Computational Materials Science 47 (2010) 985–993
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Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci
Radial mechanical properties of single-walled carbon nanotubes using modified molecular structure mechanics Wen-Hwa Chen a, Hsien-Chie Cheng b, Yang-Lun Liu a,* a b
Department of Power Mechanical Engineering, National Tsing Hua University, Hsinchu 30013, Taiwan Department of Aerospace and Systems Engineering, Feng Chia University, Taichung, Taiwan
a r t i c l e
i n f o
Article history: Received 15 September 2009 Received in revised form 10 November 2009 Accepted 26 November 2009
Keywords: Single-walled carbon nanotubes Modified molecular structure mechanics Mechanical properties Radial modulus Radial buckling Radial deformation
a b s t r a c t This study presents a modified molecular structure mechanics (MSM) model for determining the mechanical properties of carbon nanotubes, particularly in the radial direction. In the proposed model, the interactions between two carbon atoms are modeled with the second generation force field using continuum pseudo-rectangular beam elements while the non-bonded van der Waals (vdW) interactions among atoms are simulated with the Lennard–Jones (L–J) potential using spring elements. The pseudorectangular beam consists of a different bending rigidity along the two principal centroidal axes of the cross-section, the stiffness parameters of which are estimated based on the bond-angle variation energy and the weak inversion energy in molecular mechanics. By the approach, the mechanical properties of single-walled carbon nanotubes (SWCNTs), in particular the radial modulus, radial buckling load, and radial deformation, are calculated. To demonstrate the effectiveness of the proposed MSM model, the derived results are compared with those of the original MSM model and the published theoretical and experimental data. Results show that the radial elastic modulus of the zigzag SWCNTs with a radius of 0.39–2.35 nm is about 62–0.4 GPa through the proposed MSM model, and that of the armchair SWCNTs with a radius of 0.48–2.38 nm is about 30–0.3 GPa. The results are considerably less than those of the original MSM model but much more consistent with the literature data. Even the weak vdW interactions may potentially result in remarkable radial collapse or deformation of the SWCNTs as their radius is larger than about 1.1 nm. Besides, it is also demonstrated that the modification would have a little effect on the axial Young’s modulus and Poisson’s ratio. Ó 2009 Elsevier B.V. All rights reserved.
1. Introduction Carbon nanotubes (CNTs) have been an intensive and important research subject in nanotechnology ever since their discovery in 1991 [14]. Due to their many unique electrical, mechanical and thermal properties, wide-ranging research on a variety of CNTs is being undertaken by academic and industry to pursue their potential applications, including novel nano-structures, fiber-reinforcement nanocomposites, drug delivery, electromechanical sensors and nanoelectronics devices. To realize the potential of the nanotechnology, extensive studies on the mechanical properties of CNTs were carried out through experimental methods using some indispensable, high-resolution microscopes, such as transmission electron microscope [16,32,34,36] and scanning probe microscope [1,11,23,30,31,37]. By contrast, theoretical approaches including classic atomic computational approaches and multi-temporal and spatial scale * Corresponding author. Tel./fax: +886 3 5715131x33721. E-mail address:
[email protected] (Y.-L. Liu). 0927-0256/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.commatsci.2009.11.034
simulations are more cost-effective and efficient, and in addition, without the limitation of specimen size and difficulties in manipulation of the nanoscale specimens and also in the derivation of the physical insights of nanomaterials. Among the classic atomic computational approaches, ab initio calculation [3,22,38], density function theory [8,28], tight-binding formalism [12], MD simulation [2,5,9,35], and equivalent continuum modeling (ECM) [6,13,25] are the most widely-used methods for simulating the nanomechanics of CNTs. In spite of the greatly improved efficiency of computers nowadays, the ab initio calculations and MD simulations are still limited to an atomic-scale and femtosecond time-scale model, which contains atoms less than 103–104 and 106–108, respectively. On the other hand, ECM approach tends to be a more efficient modeling technique for simulating a larger scale of systems or a longer time span. In principle, the approach transforms chemical bonds between atoms in molecular mechanics into a continuum model using the finite element methods, and thus, provides a link between molecular mechanics and continuum mechanics. By the concept, several finite element models have been proposed, including the classical spring/truss [10,26], beam [17,18] and shell model
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[35]. These theoretical models have demonstrated their effectiveness and capability in simulating the deformation behaviors of SWNT at much larger length-scale and time-scales than standard MD simulation, and also challenged the immense limitations on specimen size from experimental methods. Among these ECM models, the MSM model, proposed by Li and Chou [17–20] and briefly termed the original MSM model in the investigation, has attracted great attention from academic circle because of its simplicity and effectiveness. It has been regarded as a very efficient and cost-effective method for simulating the mechanical properties of nanomaterials in contrast to the other theoretical approaches. In the approach, a frame structure is proposed for CNTs using equivalent continuum beam elements. The stiffness values of the beams, namely the bond-stretching, bondangle-variation and torsional-resistance force constants, are constructed by relating the strength of the continuum beam to the interactions between atoms or the bond potential energy, including the bond-stretching energy, the bond-angle variation energy and the dihedral-angle torsion energy. The cross-section of the beams was assumed to be round, implying that there is an equivalent sectional bending rigidity in any radial direction. The bending stiffness parameter (i.e., the bond-angle-variation force constant) of the round beam is determined simply based on the bond-angle variation energy. However, according to the potential energy of a covalent bond in molecular mechanics, as shown in Fig. 1, the beam bending rigidity should be dependent on both the bond-angle variation energy and the inversion energy. More specifically, the inversion energy would mainly determine the bending stiffness of the covalent bond in the minor principal centroidal axis of the section while the bond-angle variation energy for that in the major one since the former is considered trivial [4,7] as compared to the latter. Besides, the minor axis bending stiffness may greatly affect the radial mechanical properties of CNTs. A poor estimate of that, just as the original MSM model, would lead to a great inaccuracy in the calculated mechanical properties of SWCNTs, especially in the radial direction. For example, the difference in the calculated
radial elastic modulus of the original MSM model [21] from the MD simulation [33] is on the order of about one to two orders of magnitude. Although [15] proposed a modification of the original MSM model, the modified expression is valid only for a zigzag CNT, and in addition, its derivation is highly complex. In this study, we present a new ECM approach based on the modification of the original MSM model, termed the modified MSM model, to characterize the mechanical properties of SWCNTs, especially in the radial direction. As discussed in the paper, the proposed model turns out to be very effective in characterizing the mechanical properties of SWCNTs, in particular in the radial direction. 2. Theoretical modeling 2.1. The original MSM model The original MSM model was developed by Li and Chou [17,18], where an equivalent round beam and a nonlinear rod element were adopted to simulate the bonding force and vdW interaction between two atoms of CNTs, respectively. The procedure for deriving the original MSM model is briefly described as follows. The general expression of the potential energy Vt for a covalent bond system was presented by Cornell et al. [7],
Vt ¼
X
Vr þ
X
Vh þ
X
V/ þ
K
J J I (a) Vr
I
Δr
Δθ
θ0 θ (b) Vθ
L
L
φ
J
K K
I
Vx þ
I J (d) Vω
(c) Vφ
X
V vdW ;
ð2-1Þ
where Vr, Vh, Vu, Vx, and VvdW are the bond-stretching energy, the bond-angle variation energy, the dihedral-angle torsion energy, the inversion energy and the vdW interaction energy, respectively. Fig. 1 schematically represents the potential energy in molecular mechanics. The electrostatic interaction is herein neglected because CNTs are neutral. In addition, many previous studies [4,7] claimed that the inversion energy is considered trivial, as compared to the other energy terms, and thus neglected in the modeling. It will be, however, proved in this paper that the effect of the energy term
r r0
X
(e) Vv d W Fig. 1. Potential energy in molecular mechanics.
ω
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on the axial mechanical properties of CNTs can be indeed negligible, but not the radial ones. In summary, for a covalent bond structure, such as a CNT, the total potential energy is basically dominated by the first three terms of Eq. (2-1). Based on the harmonic expressions under the assumption of small deformation, the simple expressions for these energy terms can be written as
1 1 K r ðr r 0 Þ2 ¼ K r ðDrÞ2 ; 2 2 1 1 2 V h ¼ K h ðh h0 Þ ¼ K h ðDhÞ2 ; 2 2 1 2 V s ¼ V / ¼ K s ðD/Þ ; 2 Vr ¼
ð2-2Þ ð2-3Þ ð2-4Þ
where Kr, Kh, and Ks are the bond-stretching, bond-angle variation and torsional-resistance force constants, respectively, and Dr, Dh, and Du represent the bond-stretching, bond-angle and bond-twisting-angle variations, respectively. In principle, the bond-angle variation force constant is nothing but the sectional bending rigidity about the major principal axis of the covalent bond for a graphite sheet. To determine the stiffness and geometric parameters of the equivalent beam, including Young’s modulus E, shear modulus G, length L, sectional area A, moment of inertia I, and polar moment of inertia J, the relationships between them and the force constants in molecular mechanics need to be derived first. The stiffness and geometric parameters of the equivalent beam can be determined from the relationship between the potential energy of the covalent bond due to atomic interactions and the strain energy of the equivalent beam as a result of structural deformation. According to structural mechanics, the strain energy of a uniform beam with a length L, Young’s modulus E, and cross-section A subjected to a pure axial force N (Fig. 2a) can be expressed as:
UA ¼
Z
L
0
N2 1 N2 L 1 EA ¼ ðDLÞ2 ; dL ¼ 2 EA 2 L EA
ð2-5Þ
where DL is the axial stretching deformation. The strain energy of a uniform beam subjected to a pure bending moment M (see, Fig. 2b) is written as:
UM ¼
1 2
Z
M2 M2 L 1 EI ¼ ðhB Þ2 ; dL ¼ 2EI 2 L EI
ð2-6Þ
where hB is the angle of rotation at the end of the beam. The strain energy of a uniform beam subjected to a pure torsion T (Fig. 2c) is denoted as:
N L
ΔL
(a) Tension L θB
M
(b) Bending Δβ
Z
T2 T 2 L 1 GJ ¼ ðDbÞ2 ; dL ¼ 2GJ 2 L GJ
ð2-7Þ
where Db is the relative rotation between the two ends of the beam. In Eqs. (2-2)–(2-7), both Vr and UA represent the stretching energy, both Vh and UM indicate the bending energy, and both Vs and UT stand for the torsional energy. Accordingly, Dr is reasonably assumed to equal DL, Dh equals hB, and Du equals Db. Therefore, relating Eqs. (2-2)–(2-4) to Eqs. (2-5)–(2-7), respectively, yields the following direct relationship between the structural mechanics parameters and molecular mechanics force field constants,
Kr ¼
EA ; L
Kh ¼
EI ; L
Ks ¼
GJ : L
ð2-8Þ
Since the force constants Kr, Kh, and Ks, as well as the bond length L can be directly derived from the second generation force field [7], the stiffness and geometric parameters E, G, A, I, and J of the equivalent beam can be also determined. 2.2. The modified MSM model In the original MSM model, it is clearly assumed that there is an equivalent bond bending rigidity in both the major and minor principal centroidal axes of the cross-section of the covalent bond because of the round beam assumption. The bending rigidity of the round beam is principally derived from the bond-angle variation energy. This assumption may not be theoretically sound since the sectional bending rigidity of the covalent bond in the minor principal centroidal axis should be closely related to the weak inversion energy rather than the bond-angle variation energy according to molecular mechanics. As the minor axis bending rigidity of the covalent bond would primarily determine the radial stiffness properties of CNTs, the inaccurate modeling of the sectional bending rigidity in the minor axis of the covalent bond would result in a poor estimate of their radial mechanical properties. The following introduces the theoretical backgrounds behind the modified MSM model. The derivation starts from a graphite sheet, which is considered as a frame structure in the present MSM model. Fig. 3a displays two of the hexagons in a graphite sheet, where point O is defined as the origin of the local Cartesian coordinate system for beam (covalent bond) OA. Assume that there is a single force Fx parallel to the X axis at point B of beam OB, thus resulting in a bond-angle variation between beam OA and OB or a moment Mz directed along the Z axis. Since this moment would fully contribute to the bondangle variation energy between the covalent bond OA and OB, the relation between the sectional bending rigidity of the continuum pseudo-rectangular beam element about the Z axis (the major axis) and the bond-angle variation force constant is described as:
Kh ¼
EIz : L
ð2-9Þ
Assume that another single force Fz parallel to the Z axis is applied at the point O, thus inducing a moment My directed along the Y axis. The force would induce the inversion energy Vx between the atom O and the plane A–B–C, as shown in Fig. 3b. Thus, roughly one third of the inversion energy would contribute to the bending energy U My of the covalent bond OA because of the moment My,
U My ¼
1 1 V x ¼ K x h2y ; 3 6
ð2-10Þ
where Kx is the inversion force constant, which is 1.1 kcal/mol based on Cornell et al. [7], and hy is the bending angle of beam OA along the Y axis. Further, combining Eq. (2-6) with Eq. (2-10) results in:
L
(c) Torsion
1 2
UT ¼
T
Fig. 2. Deformation of a beam element.
Kx ¼
3EI0y : L
ð2-11Þ
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Z
Fz
Fz
C
C
A
O A
L
X
O′
B
Mz
(b) Inversion Energy
(a) Graphite sheet
ð2-13aÞ ð2-13bÞ
where I00y is the moment of inertial of beam OA directed along the Y axis due to the bond-angle variation energy Vh. By comparing the above two equations, the relation between I00y and Iz can be written as:
O
y
Fx
B
My
EðIy Þ EðI sin uÞ h; h¼ L L EðI cos uÞ EðIz Þ Mz ¼ M cos u ¼ h¼ h; L L
My ¼ M sin u ¼
Y
I00y I sin u ¼ ¼ tan u; Iz I cos u
ð2-14Þ
and thus,
Fx
X
C
Mz
O B
O
C
φ
Uy ¼
Y
M
Z
D
(d) Zigzag CNT (Top-view)
Fig. 3. Illustration of the moment of inertia of a covalent bond in the modified MSM model.
By relating Eq. (2-9) with Eq. (2-11), we find the relation of the moment of inertia about the minor ðI0y Þ and major (Iz) principal centroidal axes,
I0y ¼
Kx Iz ¼ 0:0058 Iz ¼ k1 Iz ; 3K h
M 2y L 2EI00y
;
Uz ¼
M 2z L : 2EIz
ð2-16Þ
The bond-angle variation energy Vh between the covalent bond OA and OB is assumed to be equal to the sum of the above two terms
A
(c) Zigzag CNT (Side-view)
ð2-15Þ
According to Eq. (2-6), the corresponding bending energy terms associated with these two moment components are
B
Y
φ My
φ
I00y ¼ ðtan uÞIz ¼ k2 Iz :
Fz
Z
ð2-12Þ
where Kh is bond-angle variation energy, which is 63 kcal/mol based on Cornell et al. [7]. From Eq. (2-12), it is evident that the I0y is far smaller than Iz, implying that the round beam in the original MSM model is clearly not conservative. For an SWCNT, a slight modification of the above derivation has to be done due to that it is a rolled-up graphene sheet rather than a plane one. Fig. 3c shows two of the hexagons in a zigzag CNT, and Fig. 3d is the top view of the CNT, where the circular dot line is the ‘‘virtual” tube wall of the CNT, and the black segment lines are the actual tube wall (i.e., a polygon). It is clear that these two hexagons in Fig. 3c do not lie in a plane. Assume that a local Cartesian coordinate system is also defined at point O for beam OA, where the tangent line to the circular circumferential wall at point O is defined as the Y axis; the radial line passing through the center of the circular dot line, which is perpendicular to the tangent line, denotes the Z axis; and the axial direction of beam OA represents the X axis. Noticeably, there is an angle u between the tangent line at point O and the plane of the hexagons. The angle changes with the radius of CNTs, and approaches to zero as the radius increases to infinity (i.e., a graphite sheet). Further, consider that a force Fx is acted on beam OB at point B in the X direction, which induces a moment M on beam OA. It should be noted that the moment would be no more parallel to the Z axis; as a result, the bond-angle variation energy between the covalent bond OA and OB would by no means equal the strain energy of beam OA under the bending moment M. In other words, Eq. (2-9) does not any more hold for a CNT. Basically, the moment M can be separated into two components, i.e., My and Mz directed along the Y and Z axis, respectively, as shown in Fig. 3c,
M 2y L 2EI00y
þ
M2z L ¼ V h: 2EIz
ð2-17Þ
Thus, the following relation, similar to Eq. (2-9), can be obtained,
Kh ¼
EðI00y þ Iz Þ : L
ð2-18Þ
By solving Eqs. (2-15) and (2-18) together, I00y can be obtained. Since the inversion energy and bond-angle variation energy both contribute to the energy of beam OA, the moment of inertia of this beam directed along the Y axis Iy can be calculated by the sum of I0y and I00y as
Iy ¼ I0y þ I00y ¼ ðk1 þ k2 Þ Iz :
ð2-19Þ
From Eqs. (2-18) and (2-19), the modified moment of inertia Iy and Iz are obtained. Fig. 4 displays the constants k1 and k2 varying with the radius of a zigzag SWCNT. A larger radius would result in a decrease of k2. In addition, as the radius approaches to zero, k1 becomes far less than k2, indicating that the bending energy is dominated by the bond-angle variation energy Vh. On the other hand, as the radius is larger than about 1.5 nm, k1 becomes larger than k2, implying that the bending energy Vh and the inversion energy Vx would both play an important part in the bending energy. 3. Results and discussion To verify the accuracy of the modified MSM model, the radial mechanical properties, such as radial elastic modulus, radial buckling behavior and radial deformation, are calculated and compared with the results of various other studies. The influences of the modification on the Poisson’s ratio and axial Young’s modulus of SWCNTs are also elucidated. 3.1. Radial elastic modulus of SWCNTs The definition of the radial elastic modulus [31] is similar to that of the axial Young’s modulus, and is the ratio of radial stress to radial strain
Er ¼
rr F=A ¼ ; er d=D
ð3-1Þ
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W.-H. Chen et al. / Computational Materials Science 47 (2010) 985–993
300
Radial Modulus E r (GPa)
0.6
k1
0.4
k2
0.2
MSM (Zigzag) MSM (Armchair) Modified MSM (Zigzag) Modified MSM (Armchair) MD Simulation (Wang et al., 2007)
200
100
0
0 0
1
2
3
Radius (nm) Fig. 4. The constants k1 and k2 as a function of the radius of a zigzag SWCNT.
F
δ
D
(a) Radial modulus F
F
(b) The forced area Fig. 5. Definition of the radial modulus and forced area on an SWCNT.
where F is the normal force acting on the top of an SWCNT, A the area on which F is applied, d the radial displacement, and D the diameter of the SWCNT, as shown in Fig. 5. The area A is defined based on the assumption of Li and Chou [21], where it is the triangular area shown in Fig. 5b. As shown in Fig. 6, the calculated radial elastic modulus by the modified MSM model ranges from 62 to 0.4 GPa as the radius of the zigzag SWCNTs increases from 0.39 to 2.35 nm, and from 30 to 0.3 GPa for the armchair CNTs with an increasing radius from 0.48 to 2.38 nm. Clearly, the radial stiffness of the zigzag type is lar-
0
0.4
0.8
1.2 Radius (nm)
1.6
2
2.4
Fig. 6. Radial modulus of SWCNTs as a function of radius.
ger than that of the armchair type. This is probably because of the difference of the structural arrangement of atoms. Furthermore, the radial elastic moduli of all these cases reported decrease greatly with the increase of the radius of the tubes, indicating that the SWCNTs with a larger tube radius present smaller radial stiffness. The results of radial elastic modulus based on the original MSM model [21] are also shown in Fig. 6 (i.e., 256–7.2 GPa for the zigzag type; 117–4.0 GPa for the armchair type). It is evident that there is a significant deviation in the results of the modified MSM from those of the original MSM, and in addition, the difference lessens with an increasing radius. Moreover, it is also observed that the radial elastic modulus of the SWCNTs is remarkably lower than the associated axial one (1.0 TPa). Since there is still limited experimental data on the radial elastic modulus of SWCNTs, the reported experimental data of the radial moduli of multi-walled CNTs (MWCNTs) in literature are used to indirectly demonstrate the effectiveness of the modified MSM model. For example, Shen et al. [31] performed nanoindentation tests to investigate the elastic modulus of MWCNTs in the radial direction using a scanning probe microscope. They found that the associated radial elastic modulus increases from 9.7 to 80.0 GPa with an increasing compressive stress. Yu et al. [37] characterized the radial elastic modulus of MWCNTs also using nanoindentation tests with a tapping-mode atomic force microscope, and reported the effective radial elastic modulus range of 0.3–4.0 GPa at different cross-sections. Muthaswami et al. [24] showed that the radial elastic modulus of MWCNTs ranges from 16.0 to 23.0 GPa using ultrasonic force microscopy (UFM). Besides, through MD simulation, Wang et al. [33] reported that the radial elastic modulus of SWCNTs is in the range of 2.0–0.1 GPa as their radius increases from 0.5 to 4.0 nm. As compared to the above reported experimental and simulation data, it is clear to find that the original MSM model would tend to overestimate the radial elastic stiffness of the SWCNTs in contrast to the modified MSM model. 3.2. Poisson’s ratio, axial Young’s modulus and shear modulus of SWCNTs To investigate if other mechanical properties of SWCNTs would be influenced by the present modification of the original MSM model, their Poisson’s ratio, axial Young’s modulus and shear
W.-H. Chen et al. / Computational Materials Science 47 (2010) 985–993
Poisson's Ratio-v
0.6
0.4
MSM Modified MSM Chang and Gao (2003) Chen et al. (2007) Hernandez et al. (1998)
Armchair
MSM Modified MSM Chang and Gao (2003) Chen et al. (2007) Hernandez et al. (1998)
Zigzag
1100
Young's Modulus-E (GPa)
990
0.2
1000
MSM Modified MSM Chang and Gao (2003) Popov et al. (2000) Chen et al. (2007)
900
800
0 0.4
0.8
1.2
0
1.6
0.4
Radius (nm)
0.8 1.2 Radius (nm)
1.6
2
1.6
2
(a) The zigzag type
Fig. 7. Poisson’s ratio of SWCNTs as a function of radius.
modulus are also examined. In principle, Poisson’s ratio m is defined as the negative ratio of lateral strain el to axial strain ea,
As shown in Fig. 7, the Poisson’s ratio of the SWCNTs decreases with an increasing radius. Furthermore, as the radius increases from 0.4 to 1.6 nm, the calculated Poisson’s ratios by using the modified MSM model (i.e., 0.20–0.10 for the armchair type and 0.22–0.10 for the zigzag type) are all larger than those of the original model (0.08–0.06 for both the armchair and zigzag types), implying that the present model yields significantly lower radial tube stiffness for the SWCNTs than the original one. Most importantly, the present results are more consistent with the literature data using various approaches. For example, Hernandez et al. [12] reported the Poisson’s ratio 0.25 for the armchair SWCNT with a radius of 0.41 nm and 0.28 for the zigzag with a radius 0.75 nm using the tight-binding method. Chang and Gao [4] gave an estimated range of the Poisson’s ration 0.18–0.16 for the armchair SWCNTs and 0.20–0.16 for the zigzag with a radius ranging from about 0.25 to 1.0 nm by using the molecular mechanics model. The axial Young’s modulus can be defined as the ratio of axial stress ra to axial strain ea,
E¼
ra : ea
ð3-3Þ
Fig. 8a and b show the data concerning the effect of the modification of the original MSM model on the axial Young’s modulus of the SWCNTs. Results show that there is a trivial difference in the axial Young’s modulus between the present MSM model (i.e., 1030– 1083 GPa for the armchair SWCNTs and 961–1056 GPa for the zigzag) and the original MSM model (i.e., 1096–1104 GPa for the armchair and 1054–1078 GPa for the zigzag), where the maximum distinction is less than about 8%. Based on the continuum assumption, the shear modulus of materials can be defined as,
G¼
TL ; hJ
1100
ð3-2Þ
ð3-4Þ
where T is the applied torque, L the tube’s length, h the torsional angle, and J the cross-sectional polar moment of inertia. The calculated
Young's Modulus-E (GPa)
e m¼ l: ea
1000
MSM Modified MSM Chang and Gao (2003) Popov et al. (2000) Chen et al. (2007)
900
800 0
0.4
0.8 1.2 Radius (nm)
(b) The armchair type Fig. 8. Axial Young’s modulus of SWCNTs as a function of tube radius.
shear moduli of the SWCNTs are shown in Fig. 9a and b, associated with the zigzag and armchair types, respectively, together with the literature data [5,27]. The results show that there is only about 8– 17% discrepancy in the shear modulus between the modified MSM model (i.e., 183–4268 GPa for the armchair type and 220– 425 GPa for the zigzag) and the original MSM model (i.e., 224– 477 GPa for the armchair type and 265–466 GPa for the zigzag). Furthermore, by comparing them with the published data from Popov et al. [27] using force constants of the valence force field type and Chen et al. [5] using MD simulation, close agreement can be also observed.In summary, the current modification of the bending rigidity in both the major and minor principal centroidal axes of the cross-section of the covalent bond based on the bond-angle variation energy and the weak inversion energy in molecular mechanics
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600
Original
Shear Modulus-G (GPa)
vdW forces
D
400
Collapse MSM Modified MSM Popov et al. (2000) Chen et al. (2007)
200
Fig. 10. Illustration of radial buckling of an SWCNT due to the vdW atomistic interactions.
25 0 0
0.4
0.8
1.2
1.6
2
Radius (nm)
MSM Modified MSM MSM Modified MSM
20
(a) The zigzag type
Armchair Zigzag
Shear Modulus-G (GPa)
Eigenvalue λcr
600 15
10
400 5
1 0 0.6
MSM Modified MSM Popov et al. (2000) Chen et al. (2007)
200
0.4
0.8
1.2
0.8
1 1.2 Radius (nm)
1.4
1.6
Fig. 11. Critical radial buckling load kcr as a function of tube radius.
0 0
Transition Region (Gao et al., 1998)
1.6
2
Radius (nm)
(b) The armchair type Fig. 9. Shear modulus of SWCNTs as a function of tube radius.
does have a much more significant impact on the radial elastic stiffness and Poisson’s ratio of SWCNTs than the axial Young’s modulus and shear modulus. 3.3. Radial buckling behavior and deformation of SWCNTs Radial buckling and radial deformations of the SWCNTs are also greatly associated with the radial stiffness of SWCNTs. The radial buckling refers to the collapse of a tube in the radial direction due to its instability before attaining the ultimate strength of the material. As shown in Fig. 10, the vdW atomistic interactions may result in the radial deformations of SWCNTs. As the radius of SWCNTs increases to a critical value, the radial buckling would
potentially occur because of the weakening of their stiffness. In this study, the critical buckling load Pcr of SWCNTs in the radial direction is derived from elastic instability analysis based on the following equation for elastic instability,
1 w ¼ 0; K 0 þ kK
ð3-5Þ
where w is the buckling-mode shape (eigenvector), K0 the linear 1 the geometric stiffness matrix. The scalar stiffness matrix and K factor k at which buckling occurs is designated ‘‘kcr ”, and the critical buckling load P cr ¼ kcr P, where P is the normalized vector of external loads. The vdW interactions between any two neighboring atoms are considered to be the external loads, and the critical scalar load-factor parameter kcr can be calculated from linear buckling analysis using finite element methods. The approach to unity for kcr indicates that the vdW atomistic interactions considered are close to the critical buckling load of the structure. Fig. 11 presents the critical scalar factor kcr of the armchair SWCNTs as a function of their radius, including the reported data in the literature. Note that as kcr becomes larger than 1.0, the SWCNTs are inclined to collapse simply under the vdW atomistic interactions. The figure shows that kcr monotonically decreases as the radius of the SWCNTs increases, indicating that the SWCNTs with a larger radius tend to
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δ
D
Original
cross-section is introduced for simulating the covalent bond in CNTs. Based on the bond-angle variation energy and the weak inversion energy in molecular mechanics, the stiffness parameters of the beam are analytically derived. In addition, an analytical relation between the moment of inertia of the beam and the angle u between the tangent line to the circular cross-section of a SWCNT and the plane of the hexagons is derived as a function of the tube radius. By the proposed model, not only the radial mechanical properties such as radial Young’s modulus, critical load and deformation but also the axial Young’s modulus and Poisson’s ratio are investigated. In brief, the original MSM model tends to give a significant overestimate of the radial stiffness of the SWCNTs, and as a result, may fail to provide an accurate prediction of the radial mechanical properties of SWCNTs. Some other essential conclusions are drawn in the following:
Deformed Fig. 12. Illustration of the radial deformation of two vertically overlapped SWCNTs.
Table 1 Radial deformation rate of two vertically overlapped SWCNTs. D = 2.0 nm
Deformation rate (%)
MSM Modified MSM Ruoff et al. [30] Abrams and Hanein [1]
1.95 8.30 7.50 5.00
collapse more easily due to the vdW interactions. More importantly, the calculated results from the modified MSM model are substantially lower than those of the original MSM model. Fig. 11 also shows the molecular dynamics simulation results by Gao et al. [9], where the armchair SWCNTs with a radius larger than the transition region, i.e., r 1.077–1.144 nm, would tend to collapse. The result is in a good agreement with that predicted by the present model, in which kcr becomes very close to unity as the radius approaches to the transition region. This may be the reason of the appearance of noncircular cross-section SWCNTs reported in the literature [1,11]. The radial deformation of two vertically adjacent SWCNTs due to the vdW atomistic interactions is also studied to further examine the validity of the present modification. An example of the radial deformation (d) of two vertically overlapped SWCNTs with an equal diameter (D) of 2.0 nm is shown in Fig. 12. Based on Abrams and Hanein [1], radial deformation rate is defined as the ratio of d to D. The calculated radial deformation rate of the SWCNT by using the original MSM model and the present MSM model are shown in Table 1, together with the literature experimental data [1,30]. The radial deformation rate obtained from the present and original MSM models is about 8.3% and 1.95%, respectively. It turns out that the present MSM model can yield a much larger radial deformation rate than the original MSM model. By further comparing them with the Ruoff et al.’s study (i.e., 7.5%) and the Abrams and Hanein’s work (i.e., 5.0%), the present result is clearly much closer to the experimental results. 4. Conclusions This study successfully presents a modified MSM model for easing the disadvantage of overestimation of the radial stiffness of SWCNTs. In the proposed model, a rectangular beam with a different bending rigidity in the two principal centroidal axes of the
(1) By using the proposed MSM model, the radial elastic modulus for the zigzag SWCNTs with a tube radius of 0.39– 2.35 nm is estimated about 62–0.4 GPa, and 30–0.3 GPa for the armchair with a radius of 0.48–2.38 nm. It is clear to find that the formal tends to be slightly larger than the latter under roughly the same radius. Besides, the radial elastic modulus of the SWCNTs decreases greatly with an increasing radius of the SWCNTs. Because of the lack of the literature experimental data available for a SWCNT, the effectiveness of the proposed MSM model on the radial elastic modulus prediction is indirectly confirmed through the reported experimental results for a MWCNT. (2) The calculated Poisson’s ratio for the armchair SWCNTs using the proposed MSM model is in the range of 0.20– 0.10 and 0.22–0.10 for the zigzag type with a tube radius ranging from 0.4 to 1.6 nm. The data are significantly larger than those of the original MSM model, and most importantly, are more consistent with the literature data using various theoretical approaches (e.g., [4,12]). (3) Results show that the current modification of the original MSM model would have a little impact on the axial Young’s modulus and shear modulus of SWCNTs. (4) From the linear buckling analysis, it is found that the present results agree well with the reported molecular dynamic results by Gao et al. [9], in which kcr becomes very close to unity while the radius approaches to the transition region, i.e., 1.077–1.144 nm. In other words, the SWCNTs are likely to collapse simply under the vdW atomistic interactions as kcr becomes larger than 1.0. This may be why noncircular cross-section SWCNTs are often reported in the literature [1,11]. (5) The present MSM model produces a much larger radial deformation rate than the original MSM model (8.3% versus 1.95%), and most importantly, the present result (8.3%) is much closer to the literature experimental results, i.e., 7.5% by Ruoff et al. [30] and 5.0% by Abrams and Hanein [1]. Acknowledgements The authors are grateful to the National Science Council, Taiwan, ROC, under Grants NSC98-2221-E-007-016-MY3 and NSC98-2221-E-035-024-MY3 for the partial financial supports of this work. References [1] Z.R. Abrams, Y. Hanein, Carbon 45 (2007) 738–743. [2] M. Agrawal Paras, B.S. Sudalayandi, L.M. Raff, R. Komanduri, Computational Material Science 41 (2008) 450–456.
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