Temperature-dependent elastic properties of single

APPLIED PHYSICS LETTERS 89, 081904 共2006兲

Temperature-dependent elastic properties of single-walled carbon nanotubes: Prediction from molecular dynamics simulation Chen-Li Zhang and Hui-Shen Shena兲 Department of Engineering Mechanics, Shanghai Jiao Tong University, Shanghai 200030, People’s Republic of China

共Received 18 March 2006; accepted 27 June 2006; published online 21 August 2006兲 The authors report here a method of determining the mechanical properties of single-walled carbon nanotubes by direct measurement from molecular dynamics simulation test. The authors find that single-walled carbon nanotubes exhibit obvious anisotropic, temperature-dependent properties. The value of Young’s modulus decreases with increase in temperature, whereas the shear modulus increases when the temperature is less than 700 K and remains almost constant when the temperature is greater than 700 K. By direct buckling measuring, the authors obtain the effective wall thickness of nanotubes and find that the effective wall thickness of zigzag nanotubes is larger than that of armchair nanotubes. © 2006 American Institute of Physics. 关DOI: 10.1063/1.2336622兴 The discovery of carbon nanotubes 共CNTs兲 has opened up quite a few opportunities for producing unique carbonbased materials. Due to their outstanding properties, nanotubes have attracted growing interest and are considered to be the most promising materials of applications in nanoengineering.1–3 The CNT-based nanocomposite devices may withstand high temperature during manufacture and operation. This makes it important to have a good knowledge of the thermal properties of nanotubes. Treacy et al.4 carried out the measurement of the thermally inducing vibration amplitude of cantilevered multiwalled carbon nanotubes in a transmission electron microscope. Lasjaunias5 studied thermal transport as well as heat capacity and adsorption of CNTs. An analytical method based on many-body interatomic potential was developed by Jiang et al.6 to explore the coefficient of thermal expansion for single-walled carbon nanotubes 共SWCNTs兲. These studies4–6 show that the physical property of carbon nanotubes depends strongly on temperature, from which we believe that the elastic constants of nanotubes, such as Young’s modulus and shear modulus, are also temperature dependent. However, it is remarkably difficult to directly measure the mechanical properties of individual SWCNTs in the experiment due to their extremely small size. We report here a method of determining the mechanical properties of SWCNTs by direct measurement from molecular dynamics simulation 共MDS兲 test. SWCNTs subjected to axial compression and torsion are simulated under temperature varying from 100 to 1200 K, and subjected to lateral pressure at 300 K only. The interaction between carbon atoms is calculated by using the many-body reactive empirical bond order 共REBO兲 potential,7 which can realistically describe the bond energies, lengths, and force constants for solid states of carbon molecules and has been widely used to predict the mechanical properties of CNTs.8,9 By using the adaptive intermolecular REBO potential, Ni et al.10 examined the buckling behavior of SWCNTs under axial compression at 100, 600, and 1500 K, and concluded that buckling load of the empty nanotube decreases as the temperature increases, whereas filling the carbon nanotubes with fullerenes a兲

Author to whom correspondence should be addressed; electronic mail: [email protected]

or gas molecules disrupts the temperature effect. Also, Trotter et al.11 examined the compressibility of filled and empty CNTs at 300 K and found that the butane-filled system has a unique yielding behavior prior to buckling. To account for the thermal effect, we use the Nose-Hoover thermostat12,13 to maintain the temperature of the system. In this thermostat scheme, the information of kinetic energy is fed back into the equations of motion so that the kinetic energy is kept constant by controlling the thermostatting force to dissipate excess heat. The temperature fluctuation is limited to 20– 30 K during the MDS test. In the framework of MDS, the nanotube can be considered as a congeries of individual atoms. The integration of Newtonian dynamics function is used to determine the variation of the instantaneous location and velocity of each atom. The process in our program is run by four steps: 共1兲 to determine the initial locations and velocities of all the atoms, 共2兲 to optimize and to relax full structure to ensure the equilibrium geometry being local minimum on the potential energy surfaces, 共3兲 to apply external load iteratively with the appropriate constraints, and 共4兲 to calculate deformations by using statistical physics techniques. Two pairs of SWCNTs with similar radius but different helicities are considered, i.e., 共10, 10兲 and 共17, 0兲 tubes with radius being 0.68 nm and 共12, 12兲 and 共21, 0兲 tubes having radius of 0.82 nm. Fixed boundary condition is assumed to be at one end of the tube, and axial compressive force P or torque T is applied on the other end with the appropriate constraints. The postbuckling behavior of armchair 共12, 12兲 tube and zigzag 共21, 0兲 tube subjected to axial compression and torsion at 300 K is plotted in Fig. 1. It is noted that the SWCNT exhibits initial linear elastic deformations almost up to the point of bucking under these two loading cases. In the sense of continuum mechanics, the SWCNT can be modeled as an elastic shell. It has been shown8,14 that most carbon nanotubes have low values of radius-tothickness ratio. As a result, the continuum mechanics model for carbon nanotubes requires the use of shear deformation shell theory. In the present letter, based on higher order shear deformation shell theory, an individual orthotropic shell model is adopted for SWCNT. Therefore, six independent moduli are needed to completely describe the elastic behavior, which are denoted by E11, G12, G13, G23, ␯12, and ␯21, respectively, where E11 is the longitudinal Young’s modulus,

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081904-2

Appl. Phys. Lett. 89, 081904 共2006兲

C.-L. Zhang and H.-S. Shen

FIG. 2. 共Color online兲 Variation of the surface Young’s modulus with temperature.

FIG. 1. 共Color online兲 Postbuckling behavior of SWCNTs under 共a兲 axial compression, and 共b兲 torsion at 300 K.

G12, G13, and G23 are shear moduli, and ␯12 and ␯21 are Poisson’s ratios. The stretching rigidity and flexural rigidity of an individual orthotropic shell can be expressed as E11h3 , 共1兲 C = E11h, D11 = 12共1 − ␯12␯12兲 where h is the effective wall thickness of the shell, and the Poisson’s radio can be measured by ␧22 ␯12 = − , 共2兲 ␧11 where ␧11 and ␧22 are the elastic strain in the axial and circumferential directions, respectively. From Eq. 共1兲, it can be seen that if the effective wall thickness h can be known a priori, the longitudinal Young’s modulus E11 and shear modulus G12 can then be determined by a so-called force approach14 as E11 =

P , A0␧11

G12 =

T , J 0␾

共3兲

where A0 is the area of cross section, J0 stands for the crosssectional polar inertia, and ␾ is torsion angle per unit length of the SWCNT. Since A0 and J0 are functions of h, we believe that the accurate predictions of elastic moduli depend very much on the crucial choice of the effective wall thickness h. Although several values varying from 0.066 to 0.34 nm have been adopted,8,14–17 no satisfactory explanation for these values could be found in the open literature. In the MDS test, the flexural rigidity D is hardly to be measured, but the stretching rigidity C can be, which is called surface Young’s modulus, and is found to be about 0.36 TPa nm by Yakobson et al.8 Several values of surface Young’s modulus have been reported. For example, Hernandez et al.18 gave the values for the surface Young’s modulus ranging from 0.31 to 0.43 TPa nm by using tight binding simulations. Sanchez-Portal et al.19 showed the surface Young’s modulus

values from 0.33 to 0.37 TPa nm by using ab initio calculations. From our measurement of the MDS test, the variation of the surface Young’s modulus versus the temperature is shown in Fig. 2. It can be seen that the surface Young’s modulus is chiral and size dependent and the trend is similar for both armchair and zigzag SWCNTs. The surface Young’s modulus of armchair nanotubes is more sensitive to temperature changes than that of zigzag nanotubes. When the temperature increases from 100 to 1200 K, a maximum reduction of the surface Young’s modulus for an armchair SWCNT is more than 9%, while the maximum relative difference is about 3% for zigzag tubes. At the same temperature, the surface Young’s modulus of armchair tubes is larger than that of zigzag ones. Moreover, the value of the surface Young’s modulus decreases with an increase in radii for both armchair and zigzag nanotubes. Similar trend has been reported by Natsuki et al.20,21 and Zhang et al.22 previously. Figure 3 shows the effect of temperature changes on the surface shear modulus, which is similarly defined as Gs = G12h. It can be seen that the surface shear modulus increases rapidly as the temperature increases from 100 to 700 K and then remains almost constant when the temperature is greater than 700 K. It is interesting to note that the shear modulus of zigzag tubes seems to depend more strongly on temperature as compared to armchair tubes, which is not in accordance with the characteristics of the surface Young’s modulus, as shown in Fig. 2. It can also be found that the surface shear modulus of armchair nanotubes is larger than that of zigzag tubes and the effect of nanotube radius is not significant. As argued before elastic moduli of SWCNTs depend strongly on the effective wall thickness h. For example, from the MDS results Yakobson et al.8 gave E = 5.5 TPa and h = 0.066 nm with Poisson’s ratios ␯ = 0.19. By using ab initio

FIG. 3. 共Color online兲 Variation of the surface shear modulus with temperature. Downloaded 02 Sep 2006 to 202.120.10.73. Redistribution subject to AIP license or copyright, see http://apl.aip.org/apl/copyright.jsp

081904-3

Appl. Phys. Lett. 89, 081904 共2006兲

C.-L. Zhang and H.-S. Shen

TABLE I. Comparison of Poisson’s ratios and wall thickness for armchair and zigzag SWCNTs at 300 K. Nanotube

␯12

␯21

h 共nm兲

共10,10兲 共12,12兲 共17,0兲 共21,0兲

0.18 0.17 0.27 0.28

0.22 0.21 0.17 0.19

0.067 0.067 0.088 0.087

共10, 10兲 tube

method, Kudin et al.17 obtained E = 3.895 TPa and h = 0.0894 nm with ␯ = 0.149. By using an effective continuum/finite element approach, Pantano et al.23 obtained E = 4.84 TPa and h = 0.075 nm with ␯ = 0.19. On the other hand, Halicioglu24 gave ␯ = 0.18 and E = 0.5 TPa, for which the effective wall thickness h = 0.68 nm, which is more than ten times of that of Yakobson et al.8 Consequently, a key issue here is that the wall thickness needs to be specified in order to predict the values of Young’s modulus and shear modulus. According to higher order shear deformation shell theory, and by taking G12 = G13 = G23 and ␯12E22 = ␯21E11, the compressive buckling load may be expressed as Pcr = ␭cr2␲E11h2

冋

␯21 3␯12共1 − ␯12␯21兲

册

1/2

,

TABLE II. Variations of Young’s and shear moduli 共in TPa兲 for armchair and zigzag SWCNTs in thermal environments.

共4兲

where ␭cr is a nondimensional parameter, which may be called the knockdown factor, and the detail of which can be found in Ref. 25. By direct measuring the buckling load of a SWCNT as shown in Fig. 1, and together with the surface Young’s and shear moduli calculated above, the effective wall thickness h can be determined uniquely. Typical results are listed in Table I for these two pairs of SWCNTs at 300 K. Note that Poisson’s ratio ␯12 is measured from axial compression and ␯21 is measured from lateral pressure simulation test. It can be found that the tube thickness is different for armchair and zigzag nanotubes even though their length and radius are almost the same. The wall thickness of two armchair tubes is 0.067 nm, which agrees well with the value of 0.066 nm for 共7, 7兲 tube obtained by Yakobson et al.8 It is noted that the results for 共n , 0兲 tubes are about 30% larger than armchair tubes. This is because both axial buckling load and critical strain for zigzag tubes are higher than those of armchair tubes despite the smaller Young’s modulus possessed by the zigzag nanotube, as shown in Fig. 1. It should be mentioned that the widely used value of 0.34 nm for tube wall thickness is thoroughly inappropriate to SWCNTs, as previously reported by Pantano et al.23 From Table I, it can also be found that Poisson’s ratio depends dramatically on the tube chirality. The values of ␯12 for 共n , n兲 tubes are smaller than those of 共n , 0兲 tubes. Similar trend has been predicted by different methods 共see, for example, Refs. 17 and 20兲. Again, the values of ␯12 for armchair tubes are close to that of 0.19 for 共7, 7兲 tube obtained by Yakobson et al.8 It is worth to note that since ␯12 is less than ␯21, the circumferential Young’s modulus E22 should be greater than the longitudinal Young’s modulus E11 for armchair tubes, which is in agreement with the conclusion drawn by Jin and Yuan.14 In contrast, for the zigzag tubes we still have E11 ⬎ E22. Although extensive researches concentrated on the evaluation of elastic properties of SWCNTs,8,14–16,20–22 there

共12, 12兲 tube

共17, 0兲 tube

共21, 0兲 tube

Temperature 共K兲

E11

G12

E11

G12

E11

G12

E11

G12

300 500 700 1000 1200

5.65 5.53 5.47 5.28 5.24

1.94 1.96 1.96 1.95 1.94

5.53 5.38 5.34 5.22 5.06

1.90 1.94 1.95 1.96 1.96

3.90 3.89 3.86 3.81 3.79

1.36 1.39 1.42 1.42 1.42

3.81 3.79 3.78 3.75 3.73

1.37 1.40 1.41 1.43 1.43

exists no report on the variation of these elastic moduli in thermal environments. Our MDS test results show that the buckling loads are reduced with increases in temperature, which confirms the finding of Ni et al.10 for an empty nanotube. Table II gives the longitudinal Young’s modulus E11 and shear modulus G12 over the temperature ranging from 300 to 1200 K for both armchair and zigzag nanotubes. We find that the predicted values of Young’s modulus and shear modulus are essentially influenced by the temperature, while they show different trends with the variation of environmental temperature. As expected, the Young’s modulus E11 is reduced as temperature is increased. The shear moduli vary from 1.90 to 1.96 TPa for 共n , n兲 tubes and from 1.36 to 1.43 TPa for 共n , 0兲 tubes, respectively. This work is supported in part by the National Natural Science Foundation of China under Grant No. 50375091. The authors are grateful for this financial support. 1

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