Intrinsic hardness of crystalline solids
The current status of various theoretical approaches to the prediction of material hardness has been reviewed. It is shown that the simple empirical correlation with the shear moduli generally provide very good estimates of the Vickers hardness. Semi-empirical models based solely on the strength of...
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| Cite this: | Intrinsic hardness of crystalline solids / J.S. Tse // Сверхтвердые материалы. — 2010. — № 3. — С. 46-65. — Бібліогр.: 51 назв. — англ. |
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| citation_txt | Intrinsic hardness of crystalline solids / J.S. Tse // Сверхтвердые материалы. — 2010. — № 3. — С. 46-65. — Бібліогр.: 51 назв. — англ. |
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| description | The current status of various theoretical approaches to the prediction of material hardness has been reviewed. It is shown that the simple empirical correlation with the shear moduli generally provide very good estimates of the Vickers hardness. Semi-empirical models based solely on the strength of the chemical bonds, although performed as well, are theoretically incomplete. First-principles calculations of the ideal stress and shear strength is perhaps the most reliable and theoretically sound approach available to compare theoretical predictions with experiment.
Розглянуто сучасний стан різних теоретичних підходів до прогнозування твердості матеріалів. Показано, що проста емпірична кореляція з модулем зсуву звичайно дає дуже хорошу оцінку твердості за Віккерсом. Розроблені також напівемпіричні моделі, основані виключно на міцності хімічного зв’язку, є теоретично неповними. Розрахунки з перших принципів ідеального напруження і міцності на зсув є, мабуть, найбільш надійним і теоретично обґрунтованим підходом для порівняння теоретичних прогнозів з експериментальними результатами.
Рассмотрено современное состояние различных теоретических подходов к прогнозированию твердости материалов. Показано, что простая эмпирическая корреляция с модулем сдвига обычно дает очень хорошую оценку твердости по Виккерсу. Разработанные также полуэмпирические модели, основанные исключительно на прочности химических связей, являются теоретически неполными. Расчеты из первых принципов идеального напряжения и прочности на сдвиг представляют собой, вероятно, наиболее надежный и теоретически обоснованный подход для сравнения теоретических прогнозов с экспериментальными результатами.
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www.ism.kiev.ua; www.rql.kiev.ua/almaz_j 46
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J. S. Tse (Saskatoon, Saskatchewan, Canada)
Intrinsic hardness of crystalline solids
The current status of various theoretical approaches to the prediction
of material hardness has been reviewed. It is shown that the simple empirical
correlation with the shear moduli generally provide very good estimates of the Vickers
hardness. Semi-empirical models based solely on the strength of the chemical bonds,
although performed as well, are theoretically incomplete. First-principles calculations
of the ideal stress and shear strength is perhaps the most reliable and theoretically
sound approach available to compare theoretical predictions with experiment.
Key words: hardness, crystalline solids, shear moduli, strength of the
chemical bonds, ideal stress, shear strength.
INTRODUCTION
“Hardness, like the storminess of the seas, is easily appreciated but not readily
measured”, commented H. O’Niell in the classic treatise on the measurement of the
hardness of metals in 1934 [1]. In colloquial terms, the hardness of a material is the
intrinsic resistance to deformation when a force is applied. Since force can be
applied in various means, hardness is not a property that can be quantified easily in
an absolute scale. The definition of hardness is complicated further by the
distinction between microscopic and macroscopic strength. The microscopic
hardness is dependent on the strength of atomic bonding. In contrast, the
macroscopic strength of a real material is dominated by the behavior of
dislocations or cracks. There are three common operational definitions [2]. Scratch
and indentation hardness refer, respectively, to the resistance to plastic deformation
due to a friction or a constant load from a sharp object. Rebound hardness is
measured as the height of the bounce of an object dropped on the material and is
related to the elasticity. The recent development of nanoindentation, where the
mechanical behavior of a material is probed in a nanosized region of a surface
beneath a nanoindenter, is expected to be less affected by cracks or dislocations, is
most amendable for comparison with theoretical predictions employing modern
electronic structure theory. However, special care is needed to be taken in the study
of materials close to the hardness of diamond (superhard materials with hardness >
60 GPa), since the indentation is no longer controlled by plastic deformation [3].
The possibility of brittle cracking and deformation of the indentation tip may affect
the measured results. The “measured” hardness is dependent on to the type of an
indenter, the applied loads, the time of the indentation, sample orientation and the
quality of the tested surface. In view of these difficulties, there is no consensus
how to characterize “ultrahardness” by a single number [3]. Similarly, a theoretical
definition for hardness of a bulk crystal is also not easy to establish. Recently,
several empirical models have been proposed to relate computable quantities to the
Vickers hardness scale [4]. As in the experiment, this choice is not unique. The
yield strength [5], which is the maximum stress that a crystal can attain under
uniform deformation without extrinsic effects, is indicated by the maxima of a
© J. S. TSE, 2010
ISSN 0203-3119. Сверхтвердые материалы, 2010, № 3 47
stress-strain curve. In principle, it can be simulated by electronic structure
calculations and, perhaps, is the most appropriate working definition.
Regardless of the definition of hardness, in the absence of extrinsic impurities,
the resistance of a material to deformation is determined by the strength of the
local (nearest-neighbor) interatomic (bonding) interactions. In a covalent solid, the
chemical bonds are localized and it is not unreasonable to expect that the
compressibility (resistance to volume change), which is characterized by the bulk
moduli, may be connected to the hardness [6]. In fact, diamond having the largest
bulk modulus of all covalent solids is also the hardest bulk material in existence.
This assumption, in the best scenario, is only valid when the forces are applied
isotropically. This is not the case in indentation experiments where both normal (to
the direction of force) and shear (parallel to the direction of force) stress are to be
considered. In a sense the hardness of a crystal is the ability to resist plastic
deformation from hydrostatic compression, tensile load and shear. From the
empirical observation of the relationship of glide mobility and stress strength [7], it
was suggested that the hardness of a material may be better correlated with the
shear modulus. A comparison of the Vickers hardness with the bulk and shear
moduli (G, modulus of rigidity, the ratio of shear stress to shear strain) from a
common set of covalent ceramic materials (oxides, nitrides, and metal carbides) are
shown in Figs. 1, a and b, respectively [8]. It is obvious that for the chosen set of
data, the Vickers hardness shows a quasi-linear dependence on the shear modulus
[8]. This is an important empirical relationship as it connects a computable
quantity, the shear modulus, with the experimentally determined Vickers hardness.
A similar linear correlation was also reported for metals [9]. The difference is that
the slope of the line is two orders of magnitude less than that of the covalent solids.
A fact that may be attributed to the weak delocalized metallic bonding [10]. For
practical applications, it is desirable to establish a universal function that connects
quantities predicted from First-Principles calculations with the measured hardness.
This universal function should comprise all the essential factors governing the
hardness and applicable to a wide range of materials with different bonding
properties.
0
V
ic
ke
rs
h
ar
dn
es
s,
G
P
a
0 100 200 300 400 500
Bulk modulus, GPa
80
60
40
20
100
a
Fig. 1. Correlation of Vickers hardness with (a) bulk modulus and (b) shear modulus. The dash
line is a linear fit to the data (taken from [8]).
www.ism.kiev.ua; www.rql.kiev.ua/almaz_j 48
V
ic
ke
rs
h
ar
dn
es
s,
G
P
a
0 100 200 300 400 500 600
Shear modulus, GPa
100
80
60
40
20
0
b
Fig. 1. (Contd.)
The outline of this paper is as follows. First, the basic principles of chemical
bonding and the relationship to hardness will be presented. Phenomenological
schemes connecting the properties of covalency-dominant solids to the hardness
scale are presented. This is followed by a discussion on the extension of these
schemes to nanostructured materials. First-principles calculations of ideal tensile
and shear strengths of crystals will be reviewed. The paper is concluded with a
brief summary.
CHEMICAL PRINCIPLES OF HARDNESS
The strength of a two center — two electron (Lewis) chemical bond between is
determined by the exchange energy and is dependent on the overlap of the
(hybridized) atomic orbitals of the respective atoms [10, 11]. Consider a simple
example of the interaction of two identical atoms (A) each contributing an atomic
orbital to the bonding; in a tight binding description [11] and neglecting repulsion,
the energy of the molecular orbitals is given by
AB
AB
0
A
1 S
EE
+
Δ+=+ and
AB
AB
0
A
1 S
EE
−
Δ−=− , where E+ and E– are the energies of the bonding and antibonding
orbitals, respectively. SAB is the overlap integral, 0
AE is the effective potential of an
electron in the atomic orbital of A, and ΔAB is the energy associated with the two
overlapping atomic orbitals. Taking the Taylor expansion to the first order,
)( AB
0
AAB
0
A SEEE −Δ+≈+ and )( AB
0
AAB
0
A SEEE −Δ−≈− . Furthermore, it can be
shown that ΔAB ∝ – SAB < 0. Thus, the energy gap (ΔE = E– – E+) ≈
2 )( AB
0
AAB SE−Δ ∝ SAB. Therefore, the better the overlap between the atomic
orbitals, the larger the stabilization (bond) energy and the larger the energy gap.
For a two electron bond, only the bonding orbital is occupied and the energy of the
highest occupied molecular orbital (HOMO) is E+, and that of the lowest
ISSN 0203-3119. Сверхтвердые материалы, 2010, № 3 49
unoccupied molecular orbital (LUMO) is E–. Since both E+ and ΔE are dependent
on the magnitude of the orbital overlap, it is a convenient indicator determining the
strength of the chemical bond. In a covalent crystal, a collection of two center —
two electron chemical bonds is the stabilization factor for the crystal structure and
determines the mechanical strength. Strong chemical bonds in a crystal not only
resist isotropic compression but they also help to maintain the structural integrity
from shear deformation.
The strength of a solid is determined by the yield stresses (e.g., indentation
hardness), which are related to motions of the dislocations. The mobility of
dislocation in a particular direction can be described as the gliding of two
crystallographic planes against each other. In this process dislocation “kinks” are
formed (Fig. 2) [9]. The stress needed to move a kink at a given velocity depends
W
a
b
c
d
Fig. 2. Movement of a kink developed from the gliding of lattice plane under a shear strain, (a) to
(d) show the sequence of bond breakage and formation (adapted from [9]).
www.ism.kiev.ua; www.rql.kiev.ua/almaz_j 50
on the energy of the kink and its position. Therefore, the energy involved in the
movement of a dislocation kink is not unlike the energy path connecting the
various stages of the gliding of two crystal planes shown in Fig. 2. In the same
manner, the potential energy profile can be depicted intuitively or semi-empirically
from the consideration of the breaking and formation of chemical bonds for a
molecule. The diagram correlating the modification of molecular orbitals in this
process is known as the Walsh diagram [12]. An illustration of the energy profile
for a dislocation motion in a covalent solid is depicted in Fig. 3 [9]. Initially, the
chemical bond is intact and the HOMO-LUMO energy separation is the largest. As
the “kinks” migrate, the covalent bonds between the crystal planes weaken. As a
result, the gap energy is reduced. Since the atoms on the two crystal planes have
moved away from the optimum bonding positions, overlaps between atoms in the
two planes are reduced and the stabilization energy (EHOMO) is expected to
diminish. Therefore, in shear deformation, EHOMO should increase with a
concomitant decrease in ELUMO. As the product of the stress and strain is the work
done on the system in the shear deformation, this quantity is related to the change
in the energy of the HOMO (EHOMO). Hence, the stress strength of a crystal is
governed by orbital overlaps in the initial structure and at the displaced dislocation
kink. If good overlap is maintained throughout the shearing, the solid will simply
undergo plastic deformation. If the breaking of chemical bonds occurs for low
stress deformation, the material will have low stress strength.
W/2
Position within king
W
“PN”
∂ε
∂x
εf
bεf
b
εf
a
εf
a
Transition state
E
le
ct
ro
ni
c
en
er
gy
Fig. 3. Schematic orbital correlation diagram (Walsh diagram) depicting the change in HOMO
(εa) and LUMO (εb) energy as a function of the displacement (W) of a kink (taken from [9]).
A convincing demonstration of the chemical principles presented above is the
hardness enhancement observed in transition-metal carbonitrides [13]. From
experiments it is found that the mixed alloy of cubic TiCxN1 – x with a rock-salt
structure achieves the maximum hardness when the valence electron concentration
is about 8.4 per unit cell. Figure 4 shows the correlation of the measured
microhardness in Vickers scale of TiCxN1 – x with calculated shear modulus (c44) as
a function of the valence electron concentration (VEC). The VEC was modified by
ISSN 0203-3119. Сверхтвердые материалы, 2010, № 3 51
varying the relative amount
of C and N atoms.
Theoretical shear moduli
were calculated using the
virtual crystal method where
an effective pseudopotential
was used to approximate
disordered C and N by taking
the weighted average of the
atomic C and N
pseudopotentials according
to the stoichiometry. It is
obvious that the maximum in
c44 corresponds to the
highest Vickers hardness and
occurs at a VEC of 8.4.
Figure 5 shows the band
structure of the reference
TiC (i.e. TiCxN1 – x x = 1) at
zero strain (Fig. 5, a) and
under a shear strain εxy = 0.01. The difference in the band structure is dramatic.
Atfinite shear strain, the fourth electronic band increases in energy (thick dark line
K → Γ → L), while the energy of the fifth band drops and crosses the Fermi level
along X → Γ. Inspection of the valence charge density map (Fig. 6, a) shows that
the fourth band is composed of mainly Ti 22 yxd − —C (px, py) bonding. A shear
deformation in the xy plane disrupts the bonding from the optimal configuration
(Fig. 6, b) resulting in the rise of the fourth electronic band. On the other hand, Ti
d-d interactions in the fifth electronic band is enhanced by the shear strain due to
the shortening of nearest neighbor Ti—Ti separation in the [110] direction. The
Γ X W K Γ L W
–15
–10
–5
0
5
10
E
, e
V
a
Fig. 5. Band structure of TiC (a) equilibrium and (b) at strain εxy = 0.01. Note the changes in the
dispersion of the 4th (thick line) and 5th electronic band (taken from [13]).
Hardness
c
44
c 44
, G
P
a
H
ar
dn
es
s,
H
V
8.0 8.2 8.4 8.6 8.8 9.0
VEC
TiC
x
N
1 – x
140
160
180
200
Fig. 4. Relationship of the valence electron count (VEC)
with calculated shear modulus (c44) and experimental
hardness (HV) for TiCxNx alloy of different compositions
(taken from [13]).
www.ism.kiev.ua; www.rql.kiev.ua/almaz_j 52
Γ X W K Γ L W
–15
–10
–5
0
5
10
E
, e
V
b
Fig. 5. (Contd.)
1
7
C
Ti
C
Ti Ti
1
7
C
Ti
C
Ti Ti
a b
Fig. 6. Charge density of the 4th and 5th electronic band of TiC under strain (εxy = 0.01) (taken
from [13]).
replacement of C by N increases the number of valence electrons. In a rigid band
description, the additional electrons occupy the empty states of the fourth band
near Γ, raising the Fermi level and strengthening the Ti—C interactions, which
results in a higher c44. The complete filling of the shear-resistive bonding states is
at a VEC ~ 8.4, which corresponds to the observed maximum in the experimental
microhardness. This is strong evidence indicating that this is the cause for the
enhancement of hardness in TiC via the replacement of C by N.
PHENOMENOLOGICAL SCHEMES
Bulk modulus
The first empirical scheme for the estimation of the bulk moduli of covalent
solids was developed by Cohen [14]. The scheme is based on the Phillips-Van
ISSN 0203-3119. Сверхтвердые материалы, 2010, № 3 53
Vechten energy scheme for the characterization of the covalent and ionic nature of
tetrahedral bonded solids [15, 16]. It was argued that the bulk modulus of
semiconductors with the diamond or zinc-blende structure is related to a homopolar
energy gap Eh. Using a scaling relationship between Eh and the nearest-neighbour
distance d, an empirical formula [17] that reproduced a number of tetrahedral
bonded semiconductors was found, 5.3
)20.271.19(
d
B λ−= , with B in Mbar and d in
Å. The parameter λ accounts for the ionicity and for homopolar semiconductors
λ = 0; for heteropolar Group III-V and II-VI λ = 1 and 2, respectively. As has been
noted above, bulk modulus is not an appropriate parameter for the estimation of
hardness due to the neglect of dislocations. Nevertheless, the Phillips-van Vachten
energy scheme is being employed in recent formulations relating the hardness to
the strength of a chemical bond. A notable success is the method suggested by Gao
et al. [18], which will be discussed in the next section.
Insulators
Hardness is not a computable quantity. However, for practical purposes it
would be very convenient to relate the measured hardness of a material to some of
the fundamental properties. Gao et al. proposed a scheme linking the Vickers
hardness for a broad class of covalent insulating solids to their macroscopic
properties, which can be obtained from the first-principles calculations. The basic
premise of this approach is the extension of the chemical principles where the
plastic deformation of a covalent solid creates motion of dislocations. The barrier,
or resistance force, to the plastic glide is related to the number of covalent bond
(Na), the solid and the corresponding energy gap Eg. This simple assumption is
reasonable as it relates the intrinsic hardness (Ha) to the bond strength (Eg) and the
coordination number [18],
H(GPa) = ANaEg, (1)
where A is the proportional constant, which can be determined from fitting the
expression to a standard set of materials with known Vickers hardness. Na can be
evaluated from the electron number density, Ne, as
( ) ,2/2/ 3/2
3/2
e
i
iia NVZnN =⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
= ∑ (2)
where ni is the number of the ith atom in the cell and Zi is the number of a valence
electron of the ith atom. Following Phillip, Eg for binary polar covalent ABm
crystal is separated into covalent homopolar gap Eh and an ionic or heterpolar gap
C [15, 16],
.222 CEE hg += (3)
The homopolar energy gap can be estimated from the empirical expression
5.275.39 −= dEn , where d is the interatomic distance. The ionic contribution to the
energy gap is deduced from Eh by treating ionicity as a simple screening effect.
This is accounted for by introducing a correction factor exp(–αfi) to describe the
screening effect for each bond, where 22 /1 ghi EEf −= is the ionicity of the
chemical bond in crystal in the Phillips scale. To determine the constant A in
Eq. (1), HV/EhNa is plotted against fi using a test of matereials with known
www.ism.kiev.ua; www.rql.kiev.ua/almaz_j 54
hardness. An equation relating the hardness in the Vickers scale to fi, Ne and d is
then obtained.
5.2
191.13/2
5.2
191.1
350556)GPa(
d
eN
d
eNH
ii f
e
f
a
V
−−
== . (4)
It is straightforward to extend the formula to multicomponent systems. The
average hardness is assumed to be the geometrical mean of the different types of
covalent bonds in the system. Therefore,
∑
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
=
μ
μ∏
μ
μ
n
n
VV HH
/1
)( , (5)
where μ
VH is the hardness of the μ-type bonds, nμ is the number of bonds of type 0.
A comparison of the calculated and observed Vickers hardness for selected
covalent solids is shown in Fig. 7. In view of the significant spread of measured
hardness (over 100 GPa), the overall agreement is highly satisfactory. In recent
years, Eq. (4) and its variant Eq. (5) have been used extensively for the prediction
of hardness for a large class of proposed superhard materials from the first-
principles calculation.
0 20 40 60 80 100
0
20
40
60
80
100 Gao et.al.
Quadratic
Linear
C
al
cu
la
te
d
ha
rd
ne
ss
, G
P
a
Vickers hardness, GPa
Fig. 7. Correlation of calculated and measured Vickers hardness values (see the text).
Does the success of Eq. (4) owe to the incorporation of all essential physics
contributing to the hardness or simply due to the success of fitting to a large
training set on the determination of the empirical parameters? A deeper
appreciation of the derivation of the formula reveals that the implicit assumptions
do not differ from the basic concept for the bulk modulus (vide supra), where the
hardness is attributed solely to the magnitude of the energy gap. This is obviously
not sufficient, since the hardness is also determined by the shear strength. A
complete theory must take into account the full elasticity of the material.
Unfortunately, no theoretical or empirical relationships correlating the energy gap
of a covalent solid with its elastic moduli, particularly, the shear modulus (G) [19]
has ever been established. There are several definitions for shear modulus.
ISSN 0203-3119. Сверхтвердые материалы, 2010, № 3 55
Throughout this paper, except indicated, the convention of Voigt is adopted. To
investigate this point further, in Fig. 7, the hardness of single component and
binary covalent crystals used in the determination of the empirical coefficients in
Eq. (4) were estimated from the correlation shown in [8] (Fig. 1, b) employing the
experimental shear moduli. To establish the correlation quantitatively, the data in
Fig. 1b are approximated by linear and quadratic functions. From the fitted
functional forms, the Vickers hardness can be estimated from the knowledge of the
shear modulus. The results are summarized in Fig. 7 together with the
corresponding values obtained using Eq. (4). The mutual agreement between the
two different empirical schemes and the overall agreement with experimental
hardness are striking. This observation indicates that although (shear) elasticity was
not explicitly included, after fitting to experimental data, the model of Gao [18]
apparently includes this effect implicitly. On the other hand, the strong correlation
shows that hardness can also be estimated from the shear modulus. First-principles
calculations of elastic constants are now routine [20] and conversions to shear
moduli for all crystal classes are well known [19]. The two methods (i.e. Eq. (4)
and hardness-shear modulus correlation) are complementary, both can be used to
estimate the hardness for new materials. However, for a given crystal, when using
Eq. (4), the structure of the crystal and the energy gaps Eh and Eg must be known.
On the other hand, the correlation shown in Fig. 1, b can be used with calculated or
measured shear modulus.
To explore the reliability of the empirical correlation for ternary compounds,
the hardness of β-BC2N [18] is estimated from Fig. 1, b. No experimental shear
modulus is available but the calculated G of β-BC2N is 300—320 GPa. The
corresponding Vickers hardness estimated from Fig. 1, b is 42—51 GPa. In
comparison, the hardness calculated by Eq. (5) is 78 GPa. It should be noted that
the experimental hardness of β-BC2N is not known yet. In [8], the calculated
hardness was compared to an experimental value for cBC2N of 76 GPa [21]. Using
an observed shear modulus of cBC2N of 450 GPa, the hardness was estimated to be
54—74 GPa from Fig. 1, b. The good agreement with experiment suggests that the
correlation of hardness with shear modulus is reliable.
An alternate scheme for predicting hardness without the use of empirical
parameters has been proposed by Simunek and Vackar [22]. This method is an
extension of the hardness relationship of Eqs. (4) and (5). Instead of using
empirical rules to determine ionicity and the energy gaps, a bond strength index Sij
between atoms i and j is defined,
)( ijijjiij ndeeS = , (6)
where ei = Zi/R is a reference energy, Zi is the valence electron number of atom i,
and nij is the number of bonds between atom i and its neighboring atoms j at the
nearest neighbor distance djj. Radius Ri for each atom in a crystal is determined
such that the radius Ri of the sphere contains exactly the charge Zi. Instead of
relating the hardness to bond energy gaps, it is assumed that hardness is
proportional to the bond strength Sij. In this way, no empirical parameters are
needed and the hardness can be determined solely from the first principles
calculations. Good agreement between the theory and experiment was found [22].
Nanostructures
An advantage of having an expression relating the hardness to the fundamental
properties is that it can be extended to other systems. Recent experiments have
www.ism.kiev.ua; www.rql.kiev.ua/almaz_j 56
shown that nanocrystalline solids, such as BN [23] or diamond [24], possess
hardness “higher” than bulk diamond. These observations contradicted the reversed
Hall-Petch effect that the hardness and yield stress should decrease with the grain
size. This argument is supported by a computer simulation, which suggested that in
nanocrystallites, the plastic deformation is mainly due to a large number of small
sliding at the grain boundaries rather than the motion of dislocations. The softening
in the shear stress is a result of the larger fraction of atoms at the grain boundaries
[25]. In contrast, novel phenomenon of hardening of nanocrystallites can be
explained by combining the empirical theory of hardness with the effect of
quantum confinement [26]. In nanocrystals, the conduction/valence band edges
shift generally to higher energy relative to the bulk material when the crystalline
size is decreased. According to the Kubo theory [27], the bandgap Eg,nano of a
nanocrystal should increase inversely with the volume V
δ+= bulkgnanog EE ,, , (7)
where Eg, bulk is an “effective” band gap of the bulk material and δ is the energy
shift due to quantum confinement given by [27]
3/12
22
)3(
2
eNmV π
π=δ h , (8)
where m is the atomic mass and Ne is the electron density. Equation (4) can be
extended to include quantum confinement effects, the energy shift for a
nanocrystallite δp can be written as,
3/13/
3/12
22
3/12
22
)(
)3(
2
)3(
2 −− =⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
π
π=
π
π=δ ee
e
p NDfN
mV
K
NmV
K hh , (9)
where parameter K may also be a function of the particle diameter. We can
approximate the term in the brackets of Eq. (9) as a function of cluster diameter D,
f(D). Equating the band shift ratio to experimental shift ratio, δp/Egp= δ/Eg and
using CdS as a calibrant [28], the bracketed term in Eq. (9) can be evaluated. It is
found that an empirical formula, 3/1
0.24
e
p DN
=δ with δp (in eV) inversely
proportional to cluster diameter D, gave a good fit to the experimental data of
nanocrystalline CdS. Finally, an expression for the hardness of a nanocrystal is
obtained by substituting Eq. (9) into Eq. (1) [26]
)/0.24()()GPa( 3/1
,, ebulkgapbulkga DNEANEANH +=δ+= . (10)
This equation has been used to compute the hardness of nano-3C- and 6-
diamonds and the results are summarized in table. The trend and the predicted
hardness are consistent with experimental observation.
Recently, a unique superhard aggregated boron nitride nanocomposites
(ABNNCs) showing the enhancement of hardness up to 100% in comparison with a
cBN single crystal has been synthesized and characterized [29]. The decrease of
the grain size down to nano- and subnanolevel results in enormous mechanical
property enhancement with maximum hardness of 85 GPa. This finding contradicts
the reverse Hall-Petch effect [25]. However, a fit of the measured hardness (HV) to
the crystallite size (d), according to the Hall-Petch equation [29, 30] (HV = H0 +
K/ d ), which has an inverse square root relationship with d was found to be
ISSN 0203-3119. Сверхтвердые материалы, 2010, № 3 57
unsatisfactory. However, adding the quantum effects term (Eq. (9)), which have a
1/d dependence, to the Hall-Petch equation improves the agreement with the
observed trend significantly over a large hardness range on the crystallite size
(14—2000 nm) (Fig. 8) [29].
Calculated Vickers hardness for 6H and 3C diamond nanocrystallites
Phase HV cal nano (GPa)
6H diamond 96 (5 nm)
94 (12 nm)
108 (1.245 nm)
3C diamond 97 (5 nm)
95 (12 nm)
112 (1.0 nm)
50 100 150 200 250
Crystallite size, nm
H
V
, GPa
40
50
60
70
80
90
Fig. 8. Comparison of calculated and experimental Vickers hardness (HV) using the Hall-Petch
equation including the correction for quantum confinement (modified from [29]).
Metals
The theoretical basis of “metallic” bonding is still under intense debate [31, 32].
In one school of thought [32] a metal may be described as comprised of weak
covalent (directions) bonds and delocalized electrons. The weak covalent bonds
help to define the crystal structure. This point of view is clearly demonstrated in
the elucidation of the structure and electronic properties of Li—Al alloys [33]. This
description of metal structure becomes even more relevant to hard, weakly metallic
compounds. An extension of Eq. (4) to transition metal carbides and nitrides has
been proposed [34]. The metallicity (fm) is related to the number of thermally
excited electrons (nm) above the Fermi level at temperature T and represent as a
product of the electron density of states at the Fermi level, Df and kBT, where kB is
the Boltzmann constant. At room temperature, fm = 0.026Df /ne, where ne is the total
number of valence electrons. It was postulated [34] that the existence of both ionic
and metallic contributions will reduce the covalent bond density Na (Eq. (1)). An
empirical correction factor to the ionicity fi is then added,
www.ism.kiev.ua; www.rql.kiev.ua/almaz_j 58
[ ] 635.0/||exp(1 PPPf Ci −−−= , (11)
where P is the overlap population of a bond and Pc is the overlap population of a
bond in a hypothetical pure covalent crystal with the same structure. In addition, an
empirical correction to the screening effect (
n
mfe β− ) is included. The final
expression is similar to Eq. (4) where the Vickers hardness is
h
f
aV EeNAH i
n
mβ−−= 191.1')GPa( . (12)
Here, the constants A′, β and n were determined from fitting to known hardness
from a selected set of compounds [34].
The expression (Eq. (12)) is not entirely satisfactory for a number of reasons.
Conceptually, it is not possible to define an energy gap (Eh) for a “bond” in metals.
Overlap population [11] is not a well defined quantity. It is well-known that it is
dependent on the system and then criteria on dividing the total overlap among two
atoms [11]. Finally, the constants appearing in Eq. (12) were determined from
overlap population computed with CASTEP code and are not easily transferable to
be used by other electronic codes.
The universality and usefulness of
this expression remains to be tested.
A comparison of the hardness
estimated from the shear moduli and
using Eq. (12) is shown in Fig. 9
along with experimental data. For
metallic carbides and nitrides the
range of hardness is small. It is no
surprise that Eq. (12), which was
fitted with the experimental data,
gave the best agreement. Note that
the shear moduli used in the
preparation of Fig. 9 include both
experimental and theoretical values
and the two sets of data may be
subjected to errors (> ±10%). It is
observed that the experimental trend
is correctly reproduced by both the
linear and quadratic fits to Fig. 1, b, except that the hardness values obtained from
the linear fit are somewhat higher. In passing, it is noteworthy that the Vickers
hardness for the recently synthesized “ultraincompressible” OsB2 [35] from the
hardness-shear modulus curve using the theoretical G of 206 GPa is 25—31 GPa
[36]. This value is comparable to 35 GPa [37] calculated with the parameter — free
expression Eq. (6) and the experimental estimate of 37 GPa [38]. The good
agreement between the hardness values obtained from the parameter — free Eq. (6)
with experiment may suggest that it is preferable over the semi-empirical scheme.
The simple hardness–shear modulus correlation (Fig. 1, b) can also be used as a
qualitative guide.
Ab intio calculations of ideal tensile and shear strength
The ideal strength is the stress where a dislocation-free crystal becomes
unstable and undergoes spontaneous plastic deformation. This sets the upper bound
on the mechanical strength of a material. Detailed analysis [39] of the mechanics of
10 15 20 25 30 35 40
10
15
20
25
30
35
40
Qaudratic
Linear
Guo et.al.
C
al
cu
la
te
d
ha
rd
ne
ss
, G
P
a
Vickers hardness, GPa
Fig. 9. Correlation of calculated and measured
Vickers hardness (see the text) for covalent
dominant transition metal carbides and nitrides.
ISSN 0203-3119. Сверхтвердые материалы, 2010, № 3 59
nanoindentation has shown that after proper consideration of the crystallography of
loading and the correction of the nonlinearity of the elastic response at large
strains, the measured values can be compared quantitatively to the results of the
first-principles calculations. In the calculations, the ideal strength is defined as the
maximum stress in the stress-strain curve in the weakest tensile stretch or shear slip
direction [40, 41]. Since first-principles ideal strength calculations explore the
stress-strain energy profile at large structural deformation, changes in the electronic
structure under strain are also correctly reproduced. Therefore, unlike the use of
bulk and shear moduli obtained at equilibrium to estimate the hardness, ideal
strength calculations provide accurate tensile and shear strengths and, from the
analysis of the results, the mechanism leading to structural failure can be revealed.
To compute the ideal strength of a crystal, one needs to determine the weakest
tensile direction. This is accomplished by calculation of the tensile stresses along
candidate crystallographic directions [40, 41]. For a given tensile direction, the
relevant lattice vectors are deformed in the direction of the applied strains. At each
step, all the atom positions are relaxed and the cell basis vectors are adjusted to
remove stresses orthogonal to the applied strains. Once the weakest tensile
direction is found, the critical shear stress is calculated by applying shear
deformation in the “easy”-slip plane perpendicular to this direction. A typical
theoretical calculated stress-strain curve with tensile applied at different
crystallography directions is shown in Fig. 10 [42].
0 0.1 0.2
Strein
0
20
40
60
80
100
S
tr
es
s,
G
P
a
Tensile <120>
Tensile <100>
Tensile <001>
Fig. 10. Theoretical stress-strain relationship for β-BC2N (taken from [42]).
The ideal strengths of many metals, metal alloys, hard ceramic materials and
nanostructures have been studied with ab inito calculations [41]. The method has
also been applied to potential superhard materials such as transition metal carbides
and nitrides, carbon nitrides, and boron carbides. A good example is the recently
synthesized “ultra”-incompressible OsB2 [35]. It was shown from high pressure X-
ray diffraction that the bulk modulus has an unusually high value of 365 GPa [35].
This value is to be compared with diamond (442 GPa) and Os metal (395—
462 GPa). Moreover, the as-synthesized powder has been shown to scratch the
polished surface of a sapphire crystal indicating that the hardness of this material
should exceed 20 GPa [35]. This surprising finding has stimulated many theoretical
www.ism.kiev.ua; www.rql.kiev.ua/almaz_j 60
studies. From a calculated shear modulus of 205 GPa using gradient corrected
density functional, the Vickers hardness was estimated to be 31.8 GPa [37]. The
measured hardness from nanoindentation under different loadings varies between
30 and 37 GPa [38, 43]. All indications point to OsB2 may be a superhard material.
Calculations of the ideal tensile and shear strengths of OsB2 have been reported
in [44]. OsB2 is formed from alternate layers of Os and B atoms. Examination of
the crystal structure suggests that the bonding in the (001) Os1—Os2 plane is the
weakest and the strength against shear deformation will be the lowest in the [010]
direction (Fig. 11). The calculated tensile stresses in several symmetric
crystallographic directions are shown in Fig. 11. Not surprisingly, the highest
tensile strength is predicted along [100] and the lowest peak of the stress is at
[011]. A calculated ideal tensile strength in the [011] direction of 25.9 GPa is still
higher than that of iron (12.6 GPa) [45]. The covalent Os—B bonds enhance the
resistance of OsB2 to tensile deformation in all stress directions.
Os1
Os2
B1
B2
B4B3
0 0.05 0.10 0.15 0.20 0.25 0.30
Strein
0
10
20
30
40
S
tr
es
s,
G
P
a
Tensile:
[100]
[010]
[001]
[110]
[101]
[011]
[111]
50
60
Fig. 11. Crystal structure (a) and calculated tensile stress-strain curves for OsB2 (adapted from
[44]) (b).
ISSN 0203-3119. Сверхтвердые материалы, 2010, № 3 61
Figure 12 shows calculated shear stresses in the (001) plane along the three
principal crystallographic axes. The shear strength is highly anisotropic. Maxima in
the stress-strain curves are found at 26.9, 19.4, and 9.1 GPa in the [001], [010], and
[100] directions, respectively. Therefore, the lowest ideal shear stress is in the
(001)[010] shear direction. The highly anisotropic strength in the (001) plane is
related to the crystal structure where the Os—B bonds only help to strengthen the
resistance against shear deformations in the (001)[100] direction. The inspection of
the stress-strain curves also provides information on mechanical properties of
OsB2. The discontinuities in the stress along (001)[100] and (001)[110] shear
directions are indicative of a brittle material, where the chemical bonds broke
abruptly. This contrasts with the continuous stress-strain relationship along the
(001)[010] shear direction, which is typical for a ductile material. The ductility can
be understood from the analysis of the evolution of the crystal structure during
shear deformation. Four instantaneous snapshots taken along the (001)[010] shear
are shown in the lower panel of Fig. 12. The Os—B bonds were maintained
throughout the deformation and the crystal planes glided against each other and did
not offer strong resistance against the shear strain. At first glance this conclusion is
perhaps counter-intuitive. However, this observation can be rationalized as follows.
Strong Os—B covalent interactions only provide strong repulsive force to resist the
bond compression and enhancing the tensile strengths. However, when sheared in
the (001)[100] direction, the Os—B distance is extended somewhat but the
overlaps are still maintained. The calculated ideal strength in the (001)[010] shear
direction is only slightly higher than that of iron (7.2 GPa). Incidentally, a
maximum shear strength along the (001)[100] of 26.9 GPa is comparable to a
measured Vickers hardness of 30 GPa on the (001) plane [43]. The example
0 0.1 0.2 0.3 0.4
Strein
0
10
20
30
40
S
tr
es
s,
G
P
a
Shear:
(001) [100]
Os–Os bond length 4.0
3.5
3.0
2.5
2.0
50
–10
(001) [010]
(001) [110]
4.5
S
0
S
1
S
2
S
3
B
on
d
le
ng
th
, Å
S
0
S
1
S
2
S
3
[001]
[010]
Os
Os
Fig. 12. Calculated stress-strain relationships for OsB2 at shear deformation in the [100], [010]
and [110] crystal planes. The bottom figure shows the structures at the deformation points
indicated in the corresponding (001)[010] stress-strain curve (taken from [44]).
www.ism.kiev.ua; www.rql.kiev.ua/almaz_j 62
presented above illustrates that the first-principles ideal strength calculations can
yield numerical results, which can be compared directly with nanoindentation
experiments and provide detail information to rationalize the mechanical processes
involved in shear deformations. It also highlights the importance and advantage on
the characterization of hardness from the examination of crystal and the electronic
structure accompanying the deformation processes. This information is, of course,
not available from simple correlation with the shear moduli at the equilibrium
structure or from parameter — free phenomenological or semi-empirical schemes.
CONCLUSIONS
In closing, to illustrate the usefulness and insight provided by the state-of-the-
art computational approaches, it is useful to remark the theoretical studies on a
recently discovered superhard cubic boron carbide (cBC5) [46]. A number of
theoretical techniques have been applied to characterize the structure and hardness
of this novel material. Since experimental results indicated that the B boron atoms
are not aggregated (clustered), the first theoretical investigation focused on stable
structures that may explain the diffraction pattern and the unusual hardness. Based
on this assumption, algebraic evaluation of possible structures followed by
calculations of the total energies, Raman spectra, and diffraction patterns led to the
conclusion that B is randomly distributed in the cubic diamond structure [47]. An
important result is that the ordered trigonal (P3m1) structure proposed earlier [48]
is energetically not favorable and the calculated diffraction pattern is in conflict
with experiment. A Vickers hardness of 79 GPa for the B-disordered structure was
estimated from the empirical shear moduli — hardness correlation, which is
remarkably close to an observed value of 79 GPa [47]. Subsequently, the
disordered structure is also confirmed to be the most stable and best candidate for
cBC5 from more elaborate special quasi-random structure (SQS) search procedure
[49]. The ideal mechanical strength on model BC5 structures has also been
investigated by two separate groups [50, 51]. Both calculations demonstrate that
the P3m1 structure [48] has weak slip directions along (10 1 ) and (1 2 1) in the
[111] plane and cannot be a candidate for cBC5 [50, 51]. Moreover, the calculated
ideal shear stresses are found to be much stronger (almost three times) on cubic
structural models with randomly distributed B without clustering [51], again
supporting a disordered structure.
A brief overview on the current status of the computational approaches for the
determination and prediction of the material hardness was presented. The empirical
scheme of correlating Vickers hardness with the shear modulus was found to
perform reasonably well. The correlation is by no means quantitative. However,
within the accuracy of measured hardness and shear moduli, the overall agreement
is deemed to be highly satisfactory. There is no doubt that strong covalent bonding
is a principal factor for ultra- and superhard materials. Phenomenological schemes
assuming that the hardness is primarily governed by the strength of chemical bonds
have been developed for covalent solids. These schemes that provide the conduits
for direct comparison with experimental data are easy to use and proven to be
reliable for a wide class of materials. Potential problems on extending these
phenomenological schemes to metallic solids are highlighted. The most rigorous
theoretical approach for studying hardness is through the calculations of ideal
tensile and shear strength with first-principles electronic methods. This approach
allows quantitative comparison with the results obtained from nanoindentation
experiments and provides physical insights into the deformation mechanisms. The
principal ingredient for a hard material is strong covalent chemical bonds.
ISSN 0203-3119. Сверхтвердые материалы, 2010, № 3 63
However, as demonstrated for the case of TiCxN1–x alloy [13], orbital
rehybridization can also play a crucial role in determining the resistance to shear
deformation. Ideal shear strength calculations have also shown that [42, 45]
weaknesses in crystallographic planes determine the lattice instability and,
therefore, the intrinsic hardness. To this end, the real space orbital correlation
approach (e.g., Fig. 3) [9, 12] is a very useful conceptual and qualitative guide to
predict hardness of a potential material. The search for high hardness and strength
materials is important for a wide variety of industrial applications such as cutting
tools, wear-resistant coatings and abrasive applications. Understanding the
fundamental principles governing the hardness of a material is a prerequisite for
the search and rational design of new materials. This is particularly important in
view of recent advances in the development of efficient electronic codes and
methods for structural prediction from first-principles.
Розглянуто сучасний стан різних теоретичних підходів до
прогнозування твердості матеріалів. Показано, що проста емпірична кореляція з модулем
зсуву звичайно дає дуже хорошу оцінку твердості за Віккерсом. Розроблені також
напівемпіричні моделі, основані виключно на міцності хімічного зв’язку, є теоретично
неповними. Розрахунки з перших принципів ідеального напруження і міцності на зсув є,
мабуть, найбільш надійним і теоретично обґрунтованим підходом для порівняння
теоретичних прогнозів з експериментальними результатами.
Ключові слова: твердість, кристал, модуль зсуву, міцність хімічного
зв’язку, ідеальне напруження, міцність на зсув.
Рассмотрено современное состояние различных теоретических
подходов к прогнозированию твердости материалов. Показано, что простая
эмпирическая корреляция с модулем сдвига обычно дает очень хорошую оценку
твердости по Виккерсу. Разработанные также полуэмпирические модели, основанные
исключительно на прочности химических связей, являются теоретически неполными.
Расчеты из первых принципов идеального напряжения и прочности на сдвиг
представляют собой, вероятно, наиболее надежный и теоретически обоснованный
подход для сравнения теоретических прогнозов с экспериментальными результатами.
Ключевые слова: твердость, кристалл, модуль сдвига, прочность
химических связей, идеальное напряжение, прочность на сдвиг.
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Department of Physics and Engineering Physics, Received 25.11.09
University of Saskatchewan
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| id | nasplib_isofts_kiev_ua-123456789-63473 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 0203-3119 |
| language | English |
| last_indexed | 2025-12-07T17:05:39Z |
| publishDate | 2010 |
| publisher | Інститут надтвердих матеріалів ім. В.М. Бакуля НАН України |
| record_format | dspace |
| spelling | Tse, J.S. 2014-06-02T06:42:34Z 2014-06-02T06:42:34Z 2010 Intrinsic hardness of crystalline solids / J.S. Tse // Сверхтвердые материалы. — 2010. — № 3. — С. 46-65. — Бібліогр.: 51 назв. — англ. 0203-3119 https://nasplib.isofts.kiev.ua/handle/123456789/63473 539.53 The current status of various theoretical approaches to the prediction of material hardness has been reviewed. It is shown that the simple empirical correlation with the shear moduli generally provide very good estimates of the Vickers hardness. Semi-empirical models based solely on the strength of the chemical bonds, although performed as well, are theoretically incomplete. First-principles calculations of the ideal stress and shear strength is perhaps the most reliable and theoretically sound approach available to compare theoretical predictions with experiment. Розглянуто сучасний стан різних теоретичних підходів до прогнозування твердості матеріалів. Показано, що проста емпірична кореляція з модулем зсуву звичайно дає дуже хорошу оцінку твердості за Віккерсом. Розроблені також напівемпіричні моделі, основані виключно на міцності хімічного зв’язку, є теоретично неповними. Розрахунки з перших принципів ідеального напруження і міцності на зсув є, мабуть, найбільш надійним і теоретично обґрунтованим підходом для порівняння теоретичних прогнозів з експериментальними результатами. Рассмотрено современное состояние различных теоретических подходов к прогнозированию твердости материалов. Показано, что простая эмпирическая корреляция с модулем сдвига обычно дает очень хорошую оценку твердости по Виккерсу. Разработанные также полуэмпирические модели, основанные исключительно на прочности химических связей, являются теоретически неполными. Расчеты из первых принципов идеального напряжения и прочности на сдвиг представляют собой, вероятно, наиболее надежный и теоретически обоснованный подход для сравнения теоретических прогнозов с экспериментальными результатами. en Інститут надтвердих матеріалів ім. В.М. Бакуля НАН України Сверхтвердые материалы Получение, структура, свойства Intrinsic hardness of crystalline solids Article published earlier |
| spellingShingle | Intrinsic hardness of crystalline solids Tse, J.S. Получение, структура, свойства |
| title | Intrinsic hardness of crystalline solids |
| title_full | Intrinsic hardness of crystalline solids |
| title_fullStr | Intrinsic hardness of crystalline solids |
| title_full_unstemmed | Intrinsic hardness of crystalline solids |
| title_short | Intrinsic hardness of crystalline solids |
| title_sort | intrinsic hardness of crystalline solids |
| topic | Получение, структура, свойства |
| topic_facet | Получение, структура, свойства |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/63473 |
| work_keys_str_mv | AT tsejs intrinsichardnessofcrystallinesolids |