Microscopic models of hardness
Recent developments in the field of microscopic hardness models have been reviewed. In these models, the theoretical hardness is described as a function of the bond density and bond strength. The bond strength may be characterized by energy gap, reference potential, electron-holding energy or Gibbs...
Збережено в:
| Опубліковано в: : | Сверхтвердые материалы |
|---|---|
| Дата: | 2010 |
| Автори: | , |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Інститут надтвердих матеріалів ім. В.М. Бакуля НАН України
2010
|
| Теми: | |
| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/63471 |
| 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: | Microscopic models of hardness / F.M. Gao, L.H. Gao // Сверхтвердые материалы. — 2010. — № 3. — С. 9-32. — Бібліогр.: 108 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859802251013390336 |
|---|---|
| author | Gao, F.M. Gao, L.H. |
| author_facet | Gao, F.M. Gao, L.H. |
| citation_txt | Microscopic models of hardness / F.M. Gao, L.H. Gao // Сверхтвердые материалы. — 2010. — № 3. — С. 9-32. — Бібліогр.: 108 назв. — англ. |
| collection | DSpace DC |
| container_title | Сверхтвердые материалы |
| description | Recent developments in the field of microscopic hardness models have been reviewed. In these models, the theoretical hardness is described as a function of the bond density and bond strength. The bond strength may be characterized by energy gap, reference potential, electron-holding energy or Gibbs free energy, and different expressions of bond strength may lead to different hardness models. In particular, the hardness model based on the chemical bond theory of complex crystals has been introduced in detail. The examples of the hardness calculations of typical crystals, such as spinel Si₃N₄, stishovite SiO₂, B₁₂O₂, ReB₂, OsB₂, RuB₂, and PtN₂, are presented. These microscopic models of hardness would play an important role in search for new hard materials.
Зроблено огляд останніх розробок в області мікромоделей твердості. У цих моделях теоретичну твердість описано як функцію щільності та міцності зв’язку. Міцність зв’язку може бути охарактеризована шириною забороненої зони, опорним потенціалом, енергією утримання електрона або вільною енергією Гіббса. Тому різні вирази міцності зв’язку приведуть до різних моделей міцності. Зокрема, докладно описано модель твердості, основану на теорії хімічного зв’язку складних кристалів. Наведено приклади розрахунків твердості типових кристалів, таких як шпінель Si₃N₄, стишовіт SiO₂, B₁₂O₂, ReB₂, OsB₂, RuB₂ и PtN₂. Ці мікромоделі твердості будуть грати важливу роль в пошуку нових твердих матеріалів.
Дан обзор последних разработок в области микромоделей твердости. В этих моделях теоретическая твердость описана как функция плотности и прочности связи. Прочность связи может быть охарактеризована шириной запрещенной зоны, опорным потенциалом, энергией удержания электрона или свободной энергией Гиббса. Поэтому различные выражения прочности связи приведут к различным моделям твердости. В частности, подробно описана модель твердости, основанная на теории химической связи сложных кристаллов. Привены примеры расчета твердости типичных кристаллов, таких как шпинель Si₃N₄, стишовит SiO₂, B₁₂O₂, ReB₂, OsB₂, RuB₂ и PtN₂. Эти микромодели твердости будут играть важную роль в поиске новых твердых материалов.
|
| first_indexed | 2025-12-07T15:13:41Z |
| format | Article |
| fulltext |
ISSN 0203-3119. Сверхтвердые материалы, 2010, № 3 9
USD 539.53
F. M. Gao, L. H. Gao (Qinhuangdao, China)
Microscopic models of hardness
Recent developments in the field of microscopic hardness models
have been reviewed. In these models, the theoretical hardness is described as a
function of the bond density and bond strength. The bond strength may be
characterized by energy gap, reference potential, electron-holding energy or Gibbs
free energy, and different expressions of bond strength may lead to different hardness
models. In particular, the hardness model based on the chemical bond theory of
complex crystals has been introduced in detail. The examples of the hardness
calculations of typical crystals, such as spinel Si3N4, stishovite SiO2, B12O2, ReB2, OsB2,
RuB2, and PtN2, are presented. These microscopic models of hardness would play an
important role in search for new hard materials.
Key words: hardness, bulk modulus, shear modulus, ionicity,
superhard materials.
HARDNESS TESTS
Hardness is an important mechanical property of materials. It is defined as the
resistance of a material to localized deformation [1—7]. In materials science, there
are two principal operational definitions of hardness: (i) Scratch hardness:
Resistance to fracture or plastic deformation due to friction from a sharp object; (ii)
Indentation hardness: Resistance to plastic deformation due to a constant load from
a sharp indenter. Indentation hardness test is the usual type of hardness test. It can
be done using Vickers as well as Knoop indenters. The Vickers test uses a
symmetrical square pyramidal indenter. The Knoop test uses an asymmetrical
rhombic-based pyramidal diamond indenter. In microindentation hardness testing,
a pyramid is impressed into the surface of the test specimen using a known applied
load of 1 to 1000 gf. The load per unit area of impression is taken as the measure of
hardness:
cSWH /= , (1)
where Sc is the contact area for Vickers hardness or the projected area for Knoop
hardness and W is the maximum applied load. The Knoop and Vickers hardness
values differ slightly. For hard materials, their values are close enough to be within
the measurement error.
Syntheses of new classes of hard and superhard materials provide new
challenges for the measurement of hardness [8—15]. Since the hardness is strongly
dependent on an indentation load, the load should always be reported with the
hardness outcome. Brazhkin et al. [16] pointed out that necessary criteria for
comparing hardness among very hard substances can be established by providing
the details regarding the type of indenter, applied loads, indentation time, sample
orientation, quality of the tested surface, and so on. Figure 1 shows the load
dependence of the Vickers hardness for the sample of BN nanocomposite
synthesized by Dubrovinskaia et al. [17]. As can be seen, its hardness even reached
145 GPa at low loads. A constant hardness is reached at loads from 4.9 to 9.8 N.
© F. M. GAO, L. H. GAO, 2010
www.ism.kiev.ua; www.rql.kiev.ua/almaz_j 10
60
80
100
120
140
160
0 2 4 6 8 10
H
ar
dn
es
s,
G
P
a
Load, N
Fig. 1. Load dependence of the Vickers hardness for the sample of superhard BN nanocomposite.
The loading time is 20 s. (the data from [17]).
PHYSICS OF HARDNESS
Superhard materials are defined as having a hardness of above 40 GPa. At
present the experimental hardness can vary by more than 10% even for the same
material; scientists have therefore been keen to devise a theoretical method for
predicting the hardness of a material with a more certainty. In order to design the
new superhard materials [18—30], clarifying the nature of hardness is of utmost
importance. In recent years many attempts to develop a physical definition of
hardness have been made.
Hardness and elastic modulus
For a material to be hard it must support the volume decrease created by the
applied force, therefore, it should have a high bulk modulus. For compounds with a
given bonding type, for example, the group IV elements (see Fig. 2), III—V
compounds (see Fig. 3), and II—VI compounds, the hardness correlates roughly
with the bulk modulus [31]. Thus, the bulk modulus was used to predict the
hardness of a new material.
Ideally, one would hope to use ab initio computations to determine B as a
function of applied strain, but this is computationally very expensive. Therefore,
some empirical relations of bulk modulus were proposed [32—35]. Cohen [32]
first proposed a relationship for the bulk modulus B of a compound solid, which
follows:
5.3
)2201972(
4 d
fN
B ic −
= , (2)
where Nc is the bulk coordination number, d is the bond length, and fi is an
empirical ionicity parameter.
As pointed out in the review article by Brazhkin et al. [35], the cohesive energy
(i.e. total bonding strength) is also a good correlator/predictor of bulk modulus.
There is a clear correlation between the molar volume Vm, bulk modulus K and
cohesive energy Ec,
ISSN 0203-3119. Сверхтвердые материалы, 2010, № 3 11
m
c
V
EK ∝ . (3)
0
20
40
60
80
100
120
0 100 200 300 400 500
H
ar
dn
es
s,
G
P
a
Bulk modulus, GPa
C
SiC
Si
Ge
Sn
Fig. 2. Hardness vs. bulk moduli for group IV elements.
0
20
40
60
80
0 100 200 300 400
Bulk modulus, GPa
H
ar
dn
es
s,
G
P
a
BN
BP
AlP
InSb
Fig. 3. Hardness vs. bulk moduli for cubic III—V compounds.
Zhang et al. [33] have studied the relation between the lattice energies and the
bulk moduli of binary inorganic crystals by the concept of the lattice energy
density. They have found that the lattice energy densities are in good linear relation
with the bulk moduli for the same type of crystals.
Although there is a correlation between the hardness values and the bulk moduli
for particular classes of materials, there are still limitations to the use of bulk
modulus for predicting hardness [36]. For example, the bulk modulus of α-Al2O3
exceeds that of B12O2, however, its hardness is significantly less. The bulk modulus
of Os is comparable with that of diamond, while its hardness is approximately
3.50 GPa, far smaller than that of diamond.
www.ism.kiev.ua; www.rql.kiev.ua/almaz_j 12
Teter [37] and Brazhkin et al. [35] pointed out that hardness values correlate
better with shear moduli than with bulk moduli, as shown in Fig. 4. The shear
modulus describes the resistance of a material to anisotropic shape change. It
depends on both the plane of shear and the direction of shear. It indicates how well
a material resists a tearing force. For covalent crystals, the shear modulus
correlates roughly with hardness [38], which is also applicable for metals as well as
ionic crystals. In order to be hard, the material must not deform in a direction
different from the applied load, in other words, it must have a high shear modulus.
0
20
40
60
80
100
120
0 100 200 300 400 500 600
V
ic
ke
rs
h
ar
dn
es
s,
G
P
a
Shear modulus, GPa
BN
BP Stishovite
SiC
Al
2
O
3
AlN GaN
Si
3
N
4
Diamond
Fig. 4. Hardness vs. shear moduli.
Bulk modulus B and shear modulus G can be obtained from elastic stiffness
constants Cij and elastic compliance constants Sij of considered crystal systems [39,
40]:
)(
9
2)(
9
1
231312332211 CCCCCCBV +++++= ;
)(2)(
1
231312332211 SSSSSS
BR +++++
= ;
)(
5
1)(
15
1
665544231312332211 CCCCCCCCCGV +++−−−++= ;
)(3)(4)(4
15
665544231312332211 SSSSSSSSS
GR +++++−++
= ;
)(
2
1
RV BBB += ;
ISSN 0203-3119. Сверхтвердые материалы, 2010, № 3 13
)(
2
1
RV GGG += .
Theoretical definition of hardness
Although the better correlation has been observed between the hardness and
shear modulus, the dependence is not yet unequivocal and monotonic [35]. For
example, the bulk and shear moduli of tungsten carbide are as high as 439 and
282 GPa, respectively, which are among the highest as known, but its hardness is
only 30 GPa [27]. Indeed, there is no one-to-one correspondence between hardness
and bulk modulus or shear modulus.
For theoretical study, an appropriate definition of hardness is necessary and
essential. Gao et al. [41, 42] took an important step towards the goal of theoretical
characterization of hardness by developing a semi-empirical formula for the
hardness of a material based on ionicity, bond length, and the number of electrons
available for bonding, and succeeded in estimating the hardness of a number of
covalent materials using the semi-empirical formulae.
Gao et. al. [41] pointed out that the hardness of covalent crystals is intrinsic and
equivalent to the sum of resistance of each bond per unit area to the indenter. In an
indentation test, the bond breaking can occur. According to Gilman’s assumption
[38], energetically breaking an electron-pair bond inside a crystal means that two
electrons become excited from the valence band to the conduction band. In other
words, the resistant force of bond can be characterized by energy gap Eg. Based on
this assumption, the hardness of covalent crystals should have the form:
H (GPa) = ANaEg, (4)
where A is the proportional coefficient and Na is the covalent bond number per unit
area.
Following Gao’s work, Šimůnek and Vackar [43, 44] proposed another
expression for hardness by introducing the bond strength concept
ijSCH
Ω
= , (5)
where C is the proportional coefficient and Ω is the volume of a pair of ij atoms. Sij
is the bond strength between atoms i, j.
Recently, Li et al. [45, 46] have also suggested a hardness formula:
qXpNH abV += , (6)
where p and q are the constants, NV is the bond density, Xab is the bond
electronegativity and,
b
b
a
a
N
X
N
X
abX = , (7)
where Xa and Xb are the electronegativity of atoms a and b, respectively. Na and Nb
are the coordination numbers of atoms a and b, respectively.
More recently, based on Gibbs free energy of atomization θΔ atG , Mukhanov et
al. [47] have proposed a thermodynamic formula for the hardness calculations:
αβε=
θΔ
VN
GatH 2 , (8)
www.ism.kiev.ua; www.rql.kiev.ua/almaz_j 14
where V is the molar volume, N is the maximum coordination number, α is the
coefficient of the relative plasticity, β is the coefficient allowing for a contribution
of the bond polarity, ε is the ratio of the amount of valent electrons per atom to the
amount of bonds that this atom forms with the neighboring atoms.
In addition, the work of Gilman [38] indicates that the hardness can often be
related to a shear instability such as associated with a structural phase transition in
tetrahedral semiconductors.
CALCULATIONS OF CHEMICAL BOND IONICITY AND HARDNESS
Chemical bond ionicity
As mentioned above, the ionicity of chemical bond plays a significant role in
mechanical properties such as hardness, bulk modulus. According to Pauling’s rule
[48], if we consider a crystal, in which cation A, carrying a charge + Ze, is
coordinated by N anions B, then each A—B bond is said to have a Pauling bond
valence of QAB = Z/N. In a stable coordinated structure the total bond valence of the
bonds that reach an anion from all the neighboring cations is equal to the charge of
the anion. Then, the ionicity of A—B bond can be calculated as follows:
4
)(exp1
2
BA
AB
XXQfi
−−−= , (9)
where XA and XB are the electronegativity of atoms A and B, respectively.
Another noted scale of ionicity is Phillips’ ionicity. Phillips’ criterion for the
ionicity is more accurate spectroscopically than that of Pauling, which is based on
the data on heats of the formation of solids [49, 50].
Reviews concerning the bond ionicity and its application had been made by
Phillips [51], Van Vechten [52, 53], Levine [54, 55]. However, it is known that
PVL (Phillips-Van Vechten-Levine) theory can deal with binary crystals only. In
order to calculate the bond ionicity of a multicomponent crystal, the
multicomponent crystal must be decomposed into pseudobinary crystals containing
only one type of chemical bond. A crucial method decomposing the complex
crystal into pseudobinary crystals each containing only one type of chemical bond
is proposed by Zhang [56, 57]. For the multibond crystal AaBb…, the subformula
for any kind of chemical bond A—B can be expressed as:
BB)(AAA)(B
CBCA
⎥
⎦
⎤
⎢
⎣
⎡ −
⎥
⎦
⎤
⎢
⎣
⎡ −
N
bN
N
aN , (10)
where A, B,… represent different elements or different sites of the same element in
the crystal formula, and a, b,… represent numbers of the corresponding element,
N(B—A) represents the number of B ions in the coordination group of the A ion,
and NCA represents the nearest coordination number of the A ion. These binary
crystals are related to each other, and every binary crystal includes only one type of
chemical bond. However, the properties of these pseudobinary crystals are
different from those of real binary crystals, although their chemical bond
parameters can be calculated in a similar way.
Equation (10) also can be rewritten in another form
nN
aN AB)AB(
CA
− (11)
and
ISSN 0203-3119. Сверхтвердые материалы, 2010, № 3 15
CB
CA
A)(B
B)(A
aNN
bNN
n
−
−
= , (12)
where the prefix N(B—A)a/NCA represents the relation between the number of B
ions and the total number of ions bonded to the central A ion, and the subscript
[N(A—B)bNCA]/[N(B—A)aNCB] represents the ratio of the element B to A, n.
According to Eq. (11), each type of bond has its corresponding subformula, and the
sum of all subformulae equals the crystal formula, which is called the bond-valence
equation.
After decomposing a complex crystal into different kinds of pseudobinary
crystals, which form an isotropic system, and introducing an effective charge of a
valence electron by Pauling’s bond valence method [48], PVL theory can be used
directly to calculate the chemical bond parameters in a complex crystal compound.
By analogy with the work of PVL, average energy gap μ
gE for every μ bond in
pseudobinary crystals can be separated into homopolar μ
hE and heteropolar Cμ
parts.
Homopolar gap μ
hE can be interpreted as produced by the symmetric part of the
total potential, while the ionic or charge-transfer gap Cμ results from the effect of
the antisymmetric part. The average valence-conduction band gap is given by
222 )()()( μμμ += CEE hg . (13)
The ionicity and covalency of any type of chemical bond is defined as follows:
22 )/()( μμμ = gi ECf ; (14)
22 )/()( μμμ = ghc EEf (15)
and
48.2)/(74.39 μμ = dEh (eV), (16)
where dμ is the bond length. For any binary crystal, i.e. ABn type compounds,
heteropolar Cμ part is defined as
[ ] μ
0
*μ
B
μ
A
*
A
μμ
0s
)()(4.14
r
eZnZZbC
rk μμ−
μ −Δ+= (eV); (17)
2
μ
μ
0
dr = ;
2/1
F
2/1
B
μ
F
s )(551.1
π
4 μμ =⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
= k
a
kk ; (18)
μμ = eNk 23
F π3)( ; (19)
www.ism.kiev.ua; www.rql.kiev.ua/almaz_j 16
μ
*μ
μ )(
b
e
e v
nN = ; (20)
μ
μ
μ
μ
μ +=
CB
*
B
CA
*
A* )()(
)(
N
Z
N
Z
ne ; (21)
[ ]∑ νν
μ
μ =
ν
b
b Nd
dv 3
3
)(
)( , (22)
where μ
bv is the bond volume, ( μ
en )* is the number of effective valence electrons
per μ bond, μ
eN is the number of valence electrons of μ bond per cubic centimeter.
μ
Fk and μ
sk are Fermi wave number and Thomas-Fermi screening wave number of
valence electrons in a binary crystal composed of only one type of bond μ,
respectively. aB is the Bohr radius and n is the ratio of element B to element A in
the subformula. *
A )( μZ and *
B )( μZ are the numbers of effective valence electrons
of the A and B ions, respectively, and *
A )( μZ = μμ
CAABNQ , *
A )( μZ =
μμμμ − BBCBAB )]8/()[( ZZNQ , μ
ABQ is the Pauling bond valence of A—B bonds, μ
BZ is
the number of valence electrons of the B atoms. Δ μ
AZ is the correction factors from
d electron effects such as the crystal field stable energy and Janh-Teller effect, etc.
[58, 59], bμ is proportional to the square of the average coordination number μ
cN
2)( μμ β= cNb ; (23)
n
nN
n
N
Nc +
+
+
=
μμ
μ
11
CBCA , (24)
where bμ depends on a given crystal structure. The typical value of β is
0.089 ± 10% [54].
If the dielectric constant of the crystal is known, the value of β can be deduced
from the Kramers-Kronig relation of dielectric function at the long wave limit,
which is written as
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
+−⋅=
μμ
2μ
2μ
μ
μ
2μ
2
μ
)48(
)(
4
1
)(
π4
F
g
F
g
g
e
E
E
E
E
Em
DeN
χ ; (25)
μ
μ
μ11)ε( χ+=χ+=∞ ∑ F , (26)
where χ is the macroscopic linear susceptibility, χμ is the total macroscopic
susceptibility of a binary crystal composed of only one type of bond μ, μ
FE is the
Fermi energy, Fμ is the fraction of the binary crystal composing the actual complex
crystal. Dμ is the periodic dependent constants tabulated in [54].
ISSN 0203-3119. Сверхтвердые материалы, 2010, № 3 17
Hardness of polar covalent crystal
Equation (4) is suitable for the hardness calculations of pure covalent crystals
only. For polar covalent crystals, besides covalent component, partial ionic
bonding has to be considered. Ionic bonding results from long-range electrostatic
force, which is not directly related to hardness [38]. Phillips’ homopolar band gap
Eh characterizes the activation energies of dislocation glide in polar covalent
crystals and the strength of the covalent bonding. On the other hand, the partial
ionic bonding results in the loss of covalent bond charge, further results in a
smaller effective covalent bond number per unit area (Na) in comparison with that
of pure covalent crystals. This screening effect can be described by a correction
factor, e–1.191fi. The hardness of polar covalent crystals is expressed as follows [41]:
HV (GPa) =14(Nae–1.191fi)Eh; (27)
3/2
2/
2 ⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
= ∑ VZn
V
ZnN ii
ii
a , (28)
where ni is the number of the ith atom in the cell, Zi is the valence electron number
of the ith atom. Also Eq. (27) can be expressed as:
HV (GPa) = 556 5.2
191.1
d
eN if
a
−
= 350 5.2
191.13/2
d
eN if
e
−
, (29)
where d is the bond length.
The Vickers hardness of diamond measured under 9.8 N loads is about 96 GPa,
which is in agreement with the calculated hardness [41]. The calculated hardness of
α-Al2O3, 21 GPa, is in agreement with its Vickers hardness measured under 2.5 N
loads [60]. The calculated hardness of stishovite (SiO2) is 30 GPa [41]. An
appreciable scatter in hardness measurements of stishovite was observed, ranging
from 21 to 33 GPa [61]. The hardness of a stishovite single crystal determined
from the Vickers diamond pyramid hardness tests under 2 N loads is 31.8±1.0 GPa
along the c-axis and 26.2±1.0 GPa along a perpendicular direction [62]. Therefore,
a comparison of values from different studies is only of a limited significance.
Here we suggest that calculated hardness values may correspond to a comparative
Vickers hardness tested under a comparative load that is taken approximately as
cal
Vdia
V
H
H
N8.9 , where dia
VH is the Vickers hardness of diamond under 9.8 N loads
and cal
VH is the calculated hardness.
Hardness of multicomponent compound systems
The hardness surely involves the cooperative softening of many bonds. When
there are differences in the strength among different types of bonds, the trend of
breaking the bonds will start from a softer one. The hardness of multicomponent
compound systems can be expressed as a geometric average of hardness of all
pseudobinary systems in the solid [41],
μ
μ
∑
μ
μ
⎥
⎦
⎤
⎢
⎣
⎡
Π=
n
VV
n
HH
/1
)( , (30)
www.ism.kiev.ua; www.rql.kiev.ua/almaz_j 18
where nμ is the number of bonds of type μ composing the actual complex crystal.
The hardness, μ
VH , of pseudobinary compound composed of μ-type bond can be
calculated using Eqs. (27)—(29).
Nevertheless, it should be noted that Eq. (30) has to be applied with care in
systems with large anisotropy, in which nμ will vary along the different directions
of the crystal.
γ-Si3N4. Recently, it has been reported that cubic Si3N4 (γ-Si3N4) with a cubic
spinel structure could be synthesized at pressures above 15 GPa and temperatures
exceeding 2,000 K, yet persists metastably in air at an ambient pressure to at least
700 K [63]. This new superhard cubic Si3N4 phase opens considerable industrial
scope in materials science [64, 65].
Equilibrium structural parameters for the γ-Si3N4 spinels calculated by using the
GGA are a = 7.65 Å, u = 0.382 [66]. In this structure the Si atom is in octahedral
coordination with six N atoms (denoted as Sio) and simultaneously in tetrahedral
coordination with four N atoms (denoted as Sit).
Pauling [48] pointed out that crystal structures obey the Electrostatic Valency
Principle: an ionic structure will be stable to the extent that the sum of the Puling
bond valence, μ
ABQ , of the bonds that reach an ion equal to the charge on that ion.
For example, in Fig. 5 each Sit4+ is surrounded by 4N3– ions. The Sit is thus in 4
fold coordination. Thus, μ
ABQ = 1. That’s to say each N ion contributes one
negative charge to the Sit. So the +4 charge on the Sit ion is balanced by 4 × 1 = 4
negative charge from the 4N ions.
Si
t
Si
o
N
1
N
1 2/3 2/3
N N N
N N Si
o
Si
o
N N
Fig. 5. Coordination and Pauling bond valence in γ-Si3N4.
According to Eq. (13), γ-Si3N4 can be decomposed into the sum of
pseudobinary crystals as follows:
γ-Si3N4 = SitSio
2N4 =
= NN)(SiSi)Si(N
N CSi C
⎥
⎦
⎤
⎢
⎣
⎡ −
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡ −
N
bN
N
aN t
t
t
t
+ NN)(SiSi)Si(N
N CSi C
⎥
⎦
⎤
⎢
⎣
⎡ −
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡ −
N
bN
N
aN o
o
o
o
=
= N
4
41Si
4
14 ×× t + N
4
43Si
6
26 ×× o = SitN + 2SioN3/2.
From Fig. 5 it is seen that the Pauling bond valence, μ
ABQ , of the Sit—N and
Sio—N bonds are 1 and 2/3, respectively. For SitN, *
A )( μZ = μμ
CAABNQ = 1 × 4 = 4,
*
B )( μZ = μμμμ − BBCBAB )]8/()[( ZZNQ = [(1 × 4)/(8–5)] × 5 = 20/3. For SioN3/2,
ISSN 0203-3119. Сверхтвердые материалы, 2010, № 3 19
*
A )( μZ = μμ
CAABNQ = (2/3) × 6 = 4, *
B )( μZ = μμμμ − BBCBAB )]8/()[( ZZNQ =
[((2/3) × 4)/(8–5)] × 5 = 40/9.
The bond volumes of the Sit—N and Sio—N bonds, μ
bv (SitN) and μ
bv (SioN),
may be calculated as follows:
,04.3
65.7
16686.18475.1/75.1
)Si()Si()]NSi([)Si()Si()]NSi([/)]NSi([)NSi(
3
333
33
3
b
=⎟
⎠
⎞
⎜
⎝
⎛ ××+××=
=
⎭
⎬
⎫
⎩
⎨
⎧ +=μ
V
nNdnNddv
ooottt
tt
65.3
65.7
16686.18475.1/86.1
)Si()Si()]NSi([)Si()Si()]NSi([/)]NSi([)NSi(
3
333
33
3
b
=⎟
⎠
⎞
⎜
⎝
⎛ ××+××=
=
⎭
⎬
⎫
⎩
⎨
⎧ +=μ
V
nNdnNddv
ooottt
oo
where V is the cell volume, d(SitN) and d(SioN) are the lengths of the Sit—N and
Sio—N bonds, respectively. N(Sit) and N(Sio) are the coordination numbers of the
Sit and Sio atoms, respectively. n(Sit) and n(Sio) are the numbers of the Sit and Sio
atoms in a cell, respectively.
As shown in Table 1, the calculated hardness values of the Sit—N and Sio—N
bonds are 57.9 GPa and 25.1 GPa, respectively. The average hardness of the γ-
Si3N4 spinels can be calculated as follows:
∑
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
=
μ
μ∏
μ
μ
n
n
VV HH
/1
)( =
[ ] )]Si()Si()Si()Si(/[1
)]Si()Si([)]Si()Si([
)]NSi([)NSi(
oott
oott nNnN
nNo
V
nNt
V HH
⋅+⋅
⋅μ⋅μ
⎭
⎬
⎫
⎩
⎨
⎧ =
= ( ) )16684/(116684 1.259.57
×+××× × = 30.9 (GPa).
Table 1. Chemical bond parameters and hardness of cubic γ-Si3N4
spinels, which are estimated by using β = 0.089. Structural parameters
are from the results obtained within the GGA, where HV av (average
hardness) is calculated
γ-Si3N4
(GGA)
Bond
type
μ
ABQ μ
cN *)( μ
AZ *)( μ
BZ *)( μ
en dμ,
Ǻ
μ
bv ,
Ǻ3
μ
eN ,
Ǻ–3
μ
hE ,
eV
μC ,
eV
μ
if μ
VH HV av,
GPa
SitN
1 4 4 20/3 8/3 1.75 3.04 0.87 9.91 6.04 0.27 57.9 30.9 a = 7.65 Å
[66]
u = 0.382 SioN3/2 2/3 4.8 3 40/9 16/9 1.86 3.65 0.49 8.55 8.94 0.52 25.1
The measured hardness values of γ-Si3N4 are still quite controversial. From the
correlation between the Vickers hardness and shear modulus presented by Teter
[37], Soignard et al. estimated the hardness of γ-Si3N4 at a value of 30 GPa [67].
Jiang et al. [68] reported a Vickers hardness of 32.78—37.27 GPa for a
www.ism.kiev.ua; www.rql.kiev.ua/almaz_j 20
polycrystalline sample. But the applied loading conditions were not clearly
described. Based on the nanoindentation tests under 5 mN loads, the Vickers
microhardness of dense γ-Si3N4 was estimated to be between 30 and 43 GPa [69].
Here, our calculated hardness of γ-Si3N4 spinels, 30.9GPa, should be
corresponding to the Vickers hardness under about 3N loads.
ReB2, OsB2 and PtN2. ReB2 [70, 71], OsB2 [72] and PtN2 [73] have attracted
considerable attention due to their superior mechanical properties. PtN2 with the
pyrite structure is a semiconductor. ReB2 with the hexagonal structure and OsB2
with the orthorhombic structure possess stronger covalency and weaker metallicity
like WC. In [41] the calculated hardness of WC is in agreement with its
experimental value. This implies that the method mentioned above may be
employed to estimate the hardness of ReB2 and OsB2.
According to Eq. (13), ReB2, OsB2, and PtN2 can be decomposed into the sum
of pseudobinary crystals as follows:
ReB2 = 3/4ReB8/7 + 1/4ReB8/7 + 3/7BB;
OsB2 = 1/2OsB8/7 + 1/4OsB8/7 + 1/4OsB8/7 + 2/7BB + 1/7BB;
PtN2 = PtN3/2 + 1/4NN.
From Fig. 6 it is seen that Pauling bond valence values μ
ABQ of the Re—B and
B—B bonds are 1/2 and 1/3, respectively. The ionicity of pseudobinary compound
composed of the μ-type bond in these compounds can be calculated using Eqs.
(13)—(22). The Na of pseudobinary compound ReB8/7 may be obtained using Eq.
(28) as,
3/2
2
1
7
8
2
1
3/2
CA
CBAB7
8
CAAB
82
78
2 ⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
××
××+×
=⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛ +
= μμμ
μμμμ
bb
a vvN
NQNQ
N ,
where the valence electron of B is 3. The Na of pseudobinary compound PtN3/2 may
be calculated using Eq. (28) as,
3/2
3
5
3
1
2
3
3
1
3/2
CA
B
B
CBAB2
3
CAAB
)]62/()46[(
)]2/()
8
[(
μ
μμ
μ
μ
μμμμ
×××××+×=
=
−
+=
b
ba
v
vN
Z
ZNQNQN
,
where the valence electron of N atom, μ
BZ , is 5. The calculated results are listed in
Table 2.
B
1/3 1/2 1/2
B B
B B Re
Re
B
ReB
B
B Re
Fig. 6. Coordination and Pauling bond valence in ReB2.
ISSN 0203-3119. Сверхтвердые материалы, 2010, № 3 21
Table 2. Chemical bond parameters and hardness of ReB2, OsB2, RuB2
and PtN2, which are estimated by using β = 0.089. Where HV av.
and HV exp are the calculated and experimental Vickers hardness,
respectively μ
hE
Bond
type
μ
ABQ μ
cN *)( μ
AZ *)( μ
BZ *)( μ
en dμ, Ǻ
μ
bv ,
Å3
μ
eN ,
Å–3
μ
AZ
μ
hE ,
eV
μC ,
eV
μ
if
μ
aN ,
Å–2
μ
vH HV av,
GPa
HVexp,
GPa
ReB8/7 1/2 7.47 4 35/6 4/3 2.257 2.874 0.464 2.090 5.278 2.435 0.175 0.312 18.4 23.7 26.9
[74]
ReB2
ReB8/7 1/2 7.47 4 35/6 4/3 2.226 2.757 0.483 2.045 5.462 2.703 0.196 0.320 19.1 (2.94 N)
BB 1/3 7 7/3 7/3 2/3 1.82 1.5070.442 9.000 0 0 0.366 45.5
OsB2 OsB8/7 1/2 7.47 4 35/6 4/3 2.2152.7870.478 1.978 5.530 3.0730.2360.318 18.3
OsB8/7 1/2 7.47 4 35/6 4/3 2.1482.5420.525 1.888 5.967 3.7340.2810.338 19.9
OsB8/7 1/2 7.47 4 35/6 4/3 2.3083.1540.423 2.104 4.993 2.2810.1730.293 16.4
BB 1/3 7 7/3 7/3 2/3 1.8751.6910.394 8.360 0 0 0.339 39.1
BB 1/3 7 7/3 7/3 2/3 1.8 1.4960.446 9.250 0 0 0.368 47.0
RuB2 RuB8/7 1/2 7.47 4 35/6 4/3 2.1992.7270.489 1.956 5.630 3.2230.2470.323 18.6
RuB8/7 1/2 7.47 4 35/6 4/3 2.1652.6030.512 1.911 5.852 3.5580.2700.333 19.5
RuB8/7 1/2 7.47 4 35/6 4/3 2.2522.9290.455 2.028 5.307 2.7410.2110.308 17.5
BB 1/3 7 7/3 7/3 2/3 1.9021.7650.378 8.068 0 0 0.329 36.7
BB 1/3 7 7/3 7/3 2/3 1.7741.4320.466 9.590 0 0 0.378 50.2
PtN2 PtN3/2 1/3 4.8 2 20/9 8/9 2.0874.4540.200 1.697 6.409 1.1690.0320.215 18.3
NN 2 4 8 8 4 1.4051.3592.943 17.100 1.294307.2
The calculated average hardness of ReB2 is 23.7 GPa, which is close to the
measured comparative Vickers hardness of ReB2 under 2.94 N loads, 26.9±1.3 GPa
[74]. According to the bond lengths existing in the cell, all B—B and Os—B bonds
per unit cell of the OsB2 with the orthorhombic structure could be classified into
five groups, with lengths of 1.875, 1.800, 2.215, 2.148, and 2.308 Å, respectively.
As shown by the dotted line in Fig. 7, there are only Os—B bonds with a bond
length of 2.308 Å in the (100)[010] direction. These bonds are the softest.
Therefore, the (001) plane of OsB2 could be the easiest slip planes. According to
[41], the weakest bond plays a determinative role in the hardness of materials. We
may expect that the hardness of OsB2 would be smaller in the direction parallel to
the easy-slip planes, i.e., the (001)
planes. The calculated average
hardness of OsB2 is 22.8 GPa. The
calculated hardness of the
pseudobinary compound composed
of Os—B bonds of bond length
2.308 Å, 16.4 GPa, is much closer
to the measured comparative
Vickers hardness of OsB2 under the
1.96 N load, 17.8 ± 0.7 GPa [74].
Similarly, the calculated hardness
of the pseudobinary compound
composed of Ru—B bonds of bond
length 2.252 Å, 17.5 GPa is close to
Fig. 7. Bonding structure of the (100) plane of OsB2.
The vertical dotted line indicates the (100)[010]
direction.
www.ism.kiev.ua; www.rql.kiev.ua/almaz_j 22
the measured comparative Vickers hardness of RuB2 under 1.96 N load,
15.1 ± 1.1 GPa [74].
The calculated hardness of the pseudobinary compound PtN3/2 in PtN2 is
18.6 GPa. The hardness of N—N bonds is 307 GPa, which is far greater than that
of a diamond indenter. Thus, the N—N bonds might not break in a hardness test.
Similarly, for the compounds with anionic cluster, for example CaCO3, its C—O
bond may not break in a hardness test. The anionic cluster CO3 would slip as one
unit under the indenter, and does not contribute to the hardness of CaCO3. The
hardness of CaCO3 results just from the resistance of the Ca—O bond.
Hardness of boron icosahedral structured materials
The structure of α-rhombohedral boron contains a huge hole along the c-axis
between the icosahedra. In Fig. 8 the dotted lines show the 3-centre bonds between
the 6 equatorial boron atoms in each icosahedron and 6 other icosahedra in the
same sheet at 2.025 Å. The sheets are stacked so that each icosahedron is bonded
by six 2-centre B—B bonds. B12 units in the layer above are centered over 1 and
those in the layer below are centered under 2 [75]. In the boron carbide structure
the huge hole accommodates a three-atom chain. In the structures of B12O2 the O
atoms form pairs instead of three-atom chains [76].
1
2
1
2
1
2
Fig. 8. Basal plane of α-rhombohedral boron. The dotted triangles denoted as 1 or 2 show the 3-
centre bonds between the 6 equatorial boron atoms in each icosahedron to 6 other icosahedra.
The α-rhombohedral form of boron is the simplest one in boron-rich solids. The
36 valence electrons of each B12 unit are distributed as follows: 26 electrons just
form the 10 3-centre/2-electron bonds (denoted BBB1) and 3 normal 2-centre/2-
electron bonds (denoted BB1) within the icosahedron and 6 electrons share with 6
other electrons from 6 neighboring icosahedra in adjacent planes to form the
rhombohedrally directed normal 2-centre/2-electron bonds (denoted BB2); this
leaves 4 electrons, which is just the number required for contribution to the 6
equatorial 3-centre/2-electron bonds (denoted BBB2). Thus, α-boron has 4 bond
types. B12 can be decomposed into the sum of pseudobinary crystals as follows [77,
78]:
B12 = 10 (BBB1) + 3 (BB1) + 2 (BBB2) + 3 (BB2), (31)
where the coefficient is the number of bonds in cell.
ISSN 0203-3119. Сверхтвердые материалы, 2010, № 3 23
B12O2 has the same space group R 3 m as B12 [79]. Each of the two oxygen
atoms links to three B12 icosahedral, while oxide atoms are not bonding, see Fig. 9.
It can be decomposed into the sum of pseudobinary crystals as follows:
B12O2 = 10 (BBB1) + 3 (BB1) + 3 (BB2) + 6 (BO4/3). (32)
Polar site
Equatorial
site
O atom
Fig. 9. Atomic structure of B12O2.
The distinct feature of the chemical bonding of boron solids is three-center or
icosahedral bonding. Thus, the first problem of extending Eq. (29) to boron-rich
systems is how the bond length of three-center bonds is determined. For 2-centre/2-
electron bonds, the bond length is defined by the distance between two nuclei of
bonding atoms. In Fig. 10 we employ spx hybrid orbits to show the sketch of the 2-
centre/2-electron bond and 3-centre/2-electron bond. For 3-centre/2-electron bonds,
the bond length can be taken as the diameter of a circle shown in Fig. 10. If the
distances (denoted l) between pairs of atoms in the 3-centre/2-electron bond are
equal, the bond length of 3-centre/2-electron bonds d3c2e can be
expressed as [77]:
d3c2e= l
3
2 . (33)
The calculated results of hardness and chemical bond parameters of B12O2 are
shown in Table 3.
www.ism.kiev.ua; www.rql.kiev.ua/almaz_j 24
B B
a
B B
B
b
Fig. 10. Sketch of the 2-centre/2-electron bond (a) and 3-centre/2-electron bond (b).
Table 3. Hardness and chemical bond parameters of B12O2, where HV av.
are calculated
Bond type dμ, Ǻ vb
μ, Ǻ3 μ
eN , Ǻ–3 μ
hE , eV μ
if μ
VH HV av, GPa
BB1 1.798 4.270 0.468 9.276 0.000 49.3 44.6
BBB1 2.076 6.573 0.304 6.494 0.000 25.9
BB2 1.664 3.385 0.591 11.240 0.000 69.8
BO4/3 1.476 2.362 1.073 15.132 0.460 83.7
Recently the orthorhombic B28 single crystal [9, 10] and B13N2 have been
synthesized [80]. The measured Vickers hardness of the orthorhombic B28 is about
39—61 GPa under 4.9 N loads [81]. The calculated hardness of the orthorhombic
B28 and B13N2 by Mukhanov et al. is 48.8 GPa and 40.3 GPa, respectively [47]. The
calculated hardness of B13N2 by Gou et al. is 40.8 GPa [82]. These indicate that
they are two new superhard materials.
Hardness of nanocrystals
In nanocrystals, the conduction/valence band edges shift generally to higher
energy relative to the bulk material when the crystal size is decreased [83].
According to the Kubo theory [84], the band gap (Eg nano) of a nanocrystal should
increase inversely with the volume V, the energy shift δ is given by
3/13/12
22
)3(
2
ee DN
b
NmV
=
π
π=δ h , (34)
where b is the constant, D is the cluster diameter and Ne is the electron density of
the material, and
Eg nano= Eg bulk + δ, (35)
where Eg bulk is the band gap energy of the bulk material.
An expression for the hardness including quantum confinement effects for
nanocrystals can be obtained by substituting Eq. (34) into Eq. (4) [85],
ISSN 0203-3119. Сверхтвердые материалы, 2010, № 3 25
H (GPa) = ANa (Eg bulk + δ) =
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
+ 3/1
e
bulkga DN
bEAN . (36)
Defining k = A Na b/Ne
1/3, Eq. (36) can be simplified to [86],
HV
nano = HV
bulk + k/D. (37)
Eq. (37) indicates that the hardening mechanism of nanocrystalline materials
differs from the Hall-Petch relation for coarse-grained materials where HV n = H0 +
k/D1/2 [87, 88]. However, Eq. (37) agrees with the result given in [89], showing
that the nucleation stress for nanocrystalline materials is inversely proportional to
the grain size, σn ∼ 1/D.
Hardness and mulliken overlap population
Since the first-principles calculation is the current standard model for designing
materials, it is highly desirable to estimate the hardness directly from the
information of the first-principles calculation. Segall et al. [90, 91] found
correlations of overlap population with bond strength. If we take the average
overlap population per unit volume of bond to characterize the strength of bond,
the relation between hardness and overlap population of crystals is expressed as
follow [92]:
HV = ANa(P/vb), (38)
where Na is the covalent bond number per unit area, A is a proportional coefficient,
P is the Mulliken overlap population [93] and vb is the bond volume. For crystals
with the diamond structure there are 16 bonds in a cell of the cell volume V, the
bond volume can then be expressed as vb = V/16. Na can be expressed as (16/V)2/3
or vb
–2/3. Thus, their hardness should have the following form [92]:
HV(GPa) = APvb
–5/3. (39)
For the hardness calculation of crystals with partial metallic bonding like NbN
and ReC, a correction of the formula should be considered. The DOS of ReC is
shown in Fig. 11, where the vertical line is the Fermi level (EF) [94].This phase
showed metallic behavior because of finite N(EF) at the Fermi level. There is a
deep valley (Ep) at the left of the Fermi level (EF), which is named pseudogap. So,
the electrons occupying the levels above Ep become delocalized, and the material is
said to have metallicity. The number of free electron in a cell can be estimated as
[94]
∫=
F
P
E
Efree dEENn )( , (40)
where N(E) is the density of state. The hardness is suggested as
HV(GPa) = 740 (P – P′)vb
–5/3, (41)
where, P′ is metallic Mulliken population and
P′= nfree/V. (42)
Also the metallicity of crystals may be defined as
fm = /P P′ . (43)
www.ism.kiev.ua; www.rql.kiev.ua/almaz_j 26
–20 –10 0 10 20
Energy, eV
E
p
s
p
d
Sum
0
0.4
0.8
1.2
1.6
2.0
N
, E
Density of states, electrons/eV
Fig. 11. DOS of ReC. N(E) is the density of state, Ep is the pseudogap energy.
It is to be noted that owing to the high sensitivity of Mulliken population to the
basis set [95], constant A in Eq. (38) depends also on the basis set. The typical
value of A in Eq. (38) is 740. Figure 12 shows the relationship between the
hardness calculated by Mulliken population and the experimental values of
hardness. From the inset in Fig. 12, it can be seen that although a few data appear
relative big error, the variation trend of Mulliken population hardness and
experimental hardness is accordant. Especially for individual classes of materials,
Mulliken population hardness method may provide good predictions. The
calculated hardness values of crystals with partial metallic bonding are listed in
Table 4. The results show that ReC has high hardness of 29.4 GPa, which agrees
with its measurement value [35]. Contrarily, IrC exhibits low hardness value due to
high metallicity.
0
20
40
60
80
100
0 20 40 60 80 100
E
xp
er
im
en
ta
l h
ar
dn
es
s
, G
P
a
Mulliken population hardness, GPa
0
20
40
60
80
100
0 10 20 30 40
Experimental hardness
Mulliken population
hardness
Fig. 12. Comparison between the hardness calculated by Mulliken population and the
experimental values of hardness.
ISSN 0203-3119. Сверхтвердые материалы, 2010, № 3 27
Table 4. Mulliken population hardness of transition metal compounds
[94]. The bond length, d (Ǻ), Mulliken bond overlap population, P, the
volume of bond, vb (Ǻ3), calculated hardness, HV calc (GPa), and available
experimental Vickers hardness, HV exp (GPa)
Structure D, Å P P′ fm vb, Ǻ3 HV calc HV exp
HfC NaCl 2.344 0.408 0.008 0.02 4.293 26.1 25.5
TaC NaCl 2.213 0.41 0.066 0.16 3.613 30.1 29.0
WC WC 2.223 0.38 0.019 0.05 3.590 31.8 30 ± 3
ReC WC 2.157 0.34 0.054 0.16 3.271 29.4
OsC WC 2.165 0.317 0.089 0.28 3.347 22.7
IrC WC 2.192 0.317 0.143 0.45 3.501 15.9
TiC NaCl 2.159 0.333 0.006 0.02 3.353 32.3 28.6, 29 ± 3
ZrC NaCl 2.344 0.35 0.007 0.02 4.297 22.5 25.8
TiN NaCl 2.112 0.278 0.055 0.20 3.167 24 23
ZrN NaCl 2.288 0.27 0.043 0.16 3.994 16.7 15
HfN NaCl 2.314 0.325 0.044 0.14 4.131 19.4 15.9, 17 ± 2
VN NaCl 2.067 0.262 0.128 0.49 2.944 16.3 13, 15 ± 1
NbN NaCl 2.204 0.247 0.094 0.38 3.567 13.6 13, 14 ± 1
These theoretical hardness approaches have been employed to the hardness
calculations for the hard materials [96—106]. These studies indicate that
microscopic models of hardness possess a good predictive power especially for
strong covalent solids.
Anisotropy of hardness is also a very interesting topic. Recently, Šimůnek has
suggested a method to calculate the anisotropy of hardness [107]. We may also
employ Eq. (30) to study the anisotropy of hardness in the crystals, when nμ is
taken as the number of bond of type μ along a direction of the crystal. It should be
noted that in some cases, a primary slip plane may play a dominant role in
anisotropy of hardness.
According to the above microscopic models of hardness, three conditions
should be met for a superhard material: higher bond density or electronic density,
shorter bond length, and greater degree of covalent bonding. A class of superhard
materials is thus expected to be compounds with the shortest bond lengths; these
are composed of the light elements from periods 2 and 3 of the periodic table.
Recent synthesis of BC5 [108], B28 [9, 10], and M-carbon [22] indicate that the
covalent and polar-covalent compounds formed by light elements still play an
important role in search for superhard materials. Another relatively new field of
research is borides, carbides and nitrides of dense transition metals, since dense
transition metals, such as the 5d metals, have the high valence electron densities.
The introduction of light metals into a boron network should also deserve
investigation, for example, B13N2 [80], B12N2Be [77]. Nanocrystallinity is a means
for further hardening of the traditional superhard bulk materials. However, a
further study on hardening mechanism in nanoceramics is still necessary. The
microscopic hardness models introduced here can play an important role in
designing superhard materials.
This work is supported by the National Natural Science Foundation of China
(Grant 50672080).
www.ism.kiev.ua; www.rql.kiev.ua/almaz_j 28
Зроблено огляд останніх розробок в області мікромоделей твердості. У
цих моделях теоретичну твердість описано як функцію щільності та міцності зв’язку.
Міцність зв’язку може бути охарактеризована шириною забороненої зони, опорним
потенціалом, енергією утримання електрона або вільною енергією Гіббса. Тому різні
вирази міцності зв’язку приведуть до різних моделей міцності. Зокрема, докладно описано
модель твердості, основану на теорії хімічного зв’язку складних кристалів. Наведено
приклади розрахунків твердості типових кристалів, таких як шпінель S3N4, стишовіт
SiO2, B12O2, ReB2, OsB2, RuB2 и PtN2. Ці мікромоделі твердості будуть грати важливу
роль в пошуку нових твердих матеріалів.
Ключові слова: твердість, модуль об’ємного стиснення, модуль зсуву,
ионність, надтверді матеріали.
Дан обзор последних разработок в области микромоделей твердости. В
этих моделях теоретическая твердость описана как функция плотности и прочности
связи. Прочность связи может быть охарактеризована шириной запрещенной зоны,
опорным потенциалом, энергией удержания электрона или свободной энергией Гиббса.
Поэтому различные выражения прочности связи приведут к различным моделям
твердости. В частности, подробно описана модель твердости, основанная на теории
химической связи сложных кристаллов. Привены примеры расчета твердости типичных
кристаллов, таких как шпинель S3N4, стишовит SiO2, B12O2, ReB2, OsB2, RuB2 и PtN2. Эти
микромодели твердости будут играть важную роль в поиске новых твердых материалов.
Ключевые слова: твердость, модуль объемного сжатия, модуль
сдвига, ионность, сверхтвердые материалы.
1. McColm I. J. Ceramic Hardness. — New York: Plenum Press, 1990.
2. Handbook of Ceramic Hard Materials: V. 1, 2 // Ed. R. Riedel. — Weinheim: WILEY-VCH
Verlag, Germany, 2000.
3. Shaw M. C. The fundamental basis of the hardness test. Chapter 1 // The science of hardness
testing and its research applications / Ed. J. H. Westbrook and H. Conrad. — Ohio, USA:
American Society for Metals, Metals Park, 1973.
4. Szymanski A., Szymanski. J. M. Hardness estimation of minerals rocks and ceramic materials.
— Amsterdam: Elsevier, 1989.
5. Glazov V. M., Vigdorovich V. N. Mikrotverdoct’ metallov (Microhardness of metals). —
Moskva: Metallurgiya, 1969.
6. Gilman J. J. Why silicon is hard // Science. — 1993. — 261, N 5127. — P. 1436—1439.
7. Veprek S. The search for novel superhard materials // Vac. Sci. Technol. A. — 1999. — 17,
N 5. — P. 2401—2420.
8. Levine J. B., Tolbert S. H., Kaner R. B. Advancements in the search for superhard ultra-
incompressible metal borides // Adv. Funct. Mater. — 2009. — 19. — P. 3519—3533.
9. Oganov A. R., Chen J., Gatti C. et al. Ionic high pressure form of elemental boron // Nature.
— 2009. — 457, N 7231. — P. 863—867.
10. Oganov A. R., Solozhenko V. L. Boron: a hunt for superhard polymorphs // J. Superhard
Materials. — 2009. — 31, N 5. — P. 3—11.
11. Solozhenko V. L., Andrault D., Fiiquet G. et al. Synthesis of superhard cubic BC2N // Appl.
Phys. Lett. — 2001. — 78, N 10. — P. 1385—1387.
12. Irifune T., Kurio A., Sakamoto S. et al. Ultrahard polycrystalline diamond from graphite //
Nature. — 2003. — 421, N 6923. — P. 599—600.
13. Wang Z., Zhao Y., Tait K. et al. Quenchable superhard carbon phase synthesized by cold
compression of carbon nanotubes // Proc. Nat. Acad. Sci., U.S.A. — 2004. — 101, N 38. —
P. 13699—13702.
14. Dubrovinskaia N., Dubrovinsky L. Crichton W. et al. Aggretated diamond nanorods, the
densist and least compressible form of diamond // Appl. Phys. Lett. — 2005. — 87, N 8. —
P. 083106 1—3.
15. Dubrovinsky L. S., Dubrovinskaia N. A, Swamy V. et al. The hardest known oxide // Nature.
— 2001. — 410, N 6829. — P. 653—654.
16. Brazhkin V., Dubrovinskaia N., Nicol M. et al. What does “Harder than Diamond” mean? //
Nature Materials. — 2004. — 3, N 2. — P. 576—577.
ISSN 0203-3119. Сверхтвердые материалы, 2010, № 3 29
17. Dubrovinskaia N., Solozhenko V. L., Miyajima N. et al. Superhard nanocomposite of dense
polymorphs of boron nitride: Noncarbon material has reached diamond hardness // Appl.
Phys. Lett. — 2007. — 90, N 10. — P. 101912 1—3.
18. Liu A.Y., Cohen M. L. Prediction of new low compressibily solids // Science. — 1989. —
245, N 4920. — P. 841—842.
19. Teter D. M., Hemley R. J. Low compressibility carbon nitrides // Ibid. — 1996. — 271,
N 5245. — P. 53—55.
20. Kroll P. Hafnium Nitride with thorium phosphide structure: physical properties and an
assessment of the Hf—N, Zr—N, and Ti—N phase diagrams at high pressures and
temperatures // Phys. Rev. Lett. — 2003. — 90, N 12. — P. 125501 1—4
21. Shiyou Chen, X. G. Gong, Su-Huai Wei. Superhard pseudocubic BC2N superlattices // Ibid.
— 2007. — 98, N 1. — P. 015502 1—4.
22. Q. Li, Ya. Ma, Oganov A. R. et al. Superhard monoclinic polymorph of carbon // Ibid. —
2009. — 102, N 17. — P. 175506 1—4.
23. Wang H., Li Q., Li Y. et al. Ultra-incompressible phases of tungsten dinitride predicted from
first principles // Phys. Rev. B. — 2009. — 79, N 13. — P. 132109 1—6.
24. Li. Q., Wang M., Oganov A. R. et al. Rhombohedral superhard structure of BC2N // J. Appl.
Phys. — 105, N 5. — P. 053514 1—4.
25. Zhang M., Wang M., Cui T. et al. Electronic structure, phase stability, and hardness of the
osmium borides, carbides, nitrides, and oxides: First-principles calculations // J. Phys. Chem.
Solids. — 2008. — 69, N 8. — P. 2096—2102.
26. Li Y., Wang H., Li Q. et al. Twofold coordinated ground-state and eightfold high-pressure
phases of heavy transition metal nitrides MN2 (M = Os, Ir, Ru and Rh) // Inorganic
Chemistry. — 2009. — In press.
27. Haines J., Leger J. M., Bocquillon G. Synthesis and design of superhard materials // Annu.
Rev. Mater. Res. — 2001. — 31. — P. 1—23.
28. Hong S., Jhi S. H., Roundy D. et al. Structural forms of cubic BC2N // Phys. Rev. B. — 2001.
— 64, N 9. — P. 094108.
29. He J. L., Guo L. C., Wu E. et al. First-principles study of B2CN crystals deduced from the
diamond structure // J. Physics-Condensed Matter. — 2004. — 16, N 46. — P. 8131—8138.
30. Wang Y. X., Masao A., Taizo S, Fan Z. Ab initio study of monoclinic iridium nitride as a high
bulk modulus compound // Phys. Rev. B. — 2007. — 75, N 10. — P. 104110 1—6.
31. Gilman J. J. Electronic basis of hardness and phase transformations // J. Phys. D: Appl. Phys.
— 2008. — 41, N 7. — P. 074020—074024.
32. Cohen M. L. Calculation of bulk moduli of diamond and zinc-blende solids // Phys. Rev. B.
— 1985. — 32, N 12. — P. 7988—7991.
33. Zhang S., Li H., Li H. et al. Calculation of the bulk modulus of simple and complex crystals
with the chemical bond method // J. Phys. Chem. B. — 2007. — 111, N 6. — P. 1304—1309.
34. Sung C. M., Song M. Carbon nitride and other speculative superhard materials // Mater.
Chem. Phys. — 1996. — 43, N 1. — P. 1—18.
35. Brazhkin V., Lyapin A., Hemley R. J. Harder than diamond: dreams and reality //
Philosophical Magazine A. — 2002. — 82, N 2. — P. 231—253.
36. Ceder G. Predicting Properties from Scratch // Science. — 1998. — 280, N 5366. —
P. 1099—1100.
37. Teter D. M. Computational alchemy: The search for new superhard materials // Mater. Res.
Soc. Bull. — 1998. — 23, N 1. — P. 22—27.
38. Gilman J. J. Hardness — a strength microprobe. Chapter 4 // The science of hardness testing
and its research applications / Eds. by J. H. Westbrook, H. Conrad. — Ohio, USA: American
Society for Metals, Metals Park, 1973.
39. Nye J. F. Physical properties of crystals. — London: Oxford University Press, 1957.
40. Hill A. The elastic behaviour of a crystalline aggregate // Proc Phys. Soc. —1952. — 65. —
P. 349—354.
41. Gao F. M., He J. L., Wu E. D. et al. Hardness of covalent crystals // Phys. Rev. Lett. — 2003.
— 91, N 1. — P. 015502 1—4.
42. Gao F. M. Hardness estimation of complex oxide materials // Phys. Rev. B. — 2004. — 69,
N 9. — P. 094113 1—6.
43. Simunek A., Vackar J. Hardness of covalent and ionic crystals: First-principle calculations //
Phys. Rev. Lett. — 2006. — 96, N 8. — P. 085501 1—4.
44. Simunek A. How to estimate hardness of crystals on a pocket calculator // Phys. Rev. B. —
2007. — 75, N 17. — P. 172108 1—4.
www.ism.kiev.ua; www.rql.kiev.ua/almaz_j 30
45. Li K., Wang X., Zhang K., Xue D. Electronegativity identification of novel superhard
materials // Phys. Rev. Lett. — 2008. — 100, N 23. — P. 235504 1—4.
46. Li K., Xue D. Hardness of materials: Studies at levels from atoms to crystals // Chinese
Science Bulletin. — 2009. — 54, N 1. — P. 131—136.
47. Mukhanov V. A., Kurakevych O. O., Solozhenko V. L. The Interrelation between the hardness
and compress ibility of substances and their structure and their structure and thermodynamic
properties // J. Superhard Materials. — 2008. — 30, N 6. — P. 10—22.
48. Pauling L. The nature of the chemical bond. — Ithaca, New York: Cornell Univ. Press, 1960.
— 91 p.
49. Shanker J., Verma M. P. The fractional ionic character of alkali and silver halide crystals //
Pramana. — 1973. — 1, N 6. — P. 243—246.
50. Hidaka T. Pauling’s ionicity and Phillips’ ionicity // J. Phys. Soc. Japan. — 1978. — 44, N 4.
— P. 1204—1207.
51. Phillips J. C. Ionicity of the chemical bond in crystals // Rev. Mod. Phys. — 1970. — 42,
N 3. — P. 317—356.
52. Van Vechten J. A. Quantum dielectric theory of electronegativity in covalent systems // Phys.
Rev. B. — 1969. — 182, N 3. — P. 891—905.
53. Van Vechten J. A. Quantum dielectric theory of electronegativity in covalent systems. II.
Ionization potentials and interband transition energies // Ibid. — 1969. — 187, N 3. —
P. 1007—1020.
54. Levine B. F. Bond susceptibilities and ionicities in complex crystal structures // J. Chem.
Phys. — 1973. — 59, N 3. — P. 1463—1485.
55. Levine B. F. Bond-charge calculation of nonlinear optical susceptibilities for various crystal
structures // Phys. Rev. B. — 7, N 6. — P. 2600—2626.
56. Zhang S. Y. Investigation of chemical bonds on complex crystals // Chin. J. Chem. Phys. —
1991. — 4. — P. 109—115.
57. Zhang S. Y., Gao F. M., Wu C. X. Chemical bond properties of rare earth ions in crystals // J.
Alloys Compd. — 1998. — 275—277. — P. 835—837.
58. Gao F. M., Zhang S. Y. Investigation of covalency and spectrum shifts in 3d transition-metal
compounds // Chin. J. Inorg. Chem. — 2000. — 16, N 5. — P. 751—756.
59. Gao F. M., Li D. C., Zhang S. Y. Mossbauer spectroscopy and chemical bonds in BaFe12O19
hexaferrite // J. Phys.-Cond. Matter. — 2003. — 15, N 29. — P. 5079—5084.
60. Krell A. Vickers hardness and microfracture of single and polycrystalline Al2O3 // Kristall
und Technik. — 1980. — 15, N 12. — S. 1467—1474.
61. Luo S. N., Swadenerb J. G., Ma C., Tschauner O. Examining crystallographic orientation
dependence of hardness of silica stishovite // Physica B. — 2007. — 399, N 6. — P. 138—
142.
62. Brazhkin V. V., Grimsditch M., Guedes I. et al. Elastic moduli and the mechanical properties
of stishovite single crystals // Phys. Uspekhi. — 2002. — 45, N 4. — P. 447—448.
63. Zerr A., Miehe G., Serghiou G. et al. Synthesis of cubic silicon nitride // Nature. — 1999. —
400, N 6742. — P. 340—342.
64. Mo S. D., Ouyang L., Ching W. Y. et al. Interesting Physical properties of the new spinel
phase of Si3N4 and C3N4 // Phys. Rev. Lett. — 1999. — 83, N 24. — P. 5046—5049.
65. Sekine T., Mitsuhashi T. High-temperature metastability of cubic spinel Si3N4 // Appl. Phys.
Lett. — 2001. — 79, N 17. — P. 2719 1—3.
66. Gao F. M., Xu R., Liu K. Origin of hardness in nitride spinel materials // Phys. Rev. B. —
2005. — 71, N 5. — P. 052103 1—4.
67. Soignard E., Somayazulu M., Dong J. et al. High pressure-high temperature synthesis and
elasticity of the cubic nitride spinel γ-Si3N4 // J. Phys. C. — 2001. — 13(22). — P. 557—563.
68 Jiang J. Z., Kragh F., Frost D. J. et al. Hardness and thermal stability of cubic silicon nitride
// J. Phys.: Condens. Matter. — 2001. — 13, N 22. — P. L515—L520.
69 Zerr A., Kempf M., Schwarz M. et al. Elastic moduli and hardness of cubic silicon nitride // J.
Am. Ceram. Soc. — 2002. — 85, N 1. — P. 86—90.
70. Chung H. Y., Weinberger M. B., Levine J. B. et al. Synthesis of ultra-incompressible
superhard rhenium diboride at ambient pressure // Science. — 2007. — 316, N 5823. —
P. 436—439.
71. Qin J., He D., Wang J. et al. Is rhenium diboride a superhard material? // Adv. Mater. —
2008. — 20, N 24. — P. 4780—4783.
ISSN 0203-3119. Сверхтвердые материалы, 2010, № 3 31
72. Cumberland R. W., Weinberger M. B., Gilman J. J. et al. Osmium diboride, an ultra-
incompressible, hard material // J. Amer. Chem. Soc. — 2005. — 127, N 20. — P. 7264—
7265.
73. Crowhurst J.C., Goncharov A. F., Sadigh B. et al. Synthesis and characterization of the
nitrides of platinum and iridium // Science. — 2006. — 311, N 5765. — P. 1275—1278.
74. Gu Q., Krauss G., Steurer W. Transition metal borides: superhard versus ultra-incompressible
// Adv. Mater. — 2008. — 20. — P. 3620—3626.
75. Emin D., Aselage T. L., Switendick A. C. et al. Boron-rich solids. — New York: AIP, 1990.
76. Hubert H., Devouard B., Garvie L. A. J. et al. Icosahedral packing of B12 icosahedra in boron
suboxide (B6O). // Nature. — 1998. — 391. — P. 376—378.
77. Gao F. M., Hou L., He Y. H. Origin of superhardness in icosahedral B-12 materials // J. Phys.
Chem. B. — 2004. — 108, N 35. — P. 13069—13073.
78. Gao F. M., Qin X. J., Wang L.Q. et al. Prediction of new superhard boron-rich compounds //
J. Phys. Chem. B. — 2005. — 109, N 31. — P. 14892—14895.
79. Olofsson M., Lundstrom T. Synthesis and structure of non-stoichio- metric B6O // J. Alloys.
Compounds. — 1997. — 257, N 1—2. — P. 91—95.
80. Kurakevych O. O., Solozhenko V. L. Rhombohedral boron subnitride, B13N2, by X-ray
powder diffraction // Acta Crystallogr. Sect. C. — 2007. — 63. — P. i80—i82.
81. Solozhenko V. L., Kurakevych O. O., Oganov A. R. On the hardness of a new boron phase,
orthorhombic γ-B28 // J. Superhard Materials. — 2008. — 30, N 6. — P. 428—429.
82. Gou H. Y., Zhang J. W., Gao F. M. First-principles calculations of boron-rich compounds of
B13N2 and B12C2X (X = Si, Ge) // J. Phys.-Cond. Matter. — 2008. — 20, N 50. —
P. 505211—505216.
83. Rama Krishna M. V., Friesner R. A. Quantum confinement effects in semiconductor clusters
// J. Chem. Phys. — 1991. — 95, N 11. — P. 8309—8322.
84. Halperin W. P. Quantum size effects in metal particles // Rev. Mod. Phys. — 1986. — 58,
N 3. — P. 533—606.
85. Tse J. S., Klug D. D., Gao F. M. Hardness of nanocrystalline diamonds // Phys. Rev. B. —
2006. — 73, N 14. — P. 140102 1—4.
86. Gao F. M., Klug D. D., Tse J. S. Theoretical study of new superhard materials: B4C3 // J.
Appl. Phys. — 2007. — 102, N 8. — P. 084311 1—5.
87. Andrievski R. A. Superhard materials based on nanostructured high-melting point
compounds: achievements and perspectives // Int. J. Refract. Met. Hard Mater. — 2001. —
19, N 4—6. — P. 447—452.
88. Siegel R. W., Fougere G. E. Mechanical properties of nanophase metals // Nanostruct. Mater.
— 1995. — 6, N 1—4. — P. 205—216.
89. Yamakov V., Wolf D., Phillpot S. R. et al. Deformation-mechanism map for nanocrystalline
metals by molecular-dynamics simulation // Nat. Mater. 2004. — 3, N 1. — P. 43—47
90. Segall M. D., Shah R., Pickard C. J., Payne M. C. Population analysis of plane-wave
electronic structure calculations of bulk materials // Phys. Rev. B. — 1996. — 54, N 23. —
P. 16317—16320.
91. Ching W. Y., Xu Y. N. Nonscalability and nontransferability in the electronic properties of the
Y—Al—O system // Ibid. — 1999. — 59, N 20. — P. 12815—12821.
92. Gao F. M. Theoretical model of intrinsic hardness // Ibid. — 2006. — 73, N 13. — P. 132104
1—4.
93. Mulliken R. S. Electronic Population Analysis on LCAO-MO Molecular Wave Functions. I //
J. Chem. Phys. — 1955. — 23, N 10. — P. 1833—1840.
94. Gou H. Y., Hou L., Zhang J. W., Gao F. M. Pressure-induced incompressibility of ReC and
effect of metallic bonding on its hardness // Appl. Phys. Lett. — 2008. — 92, N 24. —
P. 2419011—3.
95. Zhang X., Luo X., Han J. et al. Electronic structure, elasticity and hardness of diborides of
zirconium and hafnium: First principles calculations // Computational Mater. Sci. — 2008. —
44. — P. 411—421.
96. Gou H. Y., Hou L., Zhang J.W. et al. Theoretical hardness of PtN2 with pyrite structure //
Appl. Phys. Lett. — 2006. — 89, N 14. — P. 141910 1—3.
97. Gou H. Y., Hou L., Zhang J. W. et al. Cubic gamma-Be3N2: A superhard semiconductor
predicted from first principles // Ibid. — 2007. — 90, N 19. — P. 191905 1—3.
98. Wang Z. B., Gao F. M., Li N. et al. Density functional theory study of hexagonal carbon
phases // J. Phys.-Cond. Matter. — 2009. — 21, N 23. — P. 235401—235406.
www.ism.kiev.ua; www.rql.kiev.ua/almaz_j 32
99. Li C. L., Kuo J. L., Wang B. A. et al. A new layer compound Nb4SiC3 predicted from first-
principles theory // J. Phys. D-App. Phys. — 2009. — 42, N 7. — P. 075404—075409.
100. Liu H., Zhu J., Liu Y., Lai Z. First-principles study on the mechanical properties of
vanadium arbides VC and V4C3 // Mater. Lett. — 2008. — 62, N 17—18. — P. 3084—3086.
101. Chen Z., Gu M., Sun C. Q. et al. Ultrastiff carbides uncovered in first principles, Appl. Phys.
Lett. — 2007. — 91, N 6. — P. 061905 1—3.
102. Wang M., Li Y., Cui T. et al. Origin of Hardness in WB4 and its implications for ReB4,
TaB4, MoB4, TcB4 and OsB4 // Ibid. — 2008. — 93, N 10. — P. 101905 1—3.
103. Ribeiro F. J., Tangney P., Louie S. G., Cohen M. L. Hypothetical Hard Structures of Carbon
with Cubic Symmetry // Phys. Rev. B. — 2006. — 74, N 17. — P. 172101 1—4.
104. Smedskjaer M. M., Jensen M., Yue Y. Z. Theoretical calculation and measurement of the
hardness of diopside // J. Amer. Ceram. Soc. — 2008. — 91, N 2. — P. 514—518.
105. Gao L. H., Gao F. M. Chemical bond properties and hardness estimation of rare earth
garnets // Mater. Chem. Phys. — 2009. — 113, N 1. — P.145—149.
106. Gou H. Y., Hou L., Zhang J. W. et al. First-principles study of low compressibility osmium
borides // Appl. Phys. Lett. — 2006. — 88, N 22. — P. 221904 1—3.
107. Šimůnek A. Anisotropy of hardness from first principles: The cases of ReB2 and OsB2 //
Phys. Rev. B. — 2009. — 80, N 6. — P. 060103 1—4.
108. Solozhenko V. L., Kurakevych O., Andrault D. et al. Ultimate metastable solubility of boron
in diamond: synthesis of superhard diamondlike BC5 // Phys. Rev. Lett. — 2009. — 102. —
P. 015506 1—4.
Key Laboratory of Applied Chemistry, Received 25.11.09
Yanshan University
|
| id | nasplib_isofts_kiev_ua-123456789-63471 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 0203-3119 |
| language | English |
| last_indexed | 2025-12-07T15:13:41Z |
| publishDate | 2010 |
| publisher | Інститут надтвердих матеріалів ім. В.М. Бакуля НАН України |
| record_format | dspace |
| spelling | Gao, F.M. Gao, L.H. 2014-06-02T06:38:56Z 2014-06-02T06:38:56Z 2010 Microscopic models of hardness / F.M. Gao, L.H. Gao // Сверхтвердые материалы. — 2010. — № 3. — С. 9-32. — Бібліогр.: 108 назв. — англ. 0203-3119 https://nasplib.isofts.kiev.ua/handle/123456789/63471 539.53 Recent developments in the field of microscopic hardness models have been reviewed. In these models, the theoretical hardness is described as a function of the bond density and bond strength. The bond strength may be characterized by energy gap, reference potential, electron-holding energy or Gibbs free energy, and different expressions of bond strength may lead to different hardness models. In particular, the hardness model based on the chemical bond theory of complex crystals has been introduced in detail. The examples of the hardness calculations of typical crystals, such as spinel Si₃N₄, stishovite SiO₂, B₁₂O₂, ReB₂, OsB₂, RuB₂, and PtN₂, are presented. These microscopic models of hardness would play an important role in search for new hard materials. Зроблено огляд останніх розробок в області мікромоделей твердості. У цих моделях теоретичну твердість описано як функцію щільності та міцності зв’язку. Міцність зв’язку може бути охарактеризована шириною забороненої зони, опорним потенціалом, енергією утримання електрона або вільною енергією Гіббса. Тому різні вирази міцності зв’язку приведуть до різних моделей міцності. Зокрема, докладно описано модель твердості, основану на теорії хімічного зв’язку складних кристалів. Наведено приклади розрахунків твердості типових кристалів, таких як шпінель Si₃N₄, стишовіт SiO₂, B₁₂O₂, ReB₂, OsB₂, RuB₂ и PtN₂. Ці мікромоделі твердості будуть грати важливу роль в пошуку нових твердих матеріалів. Дан обзор последних разработок в области микромоделей твердости. В этих моделях теоретическая твердость описана как функция плотности и прочности связи. Прочность связи может быть охарактеризована шириной запрещенной зоны, опорным потенциалом, энергией удержания электрона или свободной энергией Гиббса. Поэтому различные выражения прочности связи приведут к различным моделям твердости. В частности, подробно описана модель твердости, основанная на теории химической связи сложных кристаллов. Привены примеры расчета твердости типичных кристаллов, таких как шпинель Si₃N₄, стишовит SiO₂, B₁₂O₂, ReB₂, OsB₂, RuB₂ и PtN₂. Эти микромодели твердости будут играть важную роль в поиске новых твердых материалов. This work is supported by the National Natural Science Foundation of China (Grant 50672080). en Інститут надтвердих матеріалів ім. В.М. Бакуля НАН України Сверхтвердые материалы Получение, структура, свойства Microscopic models of hardness Article published earlier |
| spellingShingle | Microscopic models of hardness Gao, F.M. Gao, L.H. Получение, структура, свойства |
| title | Microscopic models of hardness |
| title_full | Microscopic models of hardness |
| title_fullStr | Microscopic models of hardness |
| title_full_unstemmed | Microscopic models of hardness |
| title_short | Microscopic models of hardness |
| title_sort | microscopic models of hardness |
| topic | Получение, структура, свойства |
| topic_facet | Получение, структура, свойства |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/63471 |
| work_keys_str_mv | AT gaofm microscopicmodelsofhardness AT gaolh microscopicmodelsofhardness |