6 0 6 . 6 . 4 . 0
F 12 F 22
0 F 23
..
0
. F N ðN 1Þ
0 .. . .. .
3
2
7 7 7 7 7 7 7 7 0 5 F NN
w1
3
6w 7 6 27 6. 7 ¼0 6. 7 4. 5 wN N 1 N N
ð13Þ The coefficients Fij in Eq. (13) are as follows: Eh 4 2 F 11 ¼ Dða2 þ b21 Þ þ 2 a4 þ ða2 þ b21 Þ R1 2 3 !1 N X T 4 R2i ab1 þ pvdW ðzÞR1 b21 þ c5 p i¼1 2
F 12 ¼ cða2 þ b21 Þ R1 F 21 ¼ c ða2 þ b22 Þ2 R2 Eh 4 2 F 22 ¼ Dða2 þ b22 Þ þ 2 a4 þ ða2 þ b22 Þ R2 2 3 !1 N X T R 1 5 4 R2i ab2 þ c 1 þ p R2 i¼1
955
The critical torsional load can now be obtained by minimizing the right hand side of Eq. (17) with respect to a, b and z. To minimize the right hand side of Eq. (17) with respect to z, the vdW pressure pvdW in the last term of the numerator of Eq. (17), should adopt its minimum, or in other words its maximum attractive value. This suggests that the torsional buckling would initiate at a point (section) of the host CNT with maximum attractive vdW pressure. This condition is realized at point (section) A of Fig. 2 ˚. with the axial coordinate (z) equal to 7.1 A Thus, according to Eq. (17), it is concluded that filling a single-walled carbon nanotube (SWNT) with linearly arranged fullerenes, results in a decrease in the torsional stability of the nanopeapod. This will be quantitatively examined in next section. It is worthy to note that in this analysis, the possibility of new carbon–carbon bonds formation between adjacent nested fullerenes due to torsional loading of the nanopeapod is neglected. 4. Numerical results
ð14Þ
2
F 23 ¼ cða2 þ b22 Þ .. . RN 1 2 F N ðN 1Þ ¼ c ða þ b2N Þ2 RN 4 Eh 2 F NN ¼ D a2 þ b2N þ 2 a4 þ a2 þ b2N RN 2 3 !1 N X T R N 1 5 4 R2i abN þ c p R N i¼1
In single-walled carbon nanopeapods, it was shown that the critical torsional load is less than that of a SWNT under otherwise identical conditions. Unfortunately, for the case of a double-walled nanopeapod (or more generally for multi-walled ones) it is not easy to reach the same conclusion due to the complexity of corresponding equations. Thus, in order to extend our conclusion for single-walled nanopeapods to the case with double-walled host CNT, two example nanopeapods are studied numerically. The first example is C60@(10, 10) single-walled nanopeapod and the second example is C60@(10, 10)@(15, 15) double-walled nanopea˚ . Other pod. Both of the examples have a length of 81.36 A common characteristics are as follows [16]: ˚ D ¼ 0:85 eV; h ¼ 3:4 A;
In Eq. (13), the existence condition for a non-zero solution necessitates that det½F ij N N ¼ 0
c ¼ 9:9187E19
Eh ¼ 360
J ; m2
N m3
ð15Þ
Eq. (15) determines the critical torsional load of the nanopeapod. For the case of a single-walled nanopeapod (i.e. a nanopeapod with single-walled host CNT), Eq. (15) results in a linear algebraic equation which governs the torsional instability as follows: 4
2
Dða2 þ b2 Þ þ ða2 þ b2 Þ T Eh 2 ab þ pvdW ðzÞRb2 þ 2 a4 ¼ 0 pR R
ð16Þ
where T denotes the torque applied to the single-walled nanopeapod. Consequently, the torque is determined as 4
T ¼ pR2
Dða2 þ b2 Þ þ Eh a4 þ RpvdW ðzÞb2 ða2 þ b2 Þ R2 abða2 þ b2 Þ2
2
ð17Þ
Fig. 3. The applied torque versus the wavenumbers m and n for C60@(10, 10) nanopeapod.
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Fig. 4. The applied torque versus the wavenumbers for the host (10, 10) CNT.
In Fig. 3, the torsional load is rendered versus the axial and circumferential wavenumbers, m and n, for the first example nanopeapod. According to this figure, the critical torque is obtained as 6.766 nN Æ nm. In Fig. 4, the torsional load of the host (10, 10) CNT is shown versus m and n, with the critical value of 9.496 nN Æ nm. Based on molecular dynamics (MD) simulations, the critical torque of a (10, 10) CNT with almost the same aspect ratio of ours, (i.e. 12), was calculated 1.2 nN Æ nm [21], while applying the continuum surface theory with consideration of the exponential Cauchy–Born rule resulted in an approximate value of 12.78 nN Æ nm for the critical torque of (10, 10) CNT [22]. Comparison shows that our result lies reasonably well in the range of the published data mentioned in the literature. As observed in Figs. 3 and 4, the effect of C60 fullerenes encapsulation in (10, 10) carbon nanotube is a 29% decrease in the critical torque. In Figs. 5 and 6, the torsional loads for the second example nanopeapod and its host CNT are shown versus the axial and circumferential wavenumbers, respectively.
Fig. 5. The applied torque versus the wavenumbers for C60@(10, 10)@ (15, 15) nanopeapod.
Fig. 6. The applied torque versus the wavenumbers for the host (10, 10)@(15, 15) CNT.
As it is observed, there exists a 26% decrease in the critical torsional load, from 29.21 nN Æ nm down to 21.76 nN Æ nm, in the case of the second example doublewalled nanopeapod compared to its host double-walled carbon nanotube (DWNT). Based on the numerical results, it is concluded that increasing the number of layers of the host CNT reduces the weakening effect of encapsulated fullerenes on total torsional stability of the nanopeapod. Also, based on Figs. 3 and 5, it is inferred that similar to the case of a carbon nanotube [17], there exists only one combination of wavenumbers which corresponds to the torsional instability mode of a nanopeapod. Hence, the wavenumbers corresponding to the torsional buckling mode of the nanopeapod can be determined uniquely. 5. Conclusions In summary, torsional stability of nanopeapods is studied using a continuum-based shell model which assimilates non-bonded vdW interactions between nested fullerenes and the innermost layer of the host nanotube. In this model, it is assumed that during the torsional loading of the nanopeapod the host CNT retains its cylindrical shape and the nested fullerenes preserve their initial linear arrangement. Also, the possibility of formation of new carbon–carbon bonds between adjacent fullerenes is neglected. To verify the model, it is applied to the problem of torsional buckling of a (10, 10) CNT and the result is shown to be in good agreement with the results available in the literature. Using the proposed model, explicit equations for determination of the critical torques of the nanopeapods are derived based on which it is concluded that the torsional stability of a nanopeapod is less than that of a carbon nanotube under otherwise identical conditions. Also, it is shown that increasing the number of layers of the host CNT decreases the weakening effect of encapsulated fullerenes on total torsional stability of the nanopeapod.
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Since the stability of nanopeapods is shown to be greatly affected by non-bonded vdW interactions between nested fullerenes and the innermost layer of the host CNTs, it is suggested that the type of nested fullerenes would effectively influence the torsional stability of the hybrid structure of the nanopeapod. This can be regarded as an outstanding characteristic of the carbon nanopeapods, entitled mechanical tunability. In other words, one can adjust the mechanical stability of the nanopeapods by controlling the type of encapsulated fullerenes. Such unique property could find amazing and unprecedented applications in nanotechnology. For example, one can imagine adjustable nanodevices based on carbon-nanopeapods which are tailored to respond when the torsional load reaches a predefined value. References [1] Kroto HW, Heath JR, O’Brien SC, Curl RF, Smalley RE. C60: buckminsterfullerene. Nature 1985;318:162–3. [2] Iijima S. Helical microtubules of graphitic carbon. Nature 1991; 354:56–8. [3] Iijima S, Ichihashi T. Single-shell carbon nanotubes of 1-nm diameter. Nature 1993;363:603–5. [4] Smith BW, Monthioux M, Luzzi DE. Encapsulated C60 in carbon nanotubes. Nature 1998;396:323–4. [5] Hodak M, Girifalco LA. Ordered phases of fullerene molecules formed inside carbon nanotubes. Phys. Rev. B 2003;67:075419. [6] Troche KS, Coluci VR, Braga SF, Chinellato DD, Sato F, Legoas SB, et al. Prediction of ordered phases of encapsulated C60, C70, and C78 inside carbon nanotubes. Nano Lett. 2005;5(2):349–55. [7] Kholbystov A, Britz DA, Ardavan A, Briggs GAD. Observation of ordered phases of fullerenes in carbon nanotubes. Phys. Rev. Lett. 2004;92:245507. [8] Service RF. Nanotube peapods show electrifying promise. Science 2001;292:45.
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[9] Kang JW, Hwang HJ. Schematics and simulations of nanomemory device based on nanopeapods. Mater. Sci. Eng. 2005;25:843–7. [10] Farajian AA, Mikami M. Electronic and mechanical properties of C60-doped nanotubes. J. Phys.: Condens. Matter 2001;13:8049–59. [11] Ru CQ. Axially compressed buckling of a doublewall carbon nanotube embedded in an elastic medium. J. Mech. Phys. Solids 2001;49:1265–79. [12] He XQ, Kitipornchai S, Liew KM. Buckling analysis of multi-walled carbon nanotubes: a continuum model accounting for van der Waals interactions. J. Mech. Phys. Solids 2005;53:303–26. [13] Qian D, Liu WK, Ruoff RS. Mechanics of C60 in nanotubes. J. Phys. Chem. B 2001;105:10753–8. [14] Yoon J, Ru CQ, Mioduchowski A. Vibration of an embedded multiwall carbon nanotube. Compos. Sci. Technol. 2003;63: 1533–42. [15] Behfar K, Naghdabadi R. Nanoscale vibrational analysis of a multilayered graphene sheet embedded in an elastic medium. Compos. Sci. Technol. 2005;65:1159–64. [16] Han Q, Lu G. Torsional buckling of a doublewall carbon nanotube embedded in an elastic medium. Eur. J. Mech. A/Solids 2003; 22:875–83. [17] Yang HK, Wang X. Torsional buckling of multi-wall carbon nanotubes embedded in an elastic medium. Compos. Struct. 2007; 77(2):182–92. [18] Wang X, Yang HK. Bending stability of multiwalled carbon nanotubes. Phys. Rev. B 2006;73:085409. [19] Cumings J, Zettl A. Low-friction nanoscale linear bearing realized from multiwall carbon nanotubes. Science 2000;289:602–4. [20] Chen J, Dong J. Electronic properties of peapods: effects of fullerene rotation and different types of tube. J. Phys.: Condens. Matter 2004;16:1401–8. [21] Shibutani Y, Ogata S. Mechanical integrity of carbon nanotubes for bending and torsion. Modell. Simul. Mater. Sci. Eng. 2004;12: 599–610. [22] Arroyo M, Belytschko T. Finite crystal elasticity of carbon nanotubes based on the exponential Cauchy–Born rule. Phys. Rev. B 2004; 69:115415.
Torsional buckling of carbon nanopeapods A.N. Sohi a, R. Naghdabadi a
a,b,*
Department of Mechanical Engineering, Sharif University of Technology, Box 11365-3567, Tehran, Iran b Institute for Nano Science and Technology, Sharif University of Technology, Tehran, Iran Received 11 May 2006; accepted 28 December 2006 Available online 5 January 2007
Abstract Torsional buckling of carbon nanopeapods (carbon nanotubes filled with fullerenes) is studied using a continuum-based multi-layered shell model. The model takes into account non-bonded van der Waals interactions between nested fullerenes and the innermost layer of host nanotube. For nanopeapods with linearly arranged nested fullerenes, equivalent pressure distribution is proposed to model these interactions. Deriving explicit equations governing the torsional stability, it is concluded that the critical torsional load of a carbon nanopeapod is less than that of a carbon nanotube under otherwise identical geometric and mechanical conditions. Performing numerical calculations, it is also shown that increasing the number of layers of the host carbon nanotube decreases the weakening effect of encapsulated fullerenes on torsional stability of the nanopeapod. Ó 2007 Elsevier Ltd. All rights reserved.
1. Introduction The discoveries of ‘‘buckminsterfullerene’’ or simply ‘‘fullerene’’ in the mid 1980s [1] and subsequently carbon nanotubes (CNTs) in the early 1990s [2,3] have opened up opportunities for different applications of these nanostructures. Such discoveries raised a great interest in studying other nanostructures of carbon, ultimately leading to the discovery of ‘‘carbon nanopeapod’’ or simply ‘‘nanopeapod’’ in 1998 [4]. This new class of self-assembled hybrid structures is in essence the well known CNT filled with ordered arrangements of fullerenes. Based on the characteristics of host CNT and nested fullerenes, different ordered arrangements such as linear chain and zigzag have been predicted theoretically [5,6] and observed experimentally [7]. Due to the physical properties of nanopeapods, different potential applications have been proposed for them [8,9]. Some of these applications entail mechanical loading which necessitates the evaluation of the elastic properties and sta-
*
Corresponding author. Address: Department of Mechanical Engineering, Sharif University of Technology, Box 11365-3567, Tehran, Iran. Tel.: +98 21 66005716; fax: +98 21 66000021. E-mail address: [email protected] (R. Naghdabadi). 0008-6223/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.carbon.2006.12.027
bility of nanopeapod under the applied load. Hence, the elucidation of the elastic behaviors of nanopeapods is one of the critical steps towards their practical usage [10]. Treating such properties involves modeling non-bonded van der Waals (vdW) interactions between the guest fullerenes and the adjacent layer of the host CNTs. These interactions are responsible for self-assemblage of fullerenes along the host CNT and provide critical effects on mechanical behavior of the host CNT. In this analysis, the nanopeapod is characterized by a hybrid structure of encapsulated fullerenes inside an elastic cylindrical shell. The non-bonded vdW interactions between the linearly arranged nested fullerenes and the innermost layer of the host CNT are modeled by equivalent axisymmetric harmonic pressure distribution. Employing a continuum-based multi-layered shell model with proper account of the proposed vdW pressure distribution, the instability of the nanopeapod under torsional loading is investigated. 2. Modeling van der Waals interactions Different packing arrangements have been proposed for encapsulated fullerenes inside nanopeapods [5–7]. These
A.N. Sohi, R. Naghdabadi / Carbon 45 (2007) 952–957
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2.2. van der Waals interactions between nested fullerenes and the innermost layer of the host nanotube
Fig. 1. Schematic of double-walled carbon nanopeapod under torsional load T; y and z denote circumferential and axial coordinates, respectively.
arrangements are of principal importance as we seek to model the corresponding vdW interactions. The linear chain arrangement imposes an axisymmetric harmonic vdW pressure upon the innermost layer of the host CNT. In Fig. 1, schematic of a carbon nanopeapod with linear chain arrangement of nested fullerenes is shown. 2.1. van der Waals interactions between adjacent layers of the host nanotube In a multi-walled carbon nanotube (MWNT), if the vdW pressure exerted on layer i by the adjacent layer j is denoted by pij and the pressure exerted on layer j by the ith layer is denoted by pji, we can write pij Ri ¼ pji Rj
ð1Þ
where Ri and Rj express the mean radius of the ith and jth layers, respectively. Considering the infinitesimal buckling, interlayer vdW interactions are modeled by equivalent pressure distributions which are linear functions of the difference of deflections of adjacent layers [11]. Thus, the total inward vdW pressures on different layers are stated as
The non-bonded vdW interactions between nested fullerenes and the innermost layer of the host nanotube can be described using empirical potentials such as Lennard–Jones (LJ) [13]. The total radial force exerted by nested fullerenes on a specific atom of the CNT wall can be calculated by summing up the radial components of all of the vdW interacting forces between the selected atom and the atoms of neighboring nested fullerenes. In Fig. 2, the vdW radial force exerted by nested C60 fullerenes on carbon atoms of a (10, 10) host CNT wall, ˚ [4], is with average C60 center to center distance of 10 A shown. The horizontal axis represents the axial coordinate of the selected atom of the CNT wall, while the vertical axis shows the magnitude of the radial force. Positive and negative values of the radial force account for repulsive and attractive vdW interactions, respectively. In this figure, the diamonds stand for numerical data while the continuous line shows a typical Fourier series fitting. This Fourier series is in the form of X 2npz 2npz F vdW ðzÞ ¼ a0 þ an cos þ bn sin ð3Þ Df Df n In Eq. (3), Df represents the equilibrium distance between adjacent fullerenes inside the nanopeapod [Fig. 1] which is assumed to remain constant during the torsional loading of the nanopeapod. Also, (z) denotes the axial coordinate of the cylindrical shell and its origin is located at the center of the first nested fullerene of consideration. As can be seen, the periodic distribution of radial force can be precisely approximated by the Fourier series fitting. Any increase in number of terms of the Fourier series would further enhance the precision of the approximation. The Fourier series fitting of Fig. 2 has 17 terms (n = 8 in Eq. (3)).
p1 ¼ p12 ¼ c½w2 w1 p2 ¼ p23 þ p21 ¼ c½w3 w2 ðR1 =R2 Þðw2 w1 Þ .. .
ð2Þ
pN ¼ pN ðN 1Þ ¼ cðRN 1 =RN Þ½wN 1 wN where wi denotes the additional displacement of the ith layer due to the buckling along the inward normal direction. Subscripts 1, 2, . . . , N denote the quantities associated with the first innermost layer (1), second innermost layer (2), and so on up to the outermost layer (N) and c is the vdW characteristic coefficient. It should be noted that the initial vdW forces between adjacent layers are set to zero because the pre-buckling interlayer spacings are equal or very close to the equilibrium spacing between graphene layers. In addition, the van der Waals interactions between non-adjacent structures are neglected since these interactions are not significant when compared with those between adjacent structures [12].
Fig. 2. Radial force on carbon atoms of the host CNT in C60@(10, 10) nanopeapod.
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The relationship between the vdW radial force, FvdW(z) in Eq. (3), and its corresponding vdW pressure distribution over the innermost layer of the host CNT is expressed as pffiffiffi !1 3 3 2 d pvdW ðzÞ ¼ F vdW ðzÞ ð4Þ 4 cc where dcc represents the carbon–carbon bond length whose ˚ . The vdW pressure distribution of Eq. magnitude is 1.42 A (4) causes a circumferential membrane force in the innermost layer of the host CNT which, according to equilibrium conditions, can be stated as N 0y1 ¼ R1 pvdW ðzÞ
ð5Þ
In this analysis, the vdW interactions between nested fullerenes and the layers of the host CNT other than the innermost one are neglected. Hence, in the absence of any external pressure, the circumferential membrane forces in other layers of the host CNT are almost zero. In addition, since it is assumed that the average C60 center to center dis˚ and does not change with slight deformations tance is 10 A of the host CNT, the vdW pressure distribution over the innermost layer of the host CNT (Eq. (4)) remains valid during the torsional loading. 3. Torsional buckling analysis In recent years, some continuum-based shell models have been successfully applied to predict the behavior of carbon nanostructures. Using a linear approximation for the vdW interactions between adjacent layers (Eq. (2)), Ru et al. have systemically studied the axial instability and vibration behavior of CNTs [11,14]. Later on, this approach has been incorporated in studying other cases such as vibration of a multi-layered graphene sheet [15] and torsional and bending instability of CNTs [16–18]. Motivated by the idea of continuum-based modeling of CNT, a nanopeapod can be regarded as a hybrid structure of nested fullerenes inside an elastic cylindrical shell, with the vdW interactions between them. Since the nested fullerenes are bonded by weak vdW interactions, they willingly move to more stable positions along the host CNT when the nanopeapod undergoes structural instability. Thus, the total effect of these encapsulated fullerenes is summarized in a harmonic pressure distribution over the innermost layer of the host CNT (Eq. (4)). It has been shown that the friction between adjacent layers in MWNTs as well as between nested fullerenes and the innermost layer of the host CNT in nanopeapods are negligible [19,20]. Hence, in the absence of any tangential force, Donnell’s equilibrium equation governs the buckling behavior of the host CNT [11,16]. Considering the host CNT as a cylindrical shell with the mean radius R, thickness h, and Young’s modulus E, Donnell’s equation is stated as
Eh o4 w Dr8 w þ 2 4 R oz o2 w o2 w o2 w þ N 0y 2 þ r4 pðz; yÞ ¼ r4 N 0z 2 þ 2N 0zy oz oz oy oy
ð6Þ
where p(z, y) denotes the total inward vdW pressure due to the buckling (Eq. (2)), D is the effective bending stiffness of the elastic shell and N 0z , N 0y and N 0zy denote the axial, circumferential and shearing membrane forces due to the pre-buckling loading conditions, respectively. When torsional load is applied to the shell, the resultant shearing membrane force becomes of primary importance and the axial membrane force vanishes [17]. Consequently, Eq. (6) reduces to 2 2 Eh o4 w 8 4 0 o w 0o w Dr w þ 2 4 ¼ r 2N zy þ N y 2 þ r4 pðz; yÞ ozoy oy R oz ð7Þ If Ti (i = 1, 2, . . . , N) denotes the contribution of the ith layer to the total torque (T), then the total torque applied to the nanopeapod is expressed as T ¼ T1 þ T2 þ þ TN
ð8Þ
Assuming the shearing membrane forces in different layers of the host CNT are the same, we would have N 0zy1 ¼
T1 T2 TN ¼ N 0zy2 ¼ ¼ ¼ N 0zyN ¼ 2pR21 2pR22 2pR2N
ð9Þ
From Eqs. (8) and (9) we obtain a geometry-dependent relationship between the torque in each layer (Ti) and the total applied torque (T) as follows: R2 T i ¼ PN i
2 i¼1 Ri
ð10Þ
T
Similar to the case of a CNT [16,17], one expects that the torsional instability in a nanopeapod with simply supported end conditions would lead to a periodic, low-amplitude rippling of the layers of the host CNT which can be expressed as wi ¼ Ai sinðaz bi yÞ where mp ; a¼ L
bi ¼
n Ri
ð11Þ
ði ¼ 1; 2; . . . ; N Þ
ð12Þ
Ai is a constant defining the amplitude of the radial deflection of the ith layer and L is the length of the nanopeapod [Fig. 1]. Also, m and n are two positive integers representing the axial half wavenumber and circumferential wavenumber, respectively. Substituting Eqs. (10)–(12), along with the pressure distributions of Eq. (2) into Eq. (7), we obtain a set of coupled linear equations of unknown radial deflections wi (i = 1, 2, . . . , N) which can be stated in the following matrix form:
A.N. Sohi, R. Naghdabadi / Carbon 45 (2007) 952–957
2
F 11
6 6 F 21 6 6 6 6 0 6 . 6 . 4 . 0
F 12 F 22
0 F 23
..
0
. F N ðN 1Þ
0 .. . .. .
3
2
7 7 7 7 7 7 7 7 0 5 F NN
w1
3
6w 7 6 27 6. 7 ¼0 6. 7 4. 5 wN N 1 N N
ð13Þ The coefficients Fij in Eq. (13) are as follows: Eh 4 2 F 11 ¼ Dða2 þ b21 Þ þ 2 a4 þ ða2 þ b21 Þ R1 2 3 !1 N X T 4 R2i ab1 þ pvdW ðzÞR1 b21 þ c5 p i¼1 2
F 12 ¼ cða2 þ b21 Þ R1 F 21 ¼ c ða2 þ b22 Þ2 R2 Eh 4 2 F 22 ¼ Dða2 þ b22 Þ þ 2 a4 þ ða2 þ b22 Þ R2 2 3 !1 N X T R 1 5 4 R2i ab2 þ c 1 þ p R2 i¼1
955
The critical torsional load can now be obtained by minimizing the right hand side of Eq. (17) with respect to a, b and z. To minimize the right hand side of Eq. (17) with respect to z, the vdW pressure pvdW in the last term of the numerator of Eq. (17), should adopt its minimum, or in other words its maximum attractive value. This suggests that the torsional buckling would initiate at a point (section) of the host CNT with maximum attractive vdW pressure. This condition is realized at point (section) A of Fig. 2 ˚. with the axial coordinate (z) equal to 7.1 A Thus, according to Eq. (17), it is concluded that filling a single-walled carbon nanotube (SWNT) with linearly arranged fullerenes, results in a decrease in the torsional stability of the nanopeapod. This will be quantitatively examined in next section. It is worthy to note that in this analysis, the possibility of new carbon–carbon bonds formation between adjacent nested fullerenes due to torsional loading of the nanopeapod is neglected. 4. Numerical results
ð14Þ
2
F 23 ¼ cða2 þ b22 Þ .. . RN 1 2 F N ðN 1Þ ¼ c ða þ b2N Þ2 RN 4 Eh 2 F NN ¼ D a2 þ b2N þ 2 a4 þ a2 þ b2N RN 2 3 !1 N X T R N 1 5 4 R2i abN þ c p R N i¼1
In single-walled carbon nanopeapods, it was shown that the critical torsional load is less than that of a SWNT under otherwise identical conditions. Unfortunately, for the case of a double-walled nanopeapod (or more generally for multi-walled ones) it is not easy to reach the same conclusion due to the complexity of corresponding equations. Thus, in order to extend our conclusion for single-walled nanopeapods to the case with double-walled host CNT, two example nanopeapods are studied numerically. The first example is C60@(10, 10) single-walled nanopeapod and the second example is C60@(10, 10)@(15, 15) double-walled nanopea˚ . Other pod. Both of the examples have a length of 81.36 A common characteristics are as follows [16]: ˚ D ¼ 0:85 eV; h ¼ 3:4 A;
In Eq. (13), the existence condition for a non-zero solution necessitates that det½F ij N N ¼ 0
c ¼ 9:9187E19
Eh ¼ 360
J ; m2
N m3
ð15Þ
Eq. (15) determines the critical torsional load of the nanopeapod. For the case of a single-walled nanopeapod (i.e. a nanopeapod with single-walled host CNT), Eq. (15) results in a linear algebraic equation which governs the torsional instability as follows: 4
2
Dða2 þ b2 Þ þ ða2 þ b2 Þ T Eh 2 ab þ pvdW ðzÞRb2 þ 2 a4 ¼ 0 pR R
ð16Þ
where T denotes the torque applied to the single-walled nanopeapod. Consequently, the torque is determined as 4
T ¼ pR2
Dða2 þ b2 Þ þ Eh a4 þ RpvdW ðzÞb2 ða2 þ b2 Þ R2 abða2 þ b2 Þ2
2
ð17Þ
Fig. 3. The applied torque versus the wavenumbers m and n for C60@(10, 10) nanopeapod.
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Fig. 4. The applied torque versus the wavenumbers for the host (10, 10) CNT.
In Fig. 3, the torsional load is rendered versus the axial and circumferential wavenumbers, m and n, for the first example nanopeapod. According to this figure, the critical torque is obtained as 6.766 nN Æ nm. In Fig. 4, the torsional load of the host (10, 10) CNT is shown versus m and n, with the critical value of 9.496 nN Æ nm. Based on molecular dynamics (MD) simulations, the critical torque of a (10, 10) CNT with almost the same aspect ratio of ours, (i.e. 12), was calculated 1.2 nN Æ nm [21], while applying the continuum surface theory with consideration of the exponential Cauchy–Born rule resulted in an approximate value of 12.78 nN Æ nm for the critical torque of (10, 10) CNT [22]. Comparison shows that our result lies reasonably well in the range of the published data mentioned in the literature. As observed in Figs. 3 and 4, the effect of C60 fullerenes encapsulation in (10, 10) carbon nanotube is a 29% decrease in the critical torque. In Figs. 5 and 6, the torsional loads for the second example nanopeapod and its host CNT are shown versus the axial and circumferential wavenumbers, respectively.
Fig. 5. The applied torque versus the wavenumbers for C60@(10, 10)@ (15, 15) nanopeapod.
Fig. 6. The applied torque versus the wavenumbers for the host (10, 10)@(15, 15) CNT.
As it is observed, there exists a 26% decrease in the critical torsional load, from 29.21 nN Æ nm down to 21.76 nN Æ nm, in the case of the second example doublewalled nanopeapod compared to its host double-walled carbon nanotube (DWNT). Based on the numerical results, it is concluded that increasing the number of layers of the host CNT reduces the weakening effect of encapsulated fullerenes on total torsional stability of the nanopeapod. Also, based on Figs. 3 and 5, it is inferred that similar to the case of a carbon nanotube [17], there exists only one combination of wavenumbers which corresponds to the torsional instability mode of a nanopeapod. Hence, the wavenumbers corresponding to the torsional buckling mode of the nanopeapod can be determined uniquely. 5. Conclusions In summary, torsional stability of nanopeapods is studied using a continuum-based shell model which assimilates non-bonded vdW interactions between nested fullerenes and the innermost layer of the host nanotube. In this model, it is assumed that during the torsional loading of the nanopeapod the host CNT retains its cylindrical shape and the nested fullerenes preserve their initial linear arrangement. Also, the possibility of formation of new carbon–carbon bonds between adjacent fullerenes is neglected. To verify the model, it is applied to the problem of torsional buckling of a (10, 10) CNT and the result is shown to be in good agreement with the results available in the literature. Using the proposed model, explicit equations for determination of the critical torques of the nanopeapods are derived based on which it is concluded that the torsional stability of a nanopeapod is less than that of a carbon nanotube under otherwise identical conditions. Also, it is shown that increasing the number of layers of the host CNT decreases the weakening effect of encapsulated fullerenes on total torsional stability of the nanopeapod.
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Since the stability of nanopeapods is shown to be greatly affected by non-bonded vdW interactions between nested fullerenes and the innermost layer of the host CNTs, it is suggested that the type of nested fullerenes would effectively influence the torsional stability of the hybrid structure of the nanopeapod. This can be regarded as an outstanding characteristic of the carbon nanopeapods, entitled mechanical tunability. In other words, one can adjust the mechanical stability of the nanopeapods by controlling the type of encapsulated fullerenes. Such unique property could find amazing and unprecedented applications in nanotechnology. For example, one can imagine adjustable nanodevices based on carbon-nanopeapods which are tailored to respond when the torsional load reaches a predefined value. References [1] Kroto HW, Heath JR, O’Brien SC, Curl RF, Smalley RE. C60: buckminsterfullerene. Nature 1985;318:162–3. [2] Iijima S. Helical microtubules of graphitic carbon. Nature 1991; 354:56–8. [3] Iijima S, Ichihashi T. Single-shell carbon nanotubes of 1-nm diameter. Nature 1993;363:603–5. [4] Smith BW, Monthioux M, Luzzi DE. Encapsulated C60 in carbon nanotubes. Nature 1998;396:323–4. [5] Hodak M, Girifalco LA. Ordered phases of fullerene molecules formed inside carbon nanotubes. Phys. Rev. B 2003;67:075419. [6] Troche KS, Coluci VR, Braga SF, Chinellato DD, Sato F, Legoas SB, et al. Prediction of ordered phases of encapsulated C60, C70, and C78 inside carbon nanotubes. Nano Lett. 2005;5(2):349–55. [7] Kholbystov A, Britz DA, Ardavan A, Briggs GAD. Observation of ordered phases of fullerenes in carbon nanotubes. Phys. Rev. Lett. 2004;92:245507. [8] Service RF. Nanotube peapods show electrifying promise. Science 2001;292:45.
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