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Molecular dynamics simulation of pull-in phenomena in carbon nanotubes with Stone–Wales defects
Solid State Communications 157 (2013) 38–44
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Molecular dynamics simulation of pull-in phenomena in carbon nanotubes with Stone–Wales defects Mir Masoud Seyyed Fakhrabadi a,n, Pooria Khoddam Khorasani a, Abbas Rastgoo a, Mohammad Taghi Ahmadian b a b
College of Engineering, School of Mechanical Engineering, University of Tehran, Tehran, Iran Department of Mechanical Engineering, Sharif University of Technology, Tehran, Iran
a r t i c l e i n f o
a b s t r a c t
Article history: Received 7 September 2012 Received in revised form 11 December 2012 Accepted 18 December 2012 by Z. Tang Available online 23 December 2012
This paper deals with investigation of deformations and pull-in charges of the cantilever and doubly clamped carbon nanotubes (CNTs) with different geometries using molecular dynamics simulation technique. The well-known AIREBO potential for the covalent bonds between carbon atoms, LennarJones potential for the vdW interaction and the Coulomb potential for electrostatic actuation are employed to model the nano electromechanical system. The results reveal that longer CNTs with smaller diameters have smaller pull-in charges in comparison with shorter CNTs possessing larger diameters. Furthermore, the pull-in charges of the doubly clamped CNTs are higher than the pull-in charges of the cantilevered CNTs. Another important matter discussed in this paper is the effects of Stone–Wales defects on the pull-in charges. The results show the reduction of the pull-in charges in the presence of Stone–Wales defects in the nano system. & 2012 Elsevier Ltd. All rights reserved.
Keywords: A. Carbon nanotubes B. Molecular dynamics C. Stone–Wales defect D. Pull-in phenomena
1. Introduction Carbon nanotube (CNT) can be considered as one of the most influential and applicable nanostructures in the nano systems. Since its discovery by Ijima in 1991, many researchers and scientists with different disciplines have started to study its properties using theoretical and experimental techniques. In many aspects, especially in engineering fields, the studies have revealed that the CNT have exceptional characteristics. This fact has resulted in applying the CNTs in various engineering applications with the focus on the mechanical and electrical nano systems. Fakhrabadi et al. applied molecular mechanics method to study the vibrational properties of the CNTs with different geometries and boundary conditions [1]. They combined the mentioned method with the artificial neural networks in order to investigate the natural frequencies of the unmodeled CNTs. They proved that the proposed scheme could predict the natural frequencies with good agreement with the real values. In another research, Rossi and Meo computed the Young’s modulus, ultimate strength and strain of the single-walled CNTs using molecularmechanics technique [2]. They applied nonlinear and torsional
n
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spring elements to evaluate the mentioned properties as well as the tensile failure. Chowdhury et al. applied molecular mechanics method to analyze the vibrational properties of the zigzag and armchair CNTs. The natural frequencies and their corresponding mode shapes were presented in their paper for different geometries [3]. In addition, Joshi et al. investigated the effects of chiralities and atomic vacancies on the vibrational behaviors of the nanoresonators based on the CNTs via application of the molecular mechanics approach [4]. They studied the natural frequencies of the CNTs with different chiralities and atom vacancies in order to propose a more real model for the designers of the CNT-based devices. The vacancies imposed some variations in the natural frequencies. In another paper, they conducted similar research on the effects of pinhole defects on the vibrational characteristics of the CNTs [5]. Molecular mechanics method that is an effective technique in nanomechanics engineering was used in some other studies [6–9]. Another effective tool to investigate the physical and chemical properties of the nanostructures in general and carbon nano materials in particular is molecular dynamics applied in this paper. Hao et al. studied buckling of defective CNTs under axial compression by molecular dynamics simulation [10]. The densities of the defects and their positions played important roles in the final results. The outcomes revealed that vacancy defects possessed significant effects on the critical buckling loads of the CNTs. Wang et al. simulated the twist of the CNTs using molecular dynamic technique [11]. The ultimate twist angle per unit length
M.M.S. Fakhrabadi et al. / Solid State Communications 157 (2013) 38–44
and the deformation energy were estimated for different geometries. The authors claimed that formation of the structural defects was observed prior to the fracture. Kim et al. studied the vibrational behaviors of the CNT oscillators with intertube gaps using molecular dynamics simulations [12]. They introduced a model for the CNT resonator and computed the natural frequencies of the system for different parametric variations. They mentioned that CNT oscillators with intertube gaps could be employed as the frequency-controlled oscillators via manipulating the gap. In another research, Kang et al. investigated a CNT resonator encapsulating a nanocluster as another tunable resonator using molecular dynamics method and continuum modeling [13]. The natural frequency of the system with cantilever boundary conditions could be adjusted by controlling the position of the encapsulated nanocluster. They verified the results using the continuum mechanics. Ranjbartoreh and Wang studied the effects of length variations on the elastic modulus, tensile, compressive, and lateral stiffnesses, critical buckling strain, critical axial force and pressure of the armchair and zigzag CNTs using molecular dynamic technique [14]. The results revealed that both armchair and zigzag CNTs showed better tensile characteristics than compressive properties. Moreover, they proved that the buckling mode shapes of the CNTs changed with the length. Micro and nano electromechanical systems (MEMS/NEMS) are major parts in electrical circuits. They can be applied in nano sensors, nano actuators, nano switches, nano transistors, nano resonators and nano filters in electronics, communication devices, aerospace applications, etc. The CNT can be considered as an important candidate to be applied in these areas. Hence, besides its mechanical characteristics reviewed partly above, its electromechanical behaviors should be studied well. The dominant approach to investigate the pull-in behaviors of the electrostatically actuated NEMS in general and carbon nanotubes in particular is continuum modeling. The concepts of electrostatic actuation and pull-in phenomenon are described in the next lines. Aluru et al. studied the pull-in phenomena of the CNTs using a one degree of freedom model [15]. Their research was majorly about determination of the pull-in voltages of the CNTs suspended on the graphene sheets. In two other papers, Espinosa et al. investigated the application of the CNT in NEMS using continuum relations [16,17]. In the first paper, the quality of electrostatic charge distribution on the multi-walled CNTs was studied. In addition, the second paper was about the stretching effects and influences of charge concentration on the pull-in voltages. Nonlinear dynamics of the CNTs under DC and AC voltages with some additional considerations were studied recently using elasticity and vibration formulations [18–20]. Kinaret et al. studied the operational characteristics of a nanorelay based on a conducting CNT placed on a terrace in a silicon substrate and connected to a fixed source electrode [21]. They applied coupled continuum mechanics relations and electrical capacitance equations to investigate the mechanical behaviors of the CNT under electrostatic actuation. Jonsson et al. investigated the effects of surface forces and phonon dissipation in a three-terminal nanorelay built form CNT [22]. They showed that short range and vdW forces had a significant impact on the characteristics of the relay and introduced the design constraints. They also investigated the effects of dissipation due to phonon excitation in the drain contact altering the switching time scales of the system and decreasing the longest time scale by two orders of magnitude. In another research, Jonsson et al. studied the vibrational frequency of a CNT-based nano relay and showed that the electromechanical coupling presented a non-linear resonant
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behavior in GHz frequency range [23]. They discussed how the resonances might be detected and showed that the resonance frequencies could be tuned by the bias voltage. Also, they reported that the influence of external electromagnetic fields on the relay was negligible at all frequencies. Hwang and Kang applied molecular dynamics to study the electromechanical behaviors of a CNT with certain geometry and cantilever boundary conditions under electrostatic actuation [24]. In their paper, the CNT was suspended on a copper substrate and the CNT was considered perfect, i. e. without any structural defect. A similar study was conducted by Kang et al. to study the applicability of the CNTs as a memory device [25]. In this study, a CNT with certain geometry and double-clamped boundary conditions under electrostatic interaction was investigated using molecular dynamics and the corresponding potential, kinetic and total energies were presented. In another paper, Kang et al. studied the CNT-bridge nanoelectromechanical memory device using two-dimensional model based on electrostatic and elastostatic theories [26]. The nanotube-bridge memory device was operated by the electrostatic, the elastostatic, and the vdW forces acting on the CNT-bridge. They proved that the vdW interactions between the nanotube-bridge and the oxide substrate were very important for nonvolatile nanotube-bridge memory device. Kang et al. analyzed the static and dynamic behaviors of the CNT memory devices called nanotube random access memory (NRAM) [27]. In static analyses, the current–voltage curves showed the hysteresis characteristics. In addition, the dynamic analyses showed that the CNTs with small diameters were effective to operate the NRAMs with low turn-on voltages and high throughput rates. Although, several researches were conducted in investigating the electromechanical behaviors of the CNTs; they were not obviously enough and this requires deeper and more extensive studies. This is very important especially for the CNTs with short lengths because they have discrete structures and this fact may lead to the differences between the results obtained from continuum and discrete modeling approaches. As mentioned above, one of the main techniques to excite the MEMS and NEMS is electrostatic actuation. According to Fig. 1, in the electrostatic actuation, the movable electrode (upper part in the figure) is attached to the positive terminal and the fixed electrode (lower part in the figure) is attached to the negative terminal or vice versa. The distribution of the positive and negative charges on the electrodes causes them to attract towards each other. Hence, the upper movable electrode deflects up to where its elastic force balances with the attraction forces due to the distributed charges and interatomic interactions. This process continues up to a maximum limit and after it the movable electrode loses its tolerance and suddenly drops on the fixed electrode (ground plate). In this condition, the elastic properties of the upper electrode cannot tolerate the attraction force. This phenomenon is called pull-in or snap-down. In
Fig. 1. Schematic representation of an electrostatically actuated CNT.
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this paper, the movable electrode is considered a single walled CNT and the fixed electrode is composed of the graphene sheets.
simulations are performed using LAMMPS (large scale atomic/ molecular massively parallel simulator) software.
3. Results and discussion 2. Atomistic modeling approach The well-known molecular dynamics technique is applied in this paper to investigate the deflection and pull-in phenomena of the CNTs under electrostatic actuation. The concepts of molecular dynamics were described well in the literature. In this method, the interactions between the atoms are modeled via some potential functions. Different functions were provided for various materials and applications. One of the most effective potentials is adaptive intermolecular reactive bond order (AIREBO) potential modeling the carbon and hydrogen behaviors in the hydrocarbons. The detailed description of AIREBO potential was presented by Stuart et al. [28]. In this section, we are only going to present a brief explanation regarding this potential as well as Coulomb potential modeling the electrostatic interactions. The AIREBO potential consists of three terms [29]: 2 3 X X TORSION 1 X X4 REBO LJ 5 E¼ ð1Þ E þ Eij Ekijl 2 i j a i ij k a i,j l a i,j,k EREBO has the same functional form similar to the hydrocarbon ij REBO potential developed by Brenner et al. in [29]. The coefficients for EREBO are, in fact, the same as Brenner’s potential, but a ij few fitted values are slightly different. Nonetheless, in most cases, this term leads to the energies, forces and statistical averages same as the original REBO potential. It models the reactive capabilities and only describes short-ranged C–C, C–H and H–H interactions (r o2A). These interactions depend strongly on the coordinates through a bond order parameter adapting the attraction between the ith and jth atoms according to the positions of other nearby atoms. Hence, it possesses three and four body dependence. ELJ adds longer-ranged interactions (2A or ocutoff) ij via a form similar to the standard Lennard-Jones (L-J) potential. The ELJ term includes a series of switching functions so that the ij short-ranged L-J repulsion does not interfere with the energies captured by the EREBO term. However, the extent of the ELJ ij ij interactions can be determined by a cutoff operator in the formulation. Van der Waals interactions between the CNT and graphene sheets were also modeled using L-J potential. In addition, ETORSION is an explicit four body potential explaining various kijl dihedral angle preferences in hydrocarbon configurations [29]. The coulomb potential estimates the Coulombic interaction potential between the charged particles according to the following relation: E¼
Cqi qj , er
In this part the results of the simulations are presented and discussed. As mentioned before, there are some main differences between the results presented in this paper with the previously reported papers. The first one is investigation of the geometry and boundary condition effect on the pull-in charges of the CNTs using MD. Although in [24–27] the authors applied MD to investigate the electromechanical behaviors of the CNT, they dos not investigate the geometry variations of the CNTs. Moreover, they considered CNTs perfect without any structural defect that is very ideal. In most cases, the CNTs have defects such as Stone–Wales defect and this fact affects the electromechanical characteristics. Also, this paper studied the effects of thermal variations on the pull-in charges. Fig. 2(a) and (b) respectively shows two single walled CNTs with length 6 nm, chirality (10,10) and cantilever and doubly clamped boundary conditions suspended on five graphene sheets. The CNTs and the graphene sheets compose two electrodes of the NEMS. According to the figure, the CNT is initially in a straight position. The interactions between the two charged electrodes including electrostatic charges and vdW effects result in deflecting the CNTs towards the fixed graphene sheets. It is clear that the elastic force of the CNT withstands against the applied attractive force. But, if the charges are so large the CNT cannot tolerate the attractive loads and suddenly drops on the lower electrode. This fact is illustrated in the last part of Fig. 2. This phenomenon is pull-in and the corresponding charge is called pull-in charge. Of course, the pullin charge often is used for the threshold charge resulting in pullin behavior. Tip deflection of a cantilevered CNT vs. iteration at 300 K is demonstrated in Fig. 3. The horizontal axis is the iterations and the vertical one is the tip deflection of the CNT in z direction. The figure shows three distinct stages before attaching the CNT to the ground plate. In the stage 1, the CNT deflects towards the graphene sheets uniformly. In the stage 2, the CNT shows higher resistance against the attractive forces and the figure shows that there are more iterations to deflect in this stage. This fact can be attributed to the local deformation of the CNT. This deformation majorly takes place at the end of stage 2. The stage 3 is for the deformation of the CNT up to approaching to the lower electrode. The fluctuation after the stage 3 can be related to the struggle between the elastic properties of the CNT and attractive forces that will damp vs. time. 3.1. Cantilevered CNTs
r or c
ð2Þ
where, C is an energy-conversion constant, qi and qj are the charges on the two atoms, and epsilon is the dielectric constant. The cutoff r c truncates the interaction distance [29]. The molecular dynamics
In this subsection, the effects of parameter variations on the pull-in charges of the CNTs with cantilever boundary conditions are studied. Fig. 4 presents the variation of the pull-in charge vs. length of the CNT. The chilaity of the CNTs is (10,10), the gap is
Fig. 2. Pull-in snapshot of a SWCNT under electrostatic actuation.
M.M.S. Fakhrabadi et al. / Solid State Communications 157 (2013) 38–44
Fig. 3. Tip deflection of the CNT in z direction vs. iteration.
Fig. 4. Pull-in charge of the cantilevered CNTs vs. length.
Fig. 5. Pull-in charge of the cantilevered CNTs vs. radius.
2 nm and the temperature is 300 K. As shown in this figure, the pull-in charge decreases vs. length increment. This fact can be attributed to bending softening of the nanostructure with increasing the length. It is clear that the longer the CNT, the easier the bending is. The effects of radius variations on the pull-in charges of the cantilevered CNTs with the length 6 nm and gap 2 nm are depicted in Fig. 5. The figure proposes that the radius increment results in the pull-in charge increment. This matter can also be attributed to stiffening of the CNT with radius increment.
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Fig. 6. Pull-in charge of the cantilevered CNT vs. gap distance.
Fig. 7. Pull-in charge of the cantilevered CNT vs. temperature.
It is worth noting that the gap distance variation has significant effects on the pull-in charges. In order to quantize the effects, Fig. 6 is presented. According to the figure, gap distance increment results in the higher pull-in charges. This fact can be attributed to this matter that both electrostatic and vdW interactions highly depend on the distance. The distance increment, with unchanged other parameters, leads to the lower energies and it is clear that for the pull-in phenomenon, higher charge vales are required. Temperature variation is another effect influencing the pull-in charge. According to Fig. 7, the temperature increment reduces the pull-in charge. The CNT has 6 nm length, (10,10) chilaity and 2 nm gap. The variation seems linear to somehow. This variation should be investigated in detail in the design of NEMS. The main important consideration during the previous simulations and results is that the structures of the CNTs were conspired perfect. However, this consideration is not correct in many cases and the structures of the CNTs, similar to many other crystalline materials, have some defects. Here, we are going to scrutinize the effects of one of the most important defects in the CNTs, i.e. Stone– Wales defect, on the pull-in charges. In the Stone–Wales defect, a bond between two carbon atoms rotates about the vertical axis as shown in Fig. 8. Thus, in the neighbor of the defect, two carbon rings have five atoms, whereas two others have seven atoms. The influences of different longitudinal and peripheral defects are studied in detail. Fig. 9 shows the possible positions for the defects that their combinations are considered in this paper. There are three longitudinal positions (A, B, C) as well as three peripheral positions (0, 90, 180 degrees). A and C have distances
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Fig. 8. Stone–Wales defect.
180
Fig. 10. Effects of Stone–Wales defects with positions 0 degree on the pull-in charges of cantilevered CNT.
90 0
A
B
C
Fig. 9. Possible positions for the defects.
Table 1 Studied cases of the Stone–Wales defects. Case number
Defect positions
1 2 3 4 5 6 7 8
No defect C B A C,B C,A B,A C,B,A
Fig. 11. Effects of Stone–Wales defects with positions 90 degrees on the pull-in charges of cantilevered CNT.
1/3 of the length from the nearest ends and B is in the center. We suppose that the CNT is clamped from the left side and free from the right side. All of the studied cases are summarized in Table 1. The above case numbers are investigated in three respective different angles, i. e. 0, 90 and 180 degrees. The measurement point is considered from the lowest point on the periphery of the CNT. Fig. 10 illustrates the effects of Stone–Wales defects with the case numbers summarized in Table 1 and with the angle 0 degree (Fig. 8) on the pull-in charge of the CNT with length 6 nm, chirality (10,10) and gap distance 2 nm at 300 K. The upper part of the figure relates to the pull-in charge values and the lower one corresponds to the pull-in charge ratio wrt the pristine CNT. The pull-in charge ratio is defined as: 100
ðCharge of Pristine CNTCharge of Defective CNTÞ ðCharge of Pristine CNTÞ
ð3Þ
According to the figure, both the position and number of the defects affect the pull-in charge. In the cases 2, 3 and 4 belonging to one defect, the figure reveals that the longer the distance from the clamped side, the weaker the effect is. In other words, the defect on point C of the CNT has smaller effects than the defect on the point near the support (point A). This fact can be attributed to the matter that local deformation occurs near the support. Hence, the defects near the local deformation result in the structure weakness and facilitate the pull-in phenomena. It is obvious that the number of the defects has weakening effect on the nano
structure and leads to the smaller pull-in charges, as shown in Fig. 10. For the last case, the CNT possesses three defects on three longitudinal positions. The figure shows that this case is the worst condition among the modeled and investigated cases. In this case, the pull-in charge may be reduced about 15%. Hence, the result reveals the necessity of this study. This matter should be scrutinized deeply in the NEMS design and fabrication processes. Fig. 11 depicts the effects of the defects on the positions presented in Table 1 with 90 degrees (Fig. 9). The figure shows that the existence of Stone–Wales defects in the angle 90 degrees can also reduce the pull-in charges. However, a one by one comparison between the results of 0 degree with the results of 90 degrees reveals that the effects of 90 degree defects are weaker than the previous case. As an example, for the last case in Fig. 11, the defect reduces the pull-in charge about 11% whereas in the 0 degree, the last case reduces the charge about 15%. This fact mentions that the peripheral position of the defect is important as the longitudinal positions. Fig. 12 presents the effects of the Stone–Wales defects on the pull-in charges when they are situated on 180 degrees. The pattern of effects is similar to the two previous cases. The defects near the local deformation dominate to those far from it. The increment in the defect numbers has more reducing effects. The main point is that the results corresponding with 180 degrees are situated between the results corresponding with 0 degrees and those with 90 degrees. Hence, it can be concluded that the pattern of variations of the pull-in charges vs. peripheral angle is so that
M.M.S. Fakhrabadi et al. / Solid State Communications 157 (2013) 38–44
Fig. 12. Effects of Stone–Wales defects with positions 180 degrees on the pull-in charges of cantilevered CNT.
Fig. 14. Pull-in charge of the doubly clamped CNTs vs. radius.
Fig. 13. Pull-in charge of the doubly clamped CNTs vs. length.
Fig. 15. Pull-in charge of the doubly clamped CNTs vs. gap distance.
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the maximum reduction belongs to the 0 degree and reduces with the angle increment up to 90 degrees; then increases again until approaching to 180 degrees. However, the defects belonging to 180 degrees have smaller influences in comparison with those relating to 0 degrees. It may be attributed to the position of local deformation occurring in the lower atoms (close to 0 degree). 3.2. Doubly clamped CNT In this section, the results of variation in the length, radius and gap distance on the pull-in charges of the CNTs with doubly clamped boundary conditions are discussed. Furthermore, the effects of temperature changes and Stone–Wales defects on the pull-in charges of the doubly clamped CNT are investigated. Fig. 13 presents the effects of length increment on the pull-in charges of the doubly clamped CNTs with chirality (10,10) and gap distance 2 nm. The figure reveals that the length increment reduces the pull-in charges drastically. Another important fact relating to the figure can be deduced in comparison with its counterpart in Fig. 4. The comparison shows that the pull-in charges belonging to the doubly-clamped CNTs are remarkably higher than those corresponding to the cantilevered CNTs with other same parameters. Fig. 14 displays the effects of radius variations on the pull-in charges of the doubly clamped CNTs. Based on the figure, the diameter increasing results in the higher pull-in charges. The pattern of the results is similar to the cantilevered CNTs but the pull-in charge values are significantly higher.
Fig. 16. Pull-in charge of the doubly clamped CNT vs. temperature.
Lastly, the effects of gap distance on the pull-in charge values of the doubly clamped CNTs are presented in Fig. 15. According to the figure, the pull-in charges increase with the increment in the gap. This is, as described before, because of this fact that the distance increment weakens the electrostatic and vdW forces. Thus larger charges are required for pull-in phenomena. The effects of temperature variation on the pull-in charges of the doubly clamped CNT are depicted in Fig. 16. According to the figure, similar to the cantilevered CNT, the temperature increment reduces the pull-in charge of the doubly clamped CNT.
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The effects of Stone–Wales defects on the pull-in charges of the doubly clamped CNTs are presented in Figs. 17–19 respectively for the defect in 0, 90 and 180 degrees. According to the figures, the Stone–Wales defects reduce the pull-in charge of the doubly clamped CNTs but their influences are less than the case of cantilevered CNTs. The nearness of the defect to the center of the CNT increases the effects of the defects since the main
deformation occurs in the center. The figures reveal that the for one defect, the zero angle position (case 3) has the maximum effect in comparison to the cases C and A that are same due to symmetry. In addition, in two defect cases, the cases 5 and 6 that are same possess maximum effects and for the case of three defects, it can be observe that the influences are higher in comparison with the previous cases. Moreover, the defects in zero angles have the maximum effects and those in 90 degrees possess the minimum effects. This is also attributed to the place of deformation.
4. Conclusion
Fig. 17. Effects of Stone–Wales defects with positions 0 degree on the pull-in charges of doubly clamped CNT.
The paper presented the deflection and pull-in phenomena of the CNTs in the presence of vdW and electrostatic attractions. The ground plate consisted of the graphene sheets. AIREBO, LennardJones and Coulomb potentials were applied to model the nano system. The effects of various lengths, radii, gap distances, temperatures and boundary conditions on the pull-in charges of the CNTs were reported. The conclusion was that the stiffer CNTs had higher pull-in charges and any factor helping the CNT to become stiffer had the increasing effects on the pull-in charges. Moreover, the effects of Stone–Wales defects with different positions and numbers on the pull-in charges were investigated. The results revealed that the defects could reduce the pull-in charges drastically. As an example, three defects in the cantilevered CNT reduced the pull-in charge about 15%. For the future work, one may study the effects of other defects on the electromechanical behaviors of the CNTs under electrostatic actuation. References
Fig. 18. Effects of Stone–Wales defects with positions 90 degrees on the pull-in charges of doubly clamped CNT.
Fig. 19. Effects of Stone–Wales defects with positions 180 degrees on the pull-in charges of doubly clamped CNT.
[1] M.M.S. Fakhrabadi, M. Samadzadeh, A. Rastgoo, M. Haeri Yazdi, M. Mousavi Mashhadi, J. Phys. E 44 (2011) 565–578. [2] M. Rossi, M. Meo, J. Compos. Sci. Tech. 69 (2009) 1394–1398. [3] R. Chowdhury, S. Adhikari, C.Y. Wang, F. Scarpa, J. Comput. Mat. Sci. 48 (2010) 730–735. [4] A.Y. Joshi, S.C. Sharma, S.P. Harsha, J. Sensor Rev. 31 (2011) 47–57. [5] A.Y. Joshi, S.C. Sharma, S.P. Harsha, J. Phys. E 43 (2011) 1040–1045. [6] M.M.S. Fakhrabadi, N. Khani, R. Omidvar, A. Rastgoo, J. Comput. Mat. Sci. 61 (2012) 248–256. [7] M.M.S. Fakhrabadi, N. Khani, S. Pedrammehr, J. Phys. E 44 (2012) 1162–1168. [8] S. Rouhi, R. Ansari, J. Phys. E 44 (2011) 764–772. [9] J. Xia, D. Wei, European J. Mech. A/Solids 29 (2010) 440–447. [10] X. Hao, H. Qiang, Y. Xiaohu, J. Compos. Sci. Tech. 68 (2008) 1809–1814. [11] Z. Wang, M. Devel, B. Dulmet, J. Surf. Sci. 604 (2010) 496–499. [12] K.S. Kim, J. Park, H. Jung Hwang, J. Won Kang, J. Phys. E 44 (2012) 2027–2031. [13] J.W. Kang, K.S. Kim, J. Park, H.J. Hwang, J. Phys. E 43 (2011) 909–913. [14] A.R. Ranjbartoreh, G. Wang, J. Phys. E 43 (2010) 202–206. [15] M. Dequesnes, S.V. Rotkin, N.R. Aluru, J. Nanotech. 13 (2002) 120–131. [16] C. Ke, H.D. Espinosa, J. Appl. Mech. 72 (2005) 721–725. [17] C. Ke, H.D. Espinosa, N. Pugno, J. Appl. Mech. 72 (2005) 726–731. [18] H.M. Ouakad, M.I. Younis, J. Comput. Nonlin. Dyn. 5 (2010) 1–13. [19] H.M. Ouakad, M.I. Younis, J. Nonlin. Dyn. 67 (2012) 1419–1436. [20] H.M. Ouakad, M.I. Younis, J. Sound Vib. 330 (2011) 3182–3195. [21] J.M. Kinaret, T. Nord, S.J. Viefers, Appl. Phys. Lett. 82 (2003) 3182–3195. [22] L.M. Jonsson, T. Nord, J.M. Kinaret, S.J. Viefers, Appl. Phys. 96 (2004) 629–635. [23] L.M. Jonsson, S. Axelsson, T. Nord, S. Viefers, J.M. Kinaret, J. Nanotech. 15 (2004) 1497–1502. [24] H.J. Hwang, J.W. Kang, J. Phys. E 27 (2005) 163–175. [25] J.W. Kang, J.H. Lee, H.J. Lee, H. Hwang, J. Phys. E 27 (2005) 332–340. [26] J.W. Kang, J.H. Lee, H.J. Lee, O.K. Kwon, H. Hwang, J. Phys. E 28 (2005) 273–280. [27] J. Won Kang, O.K. Kwon, J.Ha Lee, H.Joo Lee, Y. Song, Y. Yoon, H. Hwang, J. Phys. E 33 (2006) 41–49. [28] S.J. Stuart, A.B. Tutein, J.A. Harrison, J. Chem. Phys. 112 (2000) 6472–6486. [29] Large-scale Atomic/Molecular Massively Parallel Simulator, LAMMPS user manual, AIREBO description, Sandia Corporation, USA, 2012.
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Molecular dynamics simulation of pull-in phenomena in carbon nanotubes with Stone–Wales defects Mir Masoud Seyyed Fakhrabadi a,n, Pooria Khoddam Khorasani a, Abbas Rastgoo a, Mohammad Taghi Ahmadian b a b
College of Engineering, School of Mechanical Engineering, University of Tehran, Tehran, Iran Department of Mechanical Engineering, Sharif University of Technology, Tehran, Iran
a r t i c l e i n f o
a b s t r a c t
Article history: Received 7 September 2012 Received in revised form 11 December 2012 Accepted 18 December 2012 by Z. Tang Available online 23 December 2012
This paper deals with investigation of deformations and pull-in charges of the cantilever and doubly clamped carbon nanotubes (CNTs) with different geometries using molecular dynamics simulation technique. The well-known AIREBO potential for the covalent bonds between carbon atoms, LennarJones potential for the vdW interaction and the Coulomb potential for electrostatic actuation are employed to model the nano electromechanical system. The results reveal that longer CNTs with smaller diameters have smaller pull-in charges in comparison with shorter CNTs possessing larger diameters. Furthermore, the pull-in charges of the doubly clamped CNTs are higher than the pull-in charges of the cantilevered CNTs. Another important matter discussed in this paper is the effects of Stone–Wales defects on the pull-in charges. The results show the reduction of the pull-in charges in the presence of Stone–Wales defects in the nano system. & 2012 Elsevier Ltd. All rights reserved.
Keywords: A. Carbon nanotubes B. Molecular dynamics C. Stone–Wales defect D. Pull-in phenomena
1. Introduction Carbon nanotube (CNT) can be considered as one of the most influential and applicable nanostructures in the nano systems. Since its discovery by Ijima in 1991, many researchers and scientists with different disciplines have started to study its properties using theoretical and experimental techniques. In many aspects, especially in engineering fields, the studies have revealed that the CNT have exceptional characteristics. This fact has resulted in applying the CNTs in various engineering applications with the focus on the mechanical and electrical nano systems. Fakhrabadi et al. applied molecular mechanics method to study the vibrational properties of the CNTs with different geometries and boundary conditions [1]. They combined the mentioned method with the artificial neural networks in order to investigate the natural frequencies of the unmodeled CNTs. They proved that the proposed scheme could predict the natural frequencies with good agreement with the real values. In another research, Rossi and Meo computed the Young’s modulus, ultimate strength and strain of the single-walled CNTs using molecularmechanics technique [2]. They applied nonlinear and torsional
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spring elements to evaluate the mentioned properties as well as the tensile failure. Chowdhury et al. applied molecular mechanics method to analyze the vibrational properties of the zigzag and armchair CNTs. The natural frequencies and their corresponding mode shapes were presented in their paper for different geometries [3]. In addition, Joshi et al. investigated the effects of chiralities and atomic vacancies on the vibrational behaviors of the nanoresonators based on the CNTs via application of the molecular mechanics approach [4]. They studied the natural frequencies of the CNTs with different chiralities and atom vacancies in order to propose a more real model for the designers of the CNT-based devices. The vacancies imposed some variations in the natural frequencies. In another paper, they conducted similar research on the effects of pinhole defects on the vibrational characteristics of the CNTs [5]. Molecular mechanics method that is an effective technique in nanomechanics engineering was used in some other studies [6–9]. Another effective tool to investigate the physical and chemical properties of the nanostructures in general and carbon nano materials in particular is molecular dynamics applied in this paper. Hao et al. studied buckling of defective CNTs under axial compression by molecular dynamics simulation [10]. The densities of the defects and their positions played important roles in the final results. The outcomes revealed that vacancy defects possessed significant effects on the critical buckling loads of the CNTs. Wang et al. simulated the twist of the CNTs using molecular dynamic technique [11]. The ultimate twist angle per unit length
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and the deformation energy were estimated for different geometries. The authors claimed that formation of the structural defects was observed prior to the fracture. Kim et al. studied the vibrational behaviors of the CNT oscillators with intertube gaps using molecular dynamics simulations [12]. They introduced a model for the CNT resonator and computed the natural frequencies of the system for different parametric variations. They mentioned that CNT oscillators with intertube gaps could be employed as the frequency-controlled oscillators via manipulating the gap. In another research, Kang et al. investigated a CNT resonator encapsulating a nanocluster as another tunable resonator using molecular dynamics method and continuum modeling [13]. The natural frequency of the system with cantilever boundary conditions could be adjusted by controlling the position of the encapsulated nanocluster. They verified the results using the continuum mechanics. Ranjbartoreh and Wang studied the effects of length variations on the elastic modulus, tensile, compressive, and lateral stiffnesses, critical buckling strain, critical axial force and pressure of the armchair and zigzag CNTs using molecular dynamic technique [14]. The results revealed that both armchair and zigzag CNTs showed better tensile characteristics than compressive properties. Moreover, they proved that the buckling mode shapes of the CNTs changed with the length. Micro and nano electromechanical systems (MEMS/NEMS) are major parts in electrical circuits. They can be applied in nano sensors, nano actuators, nano switches, nano transistors, nano resonators and nano filters in electronics, communication devices, aerospace applications, etc. The CNT can be considered as an important candidate to be applied in these areas. Hence, besides its mechanical characteristics reviewed partly above, its electromechanical behaviors should be studied well. The dominant approach to investigate the pull-in behaviors of the electrostatically actuated NEMS in general and carbon nanotubes in particular is continuum modeling. The concepts of electrostatic actuation and pull-in phenomenon are described in the next lines. Aluru et al. studied the pull-in phenomena of the CNTs using a one degree of freedom model [15]. Their research was majorly about determination of the pull-in voltages of the CNTs suspended on the graphene sheets. In two other papers, Espinosa et al. investigated the application of the CNT in NEMS using continuum relations [16,17]. In the first paper, the quality of electrostatic charge distribution on the multi-walled CNTs was studied. In addition, the second paper was about the stretching effects and influences of charge concentration on the pull-in voltages. Nonlinear dynamics of the CNTs under DC and AC voltages with some additional considerations were studied recently using elasticity and vibration formulations [18–20]. Kinaret et al. studied the operational characteristics of a nanorelay based on a conducting CNT placed on a terrace in a silicon substrate and connected to a fixed source electrode [21]. They applied coupled continuum mechanics relations and electrical capacitance equations to investigate the mechanical behaviors of the CNT under electrostatic actuation. Jonsson et al. investigated the effects of surface forces and phonon dissipation in a three-terminal nanorelay built form CNT [22]. They showed that short range and vdW forces had a significant impact on the characteristics of the relay and introduced the design constraints. They also investigated the effects of dissipation due to phonon excitation in the drain contact altering the switching time scales of the system and decreasing the longest time scale by two orders of magnitude. In another research, Jonsson et al. studied the vibrational frequency of a CNT-based nano relay and showed that the electromechanical coupling presented a non-linear resonant
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behavior in GHz frequency range [23]. They discussed how the resonances might be detected and showed that the resonance frequencies could be tuned by the bias voltage. Also, they reported that the influence of external electromagnetic fields on the relay was negligible at all frequencies. Hwang and Kang applied molecular dynamics to study the electromechanical behaviors of a CNT with certain geometry and cantilever boundary conditions under electrostatic actuation [24]. In their paper, the CNT was suspended on a copper substrate and the CNT was considered perfect, i. e. without any structural defect. A similar study was conducted by Kang et al. to study the applicability of the CNTs as a memory device [25]. In this study, a CNT with certain geometry and double-clamped boundary conditions under electrostatic interaction was investigated using molecular dynamics and the corresponding potential, kinetic and total energies were presented. In another paper, Kang et al. studied the CNT-bridge nanoelectromechanical memory device using two-dimensional model based on electrostatic and elastostatic theories [26]. The nanotube-bridge memory device was operated by the electrostatic, the elastostatic, and the vdW forces acting on the CNT-bridge. They proved that the vdW interactions between the nanotube-bridge and the oxide substrate were very important for nonvolatile nanotube-bridge memory device. Kang et al. analyzed the static and dynamic behaviors of the CNT memory devices called nanotube random access memory (NRAM) [27]. In static analyses, the current–voltage curves showed the hysteresis characteristics. In addition, the dynamic analyses showed that the CNTs with small diameters were effective to operate the NRAMs with low turn-on voltages and high throughput rates. Although, several researches were conducted in investigating the electromechanical behaviors of the CNTs; they were not obviously enough and this requires deeper and more extensive studies. This is very important especially for the CNTs with short lengths because they have discrete structures and this fact may lead to the differences between the results obtained from continuum and discrete modeling approaches. As mentioned above, one of the main techniques to excite the MEMS and NEMS is electrostatic actuation. According to Fig. 1, in the electrostatic actuation, the movable electrode (upper part in the figure) is attached to the positive terminal and the fixed electrode (lower part in the figure) is attached to the negative terminal or vice versa. The distribution of the positive and negative charges on the electrodes causes them to attract towards each other. Hence, the upper movable electrode deflects up to where its elastic force balances with the attraction forces due to the distributed charges and interatomic interactions. This process continues up to a maximum limit and after it the movable electrode loses its tolerance and suddenly drops on the fixed electrode (ground plate). In this condition, the elastic properties of the upper electrode cannot tolerate the attraction force. This phenomenon is called pull-in or snap-down. In
Fig. 1. Schematic representation of an electrostatically actuated CNT.
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this paper, the movable electrode is considered a single walled CNT and the fixed electrode is composed of the graphene sheets.
simulations are performed using LAMMPS (large scale atomic/ molecular massively parallel simulator) software.
3. Results and discussion 2. Atomistic modeling approach The well-known molecular dynamics technique is applied in this paper to investigate the deflection and pull-in phenomena of the CNTs under electrostatic actuation. The concepts of molecular dynamics were described well in the literature. In this method, the interactions between the atoms are modeled via some potential functions. Different functions were provided for various materials and applications. One of the most effective potentials is adaptive intermolecular reactive bond order (AIREBO) potential modeling the carbon and hydrogen behaviors in the hydrocarbons. The detailed description of AIREBO potential was presented by Stuart et al. [28]. In this section, we are only going to present a brief explanation regarding this potential as well as Coulomb potential modeling the electrostatic interactions. The AIREBO potential consists of three terms [29]: 2 3 X X TORSION 1 X X4 REBO LJ 5 E¼ ð1Þ E þ Eij Ekijl 2 i j a i ij k a i,j l a i,j,k EREBO has the same functional form similar to the hydrocarbon ij REBO potential developed by Brenner et al. in [29]. The coefficients for EREBO are, in fact, the same as Brenner’s potential, but a ij few fitted values are slightly different. Nonetheless, in most cases, this term leads to the energies, forces and statistical averages same as the original REBO potential. It models the reactive capabilities and only describes short-ranged C–C, C–H and H–H interactions (r o2A). These interactions depend strongly on the coordinates through a bond order parameter adapting the attraction between the ith and jth atoms according to the positions of other nearby atoms. Hence, it possesses three and four body dependence. ELJ adds longer-ranged interactions (2A or ocutoff) ij via a form similar to the standard Lennard-Jones (L-J) potential. The ELJ term includes a series of switching functions so that the ij short-ranged L-J repulsion does not interfere with the energies captured by the EREBO term. However, the extent of the ELJ ij ij interactions can be determined by a cutoff operator in the formulation. Van der Waals interactions between the CNT and graphene sheets were also modeled using L-J potential. In addition, ETORSION is an explicit four body potential explaining various kijl dihedral angle preferences in hydrocarbon configurations [29]. The coulomb potential estimates the Coulombic interaction potential between the charged particles according to the following relation: E¼
Cqi qj , er
In this part the results of the simulations are presented and discussed. As mentioned before, there are some main differences between the results presented in this paper with the previously reported papers. The first one is investigation of the geometry and boundary condition effect on the pull-in charges of the CNTs using MD. Although in [24–27] the authors applied MD to investigate the electromechanical behaviors of the CNT, they dos not investigate the geometry variations of the CNTs. Moreover, they considered CNTs perfect without any structural defect that is very ideal. In most cases, the CNTs have defects such as Stone–Wales defect and this fact affects the electromechanical characteristics. Also, this paper studied the effects of thermal variations on the pull-in charges. Fig. 2(a) and (b) respectively shows two single walled CNTs with length 6 nm, chirality (10,10) and cantilever and doubly clamped boundary conditions suspended on five graphene sheets. The CNTs and the graphene sheets compose two electrodes of the NEMS. According to the figure, the CNT is initially in a straight position. The interactions between the two charged electrodes including electrostatic charges and vdW effects result in deflecting the CNTs towards the fixed graphene sheets. It is clear that the elastic force of the CNT withstands against the applied attractive force. But, if the charges are so large the CNT cannot tolerate the attractive loads and suddenly drops on the lower electrode. This fact is illustrated in the last part of Fig. 2. This phenomenon is pull-in and the corresponding charge is called pull-in charge. Of course, the pullin charge often is used for the threshold charge resulting in pullin behavior. Tip deflection of a cantilevered CNT vs. iteration at 300 K is demonstrated in Fig. 3. The horizontal axis is the iterations and the vertical one is the tip deflection of the CNT in z direction. The figure shows three distinct stages before attaching the CNT to the ground plate. In the stage 1, the CNT deflects towards the graphene sheets uniformly. In the stage 2, the CNT shows higher resistance against the attractive forces and the figure shows that there are more iterations to deflect in this stage. This fact can be attributed to the local deformation of the CNT. This deformation majorly takes place at the end of stage 2. The stage 3 is for the deformation of the CNT up to approaching to the lower electrode. The fluctuation after the stage 3 can be related to the struggle between the elastic properties of the CNT and attractive forces that will damp vs. time. 3.1. Cantilevered CNTs
r or c
ð2Þ
where, C is an energy-conversion constant, qi and qj are the charges on the two atoms, and epsilon is the dielectric constant. The cutoff r c truncates the interaction distance [29]. The molecular dynamics
In this subsection, the effects of parameter variations on the pull-in charges of the CNTs with cantilever boundary conditions are studied. Fig. 4 presents the variation of the pull-in charge vs. length of the CNT. The chilaity of the CNTs is (10,10), the gap is
Fig. 2. Pull-in snapshot of a SWCNT under electrostatic actuation.
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Fig. 3. Tip deflection of the CNT in z direction vs. iteration.
Fig. 4. Pull-in charge of the cantilevered CNTs vs. length.
Fig. 5. Pull-in charge of the cantilevered CNTs vs. radius.
2 nm and the temperature is 300 K. As shown in this figure, the pull-in charge decreases vs. length increment. This fact can be attributed to bending softening of the nanostructure with increasing the length. It is clear that the longer the CNT, the easier the bending is. The effects of radius variations on the pull-in charges of the cantilevered CNTs with the length 6 nm and gap 2 nm are depicted in Fig. 5. The figure proposes that the radius increment results in the pull-in charge increment. This matter can also be attributed to stiffening of the CNT with radius increment.
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Fig. 6. Pull-in charge of the cantilevered CNT vs. gap distance.
Fig. 7. Pull-in charge of the cantilevered CNT vs. temperature.
It is worth noting that the gap distance variation has significant effects on the pull-in charges. In order to quantize the effects, Fig. 6 is presented. According to the figure, gap distance increment results in the higher pull-in charges. This fact can be attributed to this matter that both electrostatic and vdW interactions highly depend on the distance. The distance increment, with unchanged other parameters, leads to the lower energies and it is clear that for the pull-in phenomenon, higher charge vales are required. Temperature variation is another effect influencing the pull-in charge. According to Fig. 7, the temperature increment reduces the pull-in charge. The CNT has 6 nm length, (10,10) chilaity and 2 nm gap. The variation seems linear to somehow. This variation should be investigated in detail in the design of NEMS. The main important consideration during the previous simulations and results is that the structures of the CNTs were conspired perfect. However, this consideration is not correct in many cases and the structures of the CNTs, similar to many other crystalline materials, have some defects. Here, we are going to scrutinize the effects of one of the most important defects in the CNTs, i.e. Stone– Wales defect, on the pull-in charges. In the Stone–Wales defect, a bond between two carbon atoms rotates about the vertical axis as shown in Fig. 8. Thus, in the neighbor of the defect, two carbon rings have five atoms, whereas two others have seven atoms. The influences of different longitudinal and peripheral defects are studied in detail. Fig. 9 shows the possible positions for the defects that their combinations are considered in this paper. There are three longitudinal positions (A, B, C) as well as three peripheral positions (0, 90, 180 degrees). A and C have distances
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Fig. 8. Stone–Wales defect.
180
Fig. 10. Effects of Stone–Wales defects with positions 0 degree on the pull-in charges of cantilevered CNT.
90 0
A
B
C
Fig. 9. Possible positions for the defects.
Table 1 Studied cases of the Stone–Wales defects. Case number
Defect positions
1 2 3 4 5 6 7 8
No defect C B A C,B C,A B,A C,B,A
Fig. 11. Effects of Stone–Wales defects with positions 90 degrees on the pull-in charges of cantilevered CNT.
1/3 of the length from the nearest ends and B is in the center. We suppose that the CNT is clamped from the left side and free from the right side. All of the studied cases are summarized in Table 1. The above case numbers are investigated in three respective different angles, i. e. 0, 90 and 180 degrees. The measurement point is considered from the lowest point on the periphery of the CNT. Fig. 10 illustrates the effects of Stone–Wales defects with the case numbers summarized in Table 1 and with the angle 0 degree (Fig. 8) on the pull-in charge of the CNT with length 6 nm, chirality (10,10) and gap distance 2 nm at 300 K. The upper part of the figure relates to the pull-in charge values and the lower one corresponds to the pull-in charge ratio wrt the pristine CNT. The pull-in charge ratio is defined as: 100
ðCharge of Pristine CNTCharge of Defective CNTÞ ðCharge of Pristine CNTÞ
ð3Þ
According to the figure, both the position and number of the defects affect the pull-in charge. In the cases 2, 3 and 4 belonging to one defect, the figure reveals that the longer the distance from the clamped side, the weaker the effect is. In other words, the defect on point C of the CNT has smaller effects than the defect on the point near the support (point A). This fact can be attributed to the matter that local deformation occurs near the support. Hence, the defects near the local deformation result in the structure weakness and facilitate the pull-in phenomena. It is obvious that the number of the defects has weakening effect on the nano
structure and leads to the smaller pull-in charges, as shown in Fig. 10. For the last case, the CNT possesses three defects on three longitudinal positions. The figure shows that this case is the worst condition among the modeled and investigated cases. In this case, the pull-in charge may be reduced about 15%. Hence, the result reveals the necessity of this study. This matter should be scrutinized deeply in the NEMS design and fabrication processes. Fig. 11 depicts the effects of the defects on the positions presented in Table 1 with 90 degrees (Fig. 9). The figure shows that the existence of Stone–Wales defects in the angle 90 degrees can also reduce the pull-in charges. However, a one by one comparison between the results of 0 degree with the results of 90 degrees reveals that the effects of 90 degree defects are weaker than the previous case. As an example, for the last case in Fig. 11, the defect reduces the pull-in charge about 11% whereas in the 0 degree, the last case reduces the charge about 15%. This fact mentions that the peripheral position of the defect is important as the longitudinal positions. Fig. 12 presents the effects of the Stone–Wales defects on the pull-in charges when they are situated on 180 degrees. The pattern of effects is similar to the two previous cases. The defects near the local deformation dominate to those far from it. The increment in the defect numbers has more reducing effects. The main point is that the results corresponding with 180 degrees are situated between the results corresponding with 0 degrees and those with 90 degrees. Hence, it can be concluded that the pattern of variations of the pull-in charges vs. peripheral angle is so that
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Fig. 12. Effects of Stone–Wales defects with positions 180 degrees on the pull-in charges of cantilevered CNT.
Fig. 14. Pull-in charge of the doubly clamped CNTs vs. radius.
Fig. 13. Pull-in charge of the doubly clamped CNTs vs. length.
Fig. 15. Pull-in charge of the doubly clamped CNTs vs. gap distance.
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the maximum reduction belongs to the 0 degree and reduces with the angle increment up to 90 degrees; then increases again until approaching to 180 degrees. However, the defects belonging to 180 degrees have smaller influences in comparison with those relating to 0 degrees. It may be attributed to the position of local deformation occurring in the lower atoms (close to 0 degree). 3.2. Doubly clamped CNT In this section, the results of variation in the length, radius and gap distance on the pull-in charges of the CNTs with doubly clamped boundary conditions are discussed. Furthermore, the effects of temperature changes and Stone–Wales defects on the pull-in charges of the doubly clamped CNT are investigated. Fig. 13 presents the effects of length increment on the pull-in charges of the doubly clamped CNTs with chirality (10,10) and gap distance 2 nm. The figure reveals that the length increment reduces the pull-in charges drastically. Another important fact relating to the figure can be deduced in comparison with its counterpart in Fig. 4. The comparison shows that the pull-in charges belonging to the doubly-clamped CNTs are remarkably higher than those corresponding to the cantilevered CNTs with other same parameters. Fig. 14 displays the effects of radius variations on the pull-in charges of the doubly clamped CNTs. Based on the figure, the diameter increasing results in the higher pull-in charges. The pattern of the results is similar to the cantilevered CNTs but the pull-in charge values are significantly higher.
Fig. 16. Pull-in charge of the doubly clamped CNT vs. temperature.
Lastly, the effects of gap distance on the pull-in charge values of the doubly clamped CNTs are presented in Fig. 15. According to the figure, the pull-in charges increase with the increment in the gap. This is, as described before, because of this fact that the distance increment weakens the electrostatic and vdW forces. Thus larger charges are required for pull-in phenomena. The effects of temperature variation on the pull-in charges of the doubly clamped CNT are depicted in Fig. 16. According to the figure, similar to the cantilevered CNT, the temperature increment reduces the pull-in charge of the doubly clamped CNT.
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The effects of Stone–Wales defects on the pull-in charges of the doubly clamped CNTs are presented in Figs. 17–19 respectively for the defect in 0, 90 and 180 degrees. According to the figures, the Stone–Wales defects reduce the pull-in charge of the doubly clamped CNTs but their influences are less than the case of cantilevered CNTs. The nearness of the defect to the center of the CNT increases the effects of the defects since the main
deformation occurs in the center. The figures reveal that the for one defect, the zero angle position (case 3) has the maximum effect in comparison to the cases C and A that are same due to symmetry. In addition, in two defect cases, the cases 5 and 6 that are same possess maximum effects and for the case of three defects, it can be observe that the influences are higher in comparison with the previous cases. Moreover, the defects in zero angles have the maximum effects and those in 90 degrees possess the minimum effects. This is also attributed to the place of deformation.
4. Conclusion
Fig. 17. Effects of Stone–Wales defects with positions 0 degree on the pull-in charges of doubly clamped CNT.
The paper presented the deflection and pull-in phenomena of the CNTs in the presence of vdW and electrostatic attractions. The ground plate consisted of the graphene sheets. AIREBO, LennardJones and Coulomb potentials were applied to model the nano system. The effects of various lengths, radii, gap distances, temperatures and boundary conditions on the pull-in charges of the CNTs were reported. The conclusion was that the stiffer CNTs had higher pull-in charges and any factor helping the CNT to become stiffer had the increasing effects on the pull-in charges. Moreover, the effects of Stone–Wales defects with different positions and numbers on the pull-in charges were investigated. The results revealed that the defects could reduce the pull-in charges drastically. As an example, three defects in the cantilevered CNT reduced the pull-in charge about 15%. For the future work, one may study the effects of other defects on the electromechanical behaviors of the CNTs under electrostatic actuation. References
Fig. 18. Effects of Stone–Wales defects with positions 90 degrees on the pull-in charges of doubly clamped CNT.
Fig. 19. Effects of Stone–Wales defects with positions 180 degrees on the pull-in charges of doubly clamped CNT.
[1] M.M.S. Fakhrabadi, M. Samadzadeh, A. Rastgoo, M. Haeri Yazdi, M. Mousavi Mashhadi, J. Phys. E 44 (2011) 565–578. [2] M. Rossi, M. Meo, J. Compos. Sci. Tech. 69 (2009) 1394–1398. [3] R. Chowdhury, S. Adhikari, C.Y. Wang, F. Scarpa, J. Comput. Mat. Sci. 48 (2010) 730–735. [4] A.Y. Joshi, S.C. Sharma, S.P. Harsha, J. Sensor Rev. 31 (2011) 47–57. [5] A.Y. Joshi, S.C. Sharma, S.P. Harsha, J. Phys. E 43 (2011) 1040–1045. [6] M.M.S. Fakhrabadi, N. Khani, R. Omidvar, A. Rastgoo, J. Comput. Mat. Sci. 61 (2012) 248–256. [7] M.M.S. Fakhrabadi, N. Khani, S. Pedrammehr, J. Phys. E 44 (2012) 1162–1168. [8] S. Rouhi, R. Ansari, J. Phys. E 44 (2011) 764–772. [9] J. Xia, D. Wei, European J. Mech. A/Solids 29 (2010) 440–447. [10] X. Hao, H. Qiang, Y. Xiaohu, J. Compos. Sci. Tech. 68 (2008) 1809–1814. [11] Z. Wang, M. Devel, B. Dulmet, J. Surf. Sci. 604 (2010) 496–499. [12] K.S. Kim, J. Park, H. Jung Hwang, J. Won Kang, J. Phys. E 44 (2012) 2027–2031. [13] J.W. Kang, K.S. Kim, J. Park, H.J. Hwang, J. Phys. E 43 (2011) 909–913. [14] A.R. Ranjbartoreh, G. Wang, J. Phys. E 43 (2010) 202–206. [15] M. Dequesnes, S.V. Rotkin, N.R. Aluru, J. Nanotech. 13 (2002) 120–131. [16] C. Ke, H.D. Espinosa, J. Appl. Mech. 72 (2005) 721–725. [17] C. Ke, H.D. Espinosa, N. Pugno, J. Appl. Mech. 72 (2005) 726–731. [18] H.M. Ouakad, M.I. Younis, J. Comput. Nonlin. Dyn. 5 (2010) 1–13. [19] H.M. Ouakad, M.I. Younis, J. Nonlin. Dyn. 67 (2012) 1419–1436. [20] H.M. Ouakad, M.I. Younis, J. Sound Vib. 330 (2011) 3182–3195. [21] J.M. Kinaret, T. Nord, S.J. Viefers, Appl. Phys. Lett. 82 (2003) 3182–3195. [22] L.M. Jonsson, T. Nord, J.M. Kinaret, S.J. Viefers, Appl. Phys. 96 (2004) 629–635. [23] L.M. Jonsson, S. Axelsson, T. Nord, S. Viefers, J.M. Kinaret, J. Nanotech. 15 (2004) 1497–1502. [24] H.J. Hwang, J.W. Kang, J. Phys. E 27 (2005) 163–175. [25] J.W. Kang, J.H. Lee, H.J. Lee, H. Hwang, J. Phys. E 27 (2005) 332–340. [26] J.W. Kang, J.H. Lee, H.J. Lee, O.K. Kwon, H. Hwang, J. Phys. E 28 (2005) 273–280. [27] J. Won Kang, O.K. Kwon, J.Ha Lee, H.Joo Lee, Y. Song, Y. Yoon, H. Hwang, J. Phys. E 33 (2006) 41–49. [28] S.J. Stuart, A.B. Tutein, J.A. Harrison, J. Chem. Phys. 112 (2000) 6472–6486. [29] Large-scale Atomic/Molecular Massively Parallel Simulator, LAMMPS user manual, AIREBO description, Sandia Corporation, USA, 2012.