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Stability characteristics of single-walled boron nitride nanotubes
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Original Research Article
Stability characteristics of single-walled boron nitride nanotubes R. Ansari a, S. Rouhi b,*, M. Mirnezhad a, M. Aryayi a a b
Department of Mechanical Engineering, University of Guilan, P.O. Box 3756, Rasht, Iran Young Researchers Club, Langroud Branch, Islamic Azad University, Langroud, Guilan, Iran
article info
abstract
Article history:
Boron nitride nanotubes, like carbon nanotubes, possess extraordinary mechanical proper-
Received 8 December 2012
ties. Herein, a three-dimensional finite element model is proposed in which the nanotubes
Accepted 18 January 2014
are modeled using the principles of structural mechanics. To obtain the properties of this
Available online xxx
model, a linkage between the molecular mechanics and the density functional theory is
Keywords:
nitride nanotubes with different geometries and boundary conditions. It is shown that at the
Boron nitride nanotubes
same radius, longer nanotubes are less stable. However, for sufficiently long nanotubes the
Buckling
effect of side length decreases.
constructed. The model is utilized to study the buckling behavior of single-walled boron
Finite element model
# 2014 Politechnika Wrocławska. Published by Elsevier Urban & Partner Sp. z o.o. All
Density functional theory
1.
Introduction
The discovery of carbon nanotubes (CNTs) by Iijima [1] attracted a great deal of research community attention. This is mainly due to their unique mechanical, electrical and thermal properties [2–4]. The existence possibility of some non-carbon nanotubes such as boron nitride nanotubes (BNNTs) was also studied by some researchers [5,6]. It has been shown that BNNTs, like CNTs, possess extraordinary mechanical properties, such as high tensile rigidity [7,8], high thermal conductivity along the nanotube [9], and good resistance to oxidation at high temperature [10]. Some experimental studies have been done to obtain Young's modulus of BNNTs. Using a direct force method, Golberg et al. [11] predicted Young's modulus as 0.5 TPa. Suryavanshi et al. [12] utilized the electric-field-induced resonance method and computed Young's modulus as 0.8 TPa. Theoretically,
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Young's modulus has been predicted to be 1.2 TPa [7,13]. However, unlike carbon nanotubes which depending on chirality and diameter show a metallic, semiconductor or insulator characteristic, BNNTs behave as an insulator for low electric fields. This property is independent of their chirality, diameter and number of walls [6]. Generally, two classes of approaches have been used to study the mechanical behavior of nanotubes: atomistic approaches [14–16] and continuum mechanics approaches [17–19]. Song et al. [20] developed a finite-deformation shell theory for BNNTs from the interatomic potential to account for the effect of bending and curvature. Chowdhury et al. [21] used molecular mechanics simulations and continuum mechanics theories to study the axial, torsional, transverse and radial breathing vibrations of BNNTs. They obtained that the equivalent Young's modulus and shear modulus of BNNTs, independent of the chirality, are 1 TPa and 0.4 TPa, respectively.
* Corresponding author. Tel.: +98 1425244411; fax: +98 1425244422. E-mail addresses: [email protected], [email protected], [email protected] (S. Rouhi). 1644-9665/$ – see front matter # 2014 Politechnika Wrocławska. Published by Elsevier Urban & Partner Sp. z o.o. All rights reserved. http://dx.doi.org/10.1016/j.acme.2014.01.008 Please cite this article in press as: R. Ansari et al., Stability characteristics of single-walled boron nitride nanotubes, Archives of Civil and Mechanical Engineering (2014), http://dx.doi.org/10.1016/j.acme.2014.01.008
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To compute the properties of this model, some force constants should be obtained. On the basis of the electronic structure of molecules calculations, an accurate model is proposed by the quantum mechanics. But, despite of the accuracy, this model is computationally expensive and even with exerting the simplifications it is very time consuming. On the basis of the Born–Oppenheimer approximation, the motion of electrons can be ignored in the molecular mechanics and system energy can be described as a function of nucleus position. Even though the speed of calculation is increased by this simplification, the accuracy decreases. This disadvantage is solved through a linkage between the molecular mechanics and the density functional theory (DFT) in this article. A three dimensional finite element (FE) model, named as space frame model, is used to study the mechanical behavior of singlewalled boron nitride nanotubes (SWBNNTs). The DFT calculations are used to obtain accurate force constants which are employed in determining element properties. Based on the model developed, the buckling behavior of SWBNNTs is studied.
2.
Molecular mechanics modeling
Based on the molecular mechanics, the total potential energy can be expressed by the summation of several energies due to bonded interactions or bonded and non-bonded interactions [22,23] Et ¼ Ur þ Uu þ Uv þ Ut þ Uvdw þ Ues
(1)
in which Ur, Uu, Uv, and Ut are energies associated with bond stretching, bond angle variation, bond inversion, and torsion, respectively; Uvdw and Ues are also associated with van der Waals and electrostatic interactions respectively. These energy terms can be explained in different energy functions with respect to the material and loading conditions. For single-walled nanotubes, it can be expected that Ur, Uu, and Uv are the main components of the total potential energy and also Uvdw can be omitted. In small deformations conditions, Hooke law is assessed to be such an accurate and efficient enough to be used for describing the interaction between atoms in the system. Accordingly Eq. (1) can be written as the following form Et ¼
Fig. 2 – The components of average inversion angle 1.
X1 2
Kr ðDrÞ2 þ
X1 2
Cu ðDuÞ2 þ
X1 2
Cv ðDfÞ2
(2)
in which Dr, Du and Df are the bond elongation, bond angle variance and the change of space between two atoms, respectively. Force constants of Kr, Cu and Cv are related to the stretching energy due to bond link variation, bond angle variation and bond torsion, respectively (see Fig. 1.), and they can be calculated theoretically or experimentally.
Fig. 1 – Different bonds structure of a BN cell corresponding to each energy term.
Fig. 3 – Definition for atom position of type A and B in a chiral tube.
The average inversion angle f can be computed as 1 f ¼ ðb1 þ b2 þ b3 Þ 3
(3)
The angles b1, b2 and b3 have been represented in Fig. 2. As it has been represented in Fig. 3, because of three bond length and three bond angles associated with an atom with ij indexes in a nanotube, Eq. (2) can be written in another form as follows Et ¼
X1 X 1X1 X Kr Cu ðdri jk Þ2 þ ðdui jk Þ2 2 ij 2 2 k k ij þ
X1 ij
2
Cv
1X dbi jk 3 k
¼ 1; 2; 3
!2
k (4)
in which the coefficient 1/2 in the first term of Eq. (4) implies that the bond stretching energy is accounted only once. The axial force of F acting on the chiral single-walled nanotubes can be decomposed into two components of fp and fa which are respectively perpendicular to the bond of r3 and along it (see Fig. 4). Geometric relationship between these two forces is as follows p (5) Q f p ¼ F cos 6 f a ¼ F sin
p Q 6
(6)
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in which Cudu3 is rotation moment resulting from changes in bond angle in plane r1 r2, and Cudu3u2 cos(c) is rotation moment due to du3 in plane r3 r1. Torsion angle between planes r1 r2 and r3 r1, C, can be obtained as follows cos ðCÞ ¼
tan ðu3 =2Þ tan ðu2 Þ
(10)
The geometric relationship between the angles u2 and u3 is p u3 cos (11) cos ðu2 Þ ¼ cos nþm 2 where p/n + m is the angle of bond r3 to plane r1 r2. By differentiation of both sides of Eq. (11), the following equation is achieved du2 ¼
du3 ðsin ðu3 =2Þ cos ðp=ðn þ mÞÞÞ 2 sin ðu2 Þ
(12)
By inserting Eqs. (10) and (12) into Eq. (9), for chiral singlewalled nanotubes with assumption of r3 = r1, the relationship between du3 and dr is achieved as the following form du3 ¼ Fig. 4 – A (4, 2) chiral single-walled boron-nitride nanotube subjected to axial load.
sin ðu2 Þ cot ðu3 =2Þ 4 sin ðu2 Þcot ðu3 =2Þ 2 sin ðu3 =2Þ cot ðu2 Þ cos ðp=n þ mÞ
(14)
Axial and circumferential strain for a chiral nanotube can be defined as follows [24] (7)
where the diameter and chirality of nanotube is determined by integers n and m. According to Fig. 5, the force equation is written as follows u3 u3 (8) f a cos ¼ Kr dr1 f p sin 2 2 Also from the moment equilibrium one has: ar 1 ¼ Cu du3 þ Cu du2 cos ðCÞ f p cos 2 2
(13)
in which lA ¼
in which the chiral angle Q is computed as: ! 2n þ m Q ¼ cos1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ðn2 þ nm þ m2 Þ
dr1 cot ðu3 =2Þð2lA Kr r21 Þ Cu r1
(9)
ef ¼
d½r1 sin ðu3 =2Þ þ r2 sin ðu3 =2Þ r1 sin ðu3 =2Þ þ r2 sin ðu3 =2Þ
(15)
e0 f ¼
d r3 þ r1 cos u23 u 3 r3 þ r1 cos 2
(16)
p
p
Using Eqs. (8)–(10) and (13) and also by employing simplification of r3 = r1, strains can be obtained as ! dr1 cot ðu3 =2Þ2 lA Kr r21 p ef ¼ þ1 (17) r1 Cu
e0 f ¼ p
dr1 cos ðu3 =2Þ lA Kr r21 1 r1 1 þ cos ðu3 =2Þ Cu
(18)
According to the definition, Young's modulus and Poisson ratio of an SWBNNT can be written as follows fp
p
Yf ¼
p
2pRte f
(19)
p
e0 f y¼ p ef
Fig. 5 – Schematic of a chiral single-walled boron-nitride nanotube (a) the hexagonal units and (b) force distribution in bonds r1 and r2.
(20)
After a lengthy operation, Young's modulus and Poisson ratio for chiral SWBNNT is obtained as ! 1 ðn þ mÞKr r1 Y¼ 2pRt sin ððp=3Þ þ QÞsin ðu3 =2Þ ððlA Kr r21 Þ=ðCu tan2 ðu3 =2ÞÞþ1Þ (21)
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v¼
cos ðu3 =2Þ ð1 ðlA Kr r21 =Cu ÞÞ ð1 þ cos ðu3 =2ÞÞððlA Kr r21 Þ=ðCu tan2 ðu3 =2ÞÞ þ 1Þ
(22)
In the present work the concept of surface Young's modulus is introduced to avoid defining the effective thickness. This is because of great chaos in defining effective thickness of nanotubes [25,26]. Surface Young's modulus can be defined as the conventional Young's modulus multiplied by the thickness of tube, i.e. Ys = Yt, YS ¼ Yt
! 1 ðn þ mÞKr r1 ¼ 2pR sin ððp=3Þ þ QÞ sin ðu3 =2ÞððlA Kr r21 Þ=ðCu tan2 ðu3 =2ÞÞ þ 1Þ (23)
determined based on DFT and using Hohenberg–Kohn equations [35].
3.1.2.
Kohn–Sham equations
Set of equations relevant to a particle in the form of recessive relations were represented by Kohn and Sham to derive the energy of the ground state and density of the electrons in this state [35,36]. These relations are
1 2 r þ Veff ðrÞ C i ðrÞ ¼ ei C i ðrÞ 2 Veff ðrÞ ¼ VH ðrÞ þ Vxc ðrÞ þ V ext ðrÞ Z nðrÞ dr0 VH ðrÞ ¼ jr r0 j Vxc ðrÞ ¼
3.
Calculation of force constants nðrÞ ¼
3.1. Mechanical properties of BN using density functional theory (DFT) In this section, the elastic constants BN based on the strain energy calculations is obtained in the harmonic deformation range. These calculations are accomplished based on density functional theory (DFT) [27,28] in association with the generalized gradient approximation (GGA) and using the exchange correlation of Perdew–Burke–Ernzerhof (PBE) [29,30]. Calculations presented herein are done via the Quantum-Espresso code [31,32]. Basic concepts and relationships used in the QuantumEspresso code are expressed as follows.
3.1.1.
Band structure calculations
In this section a group of atoms containing several electrons and ions is considered in which the interactions between the ions and electrons create an electronic structure [33]. These interactions can be described by the Schrödinger many-body equation as follows HCððr1 ; r2 ; . . . ; rN Þ; fRm gÞ ¼ ECððr1 ; r2 ; . . . ; rN Þ; fRm gÞ
(24)
where ri and Rm respectively represent the place of ith electron and location of ions in which according to the Born– Oppenheimer approximation it can be considered constant [34]. Besides, C and E are the many body wave function for the N electronic eigenstates and the total energy, respectively. The corresponding Hamiltonian operator, H, can be defined as follows H¼
X 1 X 2 X 1 ri þ Vext ðri ; fRm gÞ r j ri þ 2 i i< j i
(25)
This equation respectively contains the terms of kinetic energy, interactions of electrons–electrons and potential energy. Vext is the external, local, electron – nuclear potential. The equilibrium position of ions can be obtained through minimizing the total energy with respect to Rm. The theory of density functional theory (DFT) is used to solve this problem which is defined as follows: Z Z nðrÞ ¼ N dr2 . . . drN jCððr1 ; r2 ; . . . ; rN Þ; fRm gÞj2 (26) where n(r) is the electronic density in the ground state. All the characteristics of a molecular system can be
(27)
@Exc ½nðrÞ @nðrÞ
N X jC i ðrÞj2 i¼1
in which ei is the orbital energy of the corresponding Kohn–Sham orbital. Besides, VH(r) and Vxc(r) denote Hartree terms describing the electron–electron Coulomb repulsion [37,38] and exchangecorrelation potential. Exc[n(r)] is the exchange-correlation energy of an interacting system with the density of n(r).
3.1.3.
Local density approximation (LDA)
The recessive solution of Kohn–Sham is accounted as an effective process for the solid state physical problems. However, to apply this powerful method, the exchangecorrelation functional estimation is required, which is not practically in access. Accordingly, to estimate this function, a new approximation should be applied. The simplest function is the local density approximation [36]. Based on this type of approximation, the exchange-correlation energy corresponding to an electron in a homogenous electronic gas for an inconsistent distribution is achieved as follows Z ELDA ½nðrÞ ¼ nðrÞenxc0 ½nðrÞdr (28) xc n
where exc0 ½nðrÞ is the exchange-correlation energy density.
3.1.4.
Pseudopotential approximation
Internal electrons of atoms cannot have a significant effect on the interactions between electrons. Accordingly, in the Kohn–Sham equations only valence electrons are taken into consideration. The use of this principle reduces the computational operations and increases its speed. However, the wave functions of valance electrons in the core quickly change that makes the numerical solution very hard. To overcome this problem, the potential functions of ions and internal electrons are replaced by the potential functions known as Pseudopotential [39,40]. Based on the results which had been published in scientific literature [41], an increase in the size of the hexagonal unit cell does not change the results considerably. Hence, in the current work for more simplicity, the smallest size of the unit cell is considered. The Brillouin region is integrated with 20 20 1 Monkhorst and Pack k-point mesh [42]. The h-BN has a 2D hexagonal unit cell in which all atoms are located in the same
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0.8
plane. Also, lattice constant in this structure is t = 2.51 Å and the bond length is L = 1.45 Å [41].
Young's modulus and Poisson's ratio
Surface Young's modulus can be calculated using the following relation [41] ! 1 @ES 2 (29) YS ¼ A0 @e2 where A0 is plane area in equilibrium state and ES is the strain energy in which can be calculated using the following equation ES ¼ ET ðeÞ ET ðe ¼ 0Þ
(30)
where ET(e) and ET(e = 0) are respectively the total energy in the longitudinal strain e and the total energy at zero longitudinal strain. Also the longitudinal strain can be defined as e = Da/a in which a is the lattice constant. Moreover the Poisson ratio is simply the ratio of transverse strain to longitudinal strain as follows [41] n¼
etrans eaxial
0.5 0.4 0.3 0.2 0.1 0.0 0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Bending Curvature (1/A0)
Fig. 7 – Strain energy relative to fully relaxed rolled planar boron-nitride with varying radius plotted versus a quadratic approximation of the bending curvature with the flexural modulus predicted by the DFT.
0.20
where En and Esh are the strain energy per atom of a fully relaxed rolled BN sheet and the strain energy per atom of a BN sheet, respectively. The total potential energy is minimized with the constraint of constant tube length and radius to achieve pure bending of nanosheets (with now initial strain). The periodic boundary condition is applied along the axial direction of the tube to give the former constraint is. Besides, to have the latter constraint, a 2D nanosheet with the width of a is mapped into a nanotube with the radius of R = a/2p. Then, to minimize the potential energy, internal relaxation is done between the two sublattices of nanosheet in which while one sublattice is held in place, the other is free to relax. So, during energy minimization the nanotube radius is held constant. However, a fully relaxed nanotube would not be obtained in this way. In Fig. 7, Ef is plotted against curvature. The strain energy related to fully relaxed rolled h-BN sheet with different radii is illustrated as a quadratic bending energy approximate and the flexural modulus is obtained using density functional theory (DFT). The flexural modulus value calculated for h-BN sheet is equal to 0.64 eV (1.74 eV Å2/atom) and this value is independent of the rolling chirality. In the other words it can be said that these materials are isotropic.
0.15
3.3.
@2 E f
(32)
@K2
where K = (1/R) is the only main curvature of a mono-layer structure which is not zero [43,44] and Ef is strain energy per atom which can be obtained as E f ¼ En Esh
(33)
0.25
E S (eVx10)
armchair zigzag ------- Chiral
0.6
(31)
Fig. 6 reveals that the range of 0.02 < e < 0.02 can be considered as the harmonic area which in its following there is a non harmonic area in which the higher order terms in the equation of strain energy cannot be omitted. The Poisson coefficient is equal to 0.21. The surface stiffness value calculated in the present work is 282 N/m that is very close to the value of 267 N/m obtained from experimental methods [41]. This fact confirms the reliability of this method. The flexural rigidity is defined as dependence of the strain energy of a two-dimensional sheet on its curvature in a desired direction. D¼
Strain Energy Per Atom (eV/atom)
3.2.
0.7
Force constants
By inserting the values of u3 = u2 = (2p/30) and n 1 in Eqs. 22 and 23, the elastic modulus and the Poisson coefficient relations in h-BN sheet is obtained as following [45] pffiffiffi 8 3Kr YS ¼ (34) 2 ðKr r1 =Cu Þ þ 18
0.10
0.05
0.00
- 0.05 - 0. 02
- 0. 01
0. 00
0.01
0. 02
Strain ( ε)
Fig. 6 – Variation of the strain energy ES with the strain of BN nanoplate.
y¼
ðKr r21 =Cu Þ 6 ðKr r21 =Cu Þ þ 18
(35)
The values obtained for Young's modulus, Poisson's coefficient and flexural rigidity are respectively equal to
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Fig. 8 – Schematic of the (a) (5,5) armchair and (b) (9,0) zigzag SWBNNTs.
282 N/m, 0.21 and 0.64 eV. By substituting these values into above relations, force coefficients are obtained as Kr = 620.47 nN/nm and Cu = 1.05 nN nm. The relation between Cv and D can be expressed as [46]: Cv ¼ 24D
(36)
So Cv is computed as 2.47 nN nm.
4. Atomistic finite element modeling of SWBNNTs Based on the analogy between hexagonal nanostructures and macroscopic frame structures, the space-frame model was proposed. Accordingly, a three dimensional finite element method can be used to model nanotubes as a space-frame. A schematic of the space-frame model of SWBNNTs has been shown in Fig. 8. As it can be seen the seen the B and N elements are modeled by using the concentrated masses and beam elements are employed to model the bonds between elements. Using the following equations the elastic properties of beam elements can be obtained [17–19] sffiffiffiffiffi K2r L K2r Cv L Cu ;E ¼ d¼4 ;G ¼ (37) 4pCu kr 8pC2u in which d, E, G and L are the diameter, Young's modulus, shear modulus and length of the beam element. Substituting the force constants as Kr = 620.47 nN/nm, Cu = 1.05 nN nm, Cv = 2.47 nN nm and L = 0.145 nm into Eq. (32), the elastic
Table 1 – Selected radii for nanotubes. Armchair Chairality Radius (Å) Zigzag Chairality Radius (Å)
(5,5) 3.4616
(11,11) 7.6151
(9,0) 3.5974
(19,0) 7.5945
properties of beam elements can be computed as d = 1.648 Å, E = 42,155 108 N/Å2 and G = 4.9437 108 N/Å2.
5.
Results
The following equation can be used to obtain the buckling force of an SWBNNT [47]: ðKE þ lKG ÞU ¼ lP
(38)
in which KE, KG and U are the elastic matrix, geometrically nonlinear matrix for the applied load, P*, and displacement vector, respectively [37]. By vanishing the structure stiffness at bifurcation, one would have: KE þ lK ¼ 0 (39) G The minimum value of l obtained from the above equation is the critical buckling force factor. So the critical buckling force is obtained as: Pcr ¼ lmin P
(40)
Fig. 9 – Critical compressive forces of (a) simply supported, (b) clamped and (c) clamped-free nanotubes of radii R = 3.4616 Å (armchair) and R = 3.5974 Å (zigzag) versus nanotube aspect ratio. Please cite this article in press as: R. Ansari et al., Stability characteristics of single-walled boron nitride nanotubes, Archives of Civil and Mechanical Engineering (2014), http://dx.doi.org/10.1016/j.acme.2014.01.008
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Fig. 10 – Critical compressive forces of (a) simply supported, (b) clamped and (c) clamped-free nanotubes of radii R = 7.6156 Å (armchair) and R = 7.5945 Å (zigzag) versus nanotube aspect ratio.
By modeling the single-walled boron nitride nanotubes as space-frame structures, the ANSYS commercial FE code was used to analyze the buckling behavior of SWBNNTs. The three dimensional BEAM4 element is used to model the B–N bonds which has six degrees of freedom at each node. Using the computed diameter for the beam elements, the element geometrical properties are obtained as A = 2.132 Å2, Ixx = 0.7250 Å4 and Iyy = Izz = 0.3625 Å4, where A and Ixx are the cross-sectional area and torsional moment of inertia and Iyy and Izz are area moment of inertia about y and z axis, respectively. The Eigen buckling analysis type is used to extract the critical buckling force of structures. To do this, the axial compressive force is applied uniformly at one end of nanotubes. The critical compressive force is computed for different geometries and boundary conditions, including clamped–clamped, clamped–free and simply supported–simply supported which briefly is written as clamped, clamped-free and simply supported, respectively. The selected radii are represented in Table 1. The buckling behavior of zigzag and armchair SWBNNTs for the mentioned radii are drawn in Figs. 8 and 9. From the figures it can be observed that as it is expected the critical force
reduces with increasing the side length. The slope of this reduction is higher at first. It can also be seen that the zigzag nanotubes buckle at larger loads than the armchair ones, although the difference reduces at larger aspect ratios. For example for R ffi 3.5 nm under simply supported boundary conditions, at L/R = 1, 7 and 15 the differences are 55.13%, 18.56% and 11.09%. It means that increasing aspect ratios results in decreasing the effect of length. Comparing Figs. 9 and 10 reveals that nanotubes with smaller radii are more stable than the larger ones. Fig. 11 is drawn to show the effect of boundary conditions. As it is anticipated the nanotubes with clamped ends are more stable than simply supported ones and nanotubes with clamped-free boundary conditions have the smallest critical compressive force. It can be seen that as the aspect ratio increases, the difference between the curves, specially between clamped and simply supported ones, reduces. Finally, Table 2 briefly shows the computed and utilized coefficients in this paper with their values.
Table 2 – Computed and utilized coefficients with their values. Coefficient
Fig. 11 – Critical compressive forces of an armchair nanotube with radius R = 3.4616 Å versus nanotube aspect ratio under different boundary conditions.
t L n Ys D Kr Cu Cv d E G A Ixx Iyy Izz
Value 2.51 Å 1.45 Å 0.21 282 N/m 0.64 eV 620.47 nN/nm 1.05 nN nm 2.47 nN nm 1.648 Å 4.2155 108 N/Å 2 4.9437 108 N/Å 2 2.132 Å 2 0.7250 Å 4 0.3625 Å 4 0.3625 Å 4
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6.
Concluding remarks
The critical compressive force of single-walled boron nitride nanotubes was studied in the present work. For this purpose space-frame FE model was employed. In the space frame model, the bonds and atoms of nanotubes are modeled with beams and masses. To derive the elastic properties of beams, a linkage between molecular and quantum mechanics was constructed. To this end, surface Young's modulus, Poisson's ratio and flexural rigidity were obtained by using DFT method and then substituted into the molecular mechanics relations to compute force constants, Kr, Cu and Cv. These force constants were used to compute Young's modulus, shear modulus and diameter of beams. It was observed that the zigzag nanotubes are more stable than the armchair ones. Also it was shown that the nanotubes with larger radii buckle at smaller forces. Comparing the critical force for different boundary conditions showed that increasing the lengths of nanotubes results in reducing the effect of end conditions.
references
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Available online at www.sciencedirect.com
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Original Research Article
Stability characteristics of single-walled boron nitride nanotubes R. Ansari a, S. Rouhi b,*, M. Mirnezhad a, M. Aryayi a a b
Department of Mechanical Engineering, University of Guilan, P.O. Box 3756, Rasht, Iran Young Researchers Club, Langroud Branch, Islamic Azad University, Langroud, Guilan, Iran
article info
abstract
Article history:
Boron nitride nanotubes, like carbon nanotubes, possess extraordinary mechanical proper-
Received 8 December 2012
ties. Herein, a three-dimensional finite element model is proposed in which the nanotubes
Accepted 18 January 2014
are modeled using the principles of structural mechanics. To obtain the properties of this
Available online xxx
model, a linkage between the molecular mechanics and the density functional theory is
Keywords:
nitride nanotubes with different geometries and boundary conditions. It is shown that at the
Boron nitride nanotubes
same radius, longer nanotubes are less stable. However, for sufficiently long nanotubes the
Buckling
effect of side length decreases.
constructed. The model is utilized to study the buckling behavior of single-walled boron
Finite element model
# 2014 Politechnika Wrocławska. Published by Elsevier Urban & Partner Sp. z o.o. All
Density functional theory
1.
Introduction
The discovery of carbon nanotubes (CNTs) by Iijima [1] attracted a great deal of research community attention. This is mainly due to their unique mechanical, electrical and thermal properties [2–4]. The existence possibility of some non-carbon nanotubes such as boron nitride nanotubes (BNNTs) was also studied by some researchers [5,6]. It has been shown that BNNTs, like CNTs, possess extraordinary mechanical properties, such as high tensile rigidity [7,8], high thermal conductivity along the nanotube [9], and good resistance to oxidation at high temperature [10]. Some experimental studies have been done to obtain Young's modulus of BNNTs. Using a direct force method, Golberg et al. [11] predicted Young's modulus as 0.5 TPa. Suryavanshi et al. [12] utilized the electric-field-induced resonance method and computed Young's modulus as 0.8 TPa. Theoretically,
rights reserved.
Young's modulus has been predicted to be 1.2 TPa [7,13]. However, unlike carbon nanotubes which depending on chirality and diameter show a metallic, semiconductor or insulator characteristic, BNNTs behave as an insulator for low electric fields. This property is independent of their chirality, diameter and number of walls [6]. Generally, two classes of approaches have been used to study the mechanical behavior of nanotubes: atomistic approaches [14–16] and continuum mechanics approaches [17–19]. Song et al. [20] developed a finite-deformation shell theory for BNNTs from the interatomic potential to account for the effect of bending and curvature. Chowdhury et al. [21] used molecular mechanics simulations and continuum mechanics theories to study the axial, torsional, transverse and radial breathing vibrations of BNNTs. They obtained that the equivalent Young's modulus and shear modulus of BNNTs, independent of the chirality, are 1 TPa and 0.4 TPa, respectively.
* Corresponding author. Tel.: +98 1425244411; fax: +98 1425244422. E-mail addresses: [email protected], [email protected], [email protected] (S. Rouhi). 1644-9665/$ – see front matter # 2014 Politechnika Wrocławska. Published by Elsevier Urban & Partner Sp. z o.o. All rights reserved. http://dx.doi.org/10.1016/j.acme.2014.01.008 Please cite this article in press as: R. Ansari et al., Stability characteristics of single-walled boron nitride nanotubes, Archives of Civil and Mechanical Engineering (2014), http://dx.doi.org/10.1016/j.acme.2014.01.008
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To compute the properties of this model, some force constants should be obtained. On the basis of the electronic structure of molecules calculations, an accurate model is proposed by the quantum mechanics. But, despite of the accuracy, this model is computationally expensive and even with exerting the simplifications it is very time consuming. On the basis of the Born–Oppenheimer approximation, the motion of electrons can be ignored in the molecular mechanics and system energy can be described as a function of nucleus position. Even though the speed of calculation is increased by this simplification, the accuracy decreases. This disadvantage is solved through a linkage between the molecular mechanics and the density functional theory (DFT) in this article. A three dimensional finite element (FE) model, named as space frame model, is used to study the mechanical behavior of singlewalled boron nitride nanotubes (SWBNNTs). The DFT calculations are used to obtain accurate force constants which are employed in determining element properties. Based on the model developed, the buckling behavior of SWBNNTs is studied.
2.
Molecular mechanics modeling
Based on the molecular mechanics, the total potential energy can be expressed by the summation of several energies due to bonded interactions or bonded and non-bonded interactions [22,23] Et ¼ Ur þ Uu þ Uv þ Ut þ Uvdw þ Ues
(1)
in which Ur, Uu, Uv, and Ut are energies associated with bond stretching, bond angle variation, bond inversion, and torsion, respectively; Uvdw and Ues are also associated with van der Waals and electrostatic interactions respectively. These energy terms can be explained in different energy functions with respect to the material and loading conditions. For single-walled nanotubes, it can be expected that Ur, Uu, and Uv are the main components of the total potential energy and also Uvdw can be omitted. In small deformations conditions, Hooke law is assessed to be such an accurate and efficient enough to be used for describing the interaction between atoms in the system. Accordingly Eq. (1) can be written as the following form Et ¼
Fig. 2 – The components of average inversion angle 1.
X1 2
Kr ðDrÞ2 þ
X1 2
Cu ðDuÞ2 þ
X1 2
Cv ðDfÞ2
(2)
in which Dr, Du and Df are the bond elongation, bond angle variance and the change of space between two atoms, respectively. Force constants of Kr, Cu and Cv are related to the stretching energy due to bond link variation, bond angle variation and bond torsion, respectively (see Fig. 1.), and they can be calculated theoretically or experimentally.
Fig. 1 – Different bonds structure of a BN cell corresponding to each energy term.
Fig. 3 – Definition for atom position of type A and B in a chiral tube.
The average inversion angle f can be computed as 1 f ¼ ðb1 þ b2 þ b3 Þ 3
(3)
The angles b1, b2 and b3 have been represented in Fig. 2. As it has been represented in Fig. 3, because of three bond length and three bond angles associated with an atom with ij indexes in a nanotube, Eq. (2) can be written in another form as follows Et ¼
X1 X 1X1 X Kr Cu ðdri jk Þ2 þ ðdui jk Þ2 2 ij 2 2 k k ij þ
X1 ij
2
Cv
1X dbi jk 3 k
¼ 1; 2; 3
!2
k (4)
in which the coefficient 1/2 in the first term of Eq. (4) implies that the bond stretching energy is accounted only once. The axial force of F acting on the chiral single-walled nanotubes can be decomposed into two components of fp and fa which are respectively perpendicular to the bond of r3 and along it (see Fig. 4). Geometric relationship between these two forces is as follows p (5) Q f p ¼ F cos 6 f a ¼ F sin
p Q 6
(6)
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in which Cudu3 is rotation moment resulting from changes in bond angle in plane r1 r2, and Cudu3u2 cos(c) is rotation moment due to du3 in plane r3 r1. Torsion angle between planes r1 r2 and r3 r1, C, can be obtained as follows cos ðCÞ ¼
tan ðu3 =2Þ tan ðu2 Þ
(10)
The geometric relationship between the angles u2 and u3 is p u3 cos (11) cos ðu2 Þ ¼ cos nþm 2 where p/n + m is the angle of bond r3 to plane r1 r2. By differentiation of both sides of Eq. (11), the following equation is achieved du2 ¼
du3 ðsin ðu3 =2Þ cos ðp=ðn þ mÞÞÞ 2 sin ðu2 Þ
(12)
By inserting Eqs. (10) and (12) into Eq. (9), for chiral singlewalled nanotubes with assumption of r3 = r1, the relationship between du3 and dr is achieved as the following form du3 ¼ Fig. 4 – A (4, 2) chiral single-walled boron-nitride nanotube subjected to axial load.
sin ðu2 Þ cot ðu3 =2Þ 4 sin ðu2 Þcot ðu3 =2Þ 2 sin ðu3 =2Þ cot ðu2 Þ cos ðp=n þ mÞ
(14)
Axial and circumferential strain for a chiral nanotube can be defined as follows [24] (7)
where the diameter and chirality of nanotube is determined by integers n and m. According to Fig. 5, the force equation is written as follows u3 u3 (8) f a cos ¼ Kr dr1 f p sin 2 2 Also from the moment equilibrium one has: ar 1 ¼ Cu du3 þ Cu du2 cos ðCÞ f p cos 2 2
(13)
in which lA ¼
in which the chiral angle Q is computed as: ! 2n þ m Q ¼ cos1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ðn2 þ nm þ m2 Þ
dr1 cot ðu3 =2Þð2lA Kr r21 Þ Cu r1
(9)
ef ¼
d½r1 sin ðu3 =2Þ þ r2 sin ðu3 =2Þ r1 sin ðu3 =2Þ þ r2 sin ðu3 =2Þ
(15)
e0 f ¼
d r3 þ r1 cos u23 u 3 r3 þ r1 cos 2
(16)
p
p
Using Eqs. (8)–(10) and (13) and also by employing simplification of r3 = r1, strains can be obtained as ! dr1 cot ðu3 =2Þ2 lA Kr r21 p ef ¼ þ1 (17) r1 Cu
e0 f ¼ p
dr1 cos ðu3 =2Þ lA Kr r21 1 r1 1 þ cos ðu3 =2Þ Cu
(18)
According to the definition, Young's modulus and Poisson ratio of an SWBNNT can be written as follows fp
p
Yf ¼
p
2pRte f
(19)
p
e0 f y¼ p ef
Fig. 5 – Schematic of a chiral single-walled boron-nitride nanotube (a) the hexagonal units and (b) force distribution in bonds r1 and r2.
(20)
After a lengthy operation, Young's modulus and Poisson ratio for chiral SWBNNT is obtained as ! 1 ðn þ mÞKr r1 Y¼ 2pRt sin ððp=3Þ þ QÞsin ðu3 =2Þ ððlA Kr r21 Þ=ðCu tan2 ðu3 =2ÞÞþ1Þ (21)
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v¼
cos ðu3 =2Þ ð1 ðlA Kr r21 =Cu ÞÞ ð1 þ cos ðu3 =2ÞÞððlA Kr r21 Þ=ðCu tan2 ðu3 =2ÞÞ þ 1Þ
(22)
In the present work the concept of surface Young's modulus is introduced to avoid defining the effective thickness. This is because of great chaos in defining effective thickness of nanotubes [25,26]. Surface Young's modulus can be defined as the conventional Young's modulus multiplied by the thickness of tube, i.e. Ys = Yt, YS ¼ Yt
! 1 ðn þ mÞKr r1 ¼ 2pR sin ððp=3Þ þ QÞ sin ðu3 =2ÞððlA Kr r21 Þ=ðCu tan2 ðu3 =2ÞÞ þ 1Þ (23)
determined based on DFT and using Hohenberg–Kohn equations [35].
3.1.2.
Kohn–Sham equations
Set of equations relevant to a particle in the form of recessive relations were represented by Kohn and Sham to derive the energy of the ground state and density of the electrons in this state [35,36]. These relations are
1 2 r þ Veff ðrÞ C i ðrÞ ¼ ei C i ðrÞ 2 Veff ðrÞ ¼ VH ðrÞ þ Vxc ðrÞ þ V ext ðrÞ Z nðrÞ dr0 VH ðrÞ ¼ jr r0 j Vxc ðrÞ ¼
3.
Calculation of force constants nðrÞ ¼
3.1. Mechanical properties of BN using density functional theory (DFT) In this section, the elastic constants BN based on the strain energy calculations is obtained in the harmonic deformation range. These calculations are accomplished based on density functional theory (DFT) [27,28] in association with the generalized gradient approximation (GGA) and using the exchange correlation of Perdew–Burke–Ernzerhof (PBE) [29,30]. Calculations presented herein are done via the Quantum-Espresso code [31,32]. Basic concepts and relationships used in the QuantumEspresso code are expressed as follows.
3.1.1.
Band structure calculations
In this section a group of atoms containing several electrons and ions is considered in which the interactions between the ions and electrons create an electronic structure [33]. These interactions can be described by the Schrödinger many-body equation as follows HCððr1 ; r2 ; . . . ; rN Þ; fRm gÞ ¼ ECððr1 ; r2 ; . . . ; rN Þ; fRm gÞ
(24)
where ri and Rm respectively represent the place of ith electron and location of ions in which according to the Born– Oppenheimer approximation it can be considered constant [34]. Besides, C and E are the many body wave function for the N electronic eigenstates and the total energy, respectively. The corresponding Hamiltonian operator, H, can be defined as follows H¼
X 1 X 2 X 1 ri þ Vext ðri ; fRm gÞ r j ri þ 2 i i< j i
(25)
This equation respectively contains the terms of kinetic energy, interactions of electrons–electrons and potential energy. Vext is the external, local, electron – nuclear potential. The equilibrium position of ions can be obtained through minimizing the total energy with respect to Rm. The theory of density functional theory (DFT) is used to solve this problem which is defined as follows: Z Z nðrÞ ¼ N dr2 . . . drN jCððr1 ; r2 ; . . . ; rN Þ; fRm gÞj2 (26) where n(r) is the electronic density in the ground state. All the characteristics of a molecular system can be
(27)
@Exc ½nðrÞ @nðrÞ
N X jC i ðrÞj2 i¼1
in which ei is the orbital energy of the corresponding Kohn–Sham orbital. Besides, VH(r) and Vxc(r) denote Hartree terms describing the electron–electron Coulomb repulsion [37,38] and exchangecorrelation potential. Exc[n(r)] is the exchange-correlation energy of an interacting system with the density of n(r).
3.1.3.
Local density approximation (LDA)
The recessive solution of Kohn–Sham is accounted as an effective process for the solid state physical problems. However, to apply this powerful method, the exchangecorrelation functional estimation is required, which is not practically in access. Accordingly, to estimate this function, a new approximation should be applied. The simplest function is the local density approximation [36]. Based on this type of approximation, the exchange-correlation energy corresponding to an electron in a homogenous electronic gas for an inconsistent distribution is achieved as follows Z ELDA ½nðrÞ ¼ nðrÞenxc0 ½nðrÞdr (28) xc n
where exc0 ½nðrÞ is the exchange-correlation energy density.
3.1.4.
Pseudopotential approximation
Internal electrons of atoms cannot have a significant effect on the interactions between electrons. Accordingly, in the Kohn–Sham equations only valence electrons are taken into consideration. The use of this principle reduces the computational operations and increases its speed. However, the wave functions of valance electrons in the core quickly change that makes the numerical solution very hard. To overcome this problem, the potential functions of ions and internal electrons are replaced by the potential functions known as Pseudopotential [39,40]. Based on the results which had been published in scientific literature [41], an increase in the size of the hexagonal unit cell does not change the results considerably. Hence, in the current work for more simplicity, the smallest size of the unit cell is considered. The Brillouin region is integrated with 20 20 1 Monkhorst and Pack k-point mesh [42]. The h-BN has a 2D hexagonal unit cell in which all atoms are located in the same
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0.8
plane. Also, lattice constant in this structure is t = 2.51 Å and the bond length is L = 1.45 Å [41].
Young's modulus and Poisson's ratio
Surface Young's modulus can be calculated using the following relation [41] ! 1 @ES 2 (29) YS ¼ A0 @e2 where A0 is plane area in equilibrium state and ES is the strain energy in which can be calculated using the following equation ES ¼ ET ðeÞ ET ðe ¼ 0Þ
(30)
where ET(e) and ET(e = 0) are respectively the total energy in the longitudinal strain e and the total energy at zero longitudinal strain. Also the longitudinal strain can be defined as e = Da/a in which a is the lattice constant. Moreover the Poisson ratio is simply the ratio of transverse strain to longitudinal strain as follows [41] n¼
etrans eaxial
0.5 0.4 0.3 0.2 0.1 0.0 0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Bending Curvature (1/A0)
Fig. 7 – Strain energy relative to fully relaxed rolled planar boron-nitride with varying radius plotted versus a quadratic approximation of the bending curvature with the flexural modulus predicted by the DFT.
0.20
where En and Esh are the strain energy per atom of a fully relaxed rolled BN sheet and the strain energy per atom of a BN sheet, respectively. The total potential energy is minimized with the constraint of constant tube length and radius to achieve pure bending of nanosheets (with now initial strain). The periodic boundary condition is applied along the axial direction of the tube to give the former constraint is. Besides, to have the latter constraint, a 2D nanosheet with the width of a is mapped into a nanotube with the radius of R = a/2p. Then, to minimize the potential energy, internal relaxation is done between the two sublattices of nanosheet in which while one sublattice is held in place, the other is free to relax. So, during energy minimization the nanotube radius is held constant. However, a fully relaxed nanotube would not be obtained in this way. In Fig. 7, Ef is plotted against curvature. The strain energy related to fully relaxed rolled h-BN sheet with different radii is illustrated as a quadratic bending energy approximate and the flexural modulus is obtained using density functional theory (DFT). The flexural modulus value calculated for h-BN sheet is equal to 0.64 eV (1.74 eV Å2/atom) and this value is independent of the rolling chirality. In the other words it can be said that these materials are isotropic.
0.15
3.3.
@2 E f
(32)
@K2
where K = (1/R) is the only main curvature of a mono-layer structure which is not zero [43,44] and Ef is strain energy per atom which can be obtained as E f ¼ En Esh
(33)
0.25
E S (eVx10)
armchair zigzag ------- Chiral
0.6
(31)
Fig. 6 reveals that the range of 0.02 < e < 0.02 can be considered as the harmonic area which in its following there is a non harmonic area in which the higher order terms in the equation of strain energy cannot be omitted. The Poisson coefficient is equal to 0.21. The surface stiffness value calculated in the present work is 282 N/m that is very close to the value of 267 N/m obtained from experimental methods [41]. This fact confirms the reliability of this method. The flexural rigidity is defined as dependence of the strain energy of a two-dimensional sheet on its curvature in a desired direction. D¼
Strain Energy Per Atom (eV/atom)
3.2.
0.7
Force constants
By inserting the values of u3 = u2 = (2p/30) and n 1 in Eqs. 22 and 23, the elastic modulus and the Poisson coefficient relations in h-BN sheet is obtained as following [45] pffiffiffi 8 3Kr YS ¼ (34) 2 ðKr r1 =Cu Þ þ 18
0.10
0.05
0.00
- 0.05 - 0. 02
- 0. 01
0. 00
0.01
0. 02
Strain ( ε)
Fig. 6 – Variation of the strain energy ES with the strain of BN nanoplate.
y¼
ðKr r21 =Cu Þ 6 ðKr r21 =Cu Þ þ 18
(35)
The values obtained for Young's modulus, Poisson's coefficient and flexural rigidity are respectively equal to
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Fig. 8 – Schematic of the (a) (5,5) armchair and (b) (9,0) zigzag SWBNNTs.
282 N/m, 0.21 and 0.64 eV. By substituting these values into above relations, force coefficients are obtained as Kr = 620.47 nN/nm and Cu = 1.05 nN nm. The relation between Cv and D can be expressed as [46]: Cv ¼ 24D
(36)
So Cv is computed as 2.47 nN nm.
4. Atomistic finite element modeling of SWBNNTs Based on the analogy between hexagonal nanostructures and macroscopic frame structures, the space-frame model was proposed. Accordingly, a three dimensional finite element method can be used to model nanotubes as a space-frame. A schematic of the space-frame model of SWBNNTs has been shown in Fig. 8. As it can be seen the seen the B and N elements are modeled by using the concentrated masses and beam elements are employed to model the bonds between elements. Using the following equations the elastic properties of beam elements can be obtained [17–19] sffiffiffiffiffi K2r L K2r Cv L Cu ;E ¼ d¼4 ;G ¼ (37) 4pCu kr 8pC2u in which d, E, G and L are the diameter, Young's modulus, shear modulus and length of the beam element. Substituting the force constants as Kr = 620.47 nN/nm, Cu = 1.05 nN nm, Cv = 2.47 nN nm and L = 0.145 nm into Eq. (32), the elastic
Table 1 – Selected radii for nanotubes. Armchair Chairality Radius (Å) Zigzag Chairality Radius (Å)
(5,5) 3.4616
(11,11) 7.6151
(9,0) 3.5974
(19,0) 7.5945
properties of beam elements can be computed as d = 1.648 Å, E = 42,155 108 N/Å2 and G = 4.9437 108 N/Å2.
5.
Results
The following equation can be used to obtain the buckling force of an SWBNNT [47]: ðKE þ lKG ÞU ¼ lP
(38)
in which KE, KG and U are the elastic matrix, geometrically nonlinear matrix for the applied load, P*, and displacement vector, respectively [37]. By vanishing the structure stiffness at bifurcation, one would have: KE þ lK ¼ 0 (39) G The minimum value of l obtained from the above equation is the critical buckling force factor. So the critical buckling force is obtained as: Pcr ¼ lmin P
(40)
Fig. 9 – Critical compressive forces of (a) simply supported, (b) clamped and (c) clamped-free nanotubes of radii R = 3.4616 Å (armchair) and R = 3.5974 Å (zigzag) versus nanotube aspect ratio. Please cite this article in press as: R. Ansari et al., Stability characteristics of single-walled boron nitride nanotubes, Archives of Civil and Mechanical Engineering (2014), http://dx.doi.org/10.1016/j.acme.2014.01.008
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Fig. 10 – Critical compressive forces of (a) simply supported, (b) clamped and (c) clamped-free nanotubes of radii R = 7.6156 Å (armchair) and R = 7.5945 Å (zigzag) versus nanotube aspect ratio.
By modeling the single-walled boron nitride nanotubes as space-frame structures, the ANSYS commercial FE code was used to analyze the buckling behavior of SWBNNTs. The three dimensional BEAM4 element is used to model the B–N bonds which has six degrees of freedom at each node. Using the computed diameter for the beam elements, the element geometrical properties are obtained as A = 2.132 Å2, Ixx = 0.7250 Å4 and Iyy = Izz = 0.3625 Å4, where A and Ixx are the cross-sectional area and torsional moment of inertia and Iyy and Izz are area moment of inertia about y and z axis, respectively. The Eigen buckling analysis type is used to extract the critical buckling force of structures. To do this, the axial compressive force is applied uniformly at one end of nanotubes. The critical compressive force is computed for different geometries and boundary conditions, including clamped–clamped, clamped–free and simply supported–simply supported which briefly is written as clamped, clamped-free and simply supported, respectively. The selected radii are represented in Table 1. The buckling behavior of zigzag and armchair SWBNNTs for the mentioned radii are drawn in Figs. 8 and 9. From the figures it can be observed that as it is expected the critical force
reduces with increasing the side length. The slope of this reduction is higher at first. It can also be seen that the zigzag nanotubes buckle at larger loads than the armchair ones, although the difference reduces at larger aspect ratios. For example for R ffi 3.5 nm under simply supported boundary conditions, at L/R = 1, 7 and 15 the differences are 55.13%, 18.56% and 11.09%. It means that increasing aspect ratios results in decreasing the effect of length. Comparing Figs. 9 and 10 reveals that nanotubes with smaller radii are more stable than the larger ones. Fig. 11 is drawn to show the effect of boundary conditions. As it is anticipated the nanotubes with clamped ends are more stable than simply supported ones and nanotubes with clamped-free boundary conditions have the smallest critical compressive force. It can be seen that as the aspect ratio increases, the difference between the curves, specially between clamped and simply supported ones, reduces. Finally, Table 2 briefly shows the computed and utilized coefficients in this paper with their values.
Table 2 – Computed and utilized coefficients with their values. Coefficient
Fig. 11 – Critical compressive forces of an armchair nanotube with radius R = 3.4616 Å versus nanotube aspect ratio under different boundary conditions.
t L n Ys D Kr Cu Cv d E G A Ixx Iyy Izz
Value 2.51 Å 1.45 Å 0.21 282 N/m 0.64 eV 620.47 nN/nm 1.05 nN nm 2.47 nN nm 1.648 Å 4.2155 108 N/Å 2 4.9437 108 N/Å 2 2.132 Å 2 0.7250 Å 4 0.3625 Å 4 0.3625 Å 4
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6.
Concluding remarks
The critical compressive force of single-walled boron nitride nanotubes was studied in the present work. For this purpose space-frame FE model was employed. In the space frame model, the bonds and atoms of nanotubes are modeled with beams and masses. To derive the elastic properties of beams, a linkage between molecular and quantum mechanics was constructed. To this end, surface Young's modulus, Poisson's ratio and flexural rigidity were obtained by using DFT method and then substituted into the molecular mechanics relations to compute force constants, Kr, Cu and Cv. These force constants were used to compute Young's modulus, shear modulus and diameter of beams. It was observed that the zigzag nanotubes are more stable than the armchair ones. Also it was shown that the nanotubes with larger radii buckle at smaller forces. Comparing the critical force for different boundary conditions showed that increasing the lengths of nanotubes results in reducing the effect of end conditions.
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