- Email: [email protected]
Thermal buckling of initially compressed single-walled carbon nanotubes by molecular dynamics simulation
Available online at www.sciencedirect.com
Carbon 45 (2007) 2614–2620 www.elsevier.com/locate/carbon
Thermal buckling of initially compressed single-walled carbon nanotubes by molecular dynamics simulation Chen-Li Zhang, Hui-Shen Shen
*
Department of Engineering Mechanics, Shanghai Jiao Tong University, Shanghai 200030, People’s Republic of China Received 3 February 2007; accepted 7 August 2007 Available online 15 August 2007
Abstract Thermal buckling of initially compressed single-walled carbon nanotubes subjected to a uniform temperature rise is presented by using molecular dynamics simulations. Comprehensive numerical calculations are carried out for armchair and zigzag carbon nanotubes with various geometric dimensions. The results show that thermal buckling can occur beyond a critical value of temperature when the tube is initially compressed to a point prior to buckling. The critical buckling temperature increases as the compressive load ratio parameter decreases, and varies dramatically with nanotube helicity, radius and length. Owing to strong thermal oscillations of carbon atoms, a zigzag carbon nanotube with relatively small radius can buckle at a surprisingly lower temperature than the expected one. Ó 2007 Elsevier Ltd. All rights reserved.
1. Introduction The discovery of carbon nanotubes (CNTs) has opened up opportunities for researchers to apply CNTs in composite bulk materials and as individual elements of nanometerscale devises and sensors [1,2]. Furthermore, CNTs have been used as probes in scanning probe microscopy [3,4]. Recent transmission electron microscope (TEM) observations and nanoindentation experiments indicated that carbon nanotubes buckle elastically when subjected to bending and/or compression [5–7]. Buckling is a structural instability failure mode and a major concern for structural design. It has been reported that the material properties such as conductance and effective Young’s modulus of CNTs are strongly influenced by the occurrence of buckling [8,9]. CNT-based nanocomposite devices may withstand high temperature during manufacture and operation. On this account, thermal properties of carbon nanotubes and temperature effect on the buckling behavior have received considerable attentions. Treacy et al. [10] carried
*
Corresponding author. Fax: +86 21 62933021. E-mail address: [email protected] (H.-S. Shen).
0008-6223/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.carbon.2007.08.007
out the measurement of a thermally inducing vibration amplitude of multi-walled CNTs in a TEM. Tang et al. [11] reported that the structural configuration will change and the plastic deformations will occur for CNTs subjected to high pressure and at high temperature of 1800 °C. Ni et al. [12] performed the buckling analysis of single-walled carbon nanotubes (SWCNTs) under axial compression at 100, 600 and 1500 K, and reported that the buckling load of an empty nanotube decreases as the temperature increases, whereas filling the carbon nanotubes with fullerenes or gas molecules disrupts the temperature effect. It has also been shown that the temperature change has a significant effect on the postbuckling response of SWCNTs under axial compression, but it has a small effect in the loading case of torsion. In contrast, temperature only has a less effect on the postbuckling response of SWCNTs under external pressure [13]. On the other hand, a CNT may buckle due to the compressive stress caused by elevated temperature when the end of the tube is prevented from moving in the axial direction. It has been reported that CNTs exhibit extraordinarily excellent thermal stability and can sustain extreme high temperature with no signs of damages [14–17]. The atomic configuration of CNTs may not change until the
C.-L. Zhang, H.-S. Shen / Carbon 45 (2007) 2614–2620
temperature reaches 2400 °C [15]. The double-walled CNTs with inner diameter being 0.71–0.9 nm are stable without structural transformation up to 2200 °C [16]. Even the narrow (2, 2) and (3, 3) armchair nanotubes have the bond breaking temperature around 1000–1850 K [17]. It is thus reasonable to consider whether it is possible to have a classical thermal buckling problem for such a CNT. This paper deals with thermal buckling of initially compressed SWCNTs by direct observations from molecular dynamics (MD) simulations. Our main objectives are to examine the structural changes of carbon nanotubes in response to elevated temperature before and after the onset of buckling, to explore the variation of atomic interactions induced by thermal loads, and to assess the critical buckling temperature with different compressive load ratio parameters. In addition, the effect of nanotube helicity, radius and length on thermal buckling behavior is also systematically studied. 2. Simulation model In the present MD simulations, the covalent bond interactions within CNTs are calculated by the many-body reactive empirical bond order (REBO) potential [18], which can accurately describe binding energies, bond length and lattice constants of solid state carbon molecules. To account for the thermal effect, we use the Nose–Hoover thermostat [19] to maintain the temperature of the system. Using this type of thermostat, Nardelli et al. [20] studied strain release of carbon nanotubes at temperature as high as 1800 K. Since the Nose–Hoover thermostat [19] provides good conservation of energy and results in less fluctuation in temperature through the excellent feeding back scheme, we believe that it is suitable for the present problem. In order to study the effect of nanotube helicity and radius, we choose five pairs of armchair and zigzag SWCNTs with radii are closely matched with each other. The armchair type nanotubes used in this study are (7, 7), (8, 8), (9, 9), (11, 11) and (14, 14), and the counterpart zigzag nanotubes employed are (12, 0), (14, 0), (16, 0), (19, 0) and (24, 0) respectively. All tubes are set to have roughly equal length. On the other hand, to show the effect of nanotube length, we simulate five armchair (9, 9) tubes having different length ranging from 3.58 to 7.29 nm. The various geometric dimensions of the considered nanotubes are listed in Tables 1–3. During MD simulations, both ends of each tube are held rigid and externally controlled to satTable 1 Geometries for armchair SWCNTs with different radii
2615
Table 2 Geometries for zigzag SWCNTs with different radii SWCNTs
(12, 0)Tube
(14, 0)Tube
(16, 0)Tube
(19, 0)Tube
(24, 0)Tube
Length (nm) Radius (nm)
5.27
5.27
5.27
5.27
5.27
0.48
0.55
0.63
0.75
0.94
Table 3 Geometries for armchair SWCNTs with different length SWCNTs
(9, 9)Tube
(9, 9)Tube
(9, 9)Tube
(9, 9)Tube
(9, 9)Tube
Length (nm) Radius (nm)
3.58
4.32
5.31
6.30
7.29
0.61
0.61
0.61
0.61
0.61
isfy the boundary conditions, the remaining atoms in the middle part are free to evolve in time. It is worthy to note that selection of the number of thermostat atoms is an important issue in MD simulations. Mylvaganam and Zhang [21] proved that applying a thermostat to the majority atoms of a nanotube leads to less fluctuation in temperature and results in more reasonable failure configuration when the SWCNT is subjected to axial tension. In the present study, the thermostat is applied to all atoms except for rigid ones of the carbon nanotube. From simulation runs, the average value of temperature fluctuation is found to be 20–40 K, which is within acceptable ranges, and the maximum value of the fluctuation is smaller than 50 K. A SWCNT is initially optimized and freely relaxed at room temperature 300 K by minimizing the potential energy of the entire nanotube. The axial compressive force P is then applied to one end of the tube, and the force P is maintained at the prebuckling level. Let the SWCNT fully move in the conventional MD simulation scheme with the precise displacement mode until the new equilibrium configuration is obtained. Afterwards, both ends of the tube are fixed to be restrained against displacement longitudinally. The carbon nanotube is subjected to a temperature from 300 K to exceed certain critical value in a quasi-static manner. In each thermal loading step, we set the increment of the temperature being 50 K and let the tube structure evolve in dynamic integrating for 30 ps. A time step of 0.5 fs guarantees good conservation of energy. By direct measurement from MD results, the structural changes and variations of interatomic forces due to elevated temperature are carefully recorded. It is possible to observe buckling phenomenon caused by thermal loads within the MD attainable time scale.
SWCNTs
(7, 7)Tube
(8, 8)Tube
(9, 9)Tube
(11, 11)Tube
(14, 14)Tube
Length (nm) Radius (nm)
5.31
5.31
5.31
5.31
5.31
3. Results and discussion
0.48
0.55
0.61
0.75
0.95
From our MD simulations, the thermal buckling behavior of initially compressed (11, 11) armchair nanotube is
2616
C.-L. Zhang, H.-S. Shen / Carbon 45 (2007) 2614–2620
λ p =0.9
75
-7.20
F (nN)
65 D
55 C
50
E
B
2: λ p =0.8 3: λ p =0.7
λ p =0.7
60
3
1: λ p =0.9
λ p =0.8
70
45
-7.16
b
80
Energy (eV/atom)
a
2
-7.24 1 -7.28 -7.32
Critical point
A -7.36
40 35 300
600
900
1200
1500
1800
2100
-7.40 300
600
900
T (K)
1200
1500
1800
2100
T (K)
Fig. 1. Thermal buckling behavior of initially compressed (11, 11) SWCNT: (a) Variation of internal axial force versus temperature. The buckling behavior is characterized by an abrupt drop of the force. (b) Variation of energy versus temperature. The critical points are marked by solid dots.
shown in Fig. 1, in which we introduce a load ratio parameter kp, defined by kp = P/Pcr, where Pcr is the buckling load in the case of pure axial compression, as previously reported [13], and P is the applied axial compressive load at the prebuckling level. Based on the fact that the mechanical buckling onset of CNTs under axial compression can be measured when a drastic drop for internal axial forces F are observed, as previously shown in many MD simulations [5,6,12,13]. We present here the relationship between the internal force F and temperature T to elaborate the thermomechanical responses, as shown in Fig. 1a. It is found that the curves for kp = 0.9, 0.8 and 0.7 are characterized by three significant features. The first one is that the force F goes upward slightly in the initial region, which is caused by the thermal load under fixed boundary conditions of both ends. Secondly, the nanotube experiences a sudden buckling behavior beyond a critical value of the temperature, resulting in an abrupt jump down in the axial force. This thermomechanical buckling of CNTs shows a similar response to the case of axial compression. Indeed, these two are in identical state of pure axial stress. Thirdly, the critical buckling temperature Tcr increases as parameter kp decreases. The Tcr for the example of (11, 11) nanotube is predicted to be 900, 1100 and 1800 K, respectively, corresponding to kp = 0.9, 0.8 and 0.7. We should emphasize that the thermal buckling of CNTs is very sensitive to even small changes of the compressive load. With the prebuckling load varying from 71.03 (kp = 0.9) to 63.60 nN (kp = 0.8), the increase percentage of Tcr is 18.2%, and a percentage difference of 38.9% corresponds to kp decreasing from 0.8 to 0.7. In order to pinpoint the nanotube response to thermal load, we calculate caloric curves, i.e., total energy per atom against temperature for (11, 11) SWCNT, as shown in Fig. 1b. It is noted that the energy increases almost linearly with temperature in both regions before and after the critical point, reflecting the classical equipartition theorem in MD simulations, which is in accordance with the investigation by Lo´pez et al. [17]. It is difficult to identify the buck-
ling onset from the caloric curve because there is no discontinuity in the graph, which has been previously observed in buckling under axial compression [13,22,23]. In fact, thermal buckling cannot lead to apparent release of strain energy stored in the nanotube owning to the increasing temperature. As the load ratio parameter varies from 0.9 to 0.7, the energy corresponding to the critical point increases from 7.30 to 7.19 eV/atom, from which we believe that the thermal buckling of SWCNTs is hardly to occur when kp is sufficiently small. Configurations in Fig. 2 reflect structural changes for (11, 11) nanotube in response to elevated temperature with kp = 0.7. Under fixed boundary conditions of both ends, the nanotube length remains constant during the whole simulations. It is critical to note that more and more local imperfections are displayed on the tube surface as temperature increases. For the initial stage A at 300 K, the tube exhibits nearly perfect cylindrical configuration, whereas many small dimples or bulges defects occur on the wall in the case of the critical stage D at 1750 K. This can be attributed to an increase in thermal oscillations of carbon
A
B
C
D
E
Fig. 2. Thermal buckling configurations for initially compressed (11, 11) SWCNT with kp = 0.7. A: T = 300 K, initial stage; B: T = 600 K; C: T = 1200 K; D: T = 1750 K, critical stage; E: T = 1800 K, one flattening is displayed in the structure.
C.-L. Zhang, H.-S. Shen / Carbon 45 (2007) 2614–2620
atoms at higher temperature. Due to the fact that the amplitude of the atomic oscillations becomes wider as the temperature rises, the nanotube cannot remain perfect cylindrical configuration and geometric imperfections are unavoidable, which is also one of the intrinsic characteristics of nanostructure. Moreover, such local defects lead to great degradation of a nanotube against buckling since the onset of buckling is very sensitive to the perturbation imposed by geometric imperfections. On this account, we believe that the critical temperature obtained by MD simulations is smaller than that from continuum mechanics analysis with the perfect cylindrical shell model. Panel E in Fig. 2 depicts the buckling configuration of (11, 11) tube at 1800 K. Both sides of the wall collapse inward and transform into a flattening, which is similar to that found in pure axial compression. There have been considerable studies concentrated on the size dependency of mechanical behavior of nanotubes [10,13,24–27]. However, to our knowledge, relatively few studies have been made on the thermal buckling of CNTs, including helicity, radius and length effects. With the load ratio parameter being equal to 0.8, the effects of nanotube helicity and radius are shown in Figs. 3 and 4, respectively. The percentage decrease of buckling temperature Tcr of a (19, 0) tube versus a (11, 11) tube is about 18.2% despite the higher critical force for the zigzag nanotube, which shows that the nanotube arranged in zigzag chirality has a weaker resistance to thermal buckling. From Fig. 4, it can be seen that the buckling behavior exhibits significant radius dependence for both armchair and zigzag tubes. The buckling temperature decreases with increases in radius, and the similar trend is found for critical axial stress. By taking the wall thickness as 0.067 nm [24], the critical stresses of (9, 9) and (11, 11) armchair tubes are predicted to be 0.26 and 0.21 TPa, while the (16, 0) and (19, 0) zigzag tubes have values of 0.27 and 0.25 TPa. In order to gain more insights into the relationship between buckling temperature Tcr and nanotube radius as well as the effect of chirality, systematic MD simulations
80 (11,11)-tube
75
(19,0)-tube
70
F (nN)
65 60 55 50 45 40 35 300
600
900
1200
1500
1800
T (K) Fig. 3. Effect of nanotube helicity on thermal buckling behavior of initially compressed (11, 11) and (19, 0) SWCNTs with kp = 0.8.
2617
on thermal buckling behavior of five pairs of SWCNTs mentioned above have been carried out in the case of kp = 0.8. Fig. 5 shows the variation of Tcr as a function of radius R. Obviously, the buckling temperature of armchair SWCNTs is higher than that of zigzag tubes with similar geometries. As can be seen from the results for armchair nanotubes, the larger the radius, the smaller the critical buckling temperature. The effect of nanotube radius becomes more significant when R is reduced. For example, the radius of (8, 8) tube is bigger than that of (7, 7) tube by 14.6%, and the percentage decrease of buckling temperature is about 15.4%. In contrast, with the percentage increase of R being 26.7% by comparison between (14, 14) and (11, 11) tubes, the reduction of Tcr is less than 9.1%. According to MD simulations and the smooth curve fitting for zigzag nanotubes, it is critical to note that such a monotonic relationship between Tcr and R is not displayed. The buckling temperature goes down steadily as the radius increases when R > 0.55 nm, whereas the situation is opposite outside this region. Our simulations predict that the critical temperature Tcr for (12, 0) nanotube being 950 K is smaller than that of (14, 0) nanotube with larger radius. The relationship between Tcr and radius R of the armchair nanotube can be fitted by a smooth polynomial as T cr ¼ a1 þ a2 R þ a3 R2 þ a4 R3
ð1Þ
where the coefficients a1 = 10838.70 K, a2 = 33874.72 K nm1, a3 = 39513.81 K nm2, and a4 = 15532.84 K nm3. The fitted function is plotted as solid line in Fig. 5. The smooth dash line is fitted to the data points representing for zigzag nanotubes and can be expressed as T cr ¼ b1 þ b2 R þ b3 R2 þ b4 R3 þ b5 R4
ð2Þ
where the coefficients b1 = 10508.66 K, b2 = 63469.98 K nm1, b3 = 127775.78 K nm2, b4 = 111583.55 K nm3, and b5 = 35966.64 K nm4. The unit of Tcr is K and the unit of R is nm in Eqs. (1) and (2). For a nanoscale explanation of the illogical feature of zigzag nanotubes shown in Fig. 5, we examine the structural changes of (12, 0) nanotube induced by elevated temperature throughout the simulations on thermal buckling behavior. The results are presented in Fig. 6 with kp = 0.8. Similarly, the thermomechanical response shown in panel (a) is characterized by a steady increase in the axial force F until it reaches a certain critical point beyond which the buckling occurs. However, the buckling temperature is significantly lower than expected. This can be illustrated by the changes of configurations depicted in panel (b). It shows that many local geometric imperfections are randomly distributed in the zigzag (12, 0) nanotube for the critical stage C at 900 K due to dramatically atomic oscillations. As mentioned before, SWCNTs with more defects are likely to buckle at lower temperature. One can expect that these geometric defects will have even more significant impact on thermal buckling of nanotubes with relatively small radii, thus greatly accelerating the occurrence of instabilities. The stage D in Fig. 6b corresponds to the
2618
C.-L. Zhang, H.-S. Shen / Carbon 45 (2007) 2614–2620
a
b
80 (9,9)-tube
80 (19,0)-tube
70
70
65
65
60 55
60 55
50
50
45
45
40
40
35 300
600
900
1200
1500
(16,0)-tube
75
(11,11)-tube
F (nN)
F (nN)
75
35 300
1800
600
900
T (K)
1200
1500
1800
T (K)
Fig. 4. Effect of nanotube radius on thermal buckling behavior of initially compressed SWCNTs with kp = 0.8: (a) Comparisons for (9, 9) and (11, 11) armchair nanotubes; (b) comparisons for (16, 0) and (19, 0) zigzag nanotubes.
According to our MD simulations, the randomly occurring defects on tube surface do not lead to obvious decrease in bucking temperature Tcr for armchair SWCNTs with small radii. This can be understood by taking a close look at the covalent bond connections in the nanotube structure. One-third of the bonds are parallel with the axial direction for zigzag SWCNTs, while all of the bonds are inclined to the tube axis in armchair SWCNTs. When the tube radius becomes small, the distortion of carbon atoms of armchair nanotubes is more difficult to occur. Therefore the zigzag tube is more susceptible to be impacted by the effect of high temperature. Fig. 7 shows the effect of nanotube length L on the bucking temperature in the case of kp = 0.8. We study the buckling of (9, 9) tubes under elevated temperature, with their lengths varying from 3.58 to 7.29 nm. It is evident that the predicted value of buckling temperature Tcr depends strongly on the nanotube length. The value of Tcr decreases as the length increases for the considered SWCNTs of the same radius. The slope of the fitted curve
2000 MD results for armchari tubes MD results for zigzag tubes Fitting curve for armchair tubes, eq.(1) Fitting curve for zigzag tubes, eq.(2)
1800
Tcr (K)
1600 1400 1200 1000 800
0.4
0.5
0.6
0.7
0.8
0.9
1.0
R (nm) Fig. 5. The relationships between critical buckling temperature and nanotube radius for SWCNTs with kp = 0.8.
buckling configurations of zigzag (12, 0) nanotube at Tcr = 950 K. It is observed that the tube loses its nearly cylindrical configuration and begins to rotate around the flattening in the middle position.
a
b
75 70 65
F (nN)
60 55
A
C
B
50 45
D
40 35 30 300
450
600
750
900
1050
1200
1350
T (K) A
B
C
D
Fig. 6. Thermal buckling behavior of initially compressed (12, 0) SWCNT with kp = 0.8: (a) Variation of internal axial force versus temperature; (b) corresponding configurations, A: T = 300 K, initial stage; B: T = 600 K; C: T = 900 K, critical stage; D T = 950 K, the nanotube structure begins to buckle.
C.-L. Zhang, H.-S. Shen / Carbon 45 (2007) 2614–2620 1800 1700
MD results Fitting curve, eq.(3)
Tcr (K)
1600 1500 1400 1300 1200 1100
3
4
5
6
7
8
L (nm) Fig. 7. The relationships between critical buckling temperature and nanotube length with kp = 0.8. The tubes considered are (9, 9) SWCNTs with lengths varying from 3.58 to 7.29 nm.
begins to increase when the length is larger than 5.3 nm, indicating that the effect of tube length becomes larger when L > 5.3 nm, while such dependency is insensitive outside this range. Using the polynomial fitting technique for the MD results in Fig. 7, the critical buckling temperature can be described as T cr ¼ c1 þ c2 L þ c3 L2 þ c4 L3 þ c5 L4
2619
ones. This shows that the nanotubes arranged in armchair chirality need even more strain energy for occurrence of thermal buckling and have higher buckling temperature as well. It is noted that the linear relationship between potential energy and the temperature exists for all CNTs considered in the present study. To give further understanding of the effect of structural geometries on thermal bucking, size-dependent rigidity in longitudinal direction of SWCNTs should be taken into account. Many previous experimental and theoretical studies [10,24–27] have observed that the Young’s modulus of SWCNTs decreases as the radius increases. In particular, there is a strong dependence of the elastic modulus on the radius for a nanotube when R < 0.6 nm [27]. For pairs of tubes of similar dimensions but different helicity, the effective Young’s modulus of armchair nanotubes is higher than that of zigzag tubes with nearly equal dimensions, and the difference becomes larger if the nanotube radius is smaller than 0.6 nm [24,27]. It is readily to recognize that a nanotube with higher longitudinal rigidity has a stronger resistance to thermal buckling, which is in consistent with our MD results. 4. Conclusions
ð3Þ
where Tcr and length L are in units of K and nm, respectively. We obtain the values of the coefficients as c1 = 5434.19 K, c2 = 6220.84 K nm1, c3 = 1903.22 K nm2, c4 = 241.04 K nm3, and c5 = 10.98 K nm4, respectively. To provide a better explanation of this size-dependent behavior shown in Figs. 5 and 7, we first examine the effect of geometric dimensions on the mechanical properties of SWCNTs from the view point of energy. It has been shown in Fig. 1b that the total energy per atom varies linearly with respect to the temperature to reach certain critical value which increases as the load ratio parameter decreases for a given tube. On the other hand, with an identical kp, it is found that the potential energies corresponding to the critical point vary from case to case for different nanotubes due to their different structural geometries. Denote ecr as the critical strain energy where the strain energy is defined as the differences in the potential energy per atom of heated and unheated tube (i.e., the beginning nanotube structure in MD simulations). As kp is equal to 0.8, the value of ecr is predicted to increase when the radius decreases for armchair CNTs. That is, relatively more strain energy is required for nanotube with smaller radius to change the tube from cylindrical state to buckling state. This provides a good explanation for the relationship between Tcr and R in Fig. 5 for armchair tubes. As expected, the critical strain energy drops off monotonically as the tube length increases, which is in accordance with the fact shown in Fig. 7 that the longer the nanotube, the smaller the critical buckling temperature. By comparing armchair and zigzag SWCNTs with identical dimensions, it is found that the value of ecr of armchair tubes is larger than that of zigzag
Comprehensive MD simulations are carried out to investigate thermal buckling behavior of SWCNTs under the simultaneous actions of constant axial compression and uniformly increasing temperature. The nanotube is found to experience a sudden structural change from cylindrical tube to buckling configuration as the temperature is beyond certain critical value. Simulation results show that the zigzag carbon nanotube has less resistance to thermal buckling compared to the armchair tube with similar geometries. The critical bucking temperature is found to decrease as the nanotube length increases and the dependency becomes insensitive for longer CNTs. For both armchair and zigzag nanotubes with relatively large radius, the critical buckling temperature decreases with increasing in tube radius. Remarkably, the zigzag nanotube is more susceptible to be impacted by the effect of high temperature, thereby leading to much lower buckling temperature than expected as the tube radius becomes small. Acknowledgement This work is supported in part by the National Natural Science Foundation of China under Grant 50375091. The authors are grateful for this financial support. References [1] Hadjiev VG, Live MN, Arepalli S, Nikolaev P, Files BS. Raman scattering test of single-wall carbon nanotube composites. Appl Phys Lett 2001;78:3193–5. [2] Tans SJ, Verschueren ARM, Dekker C. Room temperature transistor based on a single carbon nanotube. Nature 1998;393:49–52.
2620
C.-L. Zhang, H.-S. Shen / Carbon 45 (2007) 2614–2620
[3] Dai H, Hafner JH, Rinzler AG, Colbert DT, Smalley RE. Nanotubes as nanoprobes in scanning probe microscopy. Nature 1996;384: 147–50. [4] Kim P, Lieber CM. Nanotube nanotweezers. Science 1999;286: 2148–50. [5] Iijima S, Brabec C, Maiti A, Bernholc J. Structural flexibility of carbon nanotubes. J Chem Phys 1996;104:2089–92. [6] Wong EW, Sheehan PE, Lieber CM. Nanobeam mechanics: elasticity, strength, and toughness of nanorods and nanotubes. Science 1997;277:1971–5. [7] Waters JF, Riester L, Jouzi M, Guduru PR, Xu JM. Buckling instabilities in multiwalled carbon nanotubes under uniaxial compression. Appl Phys Lett 2004;85:1787–9. [8] Poncharal P, Wang ZL, Ugarte D, de Heer WA. Electrostatic deflections and electromechanical resonances of carbon nanotubes. Science 1999;283:1513–6. [9] Postma HWCh, de Jonge M, Yao Z, Dekker C. Electrical transport through carbon nanotube junctions created by mechanical manipulation. Phys Rev B 2000;62:R10653–6. [10] Treacy MMJ, Ebbesen TW, Gibson JM. Exceptionally high Young’s modulus observed for individual carbon nanotubes. Nature 1996;381: 678–80. [11] Tang DS, Chen LC, Wang LJ, Sun LF, Liu ZQ, Wang G, et al. Behavior of carbon nanotubes under high pressure and high temperature. J Mater Res 2000;15:560–3. [12] Ni B, Sinnott SB, Mikulski PT, Harrison JA. Compression of carbon nanotubes filled with C60, CH4, or Ne: predictions from molecular dynamics simulations. Phys Rev Lett 2002;88. Art. no. 205505. [13] Zhang CL, Shen HS. Buckling and postbuckling analysis of singlewalled carbon nanotubes in thermal environments via molecular dynamics simulation. Carbon 2006;44:2608–16. [14] Fagan SB, Mota R, Baierle RJ, Paiva G, Silva AJR, Fazzio A. Stability investigation and thermal behavior of a hypothetical silicon nanotube. J Molecular Struct 2001;539:101–6. [15] de Heer WA, Ugarte D. Carbon onions produced by heat treatment of carbon soot and their relation to the 217.5 nm interstellar absorption feature. Chem Phys Lett 1993;207:480–6.
[16] Kim YA, Muramatsu H, Hayashi T, Endo M, Terrones M, Dresselhaus MS. Thermal stability and structural changes of double-walled carbon nanotubes by heat treatment. Chem Phys Lett 2004;398:87–92. [17] Lo´pez MJ, Cabria I, March NH, Alonso JA. Structural and thermal stability of narrow and short carbon nanotubes and nanostrips. Carbon 2005;43:1371–7. [18] Brenner DW, Shenderova OA, Harrison JA, Stuart SJ, Ni B, Sinnott SB. A second-generation reactive empirical bond order (REBO) potential energy expression for hydrocarbons. J Phys: Condens Mat 2002;14:783–802. [19] Hoover WG. Canonical dynamics: equilibrium phase-space distributions. Phys Rev A 1985;31:1695–7. [20] Nardelli MB, Yakobson BI, Bernholc J. Mechanism of strain release in carbon nanotubes. Phys Rev B 1998;57:R4277–80. [21] Mylvaganam K, Zhang LC. Important issues in a molecular dynamics simulation for characterizing the mechanical properties of carbon nanotubes. Carbon 2004;42:2025–32. [22] Yakobson BI, Brabec CJ, Bernholc J. Nanomechanics of carbon tubes: instabilities beyond linear response. Phys Rev Lett 1996;76: 2511–4. [23] Zhang YY, Tan VBC, Wang CM. Effect of strain rate on the buckling behavior of single- and double-walled carbon nanotubes. Carbon 2007;45:514–23. [24] Zhang CL, Shen HS. Temperature-dependent elastic properties of single-walled carbon nanotubes: prediction from molecular dynamics simulation. Appl Phys Lett 2006;89. Art. no. 081904. [25] Volodin A, Ahlskog M, Seynaeve E, Haesendonck CV. Imaging the elastic properties of coiled carbon nanotubes with atomic force microscopy. Phys Rev Lett 2000;84:3342–5. [26] Natsuki T, Tantrakarn K, Endo M. Effects of carbon nanotube structures on mechanical properties. Appl Phys A – Mater Sci Proc 2004;79:117–24. [27] Zhang HW, Wang JB, Guo X. Predicting the elastic properties of single-walled carbon nanotubes. J Mech Phys Solids 2005;53: 1929–50.
Carbon 45 (2007) 2614–2620 www.elsevier.com/locate/carbon
Thermal buckling of initially compressed single-walled carbon nanotubes by molecular dynamics simulation Chen-Li Zhang, Hui-Shen Shen
*
Department of Engineering Mechanics, Shanghai Jiao Tong University, Shanghai 200030, People’s Republic of China Received 3 February 2007; accepted 7 August 2007 Available online 15 August 2007
Abstract Thermal buckling of initially compressed single-walled carbon nanotubes subjected to a uniform temperature rise is presented by using molecular dynamics simulations. Comprehensive numerical calculations are carried out for armchair and zigzag carbon nanotubes with various geometric dimensions. The results show that thermal buckling can occur beyond a critical value of temperature when the tube is initially compressed to a point prior to buckling. The critical buckling temperature increases as the compressive load ratio parameter decreases, and varies dramatically with nanotube helicity, radius and length. Owing to strong thermal oscillations of carbon atoms, a zigzag carbon nanotube with relatively small radius can buckle at a surprisingly lower temperature than the expected one. Ó 2007 Elsevier Ltd. All rights reserved.
1. Introduction The discovery of carbon nanotubes (CNTs) has opened up opportunities for researchers to apply CNTs in composite bulk materials and as individual elements of nanometerscale devises and sensors [1,2]. Furthermore, CNTs have been used as probes in scanning probe microscopy [3,4]. Recent transmission electron microscope (TEM) observations and nanoindentation experiments indicated that carbon nanotubes buckle elastically when subjected to bending and/or compression [5–7]. Buckling is a structural instability failure mode and a major concern for structural design. It has been reported that the material properties such as conductance and effective Young’s modulus of CNTs are strongly influenced by the occurrence of buckling [8,9]. CNT-based nanocomposite devices may withstand high temperature during manufacture and operation. On this account, thermal properties of carbon nanotubes and temperature effect on the buckling behavior have received considerable attentions. Treacy et al. [10] carried
*
Corresponding author. Fax: +86 21 62933021. E-mail address: [email protected] (H.-S. Shen).
0008-6223/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.carbon.2007.08.007
out the measurement of a thermally inducing vibration amplitude of multi-walled CNTs in a TEM. Tang et al. [11] reported that the structural configuration will change and the plastic deformations will occur for CNTs subjected to high pressure and at high temperature of 1800 °C. Ni et al. [12] performed the buckling analysis of single-walled carbon nanotubes (SWCNTs) under axial compression at 100, 600 and 1500 K, and reported that the buckling load of an empty nanotube decreases as the temperature increases, whereas filling the carbon nanotubes with fullerenes or gas molecules disrupts the temperature effect. It has also been shown that the temperature change has a significant effect on the postbuckling response of SWCNTs under axial compression, but it has a small effect in the loading case of torsion. In contrast, temperature only has a less effect on the postbuckling response of SWCNTs under external pressure [13]. On the other hand, a CNT may buckle due to the compressive stress caused by elevated temperature when the end of the tube is prevented from moving in the axial direction. It has been reported that CNTs exhibit extraordinarily excellent thermal stability and can sustain extreme high temperature with no signs of damages [14–17]. The atomic configuration of CNTs may not change until the
C.-L. Zhang, H.-S. Shen / Carbon 45 (2007) 2614–2620
temperature reaches 2400 °C [15]. The double-walled CNTs with inner diameter being 0.71–0.9 nm are stable without structural transformation up to 2200 °C [16]. Even the narrow (2, 2) and (3, 3) armchair nanotubes have the bond breaking temperature around 1000–1850 K [17]. It is thus reasonable to consider whether it is possible to have a classical thermal buckling problem for such a CNT. This paper deals with thermal buckling of initially compressed SWCNTs by direct observations from molecular dynamics (MD) simulations. Our main objectives are to examine the structural changes of carbon nanotubes in response to elevated temperature before and after the onset of buckling, to explore the variation of atomic interactions induced by thermal loads, and to assess the critical buckling temperature with different compressive load ratio parameters. In addition, the effect of nanotube helicity, radius and length on thermal buckling behavior is also systematically studied. 2. Simulation model In the present MD simulations, the covalent bond interactions within CNTs are calculated by the many-body reactive empirical bond order (REBO) potential [18], which can accurately describe binding energies, bond length and lattice constants of solid state carbon molecules. To account for the thermal effect, we use the Nose–Hoover thermostat [19] to maintain the temperature of the system. Using this type of thermostat, Nardelli et al. [20] studied strain release of carbon nanotubes at temperature as high as 1800 K. Since the Nose–Hoover thermostat [19] provides good conservation of energy and results in less fluctuation in temperature through the excellent feeding back scheme, we believe that it is suitable for the present problem. In order to study the effect of nanotube helicity and radius, we choose five pairs of armchair and zigzag SWCNTs with radii are closely matched with each other. The armchair type nanotubes used in this study are (7, 7), (8, 8), (9, 9), (11, 11) and (14, 14), and the counterpart zigzag nanotubes employed are (12, 0), (14, 0), (16, 0), (19, 0) and (24, 0) respectively. All tubes are set to have roughly equal length. On the other hand, to show the effect of nanotube length, we simulate five armchair (9, 9) tubes having different length ranging from 3.58 to 7.29 nm. The various geometric dimensions of the considered nanotubes are listed in Tables 1–3. During MD simulations, both ends of each tube are held rigid and externally controlled to satTable 1 Geometries for armchair SWCNTs with different radii
2615
Table 2 Geometries for zigzag SWCNTs with different radii SWCNTs
(12, 0)Tube
(14, 0)Tube
(16, 0)Tube
(19, 0)Tube
(24, 0)Tube
Length (nm) Radius (nm)
5.27
5.27
5.27
5.27
5.27
0.48
0.55
0.63
0.75
0.94
Table 3 Geometries for armchair SWCNTs with different length SWCNTs
(9, 9)Tube
(9, 9)Tube
(9, 9)Tube
(9, 9)Tube
(9, 9)Tube
Length (nm) Radius (nm)
3.58
4.32
5.31
6.30
7.29
0.61
0.61
0.61
0.61
0.61
isfy the boundary conditions, the remaining atoms in the middle part are free to evolve in time. It is worthy to note that selection of the number of thermostat atoms is an important issue in MD simulations. Mylvaganam and Zhang [21] proved that applying a thermostat to the majority atoms of a nanotube leads to less fluctuation in temperature and results in more reasonable failure configuration when the SWCNT is subjected to axial tension. In the present study, the thermostat is applied to all atoms except for rigid ones of the carbon nanotube. From simulation runs, the average value of temperature fluctuation is found to be 20–40 K, which is within acceptable ranges, and the maximum value of the fluctuation is smaller than 50 K. A SWCNT is initially optimized and freely relaxed at room temperature 300 K by minimizing the potential energy of the entire nanotube. The axial compressive force P is then applied to one end of the tube, and the force P is maintained at the prebuckling level. Let the SWCNT fully move in the conventional MD simulation scheme with the precise displacement mode until the new equilibrium configuration is obtained. Afterwards, both ends of the tube are fixed to be restrained against displacement longitudinally. The carbon nanotube is subjected to a temperature from 300 K to exceed certain critical value in a quasi-static manner. In each thermal loading step, we set the increment of the temperature being 50 K and let the tube structure evolve in dynamic integrating for 30 ps. A time step of 0.5 fs guarantees good conservation of energy. By direct measurement from MD results, the structural changes and variations of interatomic forces due to elevated temperature are carefully recorded. It is possible to observe buckling phenomenon caused by thermal loads within the MD attainable time scale.
SWCNTs
(7, 7)Tube
(8, 8)Tube
(9, 9)Tube
(11, 11)Tube
(14, 14)Tube
Length (nm) Radius (nm)
5.31
5.31
5.31
5.31
5.31
3. Results and discussion
0.48
0.55
0.61
0.75
0.95
From our MD simulations, the thermal buckling behavior of initially compressed (11, 11) armchair nanotube is
2616
C.-L. Zhang, H.-S. Shen / Carbon 45 (2007) 2614–2620
λ p =0.9
75
-7.20
F (nN)
65 D
55 C
50
E
B
2: λ p =0.8 3: λ p =0.7
λ p =0.7
60
3
1: λ p =0.9
λ p =0.8
70
45
-7.16
b
80
Energy (eV/atom)
a
2
-7.24 1 -7.28 -7.32
Critical point
A -7.36
40 35 300
600
900
1200
1500
1800
2100
-7.40 300
600
900
T (K)
1200
1500
1800
2100
T (K)
Fig. 1. Thermal buckling behavior of initially compressed (11, 11) SWCNT: (a) Variation of internal axial force versus temperature. The buckling behavior is characterized by an abrupt drop of the force. (b) Variation of energy versus temperature. The critical points are marked by solid dots.
shown in Fig. 1, in which we introduce a load ratio parameter kp, defined by kp = P/Pcr, where Pcr is the buckling load in the case of pure axial compression, as previously reported [13], and P is the applied axial compressive load at the prebuckling level. Based on the fact that the mechanical buckling onset of CNTs under axial compression can be measured when a drastic drop for internal axial forces F are observed, as previously shown in many MD simulations [5,6,12,13]. We present here the relationship between the internal force F and temperature T to elaborate the thermomechanical responses, as shown in Fig. 1a. It is found that the curves for kp = 0.9, 0.8 and 0.7 are characterized by three significant features. The first one is that the force F goes upward slightly in the initial region, which is caused by the thermal load under fixed boundary conditions of both ends. Secondly, the nanotube experiences a sudden buckling behavior beyond a critical value of the temperature, resulting in an abrupt jump down in the axial force. This thermomechanical buckling of CNTs shows a similar response to the case of axial compression. Indeed, these two are in identical state of pure axial stress. Thirdly, the critical buckling temperature Tcr increases as parameter kp decreases. The Tcr for the example of (11, 11) nanotube is predicted to be 900, 1100 and 1800 K, respectively, corresponding to kp = 0.9, 0.8 and 0.7. We should emphasize that the thermal buckling of CNTs is very sensitive to even small changes of the compressive load. With the prebuckling load varying from 71.03 (kp = 0.9) to 63.60 nN (kp = 0.8), the increase percentage of Tcr is 18.2%, and a percentage difference of 38.9% corresponds to kp decreasing from 0.8 to 0.7. In order to pinpoint the nanotube response to thermal load, we calculate caloric curves, i.e., total energy per atom against temperature for (11, 11) SWCNT, as shown in Fig. 1b. It is noted that the energy increases almost linearly with temperature in both regions before and after the critical point, reflecting the classical equipartition theorem in MD simulations, which is in accordance with the investigation by Lo´pez et al. [17]. It is difficult to identify the buck-
ling onset from the caloric curve because there is no discontinuity in the graph, which has been previously observed in buckling under axial compression [13,22,23]. In fact, thermal buckling cannot lead to apparent release of strain energy stored in the nanotube owning to the increasing temperature. As the load ratio parameter varies from 0.9 to 0.7, the energy corresponding to the critical point increases from 7.30 to 7.19 eV/atom, from which we believe that the thermal buckling of SWCNTs is hardly to occur when kp is sufficiently small. Configurations in Fig. 2 reflect structural changes for (11, 11) nanotube in response to elevated temperature with kp = 0.7. Under fixed boundary conditions of both ends, the nanotube length remains constant during the whole simulations. It is critical to note that more and more local imperfections are displayed on the tube surface as temperature increases. For the initial stage A at 300 K, the tube exhibits nearly perfect cylindrical configuration, whereas many small dimples or bulges defects occur on the wall in the case of the critical stage D at 1750 K. This can be attributed to an increase in thermal oscillations of carbon
A
B
C
D
E
Fig. 2. Thermal buckling configurations for initially compressed (11, 11) SWCNT with kp = 0.7. A: T = 300 K, initial stage; B: T = 600 K; C: T = 1200 K; D: T = 1750 K, critical stage; E: T = 1800 K, one flattening is displayed in the structure.
C.-L. Zhang, H.-S. Shen / Carbon 45 (2007) 2614–2620
atoms at higher temperature. Due to the fact that the amplitude of the atomic oscillations becomes wider as the temperature rises, the nanotube cannot remain perfect cylindrical configuration and geometric imperfections are unavoidable, which is also one of the intrinsic characteristics of nanostructure. Moreover, such local defects lead to great degradation of a nanotube against buckling since the onset of buckling is very sensitive to the perturbation imposed by geometric imperfections. On this account, we believe that the critical temperature obtained by MD simulations is smaller than that from continuum mechanics analysis with the perfect cylindrical shell model. Panel E in Fig. 2 depicts the buckling configuration of (11, 11) tube at 1800 K. Both sides of the wall collapse inward and transform into a flattening, which is similar to that found in pure axial compression. There have been considerable studies concentrated on the size dependency of mechanical behavior of nanotubes [10,13,24–27]. However, to our knowledge, relatively few studies have been made on the thermal buckling of CNTs, including helicity, radius and length effects. With the load ratio parameter being equal to 0.8, the effects of nanotube helicity and radius are shown in Figs. 3 and 4, respectively. The percentage decrease of buckling temperature Tcr of a (19, 0) tube versus a (11, 11) tube is about 18.2% despite the higher critical force for the zigzag nanotube, which shows that the nanotube arranged in zigzag chirality has a weaker resistance to thermal buckling. From Fig. 4, it can be seen that the buckling behavior exhibits significant radius dependence for both armchair and zigzag tubes. The buckling temperature decreases with increases in radius, and the similar trend is found for critical axial stress. By taking the wall thickness as 0.067 nm [24], the critical stresses of (9, 9) and (11, 11) armchair tubes are predicted to be 0.26 and 0.21 TPa, while the (16, 0) and (19, 0) zigzag tubes have values of 0.27 and 0.25 TPa. In order to gain more insights into the relationship between buckling temperature Tcr and nanotube radius as well as the effect of chirality, systematic MD simulations
80 (11,11)-tube
75
(19,0)-tube
70
F (nN)
65 60 55 50 45 40 35 300
600
900
1200
1500
1800
T (K) Fig. 3. Effect of nanotube helicity on thermal buckling behavior of initially compressed (11, 11) and (19, 0) SWCNTs with kp = 0.8.
2617
on thermal buckling behavior of five pairs of SWCNTs mentioned above have been carried out in the case of kp = 0.8. Fig. 5 shows the variation of Tcr as a function of radius R. Obviously, the buckling temperature of armchair SWCNTs is higher than that of zigzag tubes with similar geometries. As can be seen from the results for armchair nanotubes, the larger the radius, the smaller the critical buckling temperature. The effect of nanotube radius becomes more significant when R is reduced. For example, the radius of (8, 8) tube is bigger than that of (7, 7) tube by 14.6%, and the percentage decrease of buckling temperature is about 15.4%. In contrast, with the percentage increase of R being 26.7% by comparison between (14, 14) and (11, 11) tubes, the reduction of Tcr is less than 9.1%. According to MD simulations and the smooth curve fitting for zigzag nanotubes, it is critical to note that such a monotonic relationship between Tcr and R is not displayed. The buckling temperature goes down steadily as the radius increases when R > 0.55 nm, whereas the situation is opposite outside this region. Our simulations predict that the critical temperature Tcr for (12, 0) nanotube being 950 K is smaller than that of (14, 0) nanotube with larger radius. The relationship between Tcr and radius R of the armchair nanotube can be fitted by a smooth polynomial as T cr ¼ a1 þ a2 R þ a3 R2 þ a4 R3
ð1Þ
where the coefficients a1 = 10838.70 K, a2 = 33874.72 K nm1, a3 = 39513.81 K nm2, and a4 = 15532.84 K nm3. The fitted function is plotted as solid line in Fig. 5. The smooth dash line is fitted to the data points representing for zigzag nanotubes and can be expressed as T cr ¼ b1 þ b2 R þ b3 R2 þ b4 R3 þ b5 R4
ð2Þ
where the coefficients b1 = 10508.66 K, b2 = 63469.98 K nm1, b3 = 127775.78 K nm2, b4 = 111583.55 K nm3, and b5 = 35966.64 K nm4. The unit of Tcr is K and the unit of R is nm in Eqs. (1) and (2). For a nanoscale explanation of the illogical feature of zigzag nanotubes shown in Fig. 5, we examine the structural changes of (12, 0) nanotube induced by elevated temperature throughout the simulations on thermal buckling behavior. The results are presented in Fig. 6 with kp = 0.8. Similarly, the thermomechanical response shown in panel (a) is characterized by a steady increase in the axial force F until it reaches a certain critical point beyond which the buckling occurs. However, the buckling temperature is significantly lower than expected. This can be illustrated by the changes of configurations depicted in panel (b). It shows that many local geometric imperfections are randomly distributed in the zigzag (12, 0) nanotube for the critical stage C at 900 K due to dramatically atomic oscillations. As mentioned before, SWCNTs with more defects are likely to buckle at lower temperature. One can expect that these geometric defects will have even more significant impact on thermal buckling of nanotubes with relatively small radii, thus greatly accelerating the occurrence of instabilities. The stage D in Fig. 6b corresponds to the
2618
C.-L. Zhang, H.-S. Shen / Carbon 45 (2007) 2614–2620
a
b
80 (9,9)-tube
80 (19,0)-tube
70
70
65
65
60 55
60 55
50
50
45
45
40
40
35 300
600
900
1200
1500
(16,0)-tube
75
(11,11)-tube
F (nN)
F (nN)
75
35 300
1800
600
900
T (K)
1200
1500
1800
T (K)
Fig. 4. Effect of nanotube radius on thermal buckling behavior of initially compressed SWCNTs with kp = 0.8: (a) Comparisons for (9, 9) and (11, 11) armchair nanotubes; (b) comparisons for (16, 0) and (19, 0) zigzag nanotubes.
According to our MD simulations, the randomly occurring defects on tube surface do not lead to obvious decrease in bucking temperature Tcr for armchair SWCNTs with small radii. This can be understood by taking a close look at the covalent bond connections in the nanotube structure. One-third of the bonds are parallel with the axial direction for zigzag SWCNTs, while all of the bonds are inclined to the tube axis in armchair SWCNTs. When the tube radius becomes small, the distortion of carbon atoms of armchair nanotubes is more difficult to occur. Therefore the zigzag tube is more susceptible to be impacted by the effect of high temperature. Fig. 7 shows the effect of nanotube length L on the bucking temperature in the case of kp = 0.8. We study the buckling of (9, 9) tubes under elevated temperature, with their lengths varying from 3.58 to 7.29 nm. It is evident that the predicted value of buckling temperature Tcr depends strongly on the nanotube length. The value of Tcr decreases as the length increases for the considered SWCNTs of the same radius. The slope of the fitted curve
2000 MD results for armchari tubes MD results for zigzag tubes Fitting curve for armchair tubes, eq.(1) Fitting curve for zigzag tubes, eq.(2)
1800
Tcr (K)
1600 1400 1200 1000 800
0.4
0.5
0.6
0.7
0.8
0.9
1.0
R (nm) Fig. 5. The relationships between critical buckling temperature and nanotube radius for SWCNTs with kp = 0.8.
buckling configurations of zigzag (12, 0) nanotube at Tcr = 950 K. It is observed that the tube loses its nearly cylindrical configuration and begins to rotate around the flattening in the middle position.
a
b
75 70 65
F (nN)
60 55
A
C
B
50 45
D
40 35 30 300
450
600
750
900
1050
1200
1350
T (K) A
B
C
D
Fig. 6. Thermal buckling behavior of initially compressed (12, 0) SWCNT with kp = 0.8: (a) Variation of internal axial force versus temperature; (b) corresponding configurations, A: T = 300 K, initial stage; B: T = 600 K; C: T = 900 K, critical stage; D T = 950 K, the nanotube structure begins to buckle.
C.-L. Zhang, H.-S. Shen / Carbon 45 (2007) 2614–2620 1800 1700
MD results Fitting curve, eq.(3)
Tcr (K)
1600 1500 1400 1300 1200 1100
3
4
5
6
7
8
L (nm) Fig. 7. The relationships between critical buckling temperature and nanotube length with kp = 0.8. The tubes considered are (9, 9) SWCNTs with lengths varying from 3.58 to 7.29 nm.
begins to increase when the length is larger than 5.3 nm, indicating that the effect of tube length becomes larger when L > 5.3 nm, while such dependency is insensitive outside this range. Using the polynomial fitting technique for the MD results in Fig. 7, the critical buckling temperature can be described as T cr ¼ c1 þ c2 L þ c3 L2 þ c4 L3 þ c5 L4
2619
ones. This shows that the nanotubes arranged in armchair chirality need even more strain energy for occurrence of thermal buckling and have higher buckling temperature as well. It is noted that the linear relationship between potential energy and the temperature exists for all CNTs considered in the present study. To give further understanding of the effect of structural geometries on thermal bucking, size-dependent rigidity in longitudinal direction of SWCNTs should be taken into account. Many previous experimental and theoretical studies [10,24–27] have observed that the Young’s modulus of SWCNTs decreases as the radius increases. In particular, there is a strong dependence of the elastic modulus on the radius for a nanotube when R < 0.6 nm [27]. For pairs of tubes of similar dimensions but different helicity, the effective Young’s modulus of armchair nanotubes is higher than that of zigzag tubes with nearly equal dimensions, and the difference becomes larger if the nanotube radius is smaller than 0.6 nm [24,27]. It is readily to recognize that a nanotube with higher longitudinal rigidity has a stronger resistance to thermal buckling, which is in consistent with our MD results. 4. Conclusions
ð3Þ
where Tcr and length L are in units of K and nm, respectively. We obtain the values of the coefficients as c1 = 5434.19 K, c2 = 6220.84 K nm1, c3 = 1903.22 K nm2, c4 = 241.04 K nm3, and c5 = 10.98 K nm4, respectively. To provide a better explanation of this size-dependent behavior shown in Figs. 5 and 7, we first examine the effect of geometric dimensions on the mechanical properties of SWCNTs from the view point of energy. It has been shown in Fig. 1b that the total energy per atom varies linearly with respect to the temperature to reach certain critical value which increases as the load ratio parameter decreases for a given tube. On the other hand, with an identical kp, it is found that the potential energies corresponding to the critical point vary from case to case for different nanotubes due to their different structural geometries. Denote ecr as the critical strain energy where the strain energy is defined as the differences in the potential energy per atom of heated and unheated tube (i.e., the beginning nanotube structure in MD simulations). As kp is equal to 0.8, the value of ecr is predicted to increase when the radius decreases for armchair CNTs. That is, relatively more strain energy is required for nanotube with smaller radius to change the tube from cylindrical state to buckling state. This provides a good explanation for the relationship between Tcr and R in Fig. 5 for armchair tubes. As expected, the critical strain energy drops off monotonically as the tube length increases, which is in accordance with the fact shown in Fig. 7 that the longer the nanotube, the smaller the critical buckling temperature. By comparing armchair and zigzag SWCNTs with identical dimensions, it is found that the value of ecr of armchair tubes is larger than that of zigzag
Comprehensive MD simulations are carried out to investigate thermal buckling behavior of SWCNTs under the simultaneous actions of constant axial compression and uniformly increasing temperature. The nanotube is found to experience a sudden structural change from cylindrical tube to buckling configuration as the temperature is beyond certain critical value. Simulation results show that the zigzag carbon nanotube has less resistance to thermal buckling compared to the armchair tube with similar geometries. The critical bucking temperature is found to decrease as the nanotube length increases and the dependency becomes insensitive for longer CNTs. For both armchair and zigzag nanotubes with relatively large radius, the critical buckling temperature decreases with increasing in tube radius. Remarkably, the zigzag nanotube is more susceptible to be impacted by the effect of high temperature, thereby leading to much lower buckling temperature than expected as the tube radius becomes small. Acknowledgement This work is supported in part by the National Natural Science Foundation of China under Grant 50375091. The authors are grateful for this financial support. References [1] Hadjiev VG, Live MN, Arepalli S, Nikolaev P, Files BS. Raman scattering test of single-wall carbon nanotube composites. Appl Phys Lett 2001;78:3193–5. [2] Tans SJ, Verschueren ARM, Dekker C. Room temperature transistor based on a single carbon nanotube. Nature 1998;393:49–52.
2620
C.-L. Zhang, H.-S. Shen / Carbon 45 (2007) 2614–2620
[3] Dai H, Hafner JH, Rinzler AG, Colbert DT, Smalley RE. Nanotubes as nanoprobes in scanning probe microscopy. Nature 1996;384: 147–50. [4] Kim P, Lieber CM. Nanotube nanotweezers. Science 1999;286: 2148–50. [5] Iijima S, Brabec C, Maiti A, Bernholc J. Structural flexibility of carbon nanotubes. J Chem Phys 1996;104:2089–92. [6] Wong EW, Sheehan PE, Lieber CM. Nanobeam mechanics: elasticity, strength, and toughness of nanorods and nanotubes. Science 1997;277:1971–5. [7] Waters JF, Riester L, Jouzi M, Guduru PR, Xu JM. Buckling instabilities in multiwalled carbon nanotubes under uniaxial compression. Appl Phys Lett 2004;85:1787–9. [8] Poncharal P, Wang ZL, Ugarte D, de Heer WA. Electrostatic deflections and electromechanical resonances of carbon nanotubes. Science 1999;283:1513–6. [9] Postma HWCh, de Jonge M, Yao Z, Dekker C. Electrical transport through carbon nanotube junctions created by mechanical manipulation. Phys Rev B 2000;62:R10653–6. [10] Treacy MMJ, Ebbesen TW, Gibson JM. Exceptionally high Young’s modulus observed for individual carbon nanotubes. Nature 1996;381: 678–80. [11] Tang DS, Chen LC, Wang LJ, Sun LF, Liu ZQ, Wang G, et al. Behavior of carbon nanotubes under high pressure and high temperature. J Mater Res 2000;15:560–3. [12] Ni B, Sinnott SB, Mikulski PT, Harrison JA. Compression of carbon nanotubes filled with C60, CH4, or Ne: predictions from molecular dynamics simulations. Phys Rev Lett 2002;88. Art. no. 205505. [13] Zhang CL, Shen HS. Buckling and postbuckling analysis of singlewalled carbon nanotubes in thermal environments via molecular dynamics simulation. Carbon 2006;44:2608–16. [14] Fagan SB, Mota R, Baierle RJ, Paiva G, Silva AJR, Fazzio A. Stability investigation and thermal behavior of a hypothetical silicon nanotube. J Molecular Struct 2001;539:101–6. [15] de Heer WA, Ugarte D. Carbon onions produced by heat treatment of carbon soot and their relation to the 217.5 nm interstellar absorption feature. Chem Phys Lett 1993;207:480–6.
[16] Kim YA, Muramatsu H, Hayashi T, Endo M, Terrones M, Dresselhaus MS. Thermal stability and structural changes of double-walled carbon nanotubes by heat treatment. Chem Phys Lett 2004;398:87–92. [17] Lo´pez MJ, Cabria I, March NH, Alonso JA. Structural and thermal stability of narrow and short carbon nanotubes and nanostrips. Carbon 2005;43:1371–7. [18] Brenner DW, Shenderova OA, Harrison JA, Stuart SJ, Ni B, Sinnott SB. A second-generation reactive empirical bond order (REBO) potential energy expression for hydrocarbons. J Phys: Condens Mat 2002;14:783–802. [19] Hoover WG. Canonical dynamics: equilibrium phase-space distributions. Phys Rev A 1985;31:1695–7. [20] Nardelli MB, Yakobson BI, Bernholc J. Mechanism of strain release in carbon nanotubes. Phys Rev B 1998;57:R4277–80. [21] Mylvaganam K, Zhang LC. Important issues in a molecular dynamics simulation for characterizing the mechanical properties of carbon nanotubes. Carbon 2004;42:2025–32. [22] Yakobson BI, Brabec CJ, Bernholc J. Nanomechanics of carbon tubes: instabilities beyond linear response. Phys Rev Lett 1996;76: 2511–4. [23] Zhang YY, Tan VBC, Wang CM. Effect of strain rate on the buckling behavior of single- and double-walled carbon nanotubes. Carbon 2007;45:514–23. [24] Zhang CL, Shen HS. Temperature-dependent elastic properties of single-walled carbon nanotubes: prediction from molecular dynamics simulation. Appl Phys Lett 2006;89. Art. no. 081904. [25] Volodin A, Ahlskog M, Seynaeve E, Haesendonck CV. Imaging the elastic properties of coiled carbon nanotubes with atomic force microscopy. Phys Rev Lett 2000;84:3342–5. [26] Natsuki T, Tantrakarn K, Endo M. Effects of carbon nanotube structures on mechanical properties. Appl Phys A – Mater Sci Proc 2004;79:117–24. [27] Zhang HW, Wang JB, Guo X. Predicting the elastic properties of single-walled carbon nanotubes. J Mech Phys Solids 2005;53: 1929–50.