Tensile Behavior of Boron-Nitride Nanoribbons via a Non-Linear Spring Based Structural Mechanics Approach

Full text

6th International Conference from “Scientific Computing to Computational Engineering”
6th IC-SCCE
Athens, 9-12 July, 2014
© IC-SCCE
TENSILE BEHAVIOR OF BORON-NITRIDE NANORIBBONS VIA A NON-LINEAR
SPRING BASED STRUCTURAL MECHANICS APPROACH
G.I. Giannopoulos1, S.K. Georgantzinos2, D.-P.N. Kontoni 3 and P.A. Kakavas4
1 Department of Mechanical Engineering, 3,4 Department of Civil Engineering,
Technological Educational Institute of Western Greece,
1 M. Alexandrou Str., Koukouli, GR-26334 Patras, Greece
e-mail(1): ggiannopoulos@teipat.gr, e-mail(3): kontoni@teipat.gr , e-mail(4): kakavas@teipat.gr
web page: http://www.teipat.gr , www.teipat.gr/civil , www.teipat.gr/civil/kontoni
2 Mechanical Engineering and Aeronautics Department, University of Patras, GR-26500, Greece
e-mail(2): sgeor@mech.upatras.gr, web page: www.mead.upatras.gr, http://mdl.mech.upatras.gr
Keywords: Boron nitride, Nanoribbon, Finite element analysis, Structural mechanics, Stress-strain behavior,
Mechanical properties.
Abstract. Boron nitride, like graphene, can be formed as one-atom thick sheets or as nanotubes, then cut into
nanoribbons with their atoms arranged in a hexagonal lattice. Two-dimensional hexagonal boron-nitride
nanoribbons have been extensively investigated due to their excellent mechanical properties and high thermal
conductivity. They also resist chemical change and are unaffected by high temperatures, leading researchers to
believe that they could be consummate nanomaterials. The dimensions of boron-nitride nanoribbons as well as
the shape of their edges, which may be armchair or zigzag, may affect the overall behavior of the nanoribbons.
In the present paper, the mechanical behavior of different sized zigzag and armchair boron nitride nanoribbons
is numerically investigated and predicted by using a structural mechanics approach based on the Brenner
potential for boron nitride bonds. According to the proposed method, appropriate spring elements are combined
in nanoscale in order to simulate the interatomic interactions appearing within boron-nitride nanostructure.
The study focuses on the prediction of tensile stress-strain behavior of boron-nitride nanoribbons of different
sizes and edge shapes as well as the estimation of significant corresponding material properties such as Young’s
modulus, Poisson’s ratio, tensile strength stress and tensile failure strain. The numerical results, which are
compared with corresponding data given in the open literature where possible, demonstrate thoroughly the
important influence of size and chirality of a narrow boron nitride monolayer on its mechanical behavior.
1 INTRODUCTION
A Boron-Nitride (BN) monolayer in hexagonal state (h-BN) consists of equal numbers of boron (B) and
nitrogen (N) atoms and it has similar honeycomb structure as graphene monolayer (h-C). The properties of few-
nanometer-thick BN multilayer sheets (BN flakes) have attracted the interest of many researchers over the last
several years [1]. Although individual atomic planes of BN were also isolated [2] and investigated by transmission
electron microscopy (TEM) [3] and atomic force microscopy (AFM) [4], the scientific interest concerning BN
monolayers has been rather limited in comparison with the research activity towards its “sister” material,
graphene. There has been a lot of effort in characterizing the mechanical behavior of BN sheets as well as BN
nanotubes via theoretical studies based on density functional theory (DFT) [5-10], tight-binding approximation [11],
molecular dynamics (MD) [12], continuum mechanics (CM) [13,14], and molecular mechanics (MM) [15,16]. Due to
the complexity of conducting real measurements in nanoscale, fewer are the relevant experimental approaches
[17-20].
In the present study, the in-plane non-linear stress-strain behavior of several almost square-shaped hexagonal
BN nanoribbons (h-BNNRs) is numerically estimated by using an atomistic structural mechanics (SM)
approach. According to the adopted numerical technique, in order to represent the B-N repulsive and attractive
interatomic interaction a suitable Brenner potential function [21] is utilized. The bond angle bending and bond
angle torsion interatomic interactions are as well included in the formulation by introducing corresponding
appropriate potential functions. The positions of all B and N atoms are represented by in-plane nodes according
to the exact hexagonal molecular structure of the h-BNNR under consideration. Then, atomic nodes are suitably
interconnected via special spring-like elements of appropriate stiffness which are capable of representing

G.I. Giannopoulos, S.K. Georgantzinos, D.-P.N. Kontoni, and P.A. Kakavas.
accurately the aforementioned potential energies of each interatomic interaction. In order to convert computed
force distributions to stress responses, the thickness of all considered h-BN monolayers is assumed to be equal
to the interlayer distance in h-BN flakes [22,23]. To the authors’ best knowledge, for the first time, the non-linear
mechanical behavior of h-BNNRs is numerically estimated by using a spring-based structural mechanics
method. By the arisen non-linear tensile stress-strain curves various mechanical properties such as Young's
modulus, Poisson’s ratio, tensile strength and tensile failure strain are being predicted along both armchair and
zigzag directions of the nanomaterials under investigation. Various results are illustrated and compared with
other data published in the literature, where possible.
2 GEOMETRY OF h-BN
The honeycomb lattice of h-BN, which is assumed to have thickness nm333.0
=
t [22,23] in the present
analysis, has a unit cell represented in Figure 1 by the vectors 1
r and 2
r such that:
021 3|||| r== rr (1)
where 0
r is the equilibrium B-N bond distance taken equal to 0.133nm [21]. In this basis any lattice vector r is
represented as:
21 aar mn
+
=
(2)
where n, m are integers. In Cartesian coordinates:
T
01 ]02/32/3[r=r (3)
T
02 ]02/32/3[ −= rr (4)
Figure 1. Geometry of h-BN monolayer
The vectors connecting any atom to its nearest neighbors are:
3/)2( 211 rrδ
−
=
(5)
3/)2( 122 rrδ
−
=
(6)
3/)( 213 rrδ
+
=
(7)

G.I. Giannopoulos, S.K. Georgantzinos, D.-P.N. Kontoni, and P.A. Kakavas.
3 FROM POTENTIAL ENERGY TO FORCE FIELD
The total potential energy due to interatomic interactions within a h-BN monolayer, by ignoring the
negligible van der Waals effects, may be written as:
∑∑∑
+
+
=
τθ
UUUU r (8)
where r
U represents the energy due to bond length change,
θ
U the energy due to bond angle bending and
τ
U
the energy due to bond angle torsion. The interatomic potential r
U may be expressed as:
atrrepr UUU
+
=
(9)
where rep
U and atr
U are the repulsive and attractive potential terms, respectively, given by [21]:
)2exp(
12
10rS
S
D
Urep Δ−
−
=
β
(10)
)/2exp(
12
10rS
S
SD
Uatr Δ−
−
−=
β
(11)
where
r
Δ is the bond length change while the rest parameters are constants which are equal to
nm2nN1.01738212
0=D, 1.0769=S and -1
20.43057nm=
β
for the B-N chemical bond. Figure 2a depicts the
variation of interatomic potential energy r
U due to the bond length change. The arisen force r
F between B and
N bonded atoms due to the distance change between them may be evaluated by differentiating Equation (9), the
result of which ( )d(/d rUF rr Δ= ) is illustrated in Figure 2b.
(a) (b)
Figure 2. Variation of (a) potential energy and (b) force due to B-N bond length change
The interatomic potentials
θ
U and
τ
U, respectively, are approximated as:
2
)(
2
1
θ
θθ
Δ= kU (12)
2
)(
2
1
φ
ττ
Δ= kU (13)
where
θ
k and
τ
k are the bond angle bending and bond angle torsion force constants, respectively, while
θ
Δ
and
φ
Δ represent the bond angle bending and bond angle torsion changes, respectively. The force constants
used for the h-BN nanostructure are taken equal to -2
radnmNn6952.0=
θ
k and -2
radnmNn6255.0=
τ
k [15].
Differentiation of Equations (12) and (13), respectively, yields:

G.I. Giannopoulos, S.K. Georgantzinos, D.-P.N. Kontoni, and P.A. Kakavas.
θ
θθ
θ
θ
Δ=
Δ
=k
U
M
d
d (14)
φ
φ
τ
φ
φ
Δ=
Δ
=k
U
Md
d (15)
where
θ
M and
φ
M are the arisen bending and torsion moments due to the bond angle bending and bond angle
torsion, correspondingly.
4 FORCE FIELD SIMULATION
In order to represent the entire force field appearing within h-BN nanostructure, two types of two-noded
spring elements of specific local coordinate systems, with six degrees of freedom per node i.e., three translations
and three rotations, are utilized. Firstly, spring elements, called hereafter BN, are utilized for the simulation of
both force r
F variation (Figure 2b) and torsion moment
φ
M variation (Equation 15) due to the bond length
change
r
Δ and bond angle torsion
φ
Δ, respectively. Three of the specific spring elements, denoted by node
pairs oi ′, oj ′′ and on ′′′ , are shown in Figure 3. These elements are described according to a three dimensional
local coordinate system ( zyx ,, ). The x-axis of the specific local coordinate system coincides with the line that
connects the two bonded B and N atoms while
z
-axis is vertical to the h-BN plane. Their displacement stiffness
matrix is defined by:
⎥
⎦
⎤
⎢
⎣
⎡
−
−
=BNBN
BNBN
BN
zyxzyx
zyxzyx
zyx kk
kk
K (16)
where:
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
⎡
∞
∞
∞
∞
Δ
=
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
⎡
=
00000
00000
00000
00000
00000
00000
)d(
d
00000
00000
00000
00000
00000
00000 2
2
BN
BN
BN
BN
BN
BN
BN
τ
k
r
U
k
k
k
k
k
kr
zrot
yrot
xrot
z
y
x
zyx
k (17)
where infinite symbol is used to define a very high stiffness value capable of leading to a rigid behavior
regarding corresponding nodal translations or rotations.
Figure 3. Spring element representation of interatomic interactions

G.I. Giannopoulos, S.K. Georgantzinos, D.-P.N. Kontoni, and P.A. Kakavas.
Secondly, spring elements, called hereafter NBN, are utilized for the simulation of bending moment
θ
M
variation (Equation 14) due to the bond angle torsion
θ
Δ
. Three of the specific spring elements, denoted by
node pairs oo ′′′ , oo ′′′′′ and oo ′′′′ , are depicted in Figure 3. It should be mentioned that in the actual models these
nodes coincide with each other at the atomic positions B and N. Their displacement stiffness matrix is expressed
in the global coordinate system (
z
y
x
,, ) as:
⎥
⎦
⎤
⎢
⎣
⎡
−
−
=NBNNBN
NBNNBN
NBN
xyzxyz
xyzxyz
xyz kk
kk
K (18)
where:
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
⎡
∞
∞
∞
∞
∞
=
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
⎡
=
θ
k
k
k
k
k
k
k
rotz
roty
rotx
z
y
x
xyz
00000
00000
00000
00000
00000
00000
00000
00000
00000
00000
00000
00000
NBN
NBN
NBN
NBN
NBN
NBN
NBN
k (19)
To treat the inherent non-linearity of the problem, arisen by the highly non-linear stiffness term
22BN )d(/d rUk rx Δ= in Equation (17), an incremental-iterative scheme is adopted. According to the utilized
algorithm the load is applied in IN increments. The solution at the current load increment, denoted by
INin ,..,2,1=, is achieved by initially using the solution of the previous converged increment ( 1−in ). The
system of equations is constructed by applying the elemental stiffness equations for every BN and NBN element
and then transformed to the global coordinate system. Next, all equations are assembled according to the
requirements of nodal equilibrium and the following system of equations for increment in is obtained:
ininin FUUK
Δ
=
Δ
)( (20)
where )( in
UK , in
UΔ and in
FΔ are the assembled stiffness matrix, incremental displacement vector and
incremental force vector, respectively, for the increment in . Matrix Equation (20) can be solved via standard
numerical techniques by taking into consideration the imposed boundary conditions. The well-known Newton-
Raphson iterative procedure is applied to obtain a converged solution for the current load increment. The
iterations, during which linear-like equations are solved, are continued until the convergence criterion is
satisfied. In the present work, the convergence is checked via the calculation of the Euclidean norm of the
residual vector. At the end of the iterative process the solution of the current load increment is achieved and thus
the displacement ( ininin UUU Δ+= −1) and force ( ininin FFF
Δ
+
=
−1) distribution results are obtained.
5 RESULTS AND DISCUSSION
Three models, of almost square shaped h-BNNRs have been developed according to the proposed spring
based SM method (Figure 4). The geometry of the models considered is defined by their length x
l and y
l along
x
and y-axis, respectively.
Figure 4. Structural mechanics models of the three tested h-BNNRs

G.I. Giannopoulos, S.K. Georgantzinos, D.-P.N. Kontoni, and P.A. Kakavas.
In order to obtain the stress-strain curve along direction y
x
p,
=
, one edge of the h-BNNR at 0
=
p has
been fixed while a uniform displacement p
lΔ along p-axis has been applied on its opposite edge at p
lq =. The
stress p
σ
and strain p
ε
, with y
x
p,=, has been computed, respectively, from the following formulas:
tl
f
p
Q
np
q
p
∑
=
=1
σ
(21)
p
p
pl
l
Δ
=
ε
(22)
where p
qf is the reaction along p-axis of node q due to the imposed displacement p
l
Δ
along p-axis on edge
at p
lp =. In addition, Q is the total number of nodes that are fixed and belong on the edge at 0=p. Finally, h-
BN thickness is taken equal to the interlayer distance observed within h-BN flakes [22,23], i.e. nm333.0=t.
Figure 5a and b illustrates the )( xx
ε
σ
and )( yy
ε
σ
variations of the h-BNNRs under consideration.
(a) (b)
Figure 5. Tensile stress-strain behavior along (a) x-axis (armchair direction / zigzag edges loaded) and (b) y-axis
(zigzag direction / armchair edges loaded) for three different h-BNNRs sizes
The non-linearity of stress-strain behavior in both in-plane axes becomes evident. However, a better
mechanical response of h-BN monolayer may be observed when its armchair edges are loaded ( y / armchair
direction). In addition, the smaller the BNNR size the higher tensile strength and failure strain. Overall, h-BN
monolayer presents a brittle mechanical behavior with a superior however tensile strength that may reach
75GPa. Table 1 presents all the mechanical properties, i.e Young’s modulus
E
, Poisson’s ratio
ν
, tensile
strength u
σ
and failure strain f
ε
that have been numerically predicted via the proposed formulation in contrast
with other corresponding theoretical and experimental average estimations from the open literature. To enable
the comparisons, a common h-BN thickness, i.e. nm333.0
=
t, has been used. The good agreement between
present analysis and other approaches becomes apparent.

G.I. Giannopoulos, S.K. Georgantzinos, D.-P.N. Kontoni, and P.A. Kakavas.
Table 1. Computed properties in comparison with other corresponding average results available in the literature
6 CONLUSIONS
The mechanical behavior of a nanomaterial is difficult to be simulated due to the presence of complicated
molecular phenomena in nanoscale. Thus, a numerical efficient model is required to treat such problems. In the
present study, a three dimensional, non-linear, spring based SM method has been proposed for the prediction of
the stress-strain response of h-BN monolayer sheets. All interatomic interactions have been included in the
model while special care has been given to the interatomic force arisen due to the B-N bond length change by
adopting a suitable non-linear Brenner potential energy function. The numerical results have shown that h-BN
monolayer:
• Presents in-plane orthotropy.
• Is stronger when its armchair edges are loaded
• Is stronger for smaller sizes.
REFERENCES
[1] Golberg, D., Bando, Y., Huang, Y., Terao, T., Mitome, M., Tang, C., Zhi, C. (2010), “Boron nitride
nanotubes and nanosheets”, ACS Nano, Vol. 4(6), pp. 2979-2993.
[2] Novoselov, K.S., Jiang, D., Schedin, F., Booth, T.J., Khotkevich, V.V., Morozov, S.V., Geim, A.K. (2005),
“Two-dimensional atomic crystals”, Proceedings of the National Academy of Sciences of the United States
of America, Vol. 102 (30), pp. 10451-10453.
[3] Meyer, J.C., Chuvilin, A., Algara-Siller, G., Biskupek, J., Kaiser, U. (2009), “Selective sputtering and atomic
resolution imaging of atomically thin boron nitride membranes”, Nano Letters, Vol. 9(7), pp. 2683-2689.
[4] Lee, C., Li, Q., Kalb, W., Liu, X.-Z., Berger, H., Carpick, R.W., Hone, J. (2010), “Frictional characteristics
of atomically thin sheets”, Science, Vol. 328(5974), pp. 76-80.
[5] Kudin, K.N., Scuseria, G.E., Yakobson, B.I. (2001), “C2F, BN, and C nanoshell elasticity from ab initio
computations”, Physical Review B - Condensed Matter and Materials Physics, Vol. 64(23), art. no. 235406.
[6] Baumeier, B., Krüger, P., Pollmann, J. (2007), “Structural, elastic, and electronic properties of SiC, BN, and
BeO nanotubes”, Physical Review B - Condensed Matter and Materials Physics, Vol. 76(8), art. no. 085407.
[7] Topsakal, M., Ciraci, S. (2010), “Elastic and plastic deformation of graphene, silicene, and boron nitride
honeycomb nanoribbons under uniaxial tension: A first-principles density-functional theory study”, Physical
Review B - Condensed Matter and Materials Physics, Vol. 81(2), art. no. 024107.
Present study
Average length (nm)
Mechanical
property
2.59 5.06 9.76
DFT
[7]
DFT
[8]
DFT
[9]
DFT
[10]
MD
[11]
CM
[14]
MM
[15]
Exp.
[20]
x 901.2 792.0 783.4 1004
E (GPa)
y 846.8 814.9 801.7
774.8 838.4 828.2 801.8
1111
997.2 725.2 811
x 0.255 0.296 0.297
ν
y 0.319 0.317 0.315
0.218 0.220 0.210
0.161 0.387
x 61.33 52.03 49.46 63.06
σu (GPa)
y 75.59 72.76 68.36 93.09
78.08
x 0.113 0.107 0.101 0.169
εf
y 0.155 0.153 0.143
0.210
0.210

G.I. Giannopoulos, S.K. Georgantzinos, D.-P.N. Kontoni, and P.A. Kakavas.
[8] Peng, Q., Ji, W., De, S. (2012), “Mechanical properties of the hexagonal boron nitride monolayer: Ab initio
study”, Computational Materials Science, Vol. 56, pp. 11-17.
[9] Andrew, R.C., Mapasha, R.E., Ukpong, A.M., Chetty, N. (2012), “Mechanical properties of graphene and
boronitrene”, Physical Review B - Condensed Matter and Materials Physics, Vol. 85(12), art. no. 125428.
[10] Şahin, H., Cahangirov, S., Topsakal, M., Bekaroglu, E., Akturk, E., Senger, R.T., Ciraci, S. (2009),
“Monolayer honeycomb structures of group-IV elements and III-V binary compounds: First-principles
calculations”, Physical Review B - Condensed Matter and Materials Physics, Vol. 80(15), art. no. 155453.
[11] Hernández, E., Goze, C., Bernier, P., Rubio, A. (1998), “Elastic properties of C and BxCyNz composite
nanotubes”, Physical Review Letters, Vol. 80(20), pp. 4502-4505.
[12] Verma, V., Jindal, V.K., Dharamvir, K. (2007), “Elastic moduli of a boron nitride nanotube”,
Nanotechnology, Vol. 18(43), art. no. 435711.
[13] Song, J., Wu, J., Huang, Y., Hwang, K.C. (2008), “Continuum modeling of boron nitride nanotubes”,
Nanotechnology, Vol. 19(44), art. no. 445705.
[14] Oh, E.-S. (2011), “Elastic properties of various boron-nitride structures”, Metals and Materials
International, Vol. 17(1), pp. 21-27.
[15] Boldrin, L., Scarpa, F., Chowdhury, R., Adhikari, S. (2011), “Effective mechanical properties of hexagonal
boron nitride nanosheets”, Nanotechnology, Vol. 22(50), art. no. 505702.
[16] Jiang, L., Guo, W. (2011), “A molecular mechanics study on size-dependent elastic properties of single-
walled boron nitride nanotubes”, Journal of the Mechanics and Physics of Solids, Vol. 59(6), pp. 1204-1213.
[17] Chopra, N.G., Zettl, A. (1998), “Measurement of the elastic modulus of a multi-wall boron nitride
nanotube”, Solid State Communications, Vol. 105(5), pp. 297-300.
[18] Suryavanshi, A.P., Yu, M.-F., Wen, J., Tang, C., Bando, Y. (2004), “Elastic modulus and resonance
behavior of boron nitride nanotubes”, Applied Physics Letters, Vol. 84(14), pp. 2527-2529.
[19] Golberg, D., Costa, P.M.F.J., Lourie, O., Mitome, M., Bai, X., Kurashima, K., Zhi, C., Tang, C., Bando, Y.
(2007), “Direct force measurements and kinking under elastic deformation of individual multiwalled boron
nitride nanotubes”, Nano Letters, Vol. 7(7), pp. 2146-2151.
[20] Bosak, A., Serrano, J., Krisch, M., Watanabe, K., Taniguchi, T., Kanda, H. (2006), “Elasticity of hexagonal
boron nitride: Inelastic x-ray scattering measurements”, Physical Review B - Condensed Matter and
Materials Physics, Vol. 73(4), art. no. 041402.
[21] Albe, K., Möller, W., Heinig, K.-H. (1997), “Computer simulation and boron nitride”, Radiation Effects
and Defects in Solids, Vol. 141(1-4), pp. 85-97.
[22] Hod, O. (2012), “Graphite and hexagonal boron-nitride have the same interlayer distance. Why?”, Journal
of Chemical Theory and Computation, Vol. 8(4), pp. 1360-1369.
[23] Golla, D., Chattrakun, K., Watanabe, K., Taniguchi, T., Leroy, B.J., Sandhu, A. (2013), “Optical thickness
determination of hexagonal boron nitride flakes”, Applied Physics Letters, Vol. 102(16), art. no. 161906.