- Email: [email protected]
Surface effects on the vibrational frequency of double-walled carbon nanotubes using the nonlocal Timoshenko beam model
Composites: Part B 43 (2012) 64–69
Contents lists available at ScienceDirect
Composites: Part B journal homepage: www.elsevier.com/locate/compositesb
Surface effects on the vibrational frequency of double-walled carbon nanotubes using the nonlocal Timoshenko beam model Xiao-wen Lei a, Toshiaki Natsuki b,⇑, Jin-xing Shi a, Qing-qing Ni b a b
Interdisciplinary Graduate School of Science & Technology, Shinshu University, 3-15-1 Tokida, Ueda, Nagano 386-8567, Japan Department of Functional Machinery & Mechanics, Shinshu University, 3-15-1 Tokida, Ueda, Nagano 386-8567, Japan
a r t i c l e
i n f o
Article history: Available online 28 April 2011 Keywords: B. Surface effects Nonlocal Timoshenko beam theory A. Double-walled carbon nanotubes B. Vibration
a b s t r a c t Double-walled carbon nanotubes (DWCNTs) are being investigated for use as latent materials for drug carriers. However, the surface effects cannot be ignored when drugs or other functional materials, such as nickel or silver, adhere to the surface of the outer tube of a DWCNT. In this paper, the vibrational frequency of DWCNTs, while accounting for surface effects, is studied using the nonlocal Timoshenko beam model. The influence of the surface elasticity modulus, residual surface stress, nonlocal parameter, axial half-wave number and aspect ratio are investigated in detail. The results show that the vibrational frequency is significantly affected by the surface material, nonlocal parameter, vibration mode and aspect ratio. In short DWCNTs on condition of higher vibrational modes, the influences of the surface and nonlocal effects on vibration are more pronounced. Ó 2011 Elsevier Ltd. All rights reserved.
1. Introduction Carbon nanotubes (CNTs) have been proposed and actively explored as multipurpose carriers for drug delivery and diagnostic applications [1]. Currently, chemically functionalized CNTs have shown promise in tumor-targeted accumulation in mice. They also exhibited biocompatibility, excretion and little toxicity [2]. In this case, the active substances can be dissolved in the inner core, and may also be adsorbed as their surfaces adhere to the outer tubes. Because of this, the surface effects, including the surface energy, surface tension, surface relaxation and surface reconstruction should be accounted for when the overall elastic properties of nanostructures are investigated. Because of the very large surface-to-volume ratio of nanostructures, surface effects play important roles in determining the physical and chemical properties of these materials and cannot be neglected [3,4]. Recently, considerable attention has been given to investigate the surface effects on nanoscale materials. Feng et al. [5] studied surface effects on the elastic modulus of nanoporous materials, and found that the influence of the residual surface stress on the effective modulus was weaker than that of surface elasticity. He and Lilley [6,7] investigated the effects of surface tension on the static bending of nanowires with cantilever, simply supported and fixed–fixed three different boundary conditions based on the Young–Laplace equation, and suggested that the overall Young’s modulus of nanowires should be studied as a function of the ⇑ Corresponding author. Tel.: +81 0268 21 5421; fax: +81 0268 21 5482. E-mail address: [email protected] (T. Natsuki). 1359-8368/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.compositesb.2011.04.032
surface effects. Wang and Feng [8] analyzed the surface effects on buckling of nanowires under uniaxial compression, and derived an analytical solution for the critical load. Frequency analyses of nanotubes that consider both the internal and external surface effects under various boundary conditions have been reported by some researchers [3,9–11]. Moreover, surface stress and surface elasticity have been recognized as important factors, which may explain the experimentally measured size dependence of the elastic modulus of nanoscale materials [12–15]. Most of the studies have treated nanowires and nanotubes as simple Euler beams, which neglects the effects of transverse shear deformation and rotary inertia [16–20]. Unlike this, Wang and Feng [16] and Lee and Chang [3] used the nonlocal Timoshenko beam model integrated with surface elasticity theory to discuss the natural frequency of the nanostructures. It has been clear that nonlocal effects play a significant role for nanostructures. Furthermore, the Timoshenko beam model worked better than the Euler model when considering the effects of shear deformation and rotary inertia on transverse wave propagation in individual CNTs. Yoon et al. [21] discovered this while modeling transverse terahertz wave propagation in CNTs. Based on nonlocal Timoshenko beam theory, Wang et al. [22] proposed that the nonlocal effect was more significant when dealing with short CNTs with natural frequencies of THz or greater. Lu et al. [23] investigated the vibrational properties of CNTs and reported that the nonlocal parameter e0a, had a significant influence on the dynamic properties of the beam-like structures based on nonlocal beam models. In this study, based on nonlocal Timoshenko beam theory and the surface elasticity model, we analyzed the vibrational behavior
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X.-w. Lei et al. / Composites: Part B 43 (2012) 64–69
of double-walled carbon nanotubes (DWCNTs). We hope that these studies will be helpful for selecting DWCNTs for the delivery of drugs or for the use with functionalized chemical materials. Using the proposed theoretical approach, the influences of the surface effect, nonlocal effect, aspect ratio and the vibrational modes on the vibrational behavior are investigated.
2. Theoretical approach A schematic of a DWCNT with a thin outer surface layer is showing in Fig. 1. L and h are the length and thickness of the DWCNT, respectively. d is the thickness of the thin surface layer, which approached zero for this calculation method [8]. 2.1. Nonlocal Timoshenko beam theory The governing equation of nonlocal Timoshenko beam includes two coupled differential equations, which are expressed in terms of the lateral displacement and the angle of rotation due to bending. The governing equations of transverse vibrations of CNTs with considering of the surface effects can be expressed as [3,23]:
# " 2 @ 2 ui @wi @ 2 ui 2 @ ¼ 1 ðe þ kGA u aÞ q l i 0 i i @x2 @x @x2 @t 2
ðEIÞi
jGAi
@ 2 wi @ ui @x2 @x
"
!
"
To study the vibrational behavior of DWCNTs, a double-elastic shell model was developed. In this model, it is assumed that each of the nested tubes in a CNT is an individual elastic shell, and the adjacent tubes are coupled to each other by normal van der Waals (vdW) interactions. The pressures from vdW forces exerted on the inner and outer nanotubes through the vdW-interaction forces are given as:
P1 ¼ cðw2 w1 Þ
ð5Þ
P2 ¼ cðw1 w2 Þ
ð6Þ
where wi (i = 1, 2) are the radial displacements of the nanotubes. c is the vdW-interaction coefficient between the inner and outer nanotubes, which can be estimated from the Lennard–Jones potential [24]:
c¼
peRin Rout r6 1120 3a4
Hm ¼ ðRin þ Rout Þm
K¼
#
ð2Þ
where Wi is the transverse displacement, and ui is the rotation angle. These depend on the spatial coordinate x and the time t, for inner and outer tubes (i = 1, 2), respectively. Ii is the moment of inertia and Ai is the cross-sectional area for the inner and outer tubes. q is the mass density per unit volume, and k is the shear correction factor. E and G are the Young’s and shear moduli of the DWCNTs, respectively. pi is the distributed transverse pressure per unit axial length. e0a is the scale coefficient that incorporates the nonlocal effect. ðEIÞi is the effective flexural rigidity, which includes the surface bending elasticity and the flexural rigidity. Hi is a constant parameter which are determined by the residual surface tension and shape of the cross section. ðEIÞi and Hi are defined as [3]:
ðEIÞ1 ¼ EI1
Outer tube
ðEIÞ2 ¼ EI2 þ pES R3out
H1 ¼ 0
ð3Þ H2 ¼ 4sRout
H7 1001r6 H13
ð7Þ
where
ð1Þ
@2 @ 2 wi Hi 2 @x @x2
Inner tube
3 Z
ð4Þ
p
dh
2
m
0
@ p @x2 i " # 2 @ 2 wi 2 @ ¼ 1 ðe0 aÞ q A i @x2 @t 2 þ 1 ðe0 aÞ2
2.2. Van der Waals interaction forces
#
þ 1 ðe0 aÞ2 2
where Es and s are the surface elasticity modulus and residual surface tension per length on the outer tube, respectively.
ð1 Kcos2 hÞ 2
ðm ¼ 7; 13Þ
Rin þ Rout
ð8Þ
ð9Þ
ðRin þ Rout Þ2
where a is the carbon–carbon bond length (0.142 nm). Rin and Rout are the inner and outer radii of the DWCNTs, respectively. r and e are the vdW radius and the well depth of the Lennard–Jones potential, respectively. 2.3. Transverse vibration of DWCNTs Consider a simply supported DWCNT with length L, and the boundary conditions have the following form:
wi ð0; tÞ ¼ wi ðL; tÞ ¼ 0 ði ¼ 1; 2Þ
ð10Þ
@ uð0; tÞ @ uðL; tÞ ¼ ¼ 0 ði ¼ 1; 2Þ @x @x
ð11Þ
To satisfy boundary conditions given by Eqs. (10) and (11), the displacement field for a DWCNT is given as:
wi ¼ W i wðx; tÞ wðx; tÞ ¼ sin
mpx sin xt L
ði ¼ 1; 2Þ
ð12Þ
ui ¼ Ui uðx; tÞ uðx; tÞ ¼ cos
mpx sin xt L
ði ¼ 1; 2Þ
ð13Þ
Fig. 1. Schematic diagram showing latitudinal and longitudinal cross-sections of the DWCNT analysis model.
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X.-w. Lei et al. / Composites: Part B 43 (2012) 64–69
where W1 and W2 represent the modal amplitudes of the deflections of the inner and the outer tubes, respectively. U1 and U2 represent the modal amplitudes of the slopes of the inner and outer tubes due to bending deformations, respectively. The integer m is the axial half-wave number of the vibrational mode, and x is the circular frequency. By substituting Eqs. (3)–(6) into Eqs. (1) and (2), we obtain: For inner tube
EI1
4 @ 2 u1 @ 2 u1 @w1 2 @ u1 kGA u q I þ q I ðe aÞ þ kGA1 ¼0 1 1 1 0 1 @x2 @x @t 2 @x2 @t 2 ð14Þ 2
2
@ u1 @ w1 @ w1 @ w1 þ cðe0 aÞ2 qA1 þ kGA1 @x @x2 @x2 @t 2 4 2 @ w @ w 1 2 þ qA1 ðe0 aÞ2 2 2 cw1 cðe0 aÞ2 þ cw2 ¼ 0 @x2 @x @t
ð15Þ
For outer tube
ðEI2 þ pEs R3out Þ þ kGA2
2
ð16Þ
2 @ 2 w1 @ u2 @ 2 w2 2 @ w2 þ cw kGA þ cðe aÞ þ kGA 1 2 2 0 @x2 @x @x2 @x2 4 2 @ 2 w2 @ w @ w 2 2 qA2 þ qA2 ðe0 aÞ2 2 2 cw2 þ H2 @x2 @t 2 @x @t @ 4 w2 ðe0 aÞ2 H2 ¼0 ð17Þ @x4
cðe0 aÞ2
To simplify the calculation, Eqs. (14)–(17) can be rewritten as:
2
L11 6 6 L21 6 6L 4 31
L12
L13
L22
L23
L32
L33
L41
L42
L43
9 38 L14 > > u1 > > > > > 7> L24 7< w1 = 7 ¼0 > u2 > > L34 7 5> > > > > : ; L44 w2
where Lij are the differential operators given as:
L11 ¼ EI1
@2 @2 @4 kGA1 ql1 2 þ qI1 ðe0 aÞ2 2 2 2 @x @t @x @t
L21 ¼ kGA1
L22 ¼ kGA1
@ @x
@2 @2 @2 @4 þ cðe0 aÞ2 2 qA1 2 þ qA1 ðe0 aÞ2 2 2 c 2 @x @x @t @x @t
L24 ¼ cðe0 aÞ2
L34 ¼ kGA2
@2 þc @x2
ð18Þ
@2 @2 @4 2 kGA q I þ q I ðe aÞ 2 2 2 0 @x2 @t 2 @x2 @t 2
@ @x
L42 ¼ cðe0 aÞ2
4
@ u2 @ u2 @ u kGA2 u2 qI2 þ qI2 ðe0 aÞ2 2 22 @x2 @t2 @x @t
@w2 ¼0 @x
@ @x
L33 ¼ ðEI2 þ pE s R3out Þ
2
kGA1
2
L12 ¼ kGA1
L43 ¼ kGA2
@2 þc @x2
@ @x
@2 @2 @2 @4 þ cðe0 aÞ2 2 qA2 2 þ qA2 ðe0 aÞ2 2 2 c @x2 @x @t @x @t @2 @4 2 þ H2 2 ðe0 aÞ H2 4 @x @x
L44 ¼ kGA2
L13 ¼ L14 ¼ L23 ¼ L31 ¼ L32 ¼ L41 ¼ 0
ð19Þ
The vibrational frequency of DWCNTs considering surface effects can be determined using a nontrivial solution by substituting Eqs. (12) and (13) into Eq. (18). 3. Results and discussion In this approach, the influences of the surface material, nonlocal parameter, aspect ratio and vibrational mode on the vibrational frequency are investigated. We considered a DWCNT with inner and outer diameters of 2.2 and 3.0 nm, respectively. The DWCNTs have a Young’s modulus of 1.0 TPa, Poisson’s ratio of 0.27 and the
Fig. 2. Comparison of the vibrational frequencies of DWCNTs with various surface elasticity and surface stress values as a function of the axial half-wave number of the vibrational mode (L/2Rour = 10, e0a = 1 nm).
X.-w. Lei et al. / Composites: Part B 43 (2012) 64–69
67
Fig. 3. Comparison of the vibrational frequencies of DWCNTs with various surface elasticity and surface stress values as a function of aspect ratio (m = 5, e0a = 1 nm).
mass density of 2.3 g/cm3. The layer thickness of the DWCNT was assumed to be that of a graphite sheet with a thickness of 0.34 nm [25]. The vdW parameters in the Lennard–Jones potential are r = 0.34 nm and e = 2.967 meV, which were reported by Saito et al. [26]. The dependence of the shear coefficient j, on the radius is 0.8 [21,27]. Figs. 2 and 3 describe the natural frequency of the DWCNT when the influence of surface elasticity modulus and the residual surface stress on the outer tubes was considered. They depend on the material type and the surface crystal orientation. For the theoretical calculations using (1 1 2) and (0 0 1) nickel, the surface elasticity and surface stress values were obtained using atomistic calculations that yielded values of Es = 35.3 N/m and s = 0.31 N/ m, and Es = -43.8 N/m and s = 0.71 N/m. The surface elastic modulus Es, is calculated from surface elastic constants S1111 and S1122, which were calculated by Shenoy [28] using Es = (S1111 + 2S1122) (S1111 S1122)/(S1111 + 2S1122) [7]. The relationship between the
frequency and the vibrational mode of the DWCNTs is shown in Fig. 2. As the axial half-wave number increases, the natural frequency of the DWCNTs, which have four modes and are covered with different crystal surfaces, also increases. It can be seen that the surface effective elastic modulus and residual stress have an influence on the vibrational properties of DWCNTs. When the axial half-wave mode increases from 1 to 10, the surface effects become more and more pronounced. In mode 3, comparing the results between (1 1 2) and (0 0 1) nickel covered DWCNTs, there is an 11.7% variation in the vibrational frequency when the vibrational mode reaches 10. The relationship between the frequency and the aspect ratio of the DWCNTs is presented in Fig. 3. The influence of the surface parameters is considerably higher in lower aspect ratio CNTs. As the aspect ratio of the DWCNTs increase, the surface effects disappear and the results converge into two distinct branches which are close to natural frequency of the DWCNTs without surface material. For a short DWCNT, the surface-to-volume ratio is much
Fig. 4. Comparison of the vibrational frequencies of DWCNTs with various surface elasticity and surface stress values as a function of the nonlocal parameter (L/2Rour = 10, m = 5).
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X.-w. Lei et al. / Composites: Part B 43 (2012) 64–69
Fig. 5. Comparison of the vibrational frequency ratio of DWCNTs with various axial half-wave numbers as a function of the nonlocal parameter (L/2Rour = 10, Es = 1.22 N/m, s = 0.89 N/m).
shenko beam/local Timoshenko beam. In addition to the nonlocal effect, the nonlocal Timoshenko beam model is also dominated by the effects of the vibrational mode and aspect ratio. Figs. 5 and 6 show the relationship between the frequency ratio and the nonlocal parameter of the DWCNTs. This was achieved by considering the influence of the vibrational mode and aspect ratio, in which the surface parameters of (0 0 1) single-crystal silver (Es = 1.22 N/m and s = 0.89 N/m) are used. The frequency ratio serves as an index to assess the nonlocal effect on DWCNTs vibrational solutions quantitatively. It can be found in Fig. 5 that the frequency ratios are less than unity for all four modes with various axial half-wave numbers. This means that the frequency of the DWCNTs calculated using local theory will be the largest one. As the scale coefficient e0a, increases, the frequency obtained from the nonlocal Timoshenko beam becomes smaller than those obtained from the local counterpart. For example, consider DWCNTs with m = 5, L/2Rout = 10 in mode 1. When the scale coefficient increases from 0 to 2.0 nm, the frequency decreases from 3.28 THz to 2.32 THz, which corresponds to a reduction of over 40%. Note that at higher vibrational modes, the frequency ratio is farther from unity. Variation of the frequency ratio versus the scale coefficient for the four vibration modes of DWCNTs with different aspect ratios are plotted in Fig. 6. It can be seen that the frequency ratio decreases as the nonlocal parameter increases. Meanwhile, the vibration frequency decreases when the nonlocal Timoshenko beam model is used. The scale coefficient e0a, plays a larger role in shorter DWCNTs. Conversely, for a slender DWCNT with aspect ratio of 30, the frequency ratio approaches to unity. This indicates a reduction in the nonlocal effect. In summary, the nonlocal effect significant for short DWCNTs, and can be neglected in longer tubes. 4. Conclusions An analytical procedure based on the nonlocal Timoshenko beam mode was used to investigate the vibrational behavior of DWCNTs adhered by surface materials with simply supported boundary condition. The influences of the surface materials and the nonlocal parameter on the vibrational frequency and frequency ratio are both significant, especially in DWCNTs with higher vibrational modes and shorter lengths. The present study will be useful for understanding the vibration properties of DWCNTs. In particular, we hope that it will help guide the use of DWCNTs as carriers of drug or functionalized materials. Acknowledgments
Fig. 6. Comparison of the vibrational frequency ratio of DWCNTs with various aspect ratios as a function of the nonlocal parameter (m = 1, Es = 1.22 N/m, s = 0.89 N/m).
larger than that of a slender one. Therefore the surface effects become more obviously. However, vibrational frequency of mode 4 of the DWCNTs is not significantly affected by surface material regardless of vibrational mode or aspect ratio. The relationship between the frequency and the nonlocal parameters of DWCNTs is shown in Fig. 4. The influence of the surface material on the outer tubes is considered in these calculations. From the figure, it can be seen that the influence of nonlocal parameter in modes 3 and 4 is much larger than that in mode 1 and 2. The vibrational frequencies of the DWCNTs covered with surface materials with positive and negative surface elastic moduli surround those of the DWCNTs without surface material. To illustrate the influence of nonlocal effects on vibration, we defined the frequency ratio as: frequency ratio g = nonlocal Timo-
This work was supported by a Grant-in-Aid for Global COE Program by the Ministry of Education, Culture, Sports, Science and Technology and by CLUSTER (the second stage) by the Ministry of Education, Culture, Sports, Science and Technology (Japan). References [1] Prato M, Kostarelosk K, Bianco A. Functionalized carbon nanotubes in drug design and discovery. Acc Chem Res 2008;41:60–8. [2] Liu Z, Chen K, Davis C, Sherlock S, Cao QZ, Chen XY, et al. Drug Delivery with carbon nanotubes for In vivo cancer treatment. Cancer Res 2008;68:6652–60. [3] Lee HL, Chang WJ. Surface effects on frequency analysis of nanotubes using nonlocal Timoshenko beam theory. J Appl Phys 2010;108:093503-3. [4] Guo JG, Zhao YP. The size-dependent bending elastic properties of nanobeams with surface effects. Nanotechnology 2007;18:295701–6. [5] Feng XQ, Xia R, Li XD, Li B. Surface effects on the elastic modulus of nanoporous materials. Appl Phys Lett 2009;94:011913–6. [6] He J, Lilley CM. Surface stress effect on bending resonance of nanowires with different boundary conditions. Appl Phys Lett 2008;93:263103–8. [7] He J, Lilley CM. Surface effect on the elastic behavior of static bending nanowires. Nano Lett 2008;8:1798–802. [8] Wang GF, Feng XQ. Surface effects on buckling of nanowires under uniaxial compression. Appl Phys Lett 2009;94:141913-3.
X.-w. Lei et al. / Composites: Part B 43 (2012) 64–69 [9] Gurtina ME, Weissmullerb J, Larche F. A general theory of curved deformable interfaces in solids at equilibrium. Philos Mag A 1998;78:1093–109. [10] Wang GF, Feng XQ. Effects of surface elasticity and residual surface tension on the natural frequency of microbeams. Appl Phys Lett 2007;90:231903–4. [11] Farshi B, Assadi A, Alinia-ziazi A. Frequency analysis of nanotubes with consideration of surface effects. Appl Phys Lett 2010;96:093103–5. [12] Jing GY, Duan HL, Sun XM, Zhang ZS, Xu J, Li YD, et al. Surface effects on elastic properties of silver nanowires: contact atomic-force microscopy. Phys Rev B 2006;73:235406–9. [13] Geng Y, Park HS. Surface stress effects on the bending properties of fcc metal nanowires. Phys Rev B 2009;79:195415–21. [14] Sharm P, Ganti S, Bhate N. Effect of surfaces on the size-dependent elastic state of nano-inhomogeneities. Appl Phys Lett 2003;82:535–7. [15] Wang ZQ, Zhao YP, Huang ZP. The effects of surface tension on the elastic properties of nano structures. Int J Eng Sci 2010;48:140–50. [16] Wang GF, Feng XQ. Timoshenko beam model for buckling and vibration of nanowires with surface effects. J Phys D 2009;42:155411–5. [17] Hu N, Nunoyab K, Panc D, Okabeb T, Fukunagab H. Prediction of buckling characteristics of carbon nanotubes. Int J Solids Struct 2007;44:6535–50. [18] Hu N, Fukunaga H, Lu C, Kameyama M, Yan B. Prediction of elastic properties of carbon nanotube-reinforced composites. Proc Roy Soc A – Math Phys 2005;461:1685–710.
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[19] Li CY, Chou TW. A structural mechanics approach for the analysis of carbon nanotubes. Int J Solids Struct 2003;40:2487–99. [20] Wang ZQ, Zhao YP. Self-instability and bending behaviors of nano plates. Acta Mech Solid Sin 2009;22:630–43. [21] Yoon J, Ru CQ, Mioduchowski A. Timoshenko-beam effects on transverse wave propagation in carbon nanotubes. Composites Part B 2004;35:87–93. [22] Wang CM, Zhang YY, He XQ. Vibration of nonlocal Timoshenko beams. Nanotechnology 2007;18:105401–9. [23] Lu P, Lee HP, Lu C, Zhang PQ. Application of nonlocal beam models for carbon nanotubes. Int J Solids Struct 2007;44:5289–300. [24] He XQ, Kitipornchai S, Liew KM. Buckling analysis of multi-walled carbon nanotubes: a continuum model accounting for van der Waals interaction. J Mech Phys Solids 2005;53:303–26. [25] Natsuki T, Tsuchiya T, Ni QQ, Endo M. Torsional elastic instability of doublewalled carbon nanotubes. Carbon 2008;48:4362–8. [26] Saito R, Matsuo R, Kimura T, Dresselhaus G, Dresselhaus MS. Anomalous potential barrier of double-wall carbon nanotube. Chem Phys Lett 2001;348:187–93. [27] Yoon J, Ru CQ, Mioduchowski A. Terahertz vibration of short carbon nanotubes modeled as Timoshenko beams. J Appl Mech 2005;72:10–7. [28] Shenoy VB. Atomistic calculations of elastic properties of metallic fcc crystal surfaces. Phys Rev B 2005;71:094104–11.
Contents lists available at ScienceDirect
Composites: Part B journal homepage: www.elsevier.com/locate/compositesb
Surface effects on the vibrational frequency of double-walled carbon nanotubes using the nonlocal Timoshenko beam model Xiao-wen Lei a, Toshiaki Natsuki b,⇑, Jin-xing Shi a, Qing-qing Ni b a b
Interdisciplinary Graduate School of Science & Technology, Shinshu University, 3-15-1 Tokida, Ueda, Nagano 386-8567, Japan Department of Functional Machinery & Mechanics, Shinshu University, 3-15-1 Tokida, Ueda, Nagano 386-8567, Japan
a r t i c l e
i n f o
Article history: Available online 28 April 2011 Keywords: B. Surface effects Nonlocal Timoshenko beam theory A. Double-walled carbon nanotubes B. Vibration
a b s t r a c t Double-walled carbon nanotubes (DWCNTs) are being investigated for use as latent materials for drug carriers. However, the surface effects cannot be ignored when drugs or other functional materials, such as nickel or silver, adhere to the surface of the outer tube of a DWCNT. In this paper, the vibrational frequency of DWCNTs, while accounting for surface effects, is studied using the nonlocal Timoshenko beam model. The influence of the surface elasticity modulus, residual surface stress, nonlocal parameter, axial half-wave number and aspect ratio are investigated in detail. The results show that the vibrational frequency is significantly affected by the surface material, nonlocal parameter, vibration mode and aspect ratio. In short DWCNTs on condition of higher vibrational modes, the influences of the surface and nonlocal effects on vibration are more pronounced. Ó 2011 Elsevier Ltd. All rights reserved.
1. Introduction Carbon nanotubes (CNTs) have been proposed and actively explored as multipurpose carriers for drug delivery and diagnostic applications [1]. Currently, chemically functionalized CNTs have shown promise in tumor-targeted accumulation in mice. They also exhibited biocompatibility, excretion and little toxicity [2]. In this case, the active substances can be dissolved in the inner core, and may also be adsorbed as their surfaces adhere to the outer tubes. Because of this, the surface effects, including the surface energy, surface tension, surface relaxation and surface reconstruction should be accounted for when the overall elastic properties of nanostructures are investigated. Because of the very large surface-to-volume ratio of nanostructures, surface effects play important roles in determining the physical and chemical properties of these materials and cannot be neglected [3,4]. Recently, considerable attention has been given to investigate the surface effects on nanoscale materials. Feng et al. [5] studied surface effects on the elastic modulus of nanoporous materials, and found that the influence of the residual surface stress on the effective modulus was weaker than that of surface elasticity. He and Lilley [6,7] investigated the effects of surface tension on the static bending of nanowires with cantilever, simply supported and fixed–fixed three different boundary conditions based on the Young–Laplace equation, and suggested that the overall Young’s modulus of nanowires should be studied as a function of the ⇑ Corresponding author. Tel.: +81 0268 21 5421; fax: +81 0268 21 5482. E-mail address: [email protected] (T. Natsuki). 1359-8368/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.compositesb.2011.04.032
surface effects. Wang and Feng [8] analyzed the surface effects on buckling of nanowires under uniaxial compression, and derived an analytical solution for the critical load. Frequency analyses of nanotubes that consider both the internal and external surface effects under various boundary conditions have been reported by some researchers [3,9–11]. Moreover, surface stress and surface elasticity have been recognized as important factors, which may explain the experimentally measured size dependence of the elastic modulus of nanoscale materials [12–15]. Most of the studies have treated nanowires and nanotubes as simple Euler beams, which neglects the effects of transverse shear deformation and rotary inertia [16–20]. Unlike this, Wang and Feng [16] and Lee and Chang [3] used the nonlocal Timoshenko beam model integrated with surface elasticity theory to discuss the natural frequency of the nanostructures. It has been clear that nonlocal effects play a significant role for nanostructures. Furthermore, the Timoshenko beam model worked better than the Euler model when considering the effects of shear deformation and rotary inertia on transverse wave propagation in individual CNTs. Yoon et al. [21] discovered this while modeling transverse terahertz wave propagation in CNTs. Based on nonlocal Timoshenko beam theory, Wang et al. [22] proposed that the nonlocal effect was more significant when dealing with short CNTs with natural frequencies of THz or greater. Lu et al. [23] investigated the vibrational properties of CNTs and reported that the nonlocal parameter e0a, had a significant influence on the dynamic properties of the beam-like structures based on nonlocal beam models. In this study, based on nonlocal Timoshenko beam theory and the surface elasticity model, we analyzed the vibrational behavior
65
X.-w. Lei et al. / Composites: Part B 43 (2012) 64–69
of double-walled carbon nanotubes (DWCNTs). We hope that these studies will be helpful for selecting DWCNTs for the delivery of drugs or for the use with functionalized chemical materials. Using the proposed theoretical approach, the influences of the surface effect, nonlocal effect, aspect ratio and the vibrational modes on the vibrational behavior are investigated.
2. Theoretical approach A schematic of a DWCNT with a thin outer surface layer is showing in Fig. 1. L and h are the length and thickness of the DWCNT, respectively. d is the thickness of the thin surface layer, which approached zero for this calculation method [8]. 2.1. Nonlocal Timoshenko beam theory The governing equation of nonlocal Timoshenko beam includes two coupled differential equations, which are expressed in terms of the lateral displacement and the angle of rotation due to bending. The governing equations of transverse vibrations of CNTs with considering of the surface effects can be expressed as [3,23]:
# " 2 @ 2 ui @wi @ 2 ui 2 @ ¼ 1 ðe þ kGA u aÞ q l i 0 i i @x2 @x @x2 @t 2
ðEIÞi
jGAi
@ 2 wi @ ui @x2 @x
"
!
"
To study the vibrational behavior of DWCNTs, a double-elastic shell model was developed. In this model, it is assumed that each of the nested tubes in a CNT is an individual elastic shell, and the adjacent tubes are coupled to each other by normal van der Waals (vdW) interactions. The pressures from vdW forces exerted on the inner and outer nanotubes through the vdW-interaction forces are given as:
P1 ¼ cðw2 w1 Þ
ð5Þ
P2 ¼ cðw1 w2 Þ
ð6Þ
where wi (i = 1, 2) are the radial displacements of the nanotubes. c is the vdW-interaction coefficient between the inner and outer nanotubes, which can be estimated from the Lennard–Jones potential [24]:
c¼
peRin Rout r6 1120 3a4
Hm ¼ ðRin þ Rout Þm
K¼
#
ð2Þ
where Wi is the transverse displacement, and ui is the rotation angle. These depend on the spatial coordinate x and the time t, for inner and outer tubes (i = 1, 2), respectively. Ii is the moment of inertia and Ai is the cross-sectional area for the inner and outer tubes. q is the mass density per unit volume, and k is the shear correction factor. E and G are the Young’s and shear moduli of the DWCNTs, respectively. pi is the distributed transverse pressure per unit axial length. e0a is the scale coefficient that incorporates the nonlocal effect. ðEIÞi is the effective flexural rigidity, which includes the surface bending elasticity and the flexural rigidity. Hi is a constant parameter which are determined by the residual surface tension and shape of the cross section. ðEIÞi and Hi are defined as [3]:
ðEIÞ1 ¼ EI1
Outer tube
ðEIÞ2 ¼ EI2 þ pES R3out
H1 ¼ 0
ð3Þ H2 ¼ 4sRout
H7 1001r6 H13
ð7Þ
where
ð1Þ
@2 @ 2 wi Hi 2 @x @x2
Inner tube
3 Z
ð4Þ
p
dh
2
m
0
@ p @x2 i " # 2 @ 2 wi 2 @ ¼ 1 ðe0 aÞ q A i @x2 @t 2 þ 1 ðe0 aÞ2
2.2. Van der Waals interaction forces
#
þ 1 ðe0 aÞ2 2
where Es and s are the surface elasticity modulus and residual surface tension per length on the outer tube, respectively.
ð1 Kcos2 hÞ 2
ðm ¼ 7; 13Þ
Rin þ Rout
ð8Þ
ð9Þ
ðRin þ Rout Þ2
where a is the carbon–carbon bond length (0.142 nm). Rin and Rout are the inner and outer radii of the DWCNTs, respectively. r and e are the vdW radius and the well depth of the Lennard–Jones potential, respectively. 2.3. Transverse vibration of DWCNTs Consider a simply supported DWCNT with length L, and the boundary conditions have the following form:
wi ð0; tÞ ¼ wi ðL; tÞ ¼ 0 ði ¼ 1; 2Þ
ð10Þ
@ uð0; tÞ @ uðL; tÞ ¼ ¼ 0 ði ¼ 1; 2Þ @x @x
ð11Þ
To satisfy boundary conditions given by Eqs. (10) and (11), the displacement field for a DWCNT is given as:
wi ¼ W i wðx; tÞ wðx; tÞ ¼ sin
mpx sin xt L
ði ¼ 1; 2Þ
ð12Þ
ui ¼ Ui uðx; tÞ uðx; tÞ ¼ cos
mpx sin xt L
ði ¼ 1; 2Þ
ð13Þ
Fig. 1. Schematic diagram showing latitudinal and longitudinal cross-sections of the DWCNT analysis model.
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where W1 and W2 represent the modal amplitudes of the deflections of the inner and the outer tubes, respectively. U1 and U2 represent the modal amplitudes of the slopes of the inner and outer tubes due to bending deformations, respectively. The integer m is the axial half-wave number of the vibrational mode, and x is the circular frequency. By substituting Eqs. (3)–(6) into Eqs. (1) and (2), we obtain: For inner tube
EI1
4 @ 2 u1 @ 2 u1 @w1 2 @ u1 kGA u q I þ q I ðe aÞ þ kGA1 ¼0 1 1 1 0 1 @x2 @x @t 2 @x2 @t 2 ð14Þ 2
2
@ u1 @ w1 @ w1 @ w1 þ cðe0 aÞ2 qA1 þ kGA1 @x @x2 @x2 @t 2 4 2 @ w @ w 1 2 þ qA1 ðe0 aÞ2 2 2 cw1 cðe0 aÞ2 þ cw2 ¼ 0 @x2 @x @t
ð15Þ
For outer tube
ðEI2 þ pEs R3out Þ þ kGA2
2
ð16Þ
2 @ 2 w1 @ u2 @ 2 w2 2 @ w2 þ cw kGA þ cðe aÞ þ kGA 1 2 2 0 @x2 @x @x2 @x2 4 2 @ 2 w2 @ w @ w 2 2 qA2 þ qA2 ðe0 aÞ2 2 2 cw2 þ H2 @x2 @t 2 @x @t @ 4 w2 ðe0 aÞ2 H2 ¼0 ð17Þ @x4
cðe0 aÞ2
To simplify the calculation, Eqs. (14)–(17) can be rewritten as:
2
L11 6 6 L21 6 6L 4 31
L12
L13
L22
L23
L32
L33
L41
L42
L43
9 38 L14 > > u1 > > > > > 7> L24 7< w1 = 7 ¼0 > u2 > > L34 7 5> > > > > : ; L44 w2
where Lij are the differential operators given as:
L11 ¼ EI1
@2 @2 @4 kGA1 ql1 2 þ qI1 ðe0 aÞ2 2 2 2 @x @t @x @t
L21 ¼ kGA1
L22 ¼ kGA1
@ @x
@2 @2 @2 @4 þ cðe0 aÞ2 2 qA1 2 þ qA1 ðe0 aÞ2 2 2 c 2 @x @x @t @x @t
L24 ¼ cðe0 aÞ2
L34 ¼ kGA2
@2 þc @x2
ð18Þ
@2 @2 @4 2 kGA q I þ q I ðe aÞ 2 2 2 0 @x2 @t 2 @x2 @t 2
@ @x
L42 ¼ cðe0 aÞ2
4
@ u2 @ u2 @ u kGA2 u2 qI2 þ qI2 ðe0 aÞ2 2 22 @x2 @t2 @x @t
@w2 ¼0 @x
@ @x
L33 ¼ ðEI2 þ pE s R3out Þ
2
kGA1
2
L12 ¼ kGA1
L43 ¼ kGA2
@2 þc @x2
@ @x
@2 @2 @2 @4 þ cðe0 aÞ2 2 qA2 2 þ qA2 ðe0 aÞ2 2 2 c @x2 @x @t @x @t @2 @4 2 þ H2 2 ðe0 aÞ H2 4 @x @x
L44 ¼ kGA2
L13 ¼ L14 ¼ L23 ¼ L31 ¼ L32 ¼ L41 ¼ 0
ð19Þ
The vibrational frequency of DWCNTs considering surface effects can be determined using a nontrivial solution by substituting Eqs. (12) and (13) into Eq. (18). 3. Results and discussion In this approach, the influences of the surface material, nonlocal parameter, aspect ratio and vibrational mode on the vibrational frequency are investigated. We considered a DWCNT with inner and outer diameters of 2.2 and 3.0 nm, respectively. The DWCNTs have a Young’s modulus of 1.0 TPa, Poisson’s ratio of 0.27 and the
Fig. 2. Comparison of the vibrational frequencies of DWCNTs with various surface elasticity and surface stress values as a function of the axial half-wave number of the vibrational mode (L/2Rour = 10, e0a = 1 nm).
X.-w. Lei et al. / Composites: Part B 43 (2012) 64–69
67
Fig. 3. Comparison of the vibrational frequencies of DWCNTs with various surface elasticity and surface stress values as a function of aspect ratio (m = 5, e0a = 1 nm).
mass density of 2.3 g/cm3. The layer thickness of the DWCNT was assumed to be that of a graphite sheet with a thickness of 0.34 nm [25]. The vdW parameters in the Lennard–Jones potential are r = 0.34 nm and e = 2.967 meV, which were reported by Saito et al. [26]. The dependence of the shear coefficient j, on the radius is 0.8 [21,27]. Figs. 2 and 3 describe the natural frequency of the DWCNT when the influence of surface elasticity modulus and the residual surface stress on the outer tubes was considered. They depend on the material type and the surface crystal orientation. For the theoretical calculations using (1 1 2) and (0 0 1) nickel, the surface elasticity and surface stress values were obtained using atomistic calculations that yielded values of Es = 35.3 N/m and s = 0.31 N/ m, and Es = -43.8 N/m and s = 0.71 N/m. The surface elastic modulus Es, is calculated from surface elastic constants S1111 and S1122, which were calculated by Shenoy [28] using Es = (S1111 + 2S1122) (S1111 S1122)/(S1111 + 2S1122) [7]. The relationship between the
frequency and the vibrational mode of the DWCNTs is shown in Fig. 2. As the axial half-wave number increases, the natural frequency of the DWCNTs, which have four modes and are covered with different crystal surfaces, also increases. It can be seen that the surface effective elastic modulus and residual stress have an influence on the vibrational properties of DWCNTs. When the axial half-wave mode increases from 1 to 10, the surface effects become more and more pronounced. In mode 3, comparing the results between (1 1 2) and (0 0 1) nickel covered DWCNTs, there is an 11.7% variation in the vibrational frequency when the vibrational mode reaches 10. The relationship between the frequency and the aspect ratio of the DWCNTs is presented in Fig. 3. The influence of the surface parameters is considerably higher in lower aspect ratio CNTs. As the aspect ratio of the DWCNTs increase, the surface effects disappear and the results converge into two distinct branches which are close to natural frequency of the DWCNTs without surface material. For a short DWCNT, the surface-to-volume ratio is much
Fig. 4. Comparison of the vibrational frequencies of DWCNTs with various surface elasticity and surface stress values as a function of the nonlocal parameter (L/2Rour = 10, m = 5).
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Fig. 5. Comparison of the vibrational frequency ratio of DWCNTs with various axial half-wave numbers as a function of the nonlocal parameter (L/2Rour = 10, Es = 1.22 N/m, s = 0.89 N/m).
shenko beam/local Timoshenko beam. In addition to the nonlocal effect, the nonlocal Timoshenko beam model is also dominated by the effects of the vibrational mode and aspect ratio. Figs. 5 and 6 show the relationship between the frequency ratio and the nonlocal parameter of the DWCNTs. This was achieved by considering the influence of the vibrational mode and aspect ratio, in which the surface parameters of (0 0 1) single-crystal silver (Es = 1.22 N/m and s = 0.89 N/m) are used. The frequency ratio serves as an index to assess the nonlocal effect on DWCNTs vibrational solutions quantitatively. It can be found in Fig. 5 that the frequency ratios are less than unity for all four modes with various axial half-wave numbers. This means that the frequency of the DWCNTs calculated using local theory will be the largest one. As the scale coefficient e0a, increases, the frequency obtained from the nonlocal Timoshenko beam becomes smaller than those obtained from the local counterpart. For example, consider DWCNTs with m = 5, L/2Rout = 10 in mode 1. When the scale coefficient increases from 0 to 2.0 nm, the frequency decreases from 3.28 THz to 2.32 THz, which corresponds to a reduction of over 40%. Note that at higher vibrational modes, the frequency ratio is farther from unity. Variation of the frequency ratio versus the scale coefficient for the four vibration modes of DWCNTs with different aspect ratios are plotted in Fig. 6. It can be seen that the frequency ratio decreases as the nonlocal parameter increases. Meanwhile, the vibration frequency decreases when the nonlocal Timoshenko beam model is used. The scale coefficient e0a, plays a larger role in shorter DWCNTs. Conversely, for a slender DWCNT with aspect ratio of 30, the frequency ratio approaches to unity. This indicates a reduction in the nonlocal effect. In summary, the nonlocal effect significant for short DWCNTs, and can be neglected in longer tubes. 4. Conclusions An analytical procedure based on the nonlocal Timoshenko beam mode was used to investigate the vibrational behavior of DWCNTs adhered by surface materials with simply supported boundary condition. The influences of the surface materials and the nonlocal parameter on the vibrational frequency and frequency ratio are both significant, especially in DWCNTs with higher vibrational modes and shorter lengths. The present study will be useful for understanding the vibration properties of DWCNTs. In particular, we hope that it will help guide the use of DWCNTs as carriers of drug or functionalized materials. Acknowledgments
Fig. 6. Comparison of the vibrational frequency ratio of DWCNTs with various aspect ratios as a function of the nonlocal parameter (m = 1, Es = 1.22 N/m, s = 0.89 N/m).
larger than that of a slender one. Therefore the surface effects become more obviously. However, vibrational frequency of mode 4 of the DWCNTs is not significantly affected by surface material regardless of vibrational mode or aspect ratio. The relationship between the frequency and the nonlocal parameters of DWCNTs is shown in Fig. 4. The influence of the surface material on the outer tubes is considered in these calculations. From the figure, it can be seen that the influence of nonlocal parameter in modes 3 and 4 is much larger than that in mode 1 and 2. The vibrational frequencies of the DWCNTs covered with surface materials with positive and negative surface elastic moduli surround those of the DWCNTs without surface material. To illustrate the influence of nonlocal effects on vibration, we defined the frequency ratio as: frequency ratio g = nonlocal Timo-
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