Thermodynamic model of hardness: Particular case of boron-rich solids
A number of successful theoretical models of hardness have been developed recently. A thermodynamic model of hardness, which supposes the intrinsic character of correlation between hardness and thermodynamic properties of solids, allows one to predict hardness of known or even hypothetical solids fr...
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Інститут надтвердих матеріалів ім. В.М. Бакуля НАН України
2010
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| Цитувати: | Thermodynamic model of hardness: Particular case of boron-rich solids / V.A. Mukhanov, O.O. Kurakevych, V.L. Solozhenko // Сверхтвердые материалы. — 2010. — № 3. — С. 33-45. — Бібліогр.: 69 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860119951185018880 |
|---|---|
| author | Mukhanov, V.A. Kurakevych, O.O. Solozhenko, V.L. |
| author_facet | Mukhanov, V.A. Kurakevych, O.O. Solozhenko, V.L. |
| citation_txt | Thermodynamic model of hardness: Particular case of boron-rich solids / V.A. Mukhanov, O.O. Kurakevych, V.L. Solozhenko // Сверхтвердые материалы. — 2010. — № 3. — С. 33-45. — Бібліогр.: 69 назв. — англ. |
| collection | DSpace DC |
| container_title | Сверхтвердые материалы |
| description | A number of successful theoretical models of hardness have been developed recently. A thermodynamic model of hardness, which supposes the intrinsic character of correlation between hardness and thermodynamic properties of solids, allows one to predict hardness of known or even hypothetical solids from the data on Gibbs energy of atomization of the elements, which implicitly determine the energy density per chemical bonding. The only structural data needed is the coordination number of the atoms in a lattice. Using this approach, the hardness of known and hypothetical polymorphs of pure boron and a number of boron-rich solids has been calculated. The thermodynamic interpretation of the bonding energy allows one to predict the hardness as a function of thermodynamic parameters. In particular, the excellent agreement between experimental and calculated values has been observed not only for the room-temperature values of the Vickers hardness of stoichiometric compounds, but also for its temperature and concentration dependencies.
В останній час було запропоновано ряд вдалих теоретичних моделей твердості. Термодинамічна модель твердості, яка ґрунтується на кореляції між твердістю і термодинамічними властивостями твердих тіл, дає можливість спрогнозувати твердість відомих або навіть гіпотетичних твердих тіл, виходячи з даних по енергії Гіббса атомізації елементів, які опосередковано визначають енергетичні характеристики хімічного зв’язку. При цьому єдиною необхідною структурною характеристикою є координаційне число атомів у решітці. У рамках даного підходу була розрахована твердість відомих і гіпотетичних модифікацій елементарного бору и ряду сполук на його основі. Термодинамічна інтерпретація енергетичних характеристик хімічного зв’язку дає можливість розрахувати твердість фаз як функцію їх термодинамічних параметрів. Зокрема, хороше співвідношення між експериментальними та розрахунковими значеннями твердості за Віккерсом спостерігали не тільки для стехіометричних сполучень при кімнатній температурі, але і для температурної та концентраційної залежностей твердості.
В последнее время был предложен ряд удачных теоретических моделей твердости. Термодинамическая модель твердости, основанная на корреляции между твердостью и термодинамическими свойствами твердых тел, дает возможность спрогнозировать твердость известных или даже гипотетических твердых тел, исходя из данных по энергии Гиббса атомизации элементов, которые косвенно определяют энергетические характеристики химических связей. При этом единственной необходимой структурной характеристикой является координационное число атомов в решетке. В рамках данного подхода была рассчитана твердость известных и гипотетических модификаций элементарного бора и ряда соединений на его основе. Термодинамическая интерпретация энергетических характеристик химических связей позволяет рассчитать твердость фаз как функцию их термодинамических параметров. В частности, хорошее соответствие между экспериментальными и расчетными значениями твердости по Виккерсу наблюдали не только для стехиометрических соединений при комнатной температуре, но и для температурной и концентрационной зависимостей твердости.
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| first_indexed | 2025-12-07T17:38:23Z |
| format | Article |
| fulltext |
ISSN 0203-3119. Сверхтвердые материалы, 2010, № 3 33
USD 621.921.34; 666.233.004.14
V. A. Mukhanov (Villetaneuse, France)
O. O. Kurakevych (Paris, France)
V. L. Solozhenko (Villetaneuse, France)
Thermodynamic model of hardness: Particular
case of boron-rich solids
A number of successful theoretical models of hardness have been
developed recently. A thermodynamic model of hardness, which supposes the intrinsic
character of correlation between hardness and thermodynamic properties of solids,
allows one to predict hardness of known or even hypothetical solids from the data on
Gibbs energy of atomization of the elements, which implicitly determine the energy
density per chemical bonding. The only structural data needed is the coordination
number of the atoms in a lattice. Using this approach, the hardness of known and
hypothetical polymorphs of pure boron and a number of boron-rich solids has been
calculated. The thermodynamic interpretation of the bonding energy allows one to
predict the hardness as a function of thermodynamic parameters. In particular, the
excellent agreement between experimental and calculated values has been observed not
only for the room-temperature values of the Vickers hardness of stoichiometric
compounds, but also for its temperature and concentration dependencies.
Key words: superhard materials, boron, theory of hardness.
THERMODYNAMIC MODEL OF HARDNESS
The theory of hardness and design of novel superhard materials are great
challenge to materials scientists till now. Diamond-like and boron-rich compounds
of light elements (Fig. 1, a) take a particular place in this research, since the
hardest known phases have mainly these two structural types [1].
Many attempts have been made to predict hardness using the structural data and
such characteristics as bulk (B) and shear (G) moduli, specific bond energy, band
gap (Eg), density of valence electrons (i.e. the number of valence electrons per unit
volume), etc. [2—7]. Up to date the best correspondence between the calculated
and experimental values of hardness has been achieved in the recent works [3, 4,
8—10]. In all cases, the final accuracy is about 10% for hard and superhard phases,
i.e. at the level of experimental errors.
According to our model [8—10], the hardness of a phase with isodesmic
structure1 is proportional to the atomization energy, which may be considered as a
characteristic of the bond rigidity (for clarity, we will use the standard values of
Gibbs energy of atomization ΔG°at), and is in inverse proportion to the molar
volume of a phase and to the maximal coordination number of the atoms. The
value defined in such a way has the dimensions of pressure. The plasticity of
materials can be taken into account by empirical coefficient α. In general case the
polarity of bonds leads to the hardness decrease, which may be clearly seen in the
sequence of isoelectronic analogues of diamond, i.e. diamond (100 or 115 GPa)
[11, 12] — cubic boron nitride cBN (62 GPa) [13] — BeO (13 GPa) [3, 14] — LiF
(1.5 GPa) [3, 14]. This factor has been evaluated by empirical coefficient β, which
is the measure of the bond covalence.
1 Isodesmic structure is characterized by the similar bond strength in all direction.
www.ism.kiev.ua; www.rql.kiev.ua/almaz_j 34
B
50
C
2
B
50
N
2
B
13
N
2
cBC
5
B
6
O
B
4
CB
2
O
3
cBN
C
O N
B
a
0 20 40 60 80 100
0
20
40
60
80
100
MgAlB
24
α−B
12
α−AlB
12
B
6
Si
B
4
C
diamond
H
V
, G
P
a
(t
he
or
et
ic
al
)
H
V
, GPa (experimental)
cBC
5
cBN
γ−B
28
b
Fig. 1. Principal hard and superhard phases on the B—C—N—O concentration tetrahedron (a).
The boron-rich phases are surrounded by an oval line. A comparison of experimental values of
Vickers hardness of various phases with corresponding values calculated as a function of Gibbs
energy of atomization in the framework of the thermodynamic model of hardness (Eq. (1)) (b).
The open circles correspond to the boron-rich solids, while all other compounds/phases are
presented by solid circles.
The equation that allows calculating the Vickers hardness (HV) of crystals at
298 K is
αβε
°Δ
=
VN
G
H at
V
2
, (1)
where V is the molar (atomic) volume (cm3⋅mole–1); N is the maximal coordination
number2; α is the coefficient of relative (as compared to diamond) plasticity; β is
2 For some compounds of very complex structure, such as boron-rich solids, we will
use a mean/effective value.
ISSN 0203-3119. Сверхтвердые материалы, 2010, № 3 35
the coefficient corresponding to the bond polarity (see below); ε is the ratio
between the mean number of valence electrons per atom and the number of bonds
with neighboring atoms (N)3; ΔG°at is the standard Gibbs energy of atomization
(kJ⋅mole–1) of compound XmYn:
nmnm YXfYatXatYXat GGnGmG °Δ−°Δ+°Δ=°Δ , (2)
where
nmYXfG°Δ is the standard Gibbs energy of formation of XmYn; XatG°Δ and
YatG°Δ is the standard Gibbs energies of atomization of elements X and Y.
Coefficient α has been estimated from the experimental values of HV for
diamond, dSi, dGe and dSn. For the elementary substances and compounds of
second period elements α equals 1, while for other periods (≥ 3) α makes 0.7. This
coefficient reflects the decrease of the bond strength [4] for the elements of periods
≥ 3. The precise estimation of this coefficient, reflecting the presence of large inner
electron core and multiple non-occupied d- and f-orbitals, is outside of the purposes
of this study.
Coefficient β (square of the covalence f) has been calculated by the equation
2
2
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
χ+χ
χ=β
XY
Y , (3)
where χX, χY are the electronegativities of the elements by Pauling, χX > χY [15].
For elementary substances β = 1. In fact, even the presence of small amounts of
foreign atoms in the structure should cause the remarkable decrease of hardness, as
it can be seen by the example of boron-doped diamonds [16] (from 90—110 GPa
for pure single-crystal diamond down to 70—80 GPa for single crystals of boron-
doped diamond; i.e. down to ~ 75—85 % of the initial value, which well match the
square of bond ionicity β = 0.79 for B—C bonds).
For the refractory crystalline compounds the values of hardness calculated by
Eq. (1) are in a very good agreement (in the most cases less than 4 GPa of
discrepancy, i.e. < 7 %) with the experimental values [1, 3, 4, 11—14, 17—21]
(Fig. 1, b4).
Using Eq. (1) it is possible to calculate the hardness of dense phases with three-
dimensional structures that have not been synthesized to present time, e. g., C3N4
with the Si3N4 structure [5], CO2 with the α-SiO2 structure, hp-B2O3 with the Al2O3
structure [17], and a number of diamond-like phases of the B—C system [22]. The
advantage of the proposed method is that only the maximum coordination number
is used as a structural data [8]. In this case the molar volumes may be calculated
from the covalent radii of the elements, while ΔG°f values (usually the negligible
term as compared with ΔG°at of the elements) of the phases may be fixed to the
standard Gibbs energies of the formation of known compounds in the
corresponding binary systems, i.e. C2N2, CO2, β-B2O3, B4C, respectively [9]. The
3 The use of this coefficient allows one to establish the hardness of the AIBVII (ε = 1/N)
and AIIBVI (ε = 2/N) compounds, i.e. LiF, NaCl, BeO, ZnS, MgO, etc.
4 The considered compounds/phases are diamond, Si, Ge, dSn, SiC, cBN, wBN,
cBC2N, α-rh B, β-rh B, B4C, B6О, TiC, Si3N4, BeO, TiN, Al2O3, quartz, coesite,
stishovite, WC, ReB2, LiF, Al2SiO4F2, KAlSi3O8, Ca5(PO4)3F, CaF2, СаСО3, BAs,
BP, AlN, AlP, AlAs, AlSb, GaN, GaP, GaAs, GaSb, InN, InP, InAs, InSb, ZnS,
ZnSe, ZnTe, ZnO.
www.ism.kiev.ua; www.rql.kiev.ua/almaz_j 36
applicability of this method for estimating the hardness of hypothetical compounds
has been recently illustrated by the example of diamond-like BC5 (cBC5), a novel
superhard phase synthesized under high pressures and temperatures [18, 23].
Vickers hardness of this phase has been calculated to be 70.6 GPa (table), which is
in excellent agreement with the experimental value HV = 71 GPa.
The theoretical (Eq. (1)) and experimental values of Vickers hardness HV.
The starting data (free energy of atomization, density, coordination
number and electronegativity) as well as some intermediate values
of calculation are also given
Electronegativity
[15]
HV
Phase*
ΔGf**,
kJ/mole-at
ΔGat El
[15]
ΔGat,
kJ/mole-at
ρ,
g/cm3
V,
cm3/mole-at
N
2ΔGat/NV
GPa
El anion cation
α β
theor. exp.
Boron modifications
α-B12 0 0 518.8 2.447 4.418 5 47.0 2.04 2.04 1 1 47.0
″ — — — — — 6 39.1 — — — — 39.1
42
[39]
β-B106 0 0 518.8 2.334 4.632 5 44.8 2.04 2.04 1 1 44.8 45
[40]
T-B192 0 0 518.8 2.340 4.620 5 44.9 2.04 2.04 1 1 44.9
γ-B28
[42]
0 0 518.8 2.544 4.249 5 48.8 2.04 2.04 1 1 48.8 50
[58]
dB 0 0 518.8 2.548 4.243 [8] 4 61.1 2.04 2.04 1 1 61.1
″ — — — 2.178 4.963 [18] — 52.3 — — — 52.3
α-Ga
type
0 0 518.8 2.810 3.847
[42]
7 38.5 2.04 2.04 1 1 38.5
″ — — — — — 6 45.0 2.04 2.04 1 1 45.0
Compounds of the diamond structural type
cBN –120.15 455.563 607.3 3.489 3.555 4 85.4 3.04 3.04 2.04 1 0.645 55.1 62
[13]
cBC5
[18]
0 671.26 645.8 3.267 3.612 4 89.4 2.55 2.55 2.04 1 0.79 70.6 71
BP –47.4 278.3 446.0 2.970 7.034 4 31.7 2.19 2.19 2.04 1 0.93 29.5 33
[59]
Compounds of the α-B12 structural type
B6O –93 231.7 572.6 2.575 4.474
[60]
5 51.2 3.44 2.74 2.04 1 0.729 37.3
″ 0 — 479.6 — — — 42.9 — — — — — 31.2
38
[21]
B4C –12 671.3 561.3 2.507 4.407 5 50.9 2.55 2.30 2.04 1 0.884 45.0
″ 0 — 549.3 — — — 49.9 — — — — — 44.1
45
[46]
B9C 0 671.3 534.0 2.282 4.789 5 44.6 2.55 2.30 2.04 1 0.88 39.4
B13N2
[48]
–20 455.6 530.4 2.666 4.214 [47] 5 50.3 3.04 2.54 2.04 1 0.794 39.9
B4Si 0 411.3 497.3 2.425 5.882 5 33.8 1.90 2.04 1.97 1 0.97 32.6 27
[61]
B6P 0 278.3 484.4 2.583 5.300 5 36.6 2.19 2.12 2.04 1 0.962 35.2 37
B6As 0 261 482.0 3.570 5.593 5 34.5 2.18 2.12 2.04 1 0.962 33.2
ISSN 0203-3119. Сверхтвердые материалы, 2010, № 3 37
Contd.
Compounds of the β-B106 structural type
B25Mg2
[62]
0 127 488.7 2.488 4.747 5 41.2 1.31 2.04 1.68 1 0.81 33.5
B19.7Mg
[63]
0 127 499.9 2.416 4.744 5 42.1 1.31 2.04 1.68 1 0.81 34.2
AlB31
[64]
0 285.7 511.5 2.411 4.693 5 43.6 1.61 2.04 1.83 1 0.89 39.0
B36Si
[65]
0 411.3 515.9 2.343 4.813 5 42.9 1.90 2.04 1.97 1 0.97 41.4
Compounds of the T-B52 structural type
B50C2 0 671.3 524.7 2.395 4.533 5 46.3 2.55 2.30 2.04 1 0.884 40.9
B50N2 0 455.6 516.4 2.454 4.455 5 46.4 3.04 2.50 2.04 1 0.808 37.4
B50B2 0 0 518.8 2.383 4.536 [48] 5 45.7 2.04 2.04 1 1 45.7
Compounds of the T-B192 structural type
α-AlB12 0 285.7 500.9 2.650 4.549 5 44.0 1.61 2.04 1.83 1 0.894 39.4 37
Other boron-rich compounds
γ-AlB12 0 285.7 500.9 2.560 4.709 5 42.5 1.61 2.04 1.83 1 0.894 38.1
oB6Si
[66]
0 411.3 502.5 2.399 5.598 5 35.9 1.90 2.04 1.97 1 0.97 34.7 29
[67]
MgAlB14 0 199.4 478.9 2.660 4.761 5 40.2 1.46 2.04 1.75 1 0.85 34.3 35
[68]
WB4 0 807.1 576.5 10.193 4.456 5 51.7 2.36 2.20 2.04 1 0.93 47.9 46
[69]
* Hypothetical phases are given in italics.
** For the most of compounds the estimation of ΔGf is given using the thermodynamic data of known
phases. For the boron-rich compounds with high boron content (> 85 at% of B) it has been fixed to 0.
Equation (1) also allows one to calculate the values of hardness at various
temperatures by introducing the linear approximation of the temperature
dependence of ΔGat(T), i.e.
ΔGat(T) = ΔGat(300)·[1 – (T – 300)/(Tat – 300)], (4)
where Tat is the temperature of atomization5; as well as by introducing the
temperature dependences of molar volumes V(T). The theoretical simulation [8—
10] shows a good agreement with the experimental data on the temperature
dependences of Vickers and/or Knoop hardness for diamond, cBN, B4C, ReB2 and
Al2O3 (Fig. 2, a, b) in comparison with experimental data. The theoretical values of
hardness have been calculated by equation
)()300(
)300()()300()(
TVG
VTGHTH
at
at
Δ
Δ= . (5)
At relatively high temperatures (~ 0.3—0.5 Tat) this equation gives 10—15 %
higher values than the observed ones (bold lines in Figs. 2, a, b), that should be
5 For diamond and cBN the corresponding temperatures of sublimation are 4300 and
3300 K, respectively.
www.ism.kiev.ua; www.rql.kiev.ua/almaz_j 38
attributed to the increase of materials’ plasticity due to the intensification of the
surface and bulk diffusion [24]. The influence of the temperature on plasticity
(coefficient α) can be taken into account by the following empirical equation:
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
−α=⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
∞
−α=α
−
T
Tmelt
e
k
TkT
3/2
1)300(
)(
)(1)300()( , (6)
that supposes the Arrhenius-type temperature dependence of the dislocation
propagation constant k(T) (following Ref. [25], the activation energy was set to
2/3RTmelt). This term allows a decrease in the discrepancy between experimental
and calculated data down to the level of experimental error (dashed lines in Figs. 2,
a, b). Our model describes the lowest possible decrease of hardness in the case of a
material of a fixed microstructure. This explains the excellent agreement between
the experimental and theoretical HV(T) curves for single crystals (Fig. 2, a); while
in the case of polycrystalline ceramic materials some deviations (often non-
monotone) may occur due to the temperature-induced microstructure changes.
400 600 800 1000 1200 1400 1600
0
10
20
30
40
50
60
70
80
90
100
ReB
2
(H
V
)
diamond (H
V
)
Al
2
O
3
(H
V
)
diamond (H
K
)
cBN (H
K
)
H
, G
P
a
T, K
a
400 600 800 1000 1200 1400 1600
20
25
30
35
40
45
4 5 6
35
40
45
50
B
4
C (H
V
)
H
, G
P
a
T, K
H
, G
P
a B
4
C (H
V
)
B/C ratio
b
Fig. 2. Temperature dependence of hardness of single-crystal diamond, polycrystalline cBN
(mean particle size of 5 μm), single-crystal ReB2, and B4C-, SiC- and Al2O3-based ceramics (a,
b). The symbols represent the experimental data obtained by static indentation [52—57]. The
lines show the results of calculation using Eq. (5) under assumption that α = const (solid line)
and using Eq. (6) for α (dashed line). The concentration dependence of boron carbide hardness
(c). The symbols represent experimental data [51], while solid line shows the results of
calculation using Eq. (1). The crystallographic density of corresponding carbides has been
evaluated using the lattice parameter data reported in [51].
Our model has justified the previous suggestions about the increase of hardness
with pressure [26]. Because of the lack of reliable data on ΔGat at very high
ISSN 0203-3119. Сверхтвердые материалы, 2010, № 3 39
pressures, the prediction cannot be easily made using equations (1) (the ab initio
calculations of corresponding thermodynamic parameters could be useful in this
case). However, according to [10] where the non-monotone correlation between HV
and bulk modulus B has been explained in the framework of the same
thermodynamic model under the assumption of the similar nature of energy
stocked by chemical bonds during indentation and compression, the pressure
dependence of hardness is the same (up to a constant depending on material only)
as the pressure dependence of B, i.e.
)(const)( pBpHV = . (7)
Equation (7) allows one to suggest that the hard phases even with relatively low
bulk moduli may show a remarkable hardness increase with pressure. Each
material is expected to increase its hardness when the pressure is applied, however,
the phase transformations accompanied by an increase of coordination number
could prevent an infinite increase and cause a drop of hardness at a transformation
pressure.
Some compounds with relatively high hardness at ambient pressure and
relatively high pressure derivative of bulk modulus (as compared to diamond)
under pressure become harder more rapidly than diamond (as soon as the structural
phase transformations increasing the coordination number occurs); that allows
some of them (e. g., B6O) to reach the diamond hardness at very high pressures
[26]. It is interesting to note that graphite, a very soft material at ambient
conditions, may reach the diamond hardness at the lower pressure than many other
materials. This fact is in excellent agreement with the experimental and theoretical
results reported in [27, 28] on the formation of “superhard graphite” that can
scratch a single-crystal diamond, and allows us to suggest that other ordered [29—
31] and disordered [32—35] graphite-like phases should show similar behavior
under high pressure, even if the “compressed state” is not always recoverable at
ambient pressure.
HARDNESS OF BORON-RICH SOLIDS
One more advantage of the proposed method is the possibility to easily estimate
the hardness of various forms of boron and its compounds (B4C, B6O, B13N2, etc.,
see Fig. 3, a, Table), which is rather complicated by using other methods because
of extreme complexity of boron-related structures and a large number of atoms in a
unit cell. Usually the “non-ionic” contribution to hardness (2ΔGat/NV) is close to
that of pure boron. However, the role of ionicity is not so clear because of the
strong delocalization of chemical bonds. Thus, in our calculations for boron-rich
compounds we have taken the mean value of electronegativities of all atoms
connected to B12 icosahedron as an χ value for anion (or cation). For almost all
boron-rich compounds the mean coordination number has been fixed to <N> = 56;
and only for α-B12 to 6 because the half of its icosahedral B-atoms have
coordination number 7 due to the formation of three-center electron-deficient
bondings.
Here we should also remark that the microstructure developed during various
synthesis procedures [36] and even the influence of the single-crystal purity may
6 Since it is difficult to decide whether the B-atoms or B12 icosahedra should be
considered as structural units, the approximate mean values of coordination number
has been taken, which give the best agreement between calculated and experimental
data.
www.ism.kiev.ua; www.rql.kiev.ua/almaz_j 40
significantly affect the hardness [37]. Thus, the poor/lacking data on the hardness
of some boron-rich compounds may cause the significant under- or overestimation
of experimental HV-values.
Boron polymorphs
Boron is known to be the hardest element next to carbon [1, 38]. The
experimental values of hardness for α-B12 (HV = 42 GPa [39]) and β-B106
(HV = 45 GPa as the maximal hardness for samples remelted at ambient pressure
[40, 41] and 44(4) GPa for samples remelted at ~ 5 GPa, according to our
unpublished results) phases are in good agreement with the values (39.2 and
43.8 GPa, respectively) calculated in the framework of the thermodynamic model
of hardness. The hardness of recently synthesized superhard high-pressure boron
phase, orthorhombic γ-B28 [42, 43], was found to be 50 GPa [19], which also
agrees well with the calculated value of 48.8 GPa. Our model suggests that γ-B28
has the highest hardness among the known crystalline modifications of boron (as
well as the lowest compressibility [44]) because of its highest density. The hardest
polymorph is expected to be hypothetical diamond-like boron, a strongly
metastable covalent phase, which, probably, could be stabilized (e.g., by quenching
down to low temperatures) if the activation barrier of its transformation into
conventional boron phases is high enough. Using different estimations of atomic
volume (table), the expected hardness of dB should vary between 52 and 61 GPa.
The hardness of tetragonal polymorph T-B192 [45] has not been ever reported.
However, our calculations have shown that it should be the same as that of
rhombohedral β-B106.
Boron-rich solids of the α-B12 structural type
Although the α-B12 phase is metastable at ambient pressure [42], the small
amount of non-metal contaminations (C, O, N, Si, etc.) stabilizes the boron-rich
compounds of the α-B12 structural type. The calculated values of Vickers hardness
for B4C and B6O are 44 and 38 GPa, respectively; that is in a very good agreement
with the experimental data for single crystal B4C (HV = 45 GPa [46]) and
polycrystalline B6O (HV = 38 GPa [21]). The lower value of hardness for B6O as
compared to B4C may be explained by the higher ionicity of B—O bonds than of
B—C bonds. The estimation of hardness for the recently synthesized rhombohedral
boron subnitride B13N2 [47—49] has given HV = 40.3 GPa7 that allows ascribing
B13N2 to superhard phases. Its relatively high bulk modulus comparable to those of
B4C and B6O additionally confirms this suggestion [50].
Boron carbide B4C, a very hard substance, which may be produced at ambient
pressure, is, in fact, a kind of a solid solution of carbon in boron, i.e. B4+xC1–x,
having a wide concentration range of stability. Using our model of hardness, we
have also succeeded to calculate the concentration dependence for the Vickers
hardness of B4+xC1–x (Fig. 2, c), which is in a satisfactory agreement with
experimental data reported in [51].
The calculated hardness of the α-B12-type compounds with the elements of the
3rd and higher periods somewhat decreases due to the high concentration of the
polar (partly ionic) bonds. At the same time, most of the phases not only belong to
the hardness range assigned to the “hard phases”, but also are close to its upper
limit (Fig. 3, a).
7 Τhe 2ΔG°at/NV value has been set to a mean (~ 51 GPa) of corresponding values for
B6O and B4C; β = 0.79.
ISSN 0203-3119. Сверхтвердые материалы, 2010, № 3 41
0.01 0.1 1
0
20
40
60
80
100
cBN
cBC5
γ�B28
B
or
on
�r
ic
h Superhard
H
V
, G
P
a
Atomic fraction of foreign elements
Hard
d�B
diamond
cBC2N
B4C
Compounds of group
III and IV elements γ�B2O3
β�B2O3
α�B2O3
a
b
c
d
e
f g
Fig. 3. Theoretical (○) and experimental (×) values of Vickers hardness of boron-containing
compounds as a function of the atomic fraction of foreign element(s) (a). The approximate
boundaries of hard (HV > 10 GPa), superhard (HV > 40 GPa), and boron-rich (boron content
≥ 80 at%) phases are given by horizontal and vertical lines. Principal structural types of boron-rich
compounds related to known and hypothetical modifications of boron (b—g); i.e. rhombohedral α-
B12 (b, c) and β-B106 (d); orthorhombic γ-B28 (e); tetragonal T-B192 (f), and T-B52 (g).
Boron-rich solids of the β-B106 structural type
The β-B106 phase is the only thermodynamically stable phase of boron at
pressures up to few GPa’s. Very small amounts of foreign elements, especially
metals, give a rise to a number of boron-rich compounds of the β-B106 structural
www.ism.kiev.ua; www.rql.kiev.ua/almaz_j 42
type (table). Their hardness is expected to be lower just because of the partial
ionicity of the chemical bonds.
Boron-rich solids of the T-B52 structural type
The hypothetical T-B52 phase may be stabilized only by a small amount of
nitrogen or carbon atoms as compounds B50N2 and B50C2. The hardness of these
phases have not been experimentally studied to the present day, while our
calculations show that they should have hardness between B6O and B4C, two
common superhard phases of the α-B12 type. The recent studies of the B—BN
system under high pressure [48] have revealed the stabilization of the phase that,
most probably, is a solid solution B50N2–xBx, with x ≈ 2. Its hardness is supposed to
be very close to that predicted for T-B52, i.e. 46 GPa.
CONCLUSIONS
Thus, it has been found that the hardness of solids is directly related to their
thermodynamic and structural properties. The formulated equations may be used
for a large number of compounds with various types of chemical bonding and
structures. The proposed method allows estimating the hardness and
compressibility of various hypothetical compounds using the data on the Gibbs
energy of atomization of elements and covalent/ionic radii. The applicability of the
approach to the prediction of hardness has been illustrated by examples of the
recently synthesized superhard diamond-like BC5 and orthorhombic modification
of boron, γ-B28. In the framework of the proposed method we have calculated the
hardness of a large number of boron-rich solids and found that it strongly depends
on the electronegativity of atoms incorporated into boron lattice.
The authors are grateful to the Agence Nationale de la Recherche for the
financial support (grant ANR-05-BLAN-0141).
В останній час було запропоновано ряд вдалих теоретичних моделей
твердості. Термодинамічна модель твердості, яка ґрунтується на кореляції між
твердістю і термодинамічними властивостями твердих тіл, дає можливість
спрогнозувати твердість відомих або навіть гіпотетичних твердих тіл, виходячи з даних
по енергії Гіббса атомізації елементів, які опосередковано визначають енергетичні
характеристики хімічного зв’язку. При цьому єдиною необхідною структурною
характеристикою є координаційне число атомів у решітці. У рамках даного підходу була
розрахована твердість відомих і гіпотетичних модифікацій елементарного бору и ряду
сполук на його основі. Термодинамічна інтерпретація енергетичних характеристик
хімічного зв’язку дає можливість розрахувати твердість фаз як функцію їх
термодинамічних параметрів. Зокрема, хороше співвідношення між експериментальними
та розрахунковими значеннями твердості за Віккерсом спостерігали не тільки для
стехіометричних сполучень при кімнатній температурі, але і для температурної та
концентраційної залежностей твердості.
Ключові слова: надтверді матеріали, бор, теорія твердості.
В последнее время был предложен ряд удачных теоретических моделей
твердости. Термодинамическая модель твердости, основанная на корреляции между
твердостью и термодинамическими свойствами твердых тел, дает возможность
спрогнозировать твердость известных или даже гипотетических твердых тел, исходя из
данных по энергии Гиббса атомизации элементов, которые косвенно определяют
энергетические характеристики химических связей. При этом единственной необходимой
структурной характеристикой является координационное число атомов в решетке. В
рамках данного подхода была рассчитана твердость известных и гипотетических
модификаций элементарного бора и ряда соединений на его основе. Термодинамическая
интерпретация энергетических характеристик химических связей позволяет рассчитать
твердость фаз как функцию их термодинамических параметров. В частности, хорошее
ISSN 0203-3119. Сверхтвердые материалы, 2010, № 3 43
соответствие между экспериментальными и расчетными значениями твердости по
Виккерсу наблюдали не только для стехиометрических соединений при комнатной
температуре, но и для температурной и концентрационной зависимостей твердости.
Ключевые слова: сверхтвердые материалы, бор, теория твердости.
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LPMTM-CNRS, Université Paris Nord Received 25.11.09
IMPMC, Université P & M Curie
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| publisher | Інститут надтвердих матеріалів ім. В.М. Бакуля НАН України |
| record_format | dspace |
| spelling | Mukhanov, V.A. Kurakevych, O.O. Solozhenko, V.L. 2014-06-02T06:40:45Z 2014-06-02T06:40:45Z 2010 Thermodynamic model of hardness: Particular case of boron-rich solids / V.A. Mukhanov, O.O. Kurakevych, V.L. Solozhenko // Сверхтвердые материалы. — 2010. — № 3. — С. 33-45. — Бібліогр.: 69 назв. — англ. 0203-3119 https://nasplib.isofts.kiev.ua/handle/123456789/63472 621.921.34; 666.233.004.14 A number of successful theoretical models of hardness have been developed recently. A thermodynamic model of hardness, which supposes the intrinsic character of correlation between hardness and thermodynamic properties of solids, allows one to predict hardness of known or even hypothetical solids from the data on Gibbs energy of atomization of the elements, which implicitly determine the energy density per chemical bonding. The only structural data needed is the coordination number of the atoms in a lattice. Using this approach, the hardness of known and hypothetical polymorphs of pure boron and a number of boron-rich solids has been calculated. The thermodynamic interpretation of the bonding energy allows one to predict the hardness as a function of thermodynamic parameters. In particular, the excellent agreement between experimental and calculated values has been observed not only for the room-temperature values of the Vickers hardness of stoichiometric compounds, but also for its temperature and concentration dependencies. В останній час було запропоновано ряд вдалих теоретичних моделей твердості. Термодинамічна модель твердості, яка ґрунтується на кореляції між твердістю і термодинамічними властивостями твердих тіл, дає можливість спрогнозувати твердість відомих або навіть гіпотетичних твердих тіл, виходячи з даних по енергії Гіббса атомізації елементів, які опосередковано визначають енергетичні характеристики хімічного зв’язку. При цьому єдиною необхідною структурною характеристикою є координаційне число атомів у решітці. У рамках даного підходу була розрахована твердість відомих і гіпотетичних модифікацій елементарного бору и ряду сполук на його основі. Термодинамічна інтерпретація енергетичних характеристик хімічного зв’язку дає можливість розрахувати твердість фаз як функцію їх термодинамічних параметрів. Зокрема, хороше співвідношення між експериментальними та розрахунковими значеннями твердості за Віккерсом спостерігали не тільки для стехіометричних сполучень при кімнатній температурі, але і для температурної та концентраційної залежностей твердості. В последнее время был предложен ряд удачных теоретических моделей твердости. Термодинамическая модель твердости, основанная на корреляции между твердостью и термодинамическими свойствами твердых тел, дает возможность спрогнозировать твердость известных или даже гипотетических твердых тел, исходя из данных по энергии Гиббса атомизации элементов, которые косвенно определяют энергетические характеристики химических связей. При этом единственной необходимой структурной характеристикой является координационное число атомов в решетке. В рамках данного подхода была рассчитана твердость известных и гипотетических модификаций элементарного бора и ряда соединений на его основе. Термодинамическая интерпретация энергетических характеристик химических связей позволяет рассчитать твердость фаз как функцию их термодинамических параметров. В частности, хорошее соответствие между экспериментальными и расчетными значениями твердости по Виккерсу наблюдали не только для стехиометрических соединений при комнатной температуре, но и для температурной и концентрационной зависимостей твердости. The authors are grateful to the Agence Nationale de la Recherche for the financial support (grant ANR-05-BLAN-0141). en Інститут надтвердих матеріалів ім. В.М. Бакуля НАН України Сверхтвердые материалы Получение, структура, свойства Thermodynamic model of hardness: Particular case of boron-rich solids Article published earlier |
| spellingShingle | Thermodynamic model of hardness: Particular case of boron-rich solids Mukhanov, V.A. Kurakevych, O.O. Solozhenko, V.L. Получение, структура, свойства |
| title | Thermodynamic model of hardness: Particular case of boron-rich solids |
| title_full | Thermodynamic model of hardness: Particular case of boron-rich solids |
| title_fullStr | Thermodynamic model of hardness: Particular case of boron-rich solids |
| title_full_unstemmed | Thermodynamic model of hardness: Particular case of boron-rich solids |
| title_short | Thermodynamic model of hardness: Particular case of boron-rich solids |
| title_sort | thermodynamic model of hardness: particular case of boron-rich solids |
| topic | Получение, структура, свойства |
| topic_facet | Получение, структура, свойства |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/63472 |
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