Elastic and thermodynamic properties of potentially superhard carbon boride materials
Boron icosahedral structures are the basic building structures of many important hard borides and especially В₄С. The structural and thermodynamic properties of В₄С have been examined applying molecular dynamics simulation with the use of both ab initio and bond order Tersoff potentials. Various phy...
Збережено в:
| Дата: | 2012 |
|---|---|
| Автори: | , |
| Формат: | Стаття |
| Мова: | English |
| Опубліковано: |
Інститут надтвердих матеріалів ім. В.М. Бакуля НАН України
2012
|
| Назва видання: | Сверхтвердые материалы |
| Теми: | |
| Онлайн доступ: | http://dspace.nbuv.gov.ua/handle/123456789/125955 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Elastic and thermodynamic properties of potentially superhard carbon boride materials / T.E. Letsoalo, J.E. Lowther // Сверхтвердые материалы. — 2012. — № 1. — С. 38-48. — Бібліогр.: 48 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
irk-123456789-125955 |
|---|---|
| record_format |
dspace |
| spelling |
irk-123456789-1259552017-11-11T03:04:31Z Elastic and thermodynamic properties of potentially superhard carbon boride materials Letsoalo, T.E. Lowther, J.E. Получение, структура, свойства Boron icosahedral structures are the basic building structures of many important hard borides and especially В₄С. The structural and thermodynamic properties of В₄С have been examined applying molecular dynamics simulation with the use of both ab initio and bond order Tersoff potentials. Various physical quantities of В₄С including the elastic constants, thermal expansion coefficients, and specific heat have been examined. Ікосаедрічні структури бору є основними будівельними конструкціями багатьох важливих твердих боридів і особливо це стосується В₄С. Досліджено структурні та термодинамічні властивості В₄С при застосуванні моделювання динаміки молекул з використанням як ab initio розрахунків, так і потенціалів Терсоффа. Розглянуті різні фізичні характериcтики В₄С, включаючи пружні константи, коефіцієнти теплового розширення і теплоємність. Икосаэдрические структуры бора являются основными строительными конструкциями многих важных твердых боридов и особенно это касается В₄С. Исследованы структурные и термодинамические свойства В₄С с применением моделирования динамики молекул, используя как ab initio расчеты, так и потенциалы Терсоффа. Рассмотрены различные физические характериcтики В₄С, включая постоянные упругости, коэффициенты теплового расширения и теплоемкость. 2012 Article Elastic and thermodynamic properties of potentially superhard carbon boride materials / T.E. Letsoalo, J.E. Lowther // Сверхтвердые материалы. — 2012. — № 1. — С. 38-48. — Бібліогр.: 48 назв. — англ. 0203-3119 http://dspace.nbuv.gov.ua/handle/123456789/125955 661.665.3:536.4 en Сверхтвердые материалы Інститут надтвердих матеріалів ім. В.М. Бакуля НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| topic |
Получение, структура, свойства Получение, структура, свойства |
| spellingShingle |
Получение, структура, свойства Получение, структура, свойства Letsoalo, T.E. Lowther, J.E. Elastic and thermodynamic properties of potentially superhard carbon boride materials Сверхтвердые материалы |
| description |
Boron icosahedral structures are the basic building structures of many important hard borides and especially В₄С. The structural and thermodynamic properties of В₄С have been examined applying molecular dynamics simulation with the use of both ab initio and bond order Tersoff potentials. Various physical quantities of В₄С including the elastic constants, thermal expansion coefficients, and specific heat have been examined. |
| format |
Article |
| author |
Letsoalo, T.E. Lowther, J.E. |
| author_facet |
Letsoalo, T.E. Lowther, J.E. |
| author_sort |
Letsoalo, T.E. |
| title |
Elastic and thermodynamic properties of potentially superhard carbon boride materials |
| title_short |
Elastic and thermodynamic properties of potentially superhard carbon boride materials |
| title_full |
Elastic and thermodynamic properties of potentially superhard carbon boride materials |
| title_fullStr |
Elastic and thermodynamic properties of potentially superhard carbon boride materials |
| title_full_unstemmed |
Elastic and thermodynamic properties of potentially superhard carbon boride materials |
| title_sort |
elastic and thermodynamic properties of potentially superhard carbon boride materials |
| publisher |
Інститут надтвердих матеріалів ім. В.М. Бакуля НАН України |
| publishDate |
2012 |
| topic_facet |
Получение, структура, свойства |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/125955 |
| citation_txt |
Elastic and thermodynamic properties of potentially superhard carbon boride materials / T.E. Letsoalo, J.E. Lowther // Сверхтвердые материалы. — 2012. — № 1. — С. 38-48. — Бібліогр.: 48 назв. — англ. |
| series |
Сверхтвердые материалы |
| work_keys_str_mv |
AT letsoalote elasticandthermodynamicpropertiesofpotentiallysuperhardcarbonboridematerials AT lowtherje elasticandthermodynamicpropertiesofpotentiallysuperhardcarbonboridematerials |
| first_indexed |
2025-07-09T04:02:57Z |
| last_indexed |
2025-07-09T04:02:57Z |
| _version_ |
1837140573208707072 |
| fulltext |
www.ism.kiev.ua/stm 38
UDC 661.665.3:536.4
T. E. Letsoalo, J. E. Lowther (Johannesburg, South Africa)
Elastic and thermodynamic properties
of potentially superhard carbon boride
materials
Boron icosahedral structures are the basic building structures of
many important hard borides and especially B4C. The structural and thermodynamic
properties of B4C have been examined applying molecular dynamics simulation with
the use of both ab initio and bond order Tersoff potentials. Various physical quantities
of B4C including the elastic constants, thermal expansion coefficients, and specific heat
have been examined.
Keywords: diamond, boron, electronic structure, elastic constants,
molecular dynamics.
INTRODUCTION
Lightweight atoms such as carbon, boron, nitrogen, and oxygen
[1] are very important for hard materials. Carbides, borides, nitrides, and oxides
have high melting points, good thermal conductivity, and high chemical resistance
[2–5]. Cubic boron nitride is an important hard material, which behaves as an
electrical insulator and has several properties similar to diamond. Recent work has
focused on alloying diamond with BN to identify superhard materials with a B–C–
N structure [6–8]. The hope is that a high pressure solid solution of diamond and
BN would exhibit novel properties and be a superhard material. B–C–N phases are
expected to be harder than cBN, but still retain good thermal and chemical
stability. This therefore would be an excellent material for high-speed cutting and
polishing of ferrous alloys where diamond fails.
Another interesting class of lightweight materials are boron-rich solids. Boron
has several crystallographic allotropes, many of which are classified as semicon-
ductors and capable of forming stable covalent-bonded molecular networks with
high strength [9, 10]. The crystal structures mainly contain B12 icosahedra as
essential common structural building blocks [11–13] and these solids have similar
physical properties such as high electrical conductivity, high melting points, low
density, high chemical resistance. They also have low corrosivity and high hard-
ness. The boron icosahedra primarily determine the electronic structure and chemi-
cal bonding in these solids and this is the reason, why such materials display simi-
lar characteristics. They also have advantages of being easily synthesized and
stable up to very high temperatures [14]. Recent theoretical works [12, 15] have
concluded that the incorporation of other atoms between the icosahedra indirectly
influences the bonding between the boron atoms and is responsible for the
enhanced hardness.
There are several allotropes of boron, although three of them, α, β, and tetrago-
nal, are important. The allotrope β-boron is synthesized at a high temperature,
while α-boron is metastable at ambient conditions, unstable thermodynamically,
and synthesized only at low temperatures. Tetragonal phases are intermediate
© T. E. LETSOALO, J. E. LOWTHER, 2012
ISSN 0203-3119. Сверхтвердые материалы, 2012, № 1 39
between α and β-boron. The boundary temperature depends on the preparation
method and has been estimated to be about 1500 K [16, 17]. In the case of the B50
allotropes, a B192 structure may be more appropriate [18]. Boron carbides are
among the best studied [18, 19] for potential as a hard material. B4C has often been
considered to have a stucture similar to α-boron [18]. Technical applications of
such boron–carbon structures are limited with variable homogeneity range over a
specific stoichiometry around B4C. However, detailed investigation of fundamental
properties of such boron compounds has been impeded because it is difficult to
grow large crystals. There have been several speculations concerning the structure
of B4C. However, recently it has been shown that the polar structure (where C
enters the icosahedra) is more likely associated with a B13C2 stoichiometry [20]
than the conventional chain structure (where C does not enter the icosahedra) with
B4C [21–23].
Computer modeling techniques have also been used to study boron and boron-
rich compounds. Li and Ching [24] studied the structure and properties of four B12-
based crystals using the self-consistent orthogonalized linear combination with the
local density approximation (LDA) of the density functional theory (DFT) to
calculate the band structures and ground state properties. Their results were in good
agreement with other existing first–principle calculations using the pseudopotential
plane-wave method. Various attempts have been made to study the ground state
and finite temperature properties of α- and β-boron using ab initio pseudopoten-
tials method with LDA [15, 16]. These have clarified the features of β-boron by
comparing them to those of the α-boron phase. Electronic structures of various
other borides that hold potential for superhard material properties have also been
investigated [12]. Recently [23] the mechanical properties and structure of amor-
phous and crystalline B4C prepared using various plasma-based techniques have
been investigated and it has been suggested that carbon could eventually segregate
into clusters when the material attains an amorphous structure at elevated tempera-
tures.
In the present work we investigated structural and thermodynamic properties of
selected boron icosahedra materials and B4C using both the ab initio and molecular
dynamic simulations. The molecular dynamic simulations are employed as being a
way to investigate the behavior of large numbers of atoms and thus the material
behavior at high temperatures. The results are presented for the lattice expansion
and specific heat of B4C. Finally it is shown that the behavior of carbon appears to
play an important role at high temperatures.
CALCULATIONS
The computer simulation gives information at a microscopic atomic level [24,
25], from which experimental properties are derived or predicted. Ab initio
approaches can accurately predict many physical properties but have computational
limitations as to practical realization to deal with a relatively low number of atoms.
In the present work, first–principles calculations have used the VASP code [26],
which performs plane–wave pseudopotential total energy calculations. The elec-
tronic structure of the materials have been invetigated using several methods [27,
28] and here they are calculated using the density functional theory (DFT) in the
local density approximation (LDA) with the Ceperley and Alder [29] correlation
used to treat the exchange and correlation functional. In all calculations, plane
waves are employed with an energy cut-off to 500 eV, and the Brillouin zone sam-
pling is fixed at 8×8×8 Monkhorst-Pack [30] scheme with PAW pseudopotentials
www.ism.kiev.ua/stm 40
[31]. All the calculations carried out are fully relaxed and optimized with respect to
volume with an energy convergence criteria of 10–5 eV/atom.
For large numbers of atom simulations we have used the Tersoff bond–order
potential as implemented either through the codes DLPLOLY [32] or GULP [33].
This potential has now been applied to a large variety of different systems [34–36]
and is one of the commonly used functional forms for modeling covalent systems.
The functional form of the Tersoff potential is:
;
2
1∑∑
≠
==
ji
ij
i
i vvV (1)
)],()()[( ijAijijRijCij rfbrfrfv = (2)
where V is the total potential energy of the system, fR and fA are the repulsive and
attractive pair potentials, respectively, and fC is a cutoff function. These are given
by
ijij r
ijijijR erArf λ−= )()( ; (3)
ijij r
ijijijA erBrf λ−−= )()( ; (4)
⎥
⎦
⎤
⎢
⎣
⎡ −π
⎪
⎪
⎩
⎪⎪
⎨
⎧
−=
D
Rr
rf ij
ijC
)(
2
sin
0
2
1
2
1
1
)(
;at
;at
;at
DRr
DRrDR
DRr
ij
ij
+>
+<<−
−<
(5)
where rij is the bond distance between atoms i and j with D and R cutoff radii for
the interaction potential. The A, B and λij potential parameters and the strength of
each bond depend upon the local environment as measured by another empirical
parameter χij. In the case of the boron only system the cut off parameter D was
adjusted to simulate icosahedral boron potentials, whereas for B4C the parameter
χB-C was adjusted. The parameters we have used in this work are listed in Table 1.
Table 1. Tersoff potential parameters used in this study. Parameters for B
and C are taken from Matsunaga [37] and Munetoh [38]. Some parameters
were adjusted as discussed in the text
B C
A, eV 2.7702·102 1.3936·103
B, eV 1.8349·102 3.467·102
λ, Å–1 1.992 3.4879
μ, Å–1 1.5856 2.2119
Β 1.6000·10–6 1.5724·10–7
N 3.9929 0.72751
C 5.2629·10–1 3.8049·104
D 1.5870·10–3 4.384
H 0.500 –0.57058
R, Å 1.8 1.8
D, Å 1.7–2.1 2.1
Interactions (i–j) B–C
χij 1.0025
ωij 0.9810
ISSN 0203-3119. Сверхтвердые материалы, 2012, № 1 41
Structural properties
Initially we examined the structural properties of some boride phases as well as
B4C. The boron structures studied are α-boron, tetragonal boron and crystal
structures of these together with the B4C phase we have considered are shown in
Fig. 1. The reason for choosing these borides is to attempt some indication of the
reliability of the Tersoff potential, although we stress that our prime motivation
here is to explore properties of B4C. Details of the final geometries are given in
Table 2.
a b c
Fig. 1. Various boride phases (a) α-boron, (b) tetragonal boron, (c) B4C.
Table 2. Lattice parameter (Å) and bulk moduli of some boron-related
compounds. Bulk moduli are obtained from the Birch-Murnaghan
equation of state
Material Tersoff Ab initio Experiment
B, (B′)
(Tersoff)
B, (B′)
(ab initio)
Experiment
α-B12 (R3m) a = 4.404
c = 12.286
a = 4.844
c = 12.286
a = 5.507 [39] 212, (2.26) 226, (3.80) 213–224 [36]
(4.0)
T-B50
(P42/nnm)
a = 7.940
c = 4.718
a = 8.734
c = 5.010
a = 8.75 [40, 41]
c = 5.06 [42]
208, (1.84) 206, (3.80)
B4C (R3m) a = 5.174
c = 11.695
a = 5.577
c = 11.286
a = 5.61
c = 12.14
246, (5.380) 229, (3.50) 199–241 [37]
(1.0, 4.0)
We have then applied a uniform pressure to each of the phases and fitted the
energy results to the Birch-Murnaghan equation of state using the molecular
dynamics code DLPOLY. The pressure–volume relation is shown in Fig. 2 for
borides and the B4C structure.
It is clear that the compressibility of B4C is better than of the boron structures
and this is reflected in the value of the bulk moduli obtained from a least squares fit
to the Birch-Murnaghan equation of state that is given in Table 2. However the
bulk modulus derived in this way does not reflect the full elastic properties of the
system. To this end we specifically investigated the full elastic constants of the
B4C material and such values are given in Table 3 as obtained from the GULP
program molecular dynamics code. We have also included results of the elastic
constants obtained using ab intio calculations through the VASP code. As seen
from Table 3 overall there is quite good agreement between the Tersoff and ab
initio results, and with previous calculations, and experiment where available. This
www.ism.kiev.ua/stm 42
gives some confidence in applying the Tersoff potential to finite temperature
simulations, where large numbers of atoms are really needed and the system may
become disordered.
0 5 10 15 20 25 30
Pressure, GPa
0.88
0.90
0.92
0.94
0.96
0.98
1.00
V/V
0
Fig. 2. Pressure–volume relations of α-boron (B12) (●), tetragonal–boron (B50) (▲) and B4C
(polar) (■).
Table 3. Calculated elastic properties of B4C (GPa). Other calculation and
experimental values are listed. Voigt bulk (B), shear (G) and Young (E)
moduli obtained from the elastic moduli tensors
c11 c12 c13 c23 c33 c55 B G E
Present (Tersoff) 540 180 91 32 520 305 234 200 467
Present (ab initio) 539 181 79 82 480 139 236 178 427
Other calc. [44] 562 123 69 17 517 – 234 – –
Expt. [45] 543 132 63 534 164 241 193 457
Expt. [46] 235 197 462
Thermal properties of B4C
An increase in the temperature increases the kinetic energy of atoms and the
increased interaction between the atoms gives and effective expansion of the unit
lattice cell. The possibility of disorder in the structures at very high temperatures
cannot be ruled out. To investigate, this needs quite a large number of atoms. For
this we have used supercells consisting of up to 2880 atoms. All simulations have
been preformed using a NVT ensemble with the DLPOLY code, and employing a
Verlet algorithm, with a time step of 1 ps.
First in Fig. 3 are the plots of the lattice expansion of B4C at zero pressure. We
note small changes in the lattice occurring around T = 1500 K. For this reason we
have fitted the calculated results to a polynomial and this is shown in the figures.
Thermal expansion coefficients αa and αc along a and c axes in MD simulation can
be computed directly from the temperature derivative of the lattice parameter along
those axes
ISSN 0203-3119. Сверхтвердые материалы, 2012, № 1 43
T
ca
caca ∂
∂=α ),(
),(
1
),( , (6)
where a and c are the lattice parameters along the a or c axes. The thermal expan-
sion coefficient increases linearly with temperature and we obtained a slightly
larger thermal expansion along the c axis than along the a axis at 300 K.
0 500 1000 1500 2000 2500 3000
Temperature, K
5.19
5.20
5.21
5.22
5.23
5.24
5.25
L
at
ti
ce
p
ar
am
et
er
, Å
a
0 500 1000 1500 2000 2500 3000
Temperature, K
11.58
11.60
11.62
11.64
11.66
11.68
11.70
L
at
ti
ce
p
ar
am
et
er
, Å
11.72
b
Fig. 3. Temperature dependence of the a- and c-axes of B4C: (a) a300 K(T) = –1.437·10–9T2 +
2.517·10–5T + 5.193, a1500 K(T) = –3.800·10–9T2 + 3.574·10–5T + 5.182; (b) c300 K(T) =
–3.374·10–9T2 + 5.636·10–5T + 11.58, c1500 K(T) = –8.600·10–9T2 + 8.022·10–5T + 11.56.
An average linear thermal expansion coefficient, αl, was calculated using the
formula
).2(
3
1
cal α+α=α (7)
www.ism.kiev.ua/stm 44
The results for the average expansion coefficient of these borides starting at 300
and 1500 K computed using the above equation are listed in Table 4. These results
are compared with two strongest known hard materials diamond and boron nitride.
Although the melting of solids is one the most common observations of a phase
transition, the mechanism of melting is still an outstanding problem in condensed
matter physics [42]. On the basis of the thermodynamics theory a phase transition
is characterized by an abrupt change in the slope of the total energy against the
temperature curve. Figure 4 shows the variation of energy against the temperature
for borides.
1400 1600 1800 2000 2200 2400 2600 2800 3000
Temperature, K
–7.7
–7.6
–7.5
–7.4
–7.3
E
ne
rg
y,
e
V
/a
to
m
Fig. 4. Temperature variation of the total energy of B4C: E(T) = 2.375·10–4T – 8.027.
There is a very slight change in the slope around 1500 K. This suggests a possi-
ble change of a phase. But the energy/temperature behavior also allows some
deduction of the specific heat of the materials. The values so deduced are given in
Table 4 for B4C. These values are also compared with two other hard materials,
namely diamond and cBN. As expected, the values compare quite well.
Table 4. Thermodynamic properties of B4C compared to other hard
materials
Diamond BN B4C (calculated)
,
300Klα 10–6 K–1 1.017b 1.8d 3.116
,
1500Klα 10–6 K–1 5.181b 5.47c 4.598
6–8f 4.5
Cv(kB) 2.97a 3.02c 2.76
5.16e
Note. The data taken from the following Refs. are indicated as: [43]a, [44]b, [45]c, [46]d, [47]e,
[48]f.
We have to explore further the apparent change taking place around 1500 K. To
understand this we have examined the radial distribution function (RDF) of the C
ISSN 0203-3119. Сверхтвердые материалы, 2012, № 1 45
and B atoms as obtained from the MD calculations. Figure 5 shows the RDF for
B4C at 300, 1500 and 3000 K. The results are quite striking in the case of C.
Overall some peaks decreased, while others broadened at high temperatures, the
broadening of the larger distance peaks is characteristic of some amorphization.
The retention of the lower peaks related to B shows that the B icosahedra are still
quite stable even at elevated temperature, but the pronounced change in the C peak
suggest some mobility of C and even the formation of carbon related defects as
suggested in [23].
CONCLUSIONS
In summary we have calculated the properties of boron icosahedral structures
using molecular dynamics simulations. The Tersoff parameters used are found to
be reasonably reliable to investigate structural and thermodynamic properties of the
boron icosahedral structures studied here when compared with ab initio results and
1 2 3 4 5 r, Å
0
2
4
6
8
g(r)
10
12
14
B–B
C–C
a
1 2 3 4 5 r, Å
0
2
4
6
8
g(r)
10
C–C
B–B
b
Fig. 5. Radial distribution of various atomic interactions in B4C: 300 (a), 1500 (b), 3000 (c) K.
www.ism.kiev.ua/stm 46
1 2 3 4 5 r, Å
0
2
4
6
8
g(r)
C–C
B–B
c
Fig. 5. (Contd.)
we suggest that with proper adjustment of the parameters they can be used to give
insight into the complex structures. In the case of several thousands of atoms (as
needed for very high temperature examination) possibly the Tersoff potential
method is tentatively a way forward. Using the potential we also predicted
thermodynamic properties such as linear thermal expansion coefficient and specific
heat of the B4C structure and compared this with experiment. It has been suggested
on the basis of the Tersoff results that the structure of B icosahedra seems to be
quite stable even at high temperature, whereas C atoms are more mobile and this
could lead to some clustering of the C atoms.
Ікосаедрічні структури бору є основними будівельними конструкціями
багатьох важливих твердих боридів і особливо це стосується В4С. Досліджено структу-
рні та термодинамічні властивості В4С при застосуванні моделювання динаміки молекул
з використанням як ab initio розрахунків, так і потенціалів Терсоффа. Розглянуті різні
фізичні характериcтики В4С, включаючи пружні константи, коефіцієнти теплового
розширення і теплоємність.
Ключові слова: алмаз, бор, електронна структура, пружні константи,
молекулярна динаміка.
Икосаэдрические структуры бора являются основными строительны-
ми конструкциями многих важных твердых боридов и особенно это касается В4С. Ис-
следованы структурные и термодинамические свойства В4С с применением моделирова-
ния динамики молекул, используя как ab initio расчеты, так и потенциалы Терсоффа.
Рассмотрены различные физические характериcтики В4С, включая постоянные упруго-
сти, коэффициенты теплового расширения и теплоемкость.
Ключевые слова: алмаз, бор, электронная структура, постоянные уп-
ругости, динамика молекул.
1. Wu B. R., Sung C. M., Lee F. E., Tai M. F. A First principles study of physical properties of
monoatomic structures of B, C, N, and O // Chinese J. Phys. – 2002. – 40. – P. 187–196.
2. Oganov A. R., Solozhenko V. L. Boron: a hunt for superhard polymorphs // J. Superhard Mate-
rials. – 2009. – 31, N 5. – P. 285–291.
3. McMillan P. F. New materials from high-pressure experiments // Nature Mater. – 2002. – 1. –
P. 19–25.
ISSN 0203-3119. Сверхтвердые материалы, 2012, № 1 47
4. Gregoryanz E., Sanloup C., Soyamazulu S. et al. Synthesis and characterization of a binary
noble metal nitride // Ibid. – 2004. – 3. – P. 294–297.
5. Lowther J. E. Potential super-hard phases and the stability of diamond-like boron–carbon
structures // J. Physics: Condensed Matter. – 2005. – 17. – P. 3221–3228.
6. Chen S., Gong X. G. Superhard pseudocubic BC2N superlattices // Phys. Rev. Lett. – 2007. –
98, art. 015502.
7. Kaner R. B., Gilman J. J., Tolbert S. H. Designing superhard materials // Science. – 2005. –
308. – P. 1268–1273.
8. Mattesini M., Matar S. F. Search for ultra-hard materials: theoretical characterisation of novel
orthorhombic BC2N crystals // Int. J. Inorg. Mater. – 2001. – 3. – P. 943–947.
9. Emin D. Icosahendra boron rich solids // Phys. Today. – 1987. – 20. – P. 55–59.
10. Gao F. M., Hou L., He Y. H. Origin of superhardness in icosahedral Boron materials // J.
Phys. Chem. B. – 2004. – 108, art. 13069.
11. Lowther J. E. Possible ultra-hard materials based upon boron icosahedra // Physica B. – 2002.
– 322. – P. 173–178.
12. Letsoalo T. E., Lowther J. E. Computational Investigation into elastic properties of bulk and
defective ultra hard B6O // J. Superhard Materials. – 2011. – 33, N 1. – P. 19–25.
13. McColm I. J. Ceramic Hardness. – New York: Plenum Press, 1990. – 324 p.
14. Letsoalo T. E. and Lowther J. E. Systematic trends in boron icosahedral structured materials
// Physica B. – 2008. – 403. – P. 2760–2765.
15. Suleyman E., de Wijs G. A, Brocks G. DFT study of planar boron sheets: a new template for
hydrogen storage // J. Phys. Chem. – 2009. – 13, art. 18962.
16. Widom M., Mihalkovič M. Crystal relative stability of α and β boron // J. Physics: Conf.
Series. – 2009. – 176, art. 012024.
17. Vast N., Baroni S., Zerah G. et al. Lattice dynamics of icosahedral alpha-boron under pres-
sure // Phys. Rev. Lett. – 1997. – 78. – P. 693–696.
18. Will G., Ploog K. Crystal structure of I-tetragonal boron // Nature. – 1974. – 251. – P. 406–
408.
19. Lee H., Speyer R. F. Hardness and fracture toughness of pressureless-sintered boron carbide
(B4C) // J. Am. Ceram. Soc. – 2002. – 85. – P. 1291–1293.
20. Gao F., Qin X., Wang L. et al. Prediction of new superhard boron-rich compounds // J. Phys.
Chem. B. – 2005. – 109, art. 14892.
21. Feng Y., Seidler G. T, Cross C. et al. Role of inversion symmetry and multipole effects in
nonresonant x-ray Raman scattering from icosahedral B4C // Phys. Rev. B. – 2004. – 69,
art. 125402.
22. Oganov A. R., Solozhenko V. L. Boron: a hunt for superhard polymorphs // J. Superhard
Materials. – 2009. – 31, N 5. – P. 285–291.
23. Oganov A. R., Chen J., Gatti C. et al. Ionic high-pressure form of elemental boron // Nature.
– 2009. – 457. – P. 863–865.
24. Li D., Ching W. Y. Electronic structures and optical properties of low- and high-pressure
phases of crystalline B2O3 // Phys. Rev. B. – 1996. – 54. – P. 13616–13622.
25. Kulikovsky V., Vorlicek V., Bohac R. et al. Mechanical properties and structure of amorphous
and crystalline B4C films // Diamond Relat. Mater. – 2008. – 18. – P. 27–33.
26. Allen M. P., Tildesley D. J. Computer Simulation in Chemical Physics. – New York: Kluwer
Academic Publishers, 1993. – 521 p.
27. Heermann D. W. Computer Simulation Methods in Theoretical Physics. – Berlin: Springer-
Verlag, 1990. – 148 p.
28. Kresse G., Hafner J. Ab initio molecular dynamics for liquid metals // Phys. Rev. B. – 1993.
– 47. – P. 558–561.
29. Ceperley D., Alder B. Ground state of the electron gas by a stochastic method // Phys. Rev.
Lett. – 1980. – 45. – P. 566–570.
30. Monkhorst H. J., Pack J. D. Special points for Brillouin-zone integrations // Phys. Rev. B. –
1976. – 13. – P. 5188–5193.
31. Kresse G., Joubert D. P. From ultrasoft pseudopotentials to the projector augmented wave
method // Ibid. – 1999. – 59. – P. 1758–1767.
32. Smith W., Forester T. Molecular dynamics of structured materials // J. Molecular Graphics. –
1996. – 14. – P. 136–148.
33. Gale J. D. Gulp: A computer program for the symmetry-adapted simulation of solids // J.
Chem. Soc., Faraday Trans. – 1997. – 1. – P. 629–641.
www.ism.kiev.ua/stm 48
34. Tersoff J. Empirical interatomic potential for silicon with improved elastic constants // Phys.
Rev. B. – 1988. – 38. – P. 9902–9905.
35. Tersoff J. Modeling solid-state chemistry: interatomic potentials for multicomponent systems
// Ibid. – 1989. – 39. – P. 5566–5568.
36. Chen E. T., Barnett R. N., Landman U. Surface melting of Ni(110) // Ibid. – 1990. – 41, N 1.
– P. 439–450.
37. Matsunaga K., Fisher C., Matsubara H. Tersoff potential parameters for simulating cubic
Boron Carbonitrides // Jpn. J. Appl. Phys. – 2000. – 39, Part 2, N 1A/B. – P. L48–L54.
38. Munetoh S., Motooka T., Moriguchi K., Shintani A. Interatomic potentials for Si–O systems
using Tersoff parametization // Computational Mater. Sci. – 2007. – 39. – P. 334–340.
39. Deker B. and Kasper B. Crystallographic Structure of various phases of boron // Acta Crys-
tallograhy. – 1959. – 12. – P. 503–509.
40. Nieto-Sanz D., Loubeyre P., Crichton W., Mezouar M. X-ray study of the synthesis of boron
oxides at high pressure: phase diagram and equation of state // Phys. Rev. B. – 2004. – 70,
art. 214108.
41. Hoard J., Hughes R. E., Sands D. E. The structure of tetragonal boron // J. Am. Chem. Soc. –
1958. – 80. – P. 4507–4510.
42. Delaye J. M. Modeling of multicomponent glasses: a review // Curr. Opin. Solid State Mat.
Sci. – 2001. – 5, N 5. – P. 451–478.
43. Ivashchenko V. I., Shevchenko V. I., Turchi P. E. A. First-principles study of the atomic and
electronic structures of crystalline and amorphous B4C // Phys. Rev. B. – 2009. – 80,
art. 235208.
44. Lee S. D., Bylander M., Kleinmann L. Elastic moduli of B12 and its compounds // Ibid. –
1992. – 45. – P. 3245–3251.
45. McClellan K. J., Chu F., Roper J. M., Shingo I. Room temperature single crystal elastic
constants of boron carbide // J. Mater. Sci. – 2001. – 36. – P. 3403–3409.
46. Manghnani M. H., Wang Y., Li F. et al. Elastic and vibrational properties of B4C to 21 GPa //
Proc. 17th AIRAPT Conf.: Science and Technology of High Pressure. – Hyderabad, India:
Universities Press, 2000. – P. 945–948.
47. Pierson H. O. Handbook of Carbon, Graphite, Diamond and Fullerenes: Properties, Process-
ing and Applications. – London: Noyes Publications, 1993. – 417 p.
48. Reeber R. R., Wang K. Thermal expansion, molar volume and specific heat of diamond from
0 to 3000 K // J. Electronic Mater. – 1996. – 25. – P. 63–67.
School of Physics and DST/NRF Centre Received 04.07.11
of Excellence in Strong Materials,
University of the Witwatersrand
|