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Elastic moduli of multi-walled carbon nanotubes and the effect of van der Waals forces
Composites Science and Technology 63 (2003) 1517–1524 www.elsevier.com/locate/compscitech
Elastic moduli of multi-walled carbon nanotubes and the effect of van der Waals forces Chunyu Li, Tsu-Wei Chou* Department of Mechanical Engineering, University of Delaware, Newark, DE 19716, USA Received 4 November 2002; received in revised form 19 December 2002; accepted 10 January 2003
Abstract This paper reports a study of the elastic behavior of multi-walled carbon nanotubes (MWCNTs). The nested individual layers of an MWCNT are treated as single-walled frame-like structures and simulated by the molecular structural mechanics method. The interlayer van der Waals forces are represented by Lennard–Jones potential and simulated by a nonlinear truss rod model. The computational results show that the Young’s moduli and shear moduli of MWCNTs are in the ranges of 1.05 0.05 and 0.40 0.05 TPa, respectively. Results indicate that the tube diameter, tube chirality and number of tube layers have some noticeable effects on the elastic properties of MWCNTs. Furthermore, it has been demonstrated that the inner layers of an MWCNT can be effectively deformed only through the direct application of tensile or shear forces, not through van der Waals interactions. # 2003 Elsevier Ltd. All rights reserved. Keywords: A. Nanostructures; B. Mechanical properties; C. Computational simulation; Carbon nanotube; Atomistic modelling; Nanomechanics
1. Introduction Since the discovery of multi-walled carbon nanotubes (MWCNT) in 1991 [1], researchers worldwide have engaged in fundamental studies of this novel material and have investigated the potential of its technological applications, including carbon nanotube based composites [2]. The unique characteristics of carbon nanotubes, such as high stiffness, high aspect ratio and low density, are particularly desirable for this new generation of composites [3]. In order to fully explore the potential of carbon nanotubes for application in composites, a thorough understanding of the elastic properties of carbon nanotubes is necessary. The first experimental work on the Young’s modulus of MWCNTs was performed by Treacy and co-workers [4]. They used a transmission electron microscope to measure the mean-square vibration amplitudes of MWCNTs and deduced that the average value of the Young’s modulus is 1.8 0.9 TPa. Wong et al. [5] and
* Corresponding author. Tel.: +1-302-831-2423; fax: +1-302-8313619. E-mail address: [email protected] (T.-W. Chou). 0266-3538/03/$ - see front matter # 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0266-3538(03)00072-1
Salvetat et al. [6] used the technique of atomic force microscope and found the average Young’s modulus of MWCNTs to be 1.28 0.59 and 0.81 0.41 TPa. Recently, Yu et al. [7] also measured the tensile properties of MWCNTs using atomic force microscopy and reported Young’s modulus values ranging from 270 to 950 GPa. Due to the difficulties in experimental characterization of nanotubes, computer simulation has been regarded as a powerful tool for modeling the properties of nanotubes. Among the available modeling techniques [8–14, 28], molecular dynamics simulation has been used most extensively. However, molecular dynamics simulations need to consider the thermal vibration of atoms, and thus the time step is usually in the order of femtosecond. It is not efficient for long time or static problems such as simulating stiffness and strength properties. Being very time consuming, the application of molecular dynamics simulation to MWCNTs is very limited [8]. As for theoretical predictions, Lu [12] studied the elastic properties of MWCNTs using the empirical lattice dynamics model and reported a Young’s modulus of about 1.0 TPa and a shear modulus of slightly less than 0.5 TPa. In these studies [5–7,12], the cross-sectional areas of MWNTs are calculated by assuming that
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both the single layer wall thickness as well as the interlayer spacing are 0.34 nm. From the atomistic structural viewpoint, an MWCNT can be regarded as nested SWCNTs. The interactions among these SWCNTs are mainly due to the van der Waals forces. Compared with the covalent bonds between the neighboring carbon atoms in an SWCNT, the van der Waals interaction is rather weak. Wong et al. [5] predicted that relative slippage between adjacent layers in an MWCNT may occur due to the weak binding. This prediction was verified by observations of inner layer pullout in experiments conducted by Yu et al. [7] and Cumings and Zettl [16]. Ru [28] treated double-walled nanotubes as coaxial shells with van der Waals forces between the shells and studied their buckling under axial compression. However, it is not clear from these studies what the effects of van der Waals interactions are on the elastic properties of MWCNTs. Recently, the authors proposed a molecular structural mechanics approach for modeling single-walled carbon nanotubes [15]. Fundamental to this approach is the notion that a carbon nanotube can be regarded as a frame-like structure and the primary bonds between two nearest-neighboring atoms can be considered as loadbearing beam members, whereas an individual atom acts as the joint of the related load-bearing beam members. Establishment of a linkage between structural mechanics and molecular mechanics enables the sectional property parameters of these beam members to be obtained. The accuracy and stability of the present method have been verified through its application to several single layer graphene sheets with different specimen sizes. The computational results of the elastic properties of SWCNTs are also in good agreement with existing experimental data and theoretical predictions. In this paper, the molecular structural mechanics method is extended to model the elastic behavior of MWCNTs. The individual tube layer is still simulated as a frame-like structure as in Ref. [15]. Then, a truss rod model is proposed to simulate the interlayer van der Waals forces. The objectives of the present work are to predict the Young’s moduli and shear moduli of MWCNTs and to examine the effects of nanotube structures, such as the number of tube layers, tube diameter and tube chirality on elastic properties. A comparison between the elastic behavior of single-walled and multi-walled carbon nanotubes has also been made in light of the effect of van der Waals forces.
Fig. 1. A multi-walled carbon nanotube.
coefficients (n, m) of the lattice vectors a1 and a2 in the chiral vector Ch ¼ na1 þ ma2 is usually used to describe the chirality of the layer. The index (n, 0) denotes zigzag type nanotubes and (n, n) for armchair type nanotubes. Since the nested layers are structurally independent of one another, the chirality of the layers may be different. The distance between two neighboring layers is assumed to be the same as the spacing between adjacent graphene sheets in graphite, i.e., 0.34 nm [17] (Fig. 2). Each carbon atom within the atomic layer of a graphene sheet is covalently bonded to three neighboring carbon atoms. Three sp2 orbitals on each carbon form s-bonds to three other carbon atoms. One 2p orbital remains unhybridized on each carbon; these orbitals perpendicular to the plane of the carbon ring combine to form the p-bonding network [25]. The atomic interactions between the neighboring layers are the van der Waals forces.
2. Multi-walled carbon nanotubes An MWCNT is composed of a set of coaxially situated SWCNTs of different radii, as shown in Fig. 1. Each nested cylindrical layer of an MWCNT can be visualized as a rolled-up graphene sheet. A pair of
Fig. 2. Inter-graphene layer spacing in graphite.
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3. Extension of molecular structural mechanics model The molecular structural mechanics approach was first proposed by the authors for modeling SWCNTs [15]. Its main ideas include simulating a nanoscopic SWCNT as a macroscopic frame-like structure and then establishing the linkage between computational chemistry and computational structural mechanics approaches to obtain the required parameters for structural analysis. However, the approach proposed in Ref. [15] is suitable only for simulations of SWCNTs, and it does not consider the interaction between not covalently bonded atoms. In the following, the molecular structural mechanics method is extended to treat the elastic behavior of MWCNTs by taking into account the van der Waals forces acting between the neighboring tube layers. 3.1. Simulations of SWCNTs At the atomistic scale, carbon atoms in the hexagonal array of an SWCNT are bonded to each other by covalent bonds. These covalent bonds have their characteristic bond lengths and bond angles in a three-dimensional space. When a nanotube is subjected to external forces, the displacements of the individual atoms are constrained by these bonds, and the corresponding deformation of the nanotube is the result of the bond interactions. By considering the geometrical similarity between a space frame structure and a carbon nanotube, we treat the covalent bonds as connecting beam elements between carbon atoms and the carbon atoms as joints of the beam elements. Fig. 3 depicts this concept for an SWCNT. Based on the energy equivalence, a linkage between the molecular force field constants in computational chemistry and the cross-sectional properties of beam
Fig. 3. An SWCNT simulated as a frame-like structure.
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elements in computational structural mechanics has been established in Ref. [15], and is briefly recapitulated below. The direct relationships between these parameters are as follows: EA ¼ kr ; L
EI ¼ k ; L
GJ ¼ k L
ð1Þ
where L is the bond length, EA, EI and GJ are the crosssectional properties of a beam element in tension or compression, bending and torsion, respectively; kr , k and k are the molecular force field constants of an atomic bond in stretching, bending, and torsion, respectively. Eq. (1) forms the basis of the molecular structural mechanics model, which is applicable to all covalent molecular systems, provided that the force constants kr , k and k are known. The main steps of the molecular structural mechanics approach are as follows. First, take atoms as joints and bonds as beams in the equivalent frame-like structure. Second, determine beam sectional parameters from molecular force fields constants [Eq. (1)]. Third, establish elemental stiffness matrices and elemental load vectors and then assemble the global stiffness matrix and the global load vector. Last, solve the displacements of atoms and then compute the elastic properties of SWCNTs. 3.2. van der Waals forces between neighboring tube layers The multiple cylindrical layers of an MWCNT are held together through van der Waals forces. The van der Waals force is a non-bonded interaction, and it can be an attraction force or a repulsion force. The attraction occurs when a pair of atoms approach each other within a certain distance. The repulsion occurs when the distance between the interacting atoms becomes less than the sum of their contact radii. These interactions are often modeled using the general Lennard–Jones ‘‘6–12’’ potential [18], which provides for a smooth transition between the attraction and repulsion regions. The general Lennard–Jones ‘‘6–12’’ (LJ) potential is commonly expressed as 12 6 UðrÞ ¼ 4" ð2Þ r r where, r is the distance between interacting atoms, " and are the Lennard–Jones parameters. For carbon atoms the Lennard–Jones parameters are "=0.0556 kcal/mole and =3.4 A˚ [19]. The potential U(r) is usually truncated at an interatomic distance of 2.5 without a significant loss of accuracy, i.e., no interactions are evaluated beyond this distance. Based on the Lennard–Jones potential, the van der Waals force between interacting atoms can then be written as
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dUðrÞ " 13 7 ¼ 24 2 FðrÞ ¼ dr r r
ð3Þ
The variations of the Lennard–Jones potential and the van der Waals force with the distance between two interacting atoms are illustrated in Fig. 4. 3.3. Simulations of van der Waals forces The van der Waals force acting along the connecting line between two interacting atoms is simulated by a truss rod that connects the two interacting atoms with rotatable end joints (Fig. 5). The truss rod, thus, transmits only tensile or compressive forces as given by Eq. (3). In an MWCNT, an atom situated in one of these layers may form interacting pairs with several atoms in other layers as long as the distance between the pair of atoms is less than 2.5 . For convenience in computation, we consider the van der Waals interactions only between the nearest neighboring layers (Fig. 5). Fig. 4 clearly indicates that the van der Waals force between two atoms is highly nonlinear. Therefore, the truss rod connecting the atoms is a nonlinear element with its load–displacement relationship characterized by the van der Waals force. However, the load–displacement curve of the truss rod (Fig. 6) is opposite to that of the van der Waals force-interatomic distance relation displayed in Fig. 4. The reason is that the van der Waals force is an intrinsic force between the interacting atoms, while Fig. 6 gives the load-displacement relation when a load is applied to the truss rod to overcome the van der Waals force. The load–displacement curve of Fig. 6 consists of two distinct stages in loading and unloading. This characteristic of load-displacement curve brings about some numerical difficulties in the simulation of the nonlinear
Fig. 4. Lennard–Jones potential and van der Waals force versus atomic distance.
behavior of the truss rod. Thus the selection of a suitable nonlinear analysis method is needed. The available methods include arc-length method [26], work control method [27] and generalized displacement control method [20]. In this paper, for simulating the nonlinearity of the truss rod, we adopt the generalized displacement control method [20], which is essentially similar to the arc-length method. But the parameters in the constraint equations of these two methods are different. It is formulated in N+1 dimensional space that includes one load parameter and N displacements as the unknowns. A constraint equation is added to the N equations of equilibrium and it can be justified from the bounded nature of the load parameter. A general stiffness parameter, which is updated with the displacements in iterative procedures, is used for adjusting the incremental step size, and self-adaptively changing loading directions. This method has been demonstrated to be accurate and numerically stable at the critical and snapback points. The results of simulation of the relation between the load for overcoming the van der Waals force between two neighboring atoms and the displacement of a truss
Fig. 5. Truss rods for simulating van der Waals forces.
Fig. 6. Load-displacement curve of the nonlinear truss rod.
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rod, based on the generalized displacement control method, is given in Fig. 6. Three iterations are usually required in each load increment, and the accuracy achieved by these iterations is always up to 0.0001.
4. Results and discussions The objective of this paper is to apply the molecular structural mechanics model to predict the elastic properties of multiwalled carbon nanotubes. The computational model configurations and load conditions are illustrated in Fig. 7. The Young’s modulus and shear modulus are calculated by simulating the nanotube tensile and torsional deformation, respectively. For simplicity, the nanotube end caps are not considered. Loads are applied directly and uniformly on the end atoms. 4.1. Young’s modulus of graphite The applicability and accuracy of the molecular structural mechanics model for simulation of graphene sheets and SWCNTs have been examined in Ref. [15]. Combining the space frame model with the present model for van der Waals forces, we first simulate the multi-layered graphite because the measured elastic modulus of graphite is available. The values of force constants, 12 kr ¼ 469 (kcal/mol)/A˚2 and 12 k ¼ 63 (kcal/ mol)/rad2 [21], are adopted, and the initial carbon–carbon bond length is taken as 1.421 A˚ [22]. The thickness of a graphene sheet and the interspacing between graphene layers are assumed to be 0.34 nm. The computational results of Young’s moduli for multi-layered graphite compared with that of a graphene sheet are shown in Fig. 8. There is a negligible difference between the elastic modulus of multi-layered graphite and that of a graphene sheet. The computational results ( 1.05 TPa) based on the present model are in excellent agreement with the experimental data (1.025 TPa) [17].
Fig. 7. Computational model configurations.
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4.2. Young’s moduli of MWCNTs Having established its validity for graphite, the present approach is then applied to calculating the Young’s moduli of MWCNTs. Several approaches exist in the literature for calculating the Young’s modulus of carbon nanotubes. These include the bending momentdeflection method [6], the vibration spectrum method [4,23] and the second derivative of energy method [10,11]. In this paper, we follow the original definition of Young’s modulus and simulate the deformations of carbon nanotubes under axial tension. The Young’s modulus is then expressed as Y¼
P=A0 DL=L0
ð4Þ
where P stands for the total force acting on the atoms at one end of the nanotube, A0 ¼ 4 ½ðdo þ 3:4Þ2 ðdi 3:4Þ2 A˚2 represents the cross-sectional area of the multiwalled tube with do (A˚) as the outermost tube diameter and di (A˚) as the innermost tube diameter, L0 is the initial length of tube and DL is the elongation of tube under the force P. The thickness of each tube layer is taken as the interlayer spacing of graphite, 3.4 A˚ [17]. The simulation focuses first on two-layer MWCNTs. It is assumed that both layers may be either armchair chirality or zigzag chirality. As in the simulation of graphite, the force constants kr and k are also chosen as 1 1 ˚2 2 kr ¼ 469 (kcal/mole)/A and 2 k ¼ 63 (kcal/mole)/ 2 rad . In addition, the force constant k is adopted as 1 2 2 k ¼ 20 (kcal/mole)/rad based on Refs. [21] and [24]. Although there is no benchmark data guiding determination of the force constant k , the present calculation indicates that the Young’s modulus of carbon nanotubes is insensitive to the variation in k . Fig. 9 displays the variation of the Young’s moduli of two-layer MWCNTs with nanotube outer-layer diameter. There is a slight difference between the trends of
Fig. 8. Young’s modulus of graphite.
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variation in Young’s moduli of MWCNTs and SWCNTs [15]. For SWCNTs, the Young’s modulus is almost constant when the tube diameter is larger than 1.0 nm. But for MWCNTS, there is a small variation, and the difference between maximum and minimum Young’s modulus values is about 0.15 TPa. Fig. 9 also shows that the effect of tube chirality is very weak. The average Young’s moduli are 1.05 0.05 TPa and 1.08 0.02 TPa for armchair and zigzag MWCNTs, respectively. A general conclusion that can be drawn from Fig. 9 is that the Young’s moduli of multiwalled carbon nanotubes are generally higher ( 7% in maximum) than those of single-walled carbon nanotubes. Thus, the van der Waals force has some effect on the tensile modulus of MWCNTs. Fig. 10 illustrates the effect of the number of layers in MWCNTs on the Young’s modulus. For comparison purpose, the outermost tube diameters are taken as
roughly the same for zigzag and armchair MWNTs. The overall effects of both tube layers and chirality are noticeable. The present approach gives slightly higher values of Young’s moduli than those reported in [12] by Lu, who predicted that the Young’s modulus of MWCNTs is about 0.97 TPa and is independent of the number of tube layers, tube chirality and tube diameter. Regarding the experimental results of Young’s moduli of MWCNTs, Wong et al. [5] and Salvetat et al. [6] reported values of 1.28 0.59 and 0.81 0.41 TPa, respectively, using AFM-based experiments. These data are comparable to the present prediction of 1.05 0.05 TPa. 4.3. Shear moduli of MWCNTs In the analysis of the shear moduli, it is assumed that an MWCNT is constrained at one end and subjected to a torsional moment at the other end. The torsional moment is assumed to be only acting on the outermost layer. The following formula, which is based on the theory of elasticity, is used for obtaining the shear modulus S S¼
TL0 J0
ð18Þ
Fig. 9. The effects of tube diameter and tube chirality on Young’s moduli of two-layer MWCNTs.
where T is the applied torque, L0 is the length of the MWCNT, stands for the torsional angle at the end of tube and J0 ¼ 32 ½ðdo þ 3:4Þ4 ðdi 3:4Þ4 A˚4 is the cross-sectional polar inertia of the MWCNT. Fig. 11 shows the computational results of shear moduli for two-layered armchair and zigzag MWCNTs. For comparison, the shear moduli of SWCNTs are also displayed in Fig. 11. On average, the shear modulus of an MWCNT is about 0.4 TPa and is lower than that of an SWCNT, which is about 0.45 TPa. It is also observed that the trends of variation of shear moduli with tube
Fig. 10. The effect of the number of tube layers on Young’s modulus.
Fig. 11. The effects of tube diameter and tube chirality on shear modulus.
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Two types of boundary loading conditions have been considered above. In the calculation of Young’s modulus,
it is assumed that all the layers of an MWCNT are subjected to tensile loads. In the calculations of shear modulus, torsional loads are applied only on the outermost layer of an MWCNT. The results show that the Young’s modulus of an MWCNT is not simply the average of the modulus of the individual SWCNTs comprising the MWCNT, rather, it is a combination of individual moduli plus the small influence of the van der Waals forces. The modeling of shear deformation shows that the inner layers of an MWCNT do not interact effectively with the outermost layer when the torsional load is imposed only on the outermost layer. Overall, the above modeling work has demonstrated that the elastic behavior of MWCNTs is different from that of SWCNTs due to the presence of van der Waals forces. To further delineate the effect of van der Waals forces in the deformation of MWCNT, a two-layer MWCNT under two different tensile loading conditions is examined. One loading condition assumes that uniform forces are applied only on atoms at the end of the outer layer, while in the other loading condition, uniform forces are applied on atoms at the end of both inner and outer layers. The load-displacement relations of the atoms at the end of tubes are shown in Fig. 13. In both loading conditions, the atoms at the end of the outer layer reach nearly the same displacement. But the displacements of the atoms at the end of the inner layer show a big difference in these two loading conditions. When the forces are applied only on the outer layer, the inner layer exhibits small axial displacements. This means that the force acting on the outer layer is barely transferred to the inner layer through van der Waals interactions. This finding is consistent with the experimental observation that deformation of MWCNTs often leads to the separation and pullout of the individual layers [7].
Fig. 12. The effect of the number of tube layers on shear modulus.
Fig. 13. The effect of van de Waals forces on loading transfer.
diameter are different for MWCNTs and SWCNTs. The shear moduli of SWCNTs first increase with the increase of tube diameter and then gradually become insensitive to the tube diameter. For MWCNTS, however, the shear moduli decrease with the increase of tube diameter. The effect of tube chirality is not significant. Fig. 12 summarizes the effect of number of tube layers on shear modulus. For both zigzag and armchair MWCNTs, the shear moduli are reduced when the number of tube layers increases. This is due to the fact that in the present simulation, it is assumed that only the outermost tube layer is subjected to applied torque. Because of the weak van der Waals forces between the neighboring tube-layers, the inner tube layers do not contribute as effectively as the outermost tube layer in resisting the applied torque. However, all the layers have been taken into account in the calculation of crosssectional polar inertia, which increases with the increase of the number of tube layers. This is why, under the present assumption of a boundary loading condition, the shear modulus of MWCNTs decreases when the number of tube layers increases. There are still no reports of experimental data on the shear moduli of carbon nanotubes. Lu [12] simulated the shear modulus of MWCNTs by using empirical lattice dynamics model and predicted that the shear modulus of MWCNTs is about 0.45 TPa and is insensitive to tube diameter, tube chirality and the number of tube layers. These conclusions are somewhat different from the present results. 4.4. The effect of van der Waals forces on elastic moduli of MWCNTs
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5. Conclusions In this paper, the molecular structural mechanics approach originally developed for simulating single walled carbon nanotubes is extended to simulate the elastic behavior of multi-walled carbon nanotubes under tension and torsion. The van der Waals forces between tube layers are taken into account by introducing a nonlinear truss rod model. The results show that the Young’s moduli of MWCNTs are in the range of 1.05 0.05 TPa, slightly higher than those of SWCNTs, and the shear moduli of MWCNTs are about 0.40 0.05 TPa, which is slightly lower than those of SWCNTs. There are slight but noticeable effects of tube diameter, tube chirality and number of tube layers on MWCNT elastic properties. Finally, it has been demonstrated that the inner layer of an MWCNT can be effectively deformed only through the direct application of tensile or shear forces.
Acknowledgements This work is supported by the Army Research Office (Grant No. DAAD 19-02-1-0264). Dr. Bruce LaMattina is the Program Director. References [1] Ijima S. Helical microtubules of graphitie carbon. Nature 1991; 354(6348):56–8. [2] Thostanson ET, Ren Z, Chou TW. Advances in the science and technology of carbon nanotubes and their composites: a review. Comp Sci Technol 2001;61:1899–912. [3] Thostenson ET, Chou TW. Aligned multi-walled carbon nanotube-reinforced composites: processing and mechanical characterization. J Phys D—Appl Phys 2002;35(16):L77–80. [4] Treacy MMJ, Ebbesen TW, Gibson TM. Exceptionally high Young’s modulus observed for individual carbon nanotubes. Nature 1996;381:680–7. [5] Wong EW, Sheehan PE, Lieber CM. Nanobeam mechanics: elasticity, strength, and toughness of nanorods and nanotubes. Science 1997;277:1971–5. [6] Salvetat JP, Bonard JM, Thomson NH, Kulik AJ, Farro L, Bennit W, Zuppiroli L. Mechanical properties of carbon nanotubes. J Appl Phys A 1999;69:255–60. [7] Yu MF, Lourie O, Dyer M, Moloni K, Kelly T. Strength and breaking mechanism of multi-walled carbon nanotubes under tensile load. Science 2000;287:637–40.
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Elastic moduli of multi-walled carbon nanotubes and the effect of van der Waals forces Chunyu Li, Tsu-Wei Chou* Department of Mechanical Engineering, University of Delaware, Newark, DE 19716, USA Received 4 November 2002; received in revised form 19 December 2002; accepted 10 January 2003
Abstract This paper reports a study of the elastic behavior of multi-walled carbon nanotubes (MWCNTs). The nested individual layers of an MWCNT are treated as single-walled frame-like structures and simulated by the molecular structural mechanics method. The interlayer van der Waals forces are represented by Lennard–Jones potential and simulated by a nonlinear truss rod model. The computational results show that the Young’s moduli and shear moduli of MWCNTs are in the ranges of 1.05 0.05 and 0.40 0.05 TPa, respectively. Results indicate that the tube diameter, tube chirality and number of tube layers have some noticeable effects on the elastic properties of MWCNTs. Furthermore, it has been demonstrated that the inner layers of an MWCNT can be effectively deformed only through the direct application of tensile or shear forces, not through van der Waals interactions. # 2003 Elsevier Ltd. All rights reserved. Keywords: A. Nanostructures; B. Mechanical properties; C. Computational simulation; Carbon nanotube; Atomistic modelling; Nanomechanics
1. Introduction Since the discovery of multi-walled carbon nanotubes (MWCNT) in 1991 [1], researchers worldwide have engaged in fundamental studies of this novel material and have investigated the potential of its technological applications, including carbon nanotube based composites [2]. The unique characteristics of carbon nanotubes, such as high stiffness, high aspect ratio and low density, are particularly desirable for this new generation of composites [3]. In order to fully explore the potential of carbon nanotubes for application in composites, a thorough understanding of the elastic properties of carbon nanotubes is necessary. The first experimental work on the Young’s modulus of MWCNTs was performed by Treacy and co-workers [4]. They used a transmission electron microscope to measure the mean-square vibration amplitudes of MWCNTs and deduced that the average value of the Young’s modulus is 1.8 0.9 TPa. Wong et al. [5] and
* Corresponding author. Tel.: +1-302-831-2423; fax: +1-302-8313619. E-mail address: [email protected] (T.-W. Chou). 0266-3538/03/$ - see front matter # 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0266-3538(03)00072-1
Salvetat et al. [6] used the technique of atomic force microscope and found the average Young’s modulus of MWCNTs to be 1.28 0.59 and 0.81 0.41 TPa. Recently, Yu et al. [7] also measured the tensile properties of MWCNTs using atomic force microscopy and reported Young’s modulus values ranging from 270 to 950 GPa. Due to the difficulties in experimental characterization of nanotubes, computer simulation has been regarded as a powerful tool for modeling the properties of nanotubes. Among the available modeling techniques [8–14, 28], molecular dynamics simulation has been used most extensively. However, molecular dynamics simulations need to consider the thermal vibration of atoms, and thus the time step is usually in the order of femtosecond. It is not efficient for long time or static problems such as simulating stiffness and strength properties. Being very time consuming, the application of molecular dynamics simulation to MWCNTs is very limited [8]. As for theoretical predictions, Lu [12] studied the elastic properties of MWCNTs using the empirical lattice dynamics model and reported a Young’s modulus of about 1.0 TPa and a shear modulus of slightly less than 0.5 TPa. In these studies [5–7,12], the cross-sectional areas of MWNTs are calculated by assuming that
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both the single layer wall thickness as well as the interlayer spacing are 0.34 nm. From the atomistic structural viewpoint, an MWCNT can be regarded as nested SWCNTs. The interactions among these SWCNTs are mainly due to the van der Waals forces. Compared with the covalent bonds between the neighboring carbon atoms in an SWCNT, the van der Waals interaction is rather weak. Wong et al. [5] predicted that relative slippage between adjacent layers in an MWCNT may occur due to the weak binding. This prediction was verified by observations of inner layer pullout in experiments conducted by Yu et al. [7] and Cumings and Zettl [16]. Ru [28] treated double-walled nanotubes as coaxial shells with van der Waals forces between the shells and studied their buckling under axial compression. However, it is not clear from these studies what the effects of van der Waals interactions are on the elastic properties of MWCNTs. Recently, the authors proposed a molecular structural mechanics approach for modeling single-walled carbon nanotubes [15]. Fundamental to this approach is the notion that a carbon nanotube can be regarded as a frame-like structure and the primary bonds between two nearest-neighboring atoms can be considered as loadbearing beam members, whereas an individual atom acts as the joint of the related load-bearing beam members. Establishment of a linkage between structural mechanics and molecular mechanics enables the sectional property parameters of these beam members to be obtained. The accuracy and stability of the present method have been verified through its application to several single layer graphene sheets with different specimen sizes. The computational results of the elastic properties of SWCNTs are also in good agreement with existing experimental data and theoretical predictions. In this paper, the molecular structural mechanics method is extended to model the elastic behavior of MWCNTs. The individual tube layer is still simulated as a frame-like structure as in Ref. [15]. Then, a truss rod model is proposed to simulate the interlayer van der Waals forces. The objectives of the present work are to predict the Young’s moduli and shear moduli of MWCNTs and to examine the effects of nanotube structures, such as the number of tube layers, tube diameter and tube chirality on elastic properties. A comparison between the elastic behavior of single-walled and multi-walled carbon nanotubes has also been made in light of the effect of van der Waals forces.
Fig. 1. A multi-walled carbon nanotube.
coefficients (n, m) of the lattice vectors a1 and a2 in the chiral vector Ch ¼ na1 þ ma2 is usually used to describe the chirality of the layer. The index (n, 0) denotes zigzag type nanotubes and (n, n) for armchair type nanotubes. Since the nested layers are structurally independent of one another, the chirality of the layers may be different. The distance between two neighboring layers is assumed to be the same as the spacing between adjacent graphene sheets in graphite, i.e., 0.34 nm [17] (Fig. 2). Each carbon atom within the atomic layer of a graphene sheet is covalently bonded to three neighboring carbon atoms. Three sp2 orbitals on each carbon form s-bonds to three other carbon atoms. One 2p orbital remains unhybridized on each carbon; these orbitals perpendicular to the plane of the carbon ring combine to form the p-bonding network [25]. The atomic interactions between the neighboring layers are the van der Waals forces.
2. Multi-walled carbon nanotubes An MWCNT is composed of a set of coaxially situated SWCNTs of different radii, as shown in Fig. 1. Each nested cylindrical layer of an MWCNT can be visualized as a rolled-up graphene sheet. A pair of
Fig. 2. Inter-graphene layer spacing in graphite.
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3. Extension of molecular structural mechanics model The molecular structural mechanics approach was first proposed by the authors for modeling SWCNTs [15]. Its main ideas include simulating a nanoscopic SWCNT as a macroscopic frame-like structure and then establishing the linkage between computational chemistry and computational structural mechanics approaches to obtain the required parameters for structural analysis. However, the approach proposed in Ref. [15] is suitable only for simulations of SWCNTs, and it does not consider the interaction between not covalently bonded atoms. In the following, the molecular structural mechanics method is extended to treat the elastic behavior of MWCNTs by taking into account the van der Waals forces acting between the neighboring tube layers. 3.1. Simulations of SWCNTs At the atomistic scale, carbon atoms in the hexagonal array of an SWCNT are bonded to each other by covalent bonds. These covalent bonds have their characteristic bond lengths and bond angles in a three-dimensional space. When a nanotube is subjected to external forces, the displacements of the individual atoms are constrained by these bonds, and the corresponding deformation of the nanotube is the result of the bond interactions. By considering the geometrical similarity between a space frame structure and a carbon nanotube, we treat the covalent bonds as connecting beam elements between carbon atoms and the carbon atoms as joints of the beam elements. Fig. 3 depicts this concept for an SWCNT. Based on the energy equivalence, a linkage between the molecular force field constants in computational chemistry and the cross-sectional properties of beam
Fig. 3. An SWCNT simulated as a frame-like structure.
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elements in computational structural mechanics has been established in Ref. [15], and is briefly recapitulated below. The direct relationships between these parameters are as follows: EA ¼ kr ; L
EI ¼ k ; L
GJ ¼ k L
ð1Þ
where L is the bond length, EA, EI and GJ are the crosssectional properties of a beam element in tension or compression, bending and torsion, respectively; kr , k and k are the molecular force field constants of an atomic bond in stretching, bending, and torsion, respectively. Eq. (1) forms the basis of the molecular structural mechanics model, which is applicable to all covalent molecular systems, provided that the force constants kr , k and k are known. The main steps of the molecular structural mechanics approach are as follows. First, take atoms as joints and bonds as beams in the equivalent frame-like structure. Second, determine beam sectional parameters from molecular force fields constants [Eq. (1)]. Third, establish elemental stiffness matrices and elemental load vectors and then assemble the global stiffness matrix and the global load vector. Last, solve the displacements of atoms and then compute the elastic properties of SWCNTs. 3.2. van der Waals forces between neighboring tube layers The multiple cylindrical layers of an MWCNT are held together through van der Waals forces. The van der Waals force is a non-bonded interaction, and it can be an attraction force or a repulsion force. The attraction occurs when a pair of atoms approach each other within a certain distance. The repulsion occurs when the distance between the interacting atoms becomes less than the sum of their contact radii. These interactions are often modeled using the general Lennard–Jones ‘‘6–12’’ potential [18], which provides for a smooth transition between the attraction and repulsion regions. The general Lennard–Jones ‘‘6–12’’ (LJ) potential is commonly expressed as 12 6 UðrÞ ¼ 4" ð2Þ r r where, r is the distance between interacting atoms, " and are the Lennard–Jones parameters. For carbon atoms the Lennard–Jones parameters are "=0.0556 kcal/mole and =3.4 A˚ [19]. The potential U(r) is usually truncated at an interatomic distance of 2.5 without a significant loss of accuracy, i.e., no interactions are evaluated beyond this distance. Based on the Lennard–Jones potential, the van der Waals force between interacting atoms can then be written as
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dUðrÞ " 13 7 ¼ 24 2 FðrÞ ¼ dr r r
ð3Þ
The variations of the Lennard–Jones potential and the van der Waals force with the distance between two interacting atoms are illustrated in Fig. 4. 3.3. Simulations of van der Waals forces The van der Waals force acting along the connecting line between two interacting atoms is simulated by a truss rod that connects the two interacting atoms with rotatable end joints (Fig. 5). The truss rod, thus, transmits only tensile or compressive forces as given by Eq. (3). In an MWCNT, an atom situated in one of these layers may form interacting pairs with several atoms in other layers as long as the distance between the pair of atoms is less than 2.5 . For convenience in computation, we consider the van der Waals interactions only between the nearest neighboring layers (Fig. 5). Fig. 4 clearly indicates that the van der Waals force between two atoms is highly nonlinear. Therefore, the truss rod connecting the atoms is a nonlinear element with its load–displacement relationship characterized by the van der Waals force. However, the load–displacement curve of the truss rod (Fig. 6) is opposite to that of the van der Waals force-interatomic distance relation displayed in Fig. 4. The reason is that the van der Waals force is an intrinsic force between the interacting atoms, while Fig. 6 gives the load-displacement relation when a load is applied to the truss rod to overcome the van der Waals force. The load–displacement curve of Fig. 6 consists of two distinct stages in loading and unloading. This characteristic of load-displacement curve brings about some numerical difficulties in the simulation of the nonlinear
Fig. 4. Lennard–Jones potential and van der Waals force versus atomic distance.
behavior of the truss rod. Thus the selection of a suitable nonlinear analysis method is needed. The available methods include arc-length method [26], work control method [27] and generalized displacement control method [20]. In this paper, for simulating the nonlinearity of the truss rod, we adopt the generalized displacement control method [20], which is essentially similar to the arc-length method. But the parameters in the constraint equations of these two methods are different. It is formulated in N+1 dimensional space that includes one load parameter and N displacements as the unknowns. A constraint equation is added to the N equations of equilibrium and it can be justified from the bounded nature of the load parameter. A general stiffness parameter, which is updated with the displacements in iterative procedures, is used for adjusting the incremental step size, and self-adaptively changing loading directions. This method has been demonstrated to be accurate and numerically stable at the critical and snapback points. The results of simulation of the relation between the load for overcoming the van der Waals force between two neighboring atoms and the displacement of a truss
Fig. 5. Truss rods for simulating van der Waals forces.
Fig. 6. Load-displacement curve of the nonlinear truss rod.
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rod, based on the generalized displacement control method, is given in Fig. 6. Three iterations are usually required in each load increment, and the accuracy achieved by these iterations is always up to 0.0001.
4. Results and discussions The objective of this paper is to apply the molecular structural mechanics model to predict the elastic properties of multiwalled carbon nanotubes. The computational model configurations and load conditions are illustrated in Fig. 7. The Young’s modulus and shear modulus are calculated by simulating the nanotube tensile and torsional deformation, respectively. For simplicity, the nanotube end caps are not considered. Loads are applied directly and uniformly on the end atoms. 4.1. Young’s modulus of graphite The applicability and accuracy of the molecular structural mechanics model for simulation of graphene sheets and SWCNTs have been examined in Ref. [15]. Combining the space frame model with the present model for van der Waals forces, we first simulate the multi-layered graphite because the measured elastic modulus of graphite is available. The values of force constants, 12 kr ¼ 469 (kcal/mol)/A˚2 and 12 k ¼ 63 (kcal/ mol)/rad2 [21], are adopted, and the initial carbon–carbon bond length is taken as 1.421 A˚ [22]. The thickness of a graphene sheet and the interspacing between graphene layers are assumed to be 0.34 nm. The computational results of Young’s moduli for multi-layered graphite compared with that of a graphene sheet are shown in Fig. 8. There is a negligible difference between the elastic modulus of multi-layered graphite and that of a graphene sheet. The computational results ( 1.05 TPa) based on the present model are in excellent agreement with the experimental data (1.025 TPa) [17].
Fig. 7. Computational model configurations.
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4.2. Young’s moduli of MWCNTs Having established its validity for graphite, the present approach is then applied to calculating the Young’s moduli of MWCNTs. Several approaches exist in the literature for calculating the Young’s modulus of carbon nanotubes. These include the bending momentdeflection method [6], the vibration spectrum method [4,23] and the second derivative of energy method [10,11]. In this paper, we follow the original definition of Young’s modulus and simulate the deformations of carbon nanotubes under axial tension. The Young’s modulus is then expressed as Y¼
P=A0 DL=L0
ð4Þ
where P stands for the total force acting on the atoms at one end of the nanotube, A0 ¼ 4 ½ðdo þ 3:4Þ2 ðdi 3:4Þ2 A˚2 represents the cross-sectional area of the multiwalled tube with do (A˚) as the outermost tube diameter and di (A˚) as the innermost tube diameter, L0 is the initial length of tube and DL is the elongation of tube under the force P. The thickness of each tube layer is taken as the interlayer spacing of graphite, 3.4 A˚ [17]. The simulation focuses first on two-layer MWCNTs. It is assumed that both layers may be either armchair chirality or zigzag chirality. As in the simulation of graphite, the force constants kr and k are also chosen as 1 1 ˚2 2 kr ¼ 469 (kcal/mole)/A and 2 k ¼ 63 (kcal/mole)/ 2 rad . In addition, the force constant k is adopted as 1 2 2 k ¼ 20 (kcal/mole)/rad based on Refs. [21] and [24]. Although there is no benchmark data guiding determination of the force constant k , the present calculation indicates that the Young’s modulus of carbon nanotubes is insensitive to the variation in k . Fig. 9 displays the variation of the Young’s moduli of two-layer MWCNTs with nanotube outer-layer diameter. There is a slight difference between the trends of
Fig. 8. Young’s modulus of graphite.
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variation in Young’s moduli of MWCNTs and SWCNTs [15]. For SWCNTs, the Young’s modulus is almost constant when the tube diameter is larger than 1.0 nm. But for MWCNTS, there is a small variation, and the difference between maximum and minimum Young’s modulus values is about 0.15 TPa. Fig. 9 also shows that the effect of tube chirality is very weak. The average Young’s moduli are 1.05 0.05 TPa and 1.08 0.02 TPa for armchair and zigzag MWCNTs, respectively. A general conclusion that can be drawn from Fig. 9 is that the Young’s moduli of multiwalled carbon nanotubes are generally higher ( 7% in maximum) than those of single-walled carbon nanotubes. Thus, the van der Waals force has some effect on the tensile modulus of MWCNTs. Fig. 10 illustrates the effect of the number of layers in MWCNTs on the Young’s modulus. For comparison purpose, the outermost tube diameters are taken as
roughly the same for zigzag and armchair MWNTs. The overall effects of both tube layers and chirality are noticeable. The present approach gives slightly higher values of Young’s moduli than those reported in [12] by Lu, who predicted that the Young’s modulus of MWCNTs is about 0.97 TPa and is independent of the number of tube layers, tube chirality and tube diameter. Regarding the experimental results of Young’s moduli of MWCNTs, Wong et al. [5] and Salvetat et al. [6] reported values of 1.28 0.59 and 0.81 0.41 TPa, respectively, using AFM-based experiments. These data are comparable to the present prediction of 1.05 0.05 TPa. 4.3. Shear moduli of MWCNTs In the analysis of the shear moduli, it is assumed that an MWCNT is constrained at one end and subjected to a torsional moment at the other end. The torsional moment is assumed to be only acting on the outermost layer. The following formula, which is based on the theory of elasticity, is used for obtaining the shear modulus S S¼
TL0 J0
ð18Þ
Fig. 9. The effects of tube diameter and tube chirality on Young’s moduli of two-layer MWCNTs.
where T is the applied torque, L0 is the length of the MWCNT, stands for the torsional angle at the end of tube and J0 ¼ 32 ½ðdo þ 3:4Þ4 ðdi 3:4Þ4 A˚4 is the cross-sectional polar inertia of the MWCNT. Fig. 11 shows the computational results of shear moduli for two-layered armchair and zigzag MWCNTs. For comparison, the shear moduli of SWCNTs are also displayed in Fig. 11. On average, the shear modulus of an MWCNT is about 0.4 TPa and is lower than that of an SWCNT, which is about 0.45 TPa. It is also observed that the trends of variation of shear moduli with tube
Fig. 10. The effect of the number of tube layers on Young’s modulus.
Fig. 11. The effects of tube diameter and tube chirality on shear modulus.
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Two types of boundary loading conditions have been considered above. In the calculation of Young’s modulus,
it is assumed that all the layers of an MWCNT are subjected to tensile loads. In the calculations of shear modulus, torsional loads are applied only on the outermost layer of an MWCNT. The results show that the Young’s modulus of an MWCNT is not simply the average of the modulus of the individual SWCNTs comprising the MWCNT, rather, it is a combination of individual moduli plus the small influence of the van der Waals forces. The modeling of shear deformation shows that the inner layers of an MWCNT do not interact effectively with the outermost layer when the torsional load is imposed only on the outermost layer. Overall, the above modeling work has demonstrated that the elastic behavior of MWCNTs is different from that of SWCNTs due to the presence of van der Waals forces. To further delineate the effect of van der Waals forces in the deformation of MWCNT, a two-layer MWCNT under two different tensile loading conditions is examined. One loading condition assumes that uniform forces are applied only on atoms at the end of the outer layer, while in the other loading condition, uniform forces are applied on atoms at the end of both inner and outer layers. The load-displacement relations of the atoms at the end of tubes are shown in Fig. 13. In both loading conditions, the atoms at the end of the outer layer reach nearly the same displacement. But the displacements of the atoms at the end of the inner layer show a big difference in these two loading conditions. When the forces are applied only on the outer layer, the inner layer exhibits small axial displacements. This means that the force acting on the outer layer is barely transferred to the inner layer through van der Waals interactions. This finding is consistent with the experimental observation that deformation of MWCNTs often leads to the separation and pullout of the individual layers [7].
Fig. 12. The effect of the number of tube layers on shear modulus.
Fig. 13. The effect of van de Waals forces on loading transfer.
diameter are different for MWCNTs and SWCNTs. The shear moduli of SWCNTs first increase with the increase of tube diameter and then gradually become insensitive to the tube diameter. For MWCNTS, however, the shear moduli decrease with the increase of tube diameter. The effect of tube chirality is not significant. Fig. 12 summarizes the effect of number of tube layers on shear modulus. For both zigzag and armchair MWCNTs, the shear moduli are reduced when the number of tube layers increases. This is due to the fact that in the present simulation, it is assumed that only the outermost tube layer is subjected to applied torque. Because of the weak van der Waals forces between the neighboring tube-layers, the inner tube layers do not contribute as effectively as the outermost tube layer in resisting the applied torque. However, all the layers have been taken into account in the calculation of crosssectional polar inertia, which increases with the increase of the number of tube layers. This is why, under the present assumption of a boundary loading condition, the shear modulus of MWCNTs decreases when the number of tube layers increases. There are still no reports of experimental data on the shear moduli of carbon nanotubes. Lu [12] simulated the shear modulus of MWCNTs by using empirical lattice dynamics model and predicted that the shear modulus of MWCNTs is about 0.45 TPa and is insensitive to tube diameter, tube chirality and the number of tube layers. These conclusions are somewhat different from the present results. 4.4. The effect of van der Waals forces on elastic moduli of MWCNTs
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5. Conclusions In this paper, the molecular structural mechanics approach originally developed for simulating single walled carbon nanotubes is extended to simulate the elastic behavior of multi-walled carbon nanotubes under tension and torsion. The van der Waals forces between tube layers are taken into account by introducing a nonlinear truss rod model. The results show that the Young’s moduli of MWCNTs are in the range of 1.05 0.05 TPa, slightly higher than those of SWCNTs, and the shear moduli of MWCNTs are about 0.40 0.05 TPa, which is slightly lower than those of SWCNTs. There are slight but noticeable effects of tube diameter, tube chirality and number of tube layers on MWCNT elastic properties. Finally, it has been demonstrated that the inner layer of an MWCNT can be effectively deformed only through the direct application of tensile or shear forces.
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