Computational Materials Science 82 (2014) 257–263
Contents lists available at ScienceDirect
Computational Materials Science
journal homepage: www.elsevier.com/locate/commatsci
An assessment of finite element analysis to predict the elastic modulus
and Poisson’s ratio of singlewall carbon nanotubes
G. Domínguez-Rodríguez a, A. Tapia b, F. Avilés a,⇑
a
b
Centro de Investigación Científica de Yucatán, Calle 43 No. 130, Colonia Chuburná de Hidalgo, CP 97200 Mérida, Yucatán, Mexico
Facultad de Ingeniería, Universidad Autónoma de Yucatán, Av. Industrias no Contaminantes por Periférico Norte, Cordemex, CP 97310 Mérida, Yucatán, Mexico
a r t i c l e
i n f o
Article history:
Received 9 July 2013
Received in revised form 30 August 2013
Accepted 1 October 2013
Keywords:
FEA
DFT
CNT
Armchair
Elastic properties
Poisson’s ratio
Singlewall
a b s t r a c t
Molecular mechanics implemented in finite element codes is becoming a popular method to predict elastic properties of carbon nanotubes. However, the limits of application of this approach have not yet been
systematically assessed. Therefore, this work investigates the accuracy and range of application of finite
element analysis (FEA) to predict the elastic modulus and Poisson’s ratio of singlewall carbon nanotubes
(SWCNTs), by comparing predictions of FEA to ab initio computations based on density functional theory
(DFT). FEA predicts an elastic modulus which agrees well with DFT for SWCNTs with diameters larger
than 0.8 nm. FEA underpredicts the Poisson’s ratio with respect to DFT, unless the FEA bond force constants previously computed for benzene are adjusted. The use of adequate values of bond force constants
and the equilibrium configuration of atomic coordinates predicted by DFT as the input geometry for FEA
improves its predictions. The significance of the numerical values chosen for the bond force constants in
FEA is also discussed.
Ó 2013 Elsevier B.V. All rights reserved.
1. Introduction
Isolated singlewall carbon nanotubes (SWCNTs) present
extraordinary mechanical properties which have been the subject
of numerous investigations. The reported elastic modulus of
SWCNTs varies around 1 TPa for diameters in the range of 1–
15 nm [1–4] and typically predicted Poisson’s ratios vary between
0.1 and 0.3 [5–7]. Regarding modeling approaches for nanostructures, ab initio methods based on quantum mechanics are accepted
as an adequate tool to predict the electronic structure and mechanical properties of SWCNTs. This family of detailed atomistic methods, however, demand high computational burden, which makes
them suitable only for very small systems [8]. In order to reduce
the extensive computing time demanded by ab initio methods,
quasicontinuum methods, such as those discussed by Park et al.
[9] have been proposed. The finite element method has also been
proposed to solve the Kohm–Sham equations [10]. Finite element
analysis (FEA) provides a numerical solution to systems with specific initial value and/or boundary conditions using a variational
technique for solving the governing differential equations, wherein
the continuous problem described by the differential equation is
cast into a linear combination of approximation functions [11].
⇑ Corresponding author. Tel.: +52 9999428330.
E-mail address: [email protected] (F. Avilés).
0927-0256/$ - see front matter Ó 2013 Elsevier B.V. All rights reserved.
http://dx.doi.org/10.1016/j.commatsci.2013.10.003
As a practical alternative to ab initio methods to calculate the
elastic properties of nanostructures, the governing interatomic
potentials can be simulated in finite element codes through the
introduction of extensional, bending and torsional elastic constants
in simplified structural models based on truss or beam elements
[12–17]. This method is referred to in the literature as the ‘‘atomistic’’ FEA and is based on an energetic equivalence between the
steric force field potentials of structural mechanics [18] and the
sectional stiffness parameters of solid beams [13].
Atomistic FEA has been used to predict the elastic properties
of carbon nanotubes and graphene sheets [12–14,19]. However,
the conventional FEA method proposed by Li and Chou [13]
involves only linear elastic potentials neglecting all others
non-bonded (nonlinear) interactions. Although the inclusion of
non-bonded potentials is possible in FEA [20] this practice
may compromise one of the main advantages of the method,
i.e. its facile implementation. Furthermore, FEA is a method
based on classical mechanics and as such it does not consider
the implications of the quantum interactions in the properties
of SWCNTs of small radius. An additional issue of FEA which
has been largely ignored in the literature is the different initial
positions used for the atoms (nodes) of the SWCNTs, with
respect to ab initio methods such as density functional theory
(DFT). While FEA uses fixed atomic bond lengths which define
the atomic (nodal) positions for a given SWCNT’s architecture,
the geometrical configuration is first relaxed by DFT to find
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G. Domínguez-Rodríguez et al. / Computational Materials Science 82 (2014) 257–263
Nomenclature
a, b
aeq, beq
AT
dbeam
Ebeam
Err
Ez
Gbeam
H
kr
o
kr
kh
o
kh
k/
o
k/
L
Lbeam
n
geometric bond lengths
equilibrium bond lengths
SWCNT transverse area
beam’s diameter
beam’s axial elastic modulus
error function
SWCNT axial elastic modulus
beam’s shear elastic modulus
unit cell height
axial bond force constant
axial bond force constant for benzene
bending bond force constant
bending bond force constant for benzene
torsion bond force constant
torsion bond force constant for benzene
SWCNT’s length
beam length
integer number
the atomic coordinates that minimizes the energy prior to the
calculation of elastic properties.
Therefore, it is clear that a systematic comparison between the
computations of atomistic FEA and an ab initio method (DFT) is
needed to bound the range of applicability of FEA to predict the
elastic properties of SWCNTs, which constitutes the main aim of
this investigation.
2. Methodology
2.1. Density functional theory
In order to compute the elastic properties of SWCNTs, their
atomistic structure (Fig. 1(a)) was represented by an equivalent
continuum configuration as shown in Fig. 1(b). For this homogenization process, H = 2.46 Å is the initial height of the unit cell, R is
the SWCNT’s radius, t = 3.4 Å is the thickness of the SWCNT, and
a = b = 1.42 Å are the initial carbon–carbon (C–C) bond lengths.
The outer radius in Fig. 1(b) is R þ 2t and the inner one is R 2t , so
the homogenized transverse area of the SWCNTs (AT) is the area
of an annular region of thickness t.
The DFT calculations [21] were implemented in the ‘‘SIESTA’’
code [22,23] employing the generalized gradient approximation
(GGA) potential, which is based on the work of Perdew et al. [24]
and fulfills the DFT requirements to model carbon nanotubes
[25,26]. The electron–ion interactions are treated by means of
norm conserving pseudopotentials following the Troullier and
(a)
(b)
(n, n)
P
Po
nanotube chiral vector for armchair SWCNTs
elastic property
elastic property calculated by using the benzene bond
force constants
axial load
SWCNT’s radius
SWCNT’s equilibrium radius
SWCNT’s wall thickness
axial deformation potential
unit cell volume
cartesian coordinate system
bond angle between a and z
axial strain
transverse strain
SWCNT Poisson’s ratio in the zr plane
axial stress
Pz
R
Req
t
U(z)
V0
x, y, z
a
z
r
mzr
rz
Martins [27] procedure. For the basis set, an optimized basis based
on the work of Anglada et al. [28] with a simulated pressure of
0.2 GPa was chosen. A uniform grid was used in the real space with
a mesh cutoff of 375 Rydberg. The total energy was integrated at
80 specific points of the Brillouin zone resulting from the diagonal
Monkhorst–Pack matrix (1 1 80) [29]. Unit cells were used to
model infinite armchair SWCNTs, and selected unit cells are
sketched in Fig. 2.
The computations were conducted for (3, 3), (4, 4), (5, 5), (6, 6),
(7, 7), (8, 8) and (9, 9) armchair SWCNTs and their initial structural
parameters based on geometric coordinates are shown in Table 1.
A distinction is made here between this initial radius (which is typically used in FEA) and the equilibrium radius of the SWCNTs,
which is achieved once the system is allowed to relax and is computed by DFT herein. A molecular dynamics evolution was implemented with the conjugate gradient method to first obtain the
equilibrium (relaxed) geometry, setting a criterion of maximum
force at 0.01 eV/Å.
Once equilibrium was achieved, the longitudinal elastic modulus (Ez) was computed by DFT using the SWCNT equilibrium (relaxed) geometry. These coordinates were input in the SIESTA
code to calculate the energies for different values of applied longitudinal strain (z). z was varied from 0.1 to 0.1 in steps of 0.02
and a structural relaxation was performed with the conjugated
gradient method at each step to obtain the corresponding energy
value. An energy potential, U(z), was then constructed with the
energy values corresponding to each applied strain (z), and a polynomial fit was used to obtained an analytically differentiable
expression for U(z). The elastic modulus of the SWCNT (Ez) was
hence computed as,
!
1 @ 2 Uðz Þ
Ez ¼
V0
@ 2z
ð1Þ
z ¼0
where, V0 = AT H (see Fig. 1) is the equivalent continuum volume of
the unit cell when no axial strain is applied (z = 0).
A negative radial strain (r) is naturally produced as a consequence of the applied longitudinal tensile strain (z). The Poisson’s
ratio (mzr) was then calculated by plotting z vs. r and taking the
negative of the slope, such as,
Fig. 1. Geometric parameters of an armchair SWCNT. (a) The atomic structure of a
unit cell showing the bond lengths a and b, and the angle a between C–C bonds, and
(b) solid representation of the SWCNT as a hollow tube.
mzr ¼ r
z
ð2Þ
G. Domínguez-Rodríguez et al. / Computational Materials Science 82 (2014) 257–263
259
Fig. 2. Unit cells of (3, 3), (8, 8) and (9, 9) SWCNTs used in DFT.
Table 1
Initial radii of the SWCNTs modeled by DFT.
R (Å)
(3, 3)
(4, 4)
(5, 5)
(6, 6)
(7, 7)
(8, 8)
(9, 9)
2.034
2.712
3.390
4.068
4.747
5.424
6.103
2.2. Finite element analysis
The atomistic FEA was proposed by Li and Chou [13] to calculate
the elastic properties of SWCNTs. In this methodology, the C–C
bonds are modeled as solid beams, whose elastic (Ebeam) and shear
(Gbeam) moduli are obtained from an energy equivalence between
the structural deformation (stretching, bending and torsion) and
the strain energy of the C–C bonds. Each of these deformations is
associated to a force field (or bond force) constant, representative
of stretching (kr), bending (kh) and torsion (k/) [18]. The elastic
and geometric properties of the beams representing the C–C bonds
are given by [13],
dbeam ¼ 4
sffiffiffiffiffi
kh
kr
ð3aÞ
2
Ebeam ¼
kr Lbeam
4pkh
ð3bÞ
Gbeam ¼
2
kr kh Lbeam
2
8 kh
ð3cÞ
p
where dbeam is the diameter of the beams modeling the covalent
bonds and Lbeam is the C–C bond length (Lbeam = a = b = 0.142 nm,
[30]). Unfortunately, there are no reports in the literature dedicated
to the specific calculation of the bond force constants (kr, kh and k/)
of SWCNTs. The vast majority of works on atomistic FEA of SWCNTs
[19,12,13] have used the numerical value of kr and kh reported by
Cornell et al. [30] for benzene and k/ reported by Jorgensen and Severance [31] for the same molecule, i.e., kr = 6.52 107 N nm1,
kh = 8.76 1010 N nm1 rad2, and k/ = 2.78 1010 N nm1 rad2.
These values of bond force constants were used herein for the baseline results presented in Sections 3.1 and 3.3, and an analysis of the
sensitivity of the FEA predictions to these parameters is presented
in Section 3.4.
The carbon atoms were modeled as nodes joined by solid beams
with isotropic elastic properties and the beam diameter and properties were calculated by Eq. (3). A numerical solution was conducted by the commercial code ‘‘ANSYS’’ [32]. A beam element
(BEAM4) was employed to simulate the solid beams between
atoms (nodes), because of its bending, torsion and stretching capabilities. This three-dimensional element is made of two nodes with
six degrees of freedom at each node, three for translation and three
for rotation.
The simulated SWCNTs were loaded along the axial (z) direction
by applying a load Pz on all nodes at the upper edge of the SWCNT,
whereas all nodal degrees of freedom were constrained at the lower edge of the SWCNT, see Fig. 3.
Fig. 3. FEA model of a (9, 9) SWCNT of length L.
The axial strain (z) was calculated as the difference between
the unstrained and strained length of the SWCNT divided by its original length. The axial stress (rz) was obtained by dividing the applied load (Pz) by the effective transverse area (AT), see Fig. 1(b).
Finally, the axial elastic modulus was computed as,
Ez ¼
rz
z
ð4Þ
The Poisson’s ratio was also calculated in FEA by using its definition,
Eq. (2). For FEA, the transverse and axial average strains were calculated as discrete volume averages and used in Eq. (2).
3. Results and discussion
3.1. FEA predictions
The FEA predictions of elastic modulus (Ez, Fig. 4(a)) and Poisson’s ratio (mzr, Fig. 4(b)) for different SWCNT lengths (L) are presented in Fig. 4, where the elastic parameter is plotted as a
function of the SWCNT radius (R). Data points represent the actual
FEA-computed values, while the solid lines joining the data are only
trend lines. FEA predicts a slightly higher Ez for longer SWCNTs and
a minor dependency of Ez with the SWCNT radius, unless the radius
is overly small (<0.4 nm). Nevertheless, for the whole range of
SWCNT lengths and radii examined, the elastic modulus varies from
1.033 to 1.056 TPa, which differs only by 2.2%. This small difference is not likely captured by any current experimental measurements (see e.g. Treacy et al. [33] and Yu et al. [4]). The FEA
predictions of Ez are within the range of the results of other works
for armchair SWCNTs with similar dimensions [12,13].
Two main factors produce the small variation of the elastic modulus with the SWCNT radius and length in our FEA, viz. the effect of
the reduced number of bonds at the loaded edge and the influence of
the angle between the C–C bonds a in (Fig. 1(a)). As seen from Fig. 3,
each atom (node) at the loaded end has only two bonds (beams)
linking each edge atom with two neighbour atoms, while the rest
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G. Domínguez-Rodríguez et al. / Computational Materials Science 82 (2014) 257–263
(a)
(b)
1.06
0.11
SWCNT length (L)
2.46 nm
3.69 nm
4.92 nm
6.15 nm
7.38 nm
12.3 nm
24.6 nm
1.05
0.09
νzr
Ez (TPa)
0.10
1.04
1.03
SWCNT Length (L)
2.46 nm
3.69 nm
4.92 nm
6.15 nm
7.38 nm
12.3 nm
14.8 nm
24.6 nm
0.2
0.4
0.6
0.8
1.0
0.08
0.07
0.06
0.2
0.4
0.6
0.8
1.0
R (nm)
R (nm)
Fig. 4. Elastic properties of SWCNTs as function of radius predicted by FEA for various SWCNT’s lengths. (a) Elastic modulus, and (b) Poisson’s ratio.
of the atoms in the SWCNT are bonded to three atoms by three beam
elements. This renders more freedom (less elastic constrain) to the
end atoms than to the rest of the inner atoms in the SWCNT. The
fraction of end atoms with respect to the total number of atoms in
the SWCNT is higher for shorter SWCNTs, decreasing the elastic
modulus for shorter SWCNTs. For example, for a 2.36 nm long
SWCNT the relative amount of atoms at the loaded end is 5 %,
while for a 24.6 nm long this amount is 0.5%. The angle a
(Fig. 1(a)) represents a geometric contribution to the axial deformation produced on the beams. The smaller this angle the more aligned
is the beam element with the load application (z) axis and hence, the
SWCNT stiffness is increased. This angle is smaller for SWCNTs with
smaller radius (a = 15.3° for R = 0.2 nm while a = 15.4° for
R = 0.6 nm), which may also contribute to the observed increase in
the elastic modulus, Fig. 4(a). The contribution of the angle a has
more influence on short SWCNTs (L < 7 nm). A similar behavior
was found in the work of Ferreira and Rachid [12] for armchair
SWCNTs of short lengths (2.34 nm). Thus, the FEA predictions of
the SWCNT elastic modulus have a C–C angular contribution (a)
and an end contribution, being the SWCNT length which determines
which factor has more influence on the overall elastic response.
According to FEA, when the SWCNT is long enough (L > 7 nm) its
elastic modulus becomes almost independent of its radius (R).
Fig. 4(b) presents a decreasing trend for the Poisson’s ratio predicted by FEA as the SWCNT radius increases, which is more pronounced for shorter SWCNTs. The computed Poisson’s ratio varies
from 0.06 to 0.11, and is lower than the values predicted by other
authors using FEA [12,34] and ab initio [35,36] computations, which
report values around 0.15–0.35. The reasons for this discrepancy
can be various, including the selection of the boundary conditions,
the presence or absence of non-bonded interactions and the numerical values used for the bond force constants, as will be further discussed. FEA predicts that longer SWCNTs have lower Poisson’s
ratios, specially for SWCNTs of small radius. As for the elastic modulus, this feature is mainly produced by the fraction of end atoms in
the SWCNT. The end atoms, having less elastic constrain, contract
more freely in the radial direction. This increases the Poisson’s ratio
for short SWCNTs which have a larger fraction of end atoms. However, the variation of mzr with R is minor (0.01), except for very
short SWCNTs (L < 5 nm) with small radius (R < 0.6 nm).
Table 2
Equilibrium parameters and elastic properties obtained by DFT for different armchair
SWCNTs.
Chirality
Parameter
(3, 3)
(4, 4)
(5, 5)
(6, 6)
(7, 7)
(8, 8)
(9, 9)
Req (Å)
aeq (Å)
beq (Å)
Ez (TPa)
2.111
1.436
1.445
0.9464
0.179
2.775
1.432
1.436
1.004
0.180
3.445
1.430
1.432
1.018
0.184
4.119
1.428
1.430
1.031
0.188
4.794
1.428
1.429
1.036
0.188
5.472
1.428
1.428
1.040
0.188
6.150
1.428
1.428
1.040
0.188
mzr
3.2. DFT predictions
Periodic unit cells of SWCNTs were simulated with the DFT
methodology previously discussed to compute their elastic modulus and Poisson’s ratio. The simulations were conducted for seven
armchair SWCNTs with chiral parameters (3, 3), (4, 4), (5, 5), (6, 6),
(7, 7), (8, 8) and (9, 9), see Table 1. The equilibrium atomic position
(relaxed coordinates) of the unit cell were first computed, defining
the SWCNT’s equilibrium radius (Req) and equilibrium bond
lengths (aeq and beq). The equilibrium bond lengths (aeq and beq)
are functions of H and Req. After achieving an equilibrium configuration which minimizes the energy (relaxed state), the DFT computations were repeated varying the applied strain and taking the
relaxed state as the baseline to obtain the elastic modulus and
Poisson’s ratio. As seen in Table 2, the elastic modulus of all
SWCNTs predicted by DFT are close to 1.0 TPa, varying from
0.95 TPa for the SWCNT with the smallest radius to 1.04 TPa
for the one with the largest radius. These results are within the
range of predictions by other authors based on ab initio methods
[35–37] who report an elastic modulus within the range of 0.7–
1.2 TPa. The (3, 3) SWCNT has a markedly lower elastic modulus
than the rest of the SWCNTs, which may be attributed to increased
repulsive interactions induced by the proximity of the carbon
atoms in the wall. The reduction of the SWCNT radius yields an increase in curvature which promotes a transition of hybridization in
the C–C bonds from sp2 to sp2-sp3 mixing [38,39]. The typical bond
length for sp2 C–C bonds is 1.42 Å with a standard bond angle of
120° between C atoms. In comparison, in the sp3 hybridization,
the standard value of the C–C bond length is 1.54 Å and the angle
changes to 109.5°. The mixed sp2-sp3 hybridization for reduced
diameters causes an increases in the SWCNT bond length and a
reduction in the bond angle, see Table 2 and Kürti et al. [39], which
contributes to the reduced SWCNT elastic modulus. Furthermore,
as shown by Peng et al. [40] there is a correlation between the
SWCNTs elastic modulus and its electronic structure. For SWCNTs
of small diameter the changes in electronic structure with applied
strain yield a reduction in the concavity of the strain energy function, which according to Eq. (1) yields a reduction in Ez. The DFT
predictions of Poisson’s ratio (last row of Table 2) varied from
0.180 for the (3, 3) and (4, 4) SWCNTs to 0.188 for the (6, 6) to
the (9, 9) SWCNTs (Req > 4.12 Å). Several works based on ab initio
methods have predicted Poisson’s ratios of SWCNTs from 0.15 to
0.20 [35,36]. According to Table 2, the elastic properties of armchair SWCNTs become practically size-independent for SWCNTs
with radius larger than 0.4 nm, i.e. from a (6, 6) configuration.
3.3. Comparison between FEA and DFT
A direct comparison between the results predicted by DFT and
FEA is presented in Fig. 5. The longest SWCNT (L = 24.6 nm) was selected for the FEA results in Fig. 5. The FEA results are presented for
261
G. Domínguez-Rodríguez et al. / Computational Materials Science 82 (2014) 257–263
(b) 0.20
(a)
1.025
(6,6)
(5,5)
1.000
FEA-GC
DFT
FEA-RC
(4,4)
0.975
0.950
(3,3)
0.2
(5,5)
(3,3) (4,4)
(7,7) (8,8) (9,9)
(6,6)
(7,7) (8,8)
(9,9)
0.15
FEA-GC
DFT
FEA-RC
νzr
Εz (TPa)
1.050
0.10
0.05
0.3
0.4
0.5
0.6
R (nm)
0.2
0.3
0.4
0.5
0.6
R (nm)
Fig. 5. Elastic properties of SWCNTs predicted by DFT and FEA (FEA using ‘‘geometric’’ and relaxed coordinates). (a) Elastic modulus, and (b) Poisson’s ratio.
two initial atomic coordinates, those of Table 1 (referred to as
‘‘geometric’’ herein, and labeled as ‘‘GC’’) and the relaxed coordinates (labeled as ‘‘RC’’), defined by the equilibrium parameters
listed in Table 2.
For the elastic modulus, Fig. 5(a), the difference between DFT
and both FEA curves is more pronounced for SWCNTs with smaller
radius, which may be a consequence of the electronic interactions
that increase when the neighbor atoms are very close. These nonbonded interactions (not accounted for in FEA) diminishes for
thicker SWCNTs because the distance between opposite atoms in
the wall’s circumference increases. Thus, non-bonded atomic interactions that affect the DFT prediction of Ez are significant only for
SWCNTs of R < 0.4 nm.
The FEA predictions of elastic modulus using the relaxed coordinates from DFT (FEA-RC, Table 2) in Fig. 5(a) are lower than those
obtained using the geometric coordinates (FEA-GC, Table 1 with
a = b = 1.42 Å). The difference between both FEA curves is only
structural, since the difference between them is only the variation
of the bond distances a and b. A variation of the input values for a
and b yields a variation on the SWCNT initial radius (and hence the
transverse area, AT) and also a slight variation in the angle a, see
Fig. 1(a). In this case, the equilibrium values of aeq and beq in Table 2
are larger than a = b = 1.42 Å (the value used for the geometric
coordinates). This yields a SWCNT of larger diameter, larger transverse area and increased angle a for the ‘‘relaxed’’ configuration
with respect to the ‘‘geometric’’ one. This in turn renders that all
FEA predictions of Ez using the relaxed coordinates are lower than
those using the geometric ones, and better capture the decaying
trend of DFT for SWCNTs of small radius.
In spite of the better accuracy of the FEA predictions for Ez using
the equilibrium coordinates of the SWCNTs, there is yet an appreciable difference between the DFT and FEA-RC curves in Fig. 5(a)
for SWCNTs with radii lower than 0.4 nm, which means that below such a radius there are yet non-bonded interactions that the
linear elastic FEA is not taking into account. The maximum difference between DFT and FEA-RC, however, is of the order of 6% and
occurs only for the (3, 3) SWCNT.
A similar analysis was conducted for the SWCNT’s Poisson’s ratio. Fig. 5(b) shows the predicted Poisson’s ratio for the longest
SWCNT (L = 24.6 nm) using FEA with both (geometric and relaxed)
coordinates and DFT. For R > 0.211 nm, DFT predicts Poisson’s ratios in the range of 0.179–0.188, while FEA-GC predictions are significantly lower (0.061–0.062). The computations were also
conducted for the relaxed coordinates, as shown in Fig. 5(b), but
even when the Poisson’s ratio yielded slightly higher values by
using the relaxed coordinates, this improvement was not sufficient
to achieve a good agreement between DFT and FEA. This means
that the mismatch between the FEA and DFT predictions of the
Poisson’s ratio are not ruled by the geometrical details of the initial
SWCNT configuration. Therefore, the reason for such a mismatch is
either the numerical values assumed for the bond force constants
(kr, kh and k/), which were originally derived for benzene [30,31],
or the influence of non-bonded interactions that are not considered
in FEA. In the case of non-bonded interactions, such an influence
must be more relevant for SWCNTs of small diameter (as in the
case of Ez), which is not observed for mzr.
3.4. Sensitivity of FEA predictions to the bond force constants
Using molecular mechanics and Monte Carlo simulations, Cornell et al. [30] and Jorgensen and Severance [31] obtained a set
of bond force constants kr, kh and k/ for the benzene molecule. Cornell et al. [30] obtained kr and kh by fitting those parameters to
ab initio computations and Jorgensen and Severance [31] obtained
k/ via Monte Carlo simulations. These numerical values of the bond
constants are commonly used in FEA to model graphene sheets and
carbon nanotubes [13,16,19,41,42] due to the lack of more accurate
force constants specifically computed for SWCNTs.
In order to systematically investigate the sensitivity of the FEA
predictions to the bond force constants, the FEA computations
were conducted for various numerical values of kr, kh and k/, taking
as baseline the set of bond force constants for benzene reported by
Cornell et al. [30] and Jorgensen and Serverance [31], i.e.
o
o
kr ¼ 6:52 107 N nm1 ; kh ¼ 8:76 1010 N nm1 rad2
and
o
10
1
k/ ¼ 2:78 10
N nm rad2. The parameter k/ did not show an
appreciable influence neither on the elastic modulus nor on the
Poisson’s
ratio,
and
therefore
k/
was
fixed
at
o
k/ ¼ 2:78 1010 N nm1 rad2. However, both kr and kh showed
an influence on the elastic modulus, and most importantly, on
the Poisson’s ratio of the investigated SWCNTs. Fig. 6 shows the
normalized curves for the elastic property P (elastic modulus or
o
o
Poisson’s ratio) as a function of kr =kr (Fig. 6(a)) and kh =kh
(Fig. 6(b)) for a (6, 6) SWCNT. The curves for the normalized elastic
modulus and Poisson’s ratio using SWCNTs with other radii are
similar, so only the curves for (6, 6) are included in Fig. 6. The elastic property P in Fig. 6 is normalized by P0, which is the elastic
o
o
property (Eoz or mozr ) calculated by using the set kr and kh . Both,
the elastic modulus and Poisson’s ratio in Fig. 6(a) show a linear
relationship with kr, but the influence of kr is significantly more
o
pronounced on mzr. Ez =Eoz increases with increased kr =kr with a
o
slope of 0.706, while mzr =mzr does it with a slope of 4.50. The normalized elastic modulus in Fig. 6(b) also shows a linearly increaso
ing relationship with increased kh =kh with a small slope of 0.296.
o
The normalized Poisson’s ratio mzr =mzr shows a linearly decreasing
o
relationship with increased kh =kh with a slope of 4.51.
An increase in kr stiffens the structure and increases both Ez and
mzr. The influence, however, is more marked for mzr since a 10% increase on kr yields a 44% increase in mzr but only a 7% increase in Ez,
Fig. 6(a). The influence of kh in Ez is only minor, but it does largely
affect mzr. An increase in kh of 10% yields a decrease in mzr of 42%.
An increase in kh stiffens the C–C bond under bending, which reduces the radial strain (r) in the SWCNT and hence reduces the
Poisson’s ratio according to Eq. (2). Therefore, the results of Fig. 6
highlight the importance of the proper selection of the set of bond
262
G. Domínguez-Rodríguez et al. / Computational Materials Science 82 (2014) 257–263
(a)
1.6
0
νzr/νzr
1.4
(b) 1.6
(6, 6) SWCNT
0
Ez/Ez
1.2
P/P
0
0
1.2
P/P
0
νzr/νzr
(6, 6) SWCNT
1.4
0
Ez/Ez
1.0
0
2
kr =6.52 ×10 N/m
0.8
1.0
0.6
0.4
0
0.95
1.00
1.05
-1
2
kθ=8.7 ×10 N/m rad
0.6
Ez =1.057 TPa
0
νzr=0.0616
0.90
0
0.8
0
Ez =1.057 TPa
0.4
1.10
0
νzr=0.0616
0.90
0.95
1.00
0
1.05
1.10
0
kθ/kθ
kr/kr
Fig. 6. Sensitivity of the elastic modulus and Poisson’s ratio to kr and kh for a (6, 6) SWCNT.
(a)
1.15
0
0
0
0
(b)
RC (kr/kr = kθ/kθ = 1)
1.0
1.05
DFT
Εz/Εz
DFT
0
0
RC (kr/kr = 1.14, kθ/kθ = 0.76)
0
0
GC (kr/kr = 1.12, kθ/kθ = 0.74)
νzr /νzr
1.10
0
0
0
0
RC (kr/kr = kθ/kθ = 1)
GC (kr/kr = kθ/kθ = 1)
GC (kr/kr = kθ/kθ = 1)
0.8
0
0
0
0
RC (kr/kr = 1.14, kθ/kθ = 0.76)
0.6
GC (kr/kr = 1.12, kθ/kθ = 0.74)
0.4
1.00
0.2
0.3
0.4
0.5
0.6
0.2
0.3
0.4
0.5
0.6
R (nm)
R (nm)
Fig. 7. Normalized elastic properties of SWCNTs of different radius using the calibrated and benzene bond force constants. (a) Elastic modulus, and (b) Poisson’s ratio.
force constants in order to yield accurate predictions by FEA. The
computation of such constants using first principles is out of the
scope of this work.
An attempt to produce a set of constants (kr, kh) which yields
FEA results that match the DFT predictions for Ez and mzr was further conducted by defining an error function Err(kr, kh) which
was then minimized using a least squares procedure [43].
The error function to be minimized was defined by the summation of the square error of the elastic modulus and the square error
of the Poisson’s ratio for the SWCNTs modeled by DFT, considering
all from (4, 4) to (9, 9), i.e.,
2
Errðkr ; kh Þ ¼
9 h
X
DFT
EFEA
z;n ðkr ; kh Þ Ez;n
i2
þ
h
DFT
mFEA
zr;n ðkr ; kh Þ mzr;n
i2 n¼4
ð5Þ
EFEA
z;n ðkr ; kh Þ
where
and mFEA
zr;n ðkr ; kh Þ are the elastic modulus and Poisson’s ratio of the (n, n) SWCNT obtained by FEA as a function of kr
DFT
and kh, whereas EDFT
z;n and mzr;n are the elastic properties calculated
by DFT for the same SWCNT. This method was applied to both
SWCNT coordinates investigated in Fig. 5, the one corresponding
to the geometric coordinates (Table 1, where a = b) and the one corresponding to the relaxed coordinates (Table 2). Using this method,
the calibrated bond force constants obtained for geometric coordi
o
nates (Table 1) were kr = 7.417 107 N nm1 kr =kr ¼ 1:12 and
o
10
1
2
kh = 6.526 10
N nm rad kh =kh ¼ 0:74 , whereas kr = 7.498 o
7
1
10
N nm1 rad2
10 o N nm kr =kr ¼ 1:14 and kh = 6.658 10
kh =kh ¼ 0:76 were the best fit parameters for the relaxed geometry. Fig. 7(a) shows the elastic modulus obtained by FEA using the
geometric (GC) and relaxed coordinates (RC) using the calibrated
set of bond force constants and the benzene bond force constants
calculated by Cornell et al. [30]. The elastic property has been normalized by the one predicted by DFT. As seen from this figure, a
very good agreement is found between FEA and DFT, specially when
the equilibrium coordinates are used. Fig. 7(b) shows a similar curve
to that shown in Fig. 7(a) but corresponding to the normalized
Poisson’s ratio. As seen from this figure, the large discrepancies between the FEA and the DFT for mzr obtained when the bond force
constants for benzene are used disappear when the calibrated bond
force constants are used. The use of geometric or equilibrium
parameters describing the initial morphology of the SWCNT has a
negligible influence on mzr.
4. Conclusions
The elastic modulus and Poisson’s ratio of armchair SWCNTs
ranging from 0.4 to 1.2 nm in diameter have been simulated using
a classical mechanics numerical method (FEA) and density functional theory (DFT), which includes quantum effects. Using DFT
as reference, a comparison between both approaches has been conducted to investigate the limits of applicability of FEA for the prediction of elastic properties of SWCNTs. The FEA calculations first
used the bond force constants for benzene calculated by Cornell
et al. [30] and Jorgensen and Severance [31] as a baseline, and further FEA calculations were then conducted by using calibrated
bond force constants kr and kh which minimize the difference between the predictions of DFT and FEA. These calibrated constants
differed by +14.4 % (kr) and 26.1% (kh) with respect to the values
predicted by Cornell et al. [30] for benzene, and such values significantly improved the predictions of FEA for Poisson’s ratio. The
elastic modulus showed a good correlation between both approaches (DFT and FEA) using either set of constants (for benzene
or the calibrated ones), unless the SWCNT diameter is less than
0.8 nm. However, the Poisson’s ratio predicted by FEA here using
the bond force constant for benzene is almost one third of that predicted by DFT. The use of the equilibrium atomic bond distances
obtained by DFT as an input to FEA further improves the precision
of the elastic modulus, but it is not very influential on the FEA predictions of the Poisson’s ratio.
The high sensitivity of the FEA predictions, specially on the
Poisson’s ratio, to the numerical value of the bond force constants
G. Domínguez-Rodríguez et al. / Computational Materials Science 82 (2014) 257–263
suggests that more ab initio efforts need to be devoted to accurate
computations of these constants, since they seem to depend on the
structural details and sp-hybridization of the specific molecule.
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