Strain energy calculations of hexagonal boron nanotubes: An ab-initio approach
An ab initio calculations have been carried out for examining the curvature effect of small diameter hexagonal boron nanotubes. The considered conformations of boron nanotubes are namely armchair (3,3), zigzag (5,0) and chiral (4,2), and consist of 12, 20, and 56 atoms, respectively. The strain ener...
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irk-123456789-1208222017-06-14T03:04:17Z Strain energy calculations of hexagonal boron nanotubes: An ab-initio approach Jain, S.K. Srivastava, P. An ab initio calculations have been carried out for examining the curvature effect of small diameter hexagonal boron nanotubes. The considered conformations of boron nanotubes are namely armchair (3,3), zigzag (5,0) and chiral (4,2), and consist of 12, 20, and 56 atoms, respectively. The strain energy is evaluated in order to examine the curvature effect. It is found that the strain energy of hexagonal BNT strongly depends upon the radius, whereas the strain energy of triangular BNTs depends on both radius and chirality. Здiйснено першопринципнi обчислення для вивчення впливу кривизни малого дiаметра гексагональної борової нанотрубки. Розглянутi конформацiї борової нанотрубки є майже крiслоподiбними (armchair) (3,3), зигзагоподiбними (zigzag) (5,0) i хiральними (4,2), складаючись, вiдповiдно, з 12, 20 та 56 атомiв. Для того, щоб дослiдити вплив кривизни, оцiнено енергiю деформацiї. Знайдено, що енергiя деформацiї гексагональної борової нанотрубки строго залежить вiд радiуса, тодi як енергiя деформацiї трикутних борових нанотрубок залежить як вiд радiуса, так i вiд хiральностi. 2013 Article Strain energy calculations of hexagonal boron nanotubes: An ab-initio approach / S.K. Jain, P. Srivastava // Condensed Matter Physics. — 2013. — Т. 16, № 3. — С. 33802:1-7 . — Бібліогр.: 46 назв. — англ. 1607-324X PACS: 31.15.A-, 06.30.Bp, 61.48.De, 62.25.-g, 71.15.Mb DOI:10.5488/CMP.16.33802 arXiv:1310.1237 http://dspace.nbuv.gov.ua/handle/123456789/120822 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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An ab initio calculations have been carried out for examining the curvature effect of small diameter hexagonal boron nanotubes. The considered conformations of boron nanotubes are namely armchair (3,3), zigzag (5,0) and chiral (4,2), and consist of 12, 20, and 56 atoms, respectively. The strain energy is evaluated in order to examine the curvature effect. It is found that the strain energy of hexagonal BNT strongly depends upon the radius, whereas the strain energy of triangular BNTs depends on both radius and chirality. |
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Article |
| author |
Jain, S.K. Srivastava, P. |
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Jain, S.K. Srivastava, P. Strain energy calculations of hexagonal boron nanotubes: An ab-initio approach Condensed Matter Physics |
| author_facet |
Jain, S.K. Srivastava, P. |
| author_sort |
Jain, S.K. |
| title |
Strain energy calculations of hexagonal boron nanotubes: An ab-initio approach |
| title_short |
Strain energy calculations of hexagonal boron nanotubes: An ab-initio approach |
| title_full |
Strain energy calculations of hexagonal boron nanotubes: An ab-initio approach |
| title_fullStr |
Strain energy calculations of hexagonal boron nanotubes: An ab-initio approach |
| title_full_unstemmed |
Strain energy calculations of hexagonal boron nanotubes: An ab-initio approach |
| title_sort |
strain energy calculations of hexagonal boron nanotubes: an ab-initio approach |
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Інститут фізики конденсованих систем НАН України |
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2013 |
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http://dspace.nbuv.gov.ua/handle/123456789/120822 |
| citation_txt |
Strain energy calculations of hexagonal boron nanotubes: An ab-initio approach / S.K. Jain, P. Srivastava // Condensed Matter Physics. — 2013. — Т. 16, № 3. — С. 33802:1-7 . — Бібліогр.: 46 назв. — англ. |
| series |
Condensed Matter Physics |
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AT jainsk strainenergycalculationsofhexagonalboronnanotubesanabinitioapproach AT srivastavap strainenergycalculationsofhexagonalboronnanotubesanabinitioapproach |
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2025-07-08T18:40:39Z |
| last_indexed |
2025-07-08T18:40:39Z |
| _version_ |
1837105199639953408 |
| fulltext |
Condensed Matter Physics, 2013, Vol. 16, No 3, 33802: 1–7
DOI: 10.5488/CMP.16.33802
http://www.icmp.lviv.ua/journal
Strain energy calculations of hexagonal boron
nanotubes: An ab-initio approach
S.K. Jain∗, P. Srivastava
Nanomaterials Research Group, Computational Nanoscience and Technology Lab (CNTL),
ABV-Indian Institute of Information Technology and Management, Gwalior–474015, India
Received September 3, 2012, in final form February 21, 2013
An ab initio calculations have been carried out for examining the curvature effect of small diameter hexagonal
boron nanotubes. The considered conformations of boron nanotubes are namely armchair (3,3), zigzag (5,0)
and chiral (4,2), and consist of 12, 20, and 56 atoms, respectively. The strain energy is evaluated in order to
examine the curvature effect. It is found that the strain energy of hexagonal BNT strongly depends upon the
radius, whereas the strain energy of triangular BNTs depends on both radius and chirality.
Key words: cohesive energy, curvature effect, strain energy, ab-initio calculations, boron nanotube
PACS: 31.15.A-, 06.30.Bp, 61.48.De, 62.25.-g, 71.15.Mb
1. Introduction
Boron is an electron deficient element [1] in the periodic table and possesses a richness of chemistry
next to carbon. Boron has a rather fascinating chemistry. Pure boron compounds have neither purely
covalent nor purely metallic character. This results in a chemical versatility, which is unique among
the elements of the periodic table. Various nanostructures of boron are as follows: boron nanotubes
(BNTs) [2–9] boron sheet (BS) [10, 11], boron clusters [6], boron nanowires [12–15], boron nanoribbon [16]
and boron buckyball [17]. The boron sheets of various morphologies viz. hexagonal, triangular, and α-
boron sheet have been predicted based on ab initio calculations. However, boron sheets have not been yet
synthesized experimentally. Further it has been found thatα-boron sheet is highly stable while hexagonal
boron sheet is least stable among these conformations. The boron nanotubes rolled from these BS have
also been investigated and consequently it has been predicted that all the BNTs are metallic independent
of diameter and chirality. The experimental synthesis of pure BNTs [18, 19] has confirmed the existence
of BNTs. Moreover, BNTs have a lot of morphologies such as hexagonal, triangular andα-BNTs in contrast
to only one morphology of CNTs i.e., hexagonal.
The Aufbau principle was proposed for elemental boron by Boustani [20] which states that stable
boron cluster can be constructed of two basic units only: a pentagonal pyramidal B6 unit and a hexago-
nal pyramidal B7 unit. Further, it implies that quasiplanar [21], tubular [2, 22], convex and spherical [23]
boron clusters are made of these units. Moreover, the stable BNTs and BS can be constructed of only
one unit i.e. hexagonal pyramidal B7. The existence of quasiplanar clusters or sheet was recently con-
firmed by experiment [24], in perfect agreement with earlier theoretical predictions [25]. Beyond the
Aufbau principle, hexagonal structures are also possible. Recently, it has been revealed that the thinnest
CNT-like-BNTs are stable as predicted by Dongju [26] which confirmed the existence of hexagonal BNTs.
Bezugly et al. [27] have predicted the boron nanotube to be highly conductive. The boron nanotubes that
we have considered are armchair (3,3), zigzag (5,0), and chiral (4,2) because the CNT of the same chirality
has been investigated [28–30] theoretically as well as experimentally. Therefore, the comparison in the
properties of the tubes can be studied in order to distinguish them. It is brought out that zigzag (5,0) CNT
∗E-mail: sandeepjain@iiitm.ac.in, phone: +91-751-2449814, Fax: +91-751-2449814
© S.K. Jain, P. Srivastava, 2013 33802-1
http://dx.doi.org/10.5488/CMP.16.33802
http://www.icmp.lviv.ua/journal
S.K. Jain, P. Srivastava
is found to bemetallic in contrast to semiconducting (as per zone folding prediction). This transition is at-
tributed to the curvature effect. To our best knowledge, curvature effect is also a prominent one at small
diameters which can be examined through strain energy calculations. The strain energy is a function of
diameter and/or chirality. As the diameter increases, the curvature energy decreases. Consequently, cur-
vature energy is more dominant at a small diameter. In this paper we have studied the effect of curvature
in the hexagonal BNTs of small diameters.
2. Computational methodology
We have performed pseudopotential plane wave calculations for determining the curvature effect
of small diameter BNTs. The geometrical structures of BNTs are optimized and then the total energy of
ground state is calculated by using DFT based CASTEP [31] simulation tool. The plane wave cutoff energy
is set to be 320 eV and exchange correlation effects are described by a generalized gradient approxi-
mation (GGA) proposed by Perdew-Burke-Ernzehrof (PBE) [32]. The integration is performed in the first
brillouin zone [33] by using the k-points generated by 1×1×8 grid parameters. These indicated param-
eters are sufficient to optimize the structures until the force on each atom becomes less than 0.03 eV/Å.
Moreover, self-consistency has been obtained on these parameters for other calculations. The ultrasoft
pseudopotential [34] is used in the reciprocal space. Similar calculations are also separately performed
byMao et al. [35] , Wang et al. [9], and Tian et al. [36]. The intertubular distance is kept fixed 10 Å in order
to avoid a periodic image interaction. The bond length B-B is chosen to be 1.67 Å before optimization.
The considered conformations of BNTs include armchair (3,3), zigzag (5,0) , and chiral (4,2) consisting of
12, 20, and 56 atoms, respectively. These tubes are periodic along z-axis. The diameters of these tubes
are 4.60 Å, 4.78 Å, and 4.87 Å for (5,0), (3,3), and (4,2) BNTs, respectively. In addition, non-spin-polarized
calculations are performed.
3. Results and discussion
The ab-initio calculations are carried out for boron nanotubes (BNTs) viz. zigzag (5,0), armchair (3,3),
and chiral (4,2) of diameters 4.60 Å, 4.78 Å, and 4.87 Å, respectively, with the number of atoms 20, 12,
and 56, respectively. The strain energy of a nanotube is defined as the amount of energy required to
bend a sheet into nanotube. Thus, strain energy is the difference of cohesive energies between the sheet
and nanotube i.e., Estrain = Esheet−ENT (R,θ), where Esheet and ENT are the cohesive energies [33] of the
sheet and tube, respectively. Strain energy Estrain quantifies (1) the difference in cohesive energy among
different (R,θ) nanotubes, (2) the deformation energy per atom, which is necessary to roll up a single
sheet into a nanotube of certain radius and chiral angle (θ), and (3) it is a measure of the mechanical
tension of a nanotube. This tension stabilizes the tubular shape.
For CNTs the strain energy effectively depends on the radius R rather than on the chirality (θ)
Estrain = Estrain(R). This radial dependence makes it possible to understand that the radius is just a mea-
sure for the curvature of a CNT, and the smaller its radius the more energy is needed to bend a Graphene
sheet into CNT. The strain energy is independent of chirality which is attributed to the nearly isotropic
in-plane mechanical properties of the Graphene sheet. It was found for CNTs that the diagonal elements
of the elastic tensors are the same due to hexagonal symmetry of the honeycomb lattice [37]. There-
fore, when stretching a Graphene sheet along different in-plane lattice directions, one observes the same
stiffness, and the systems behave like a homogeneous 2D continuum. From a chemical point of view
this mechanical isotropy is caused by a hexagonal network of stiff sp2
σ bonds. Thus, when rolling up
a graphene sheet along different in-plane directions to form various nanotubes with similar radii, this
process requires similar deformation energies. Therefore, Estrain will be independent of the chirality of
the CNT. This mechanical behavior is analogous to a simple sheet of paper that is rolled up to form a
tube. This process will require little energy for big radii and more energy for small radii. However, due
to the isotropic in-plane mechanical properties of the paper sheet, the energy needed to roll up a paper
tube is independent of the roll up direction. A similar behavior is also known for BN [38, 39], BC3 [40] and
MoS2 [41] nanotubes.
33802-2
Strain energy calculations of hexagonal boron nanotubes
Further, it had been reported for triangular boron sheet that stretching the sheet along its armchair
direction will be much harder than stretching it along its zigzag direction [42]. The bending of the sheet
along the armchair direction, which involves the bending of rather stiff σ bonds, will take more energy
(strain energy) than bending the BNTs along their zigzag direction, where no σ bonds will be affected.
It is investigated that the armchair BNTs have high strain energy, whereas zigzag BNTs have nearly van-
ishing strain energies. Thus, the strain energy of triangular BNTs depends on their radii and on their
chiral angles i.e., Estrain = Estrain(R,θ). Thus, the triangular boron sheet basically behaves like a piece of
cloth that is reinforced along one direction with parallel chains of stiffeners (the σ bonds). Bending the
cloth along the lines of stiffeners (armchair direction) takes much more energy than bending the cloth
perpendicular to it (zigzag direction). The calculations further predict that BNTs are always metallic, in-
dependent of their radii and chiralities. The Fermi surface of the triangular boron sheet has some well
pronounced contours in the 2D Brillouin zone, and backfolding of the Fermi surface into the 1D Brillouin
zone of a BNT is possible for any radius and any chirality [4]. For Graphene, on the other hand, the Fermi
surface just exists at the k-points of the Brillouin zone, and backfolding of these special points into the 1D
Brillouin zone of a (n,m) CNT is possible, but only if (n −m) turns out to be a multiple of 3 [37]. Thus,
for CNTs their electronic properties (semiconducting versus metallic) vary quite strongly with radius and
chiral angle, but their energies are independent of chirality. BNTs are just the opposite, in the sense that
their electronic properties will not depend on the structure type but their total energies actually do. To
our knowledge, this is the first nanotubular system for which the theory predicts a direct control over its
basic structural and electronic properties. The spectrum of nanotube radii obtained during the synthe-
sis of CNTs will depend on the specific reaction conditions (temperature, pressure, catalyst, reaction gas,
etc.), and it can be shifted and/or broadened by changing these conditions. Nevertheless, the CNT chirali-
ties remain random and rather uncontrollable. This was demonstrated by Iijima et al., Bethune et al. and
Journet et al. [43–45] , who were all synthesizing single walled CNTs using the arc-discharge method, but
they reported different mean diameters of 1.0, 1.2 and 1.4 nm, respectively.
Furthermore, they noted that the chiral angles varied quite strongly for a given tube diameter. This
is a direct consequence of the E = E (R) dependence of CNTs, because during synthesis the reaction con-
ditions just determine a certain energy range for the resulting nanotubes which gives a certain range of
radii but leaves the chirality totally unspecified. In contrast to this, the energies of nanotubes like BNTs,
which are derived from a sheet with anisotropic in-planemechanical properties, strongly depend on their
chiralities and radii E = E (R,θ), and the reaction conditions will effect both structural parameters. Such a
behavior might ultimately allow for better structure control among nanotubular materials, because now
the different chiral angles should be energetically separable and thus experimentally accessible.
Figure 1. (Color online) (a) Cohesive energy is shown as a function of diameter, and (b) Strain energy is
shown as a function of diameter.
In this study, we have considered boron nanotubes (BNTs) of hexagonal lattice structure. Therefore,
the chiral angle θ of BNTs ranges from 0◦ to 30◦. Thus, BNTs and CNTs relate to reference lattices of
the same symmetry, and, therefore, one has to use the same chiral indices i.e., (n,m) for both CNTs and
33802-3
S.K. Jain, P. Srivastava
BNTs. When stretching a boron sheet along different in-plane lattice directions, the same stiffness is ob-
served, and the system behaves like a homogeneous 2D continuum. From a chemical point of view this
mechanical isotropy is caused by a hexagonal network of stiff sp2
σ bonds. Thus, the rolling up of a boron
sheet along different in-plane directions to form various nanotubes with similar radii requires similar
deformation energies. Therefore, Estrain will be independent of the chirality of the hexagonal BNT. This
mechanical behavior is analogous to a simple sheet of paper that is rolled up to form a tube. This process
will require little energy for big radii, and it is becoming more and more costly with decreasing radii.
But due to the isotropic in-plane mechanical properties of the paper sheet, the energy needed to roll up a
paper tube is independent of the roll up direction.
It is evident from figure 1 that the structural stability depends on diameter. The cohesive energy in-
creases with the diameter. Therefore, it reveals that chiral BNT (4,2) is the most stable and zigzag (5,0)
is the least stable while armchair (3,3) is in between the two. The cohesive energies of hexagonal boron
sheet and BNTs have been taken (6.03 eV) from references [27] and [46], respectively. It is clearly illus-
trated that the effect of curvature decreases as the diameter increases and strain energy tends to zero
for BNTs of large diameter. Thus, it is evident from figure 1 that curvature plays a significant role at
small diameter and the zigzag (5,0) BNT possesses high curvature energy followed by armchair (3,3) and
chiral (4,2).
Furthermore, the charge density distribution of the considered BNTs has been studied in the frame-
work of DFT. The bond length of optimized armchair (3,3) BNT is explicitly shown in figure 2 and brings
out that the circular cross section is extended as the bond length (1.80 Å) is increased while along the
axial direction, the bond lengths are contracted to 1.516 Å showing strong bonding. The Charge density
distribution of the same BNT shows more density along axial direction which results in strong bonding.
The charge accumulated between two atoms indicates strong and covalent bonding between them.
(a) (b)
Figure 2. (Color online) (a) Optimized structure of armchair (3,3) BNT with bond lengths, and (b) Charge
density is shown in armchair (3,3) BNT.
(a) (b)
Figure 3. (Color online) (a) Optimized structure of zigzag (5,0) BNT with bond lengths, and (b) Charge
density is shown in zigzag (5,0) BNT.
In the case of zigzag BNT, the bond length between atoms is observed to be almost the same i.e., 1.67 Å
as depicted in figure 3. The charge density distribution showing the high density between the atoms
33802-4
Strain energy calculations of hexagonal boron nanotubes
(a) (b)
Figure 4. (Color online) (a) Optimized structure of chiral (4,2) BNT with bond lengths, and (b) Charge
density is shown in chiral (4,2) BNT.
representing the covalent nature of bonding in zigzag (5,0) BNT. The bond lengths remain the same after
optimization.
The bond lengths along circular cross section are extended which results in an increased diameter
and in a curving effect as demonstrated in figure 4. The charge density distribution is also consistent with
the optimized structure. The density is greater along the axial direction and results in strong bonding.
4. Conclusion
We have carried out ab initio calculations in order to evaluate the strain energy of small diameter
boron nanotubes in the framework of density functional theory. We have considered three conforma-
tions of BNTs viz. armchair (3,3), zigzag (5,0), and chiral (4,2). All these tubes have nearly the same diam-
eters (i.e., < 0.5 nm). Moreover, zigzag (5,0) BNT is the smallest and chiral (4,2) BNT is the largest while
armchair (3,3) is in between the two in size. In this paper, we have calculated the strain energy of small
BNTs of diameter < 0.5 nm and it turns out that the strain energy of chiral BNT is the least followed by
armchair and zigzag BNTs. The curvature energy decreases with an increasing diameter. Thus, it is evi-
dent that the effect of curvature is noticeable at a small diameter. It is also found that the charge density
distribution is also consistent with bond lengths.
Acknowledgement
We are very thankful to the Computational Nanoscience & Technology Lab (CNTL), ABV-Indian Insti-
tute of Information Technology & Management, Gwalior (India) for providing computational facilities.
We are also thankful to Accelery Inc for providing the perpetual license of CASTEP 4.4 simulation tool.
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Strain energy calculations of hexagonal boron nanotubes
Дослiдження енергiї деформацiї гексагональних борових
нанотрубок: першопринципний пiдхiд
С.К. Джаiн, П. Срiвастава
Група дослiджень наноматерiалiв, лабораторiя обчислювальної нанофiзики i технологiї,
ABV-Iндiйський iнститут iнформацiйної технологiї i менеджменту, Гвалiяр–474015, Iндiя
Здiйснено першопринципнi обчислення для вивчення впливу кривизни малого дiаметра гексагональ-
ної борової нанотрубки. Розглянутi конформацiї борової нанотрубки є майже крiслоподiбними (armchair)
(3,3), зигзагоподiбними (zigzag) (5,0) i хiральними (4,2), складаючись, вiдповiдно, з 12, 20 та 56 атомiв.
Для того, щоб дослiдити вплив кривизни, оцiнено енергiю деформацiї. Знайдено, що енергiя деформа-
цiї гексагональної борової нанотрубки строго залежить вiд радiуса, тодi як енергiя деформацiї трикутних
борових нанотрубок залежить як вiд радiуса, так i вiд хiральностi.
Ключовi слова: енергiя когезiї, вплив кривизни, енергiя деформацiї, першопринципнi обчислення,
борова нанотрубка
33802-7
Introduction
Computational methodology
Results and discussion
Conclusion
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