Article pubs.acs.org/IC
Simple Method for the Hardness Estimation of Inorganic Crystals by the Bond Valence Model Xiao Liu, Hao Wang,* Weimin Wang, and Zhengyi Fu †
State Key Laboratory of Advanced Technology for Materials Synthesis and Processing, Wuhan University of Technology, Wuhan 430070, P. R. China S Supporting Information *
ABSTRACT: On the basis of the bond valence model, an empirical hardness estimation of inorganic crystals using a simple formula is presented. A new scale, the resistant force per unit area of chemical bond by bond valence, is proposed to be closely related to the hardness of crystals. For multibond crystal systems, the hardness can be equivalent to an average of the hardness of all binary systems in inorganic crystals. Applied to γ-A3N4 (A = C, Si, Ge) nitride spinels and BC2N compounds, their calculated hardness agrees well with the available experimental results. Our empirical hardness method can offer a simple and reliable hardness prediction resulting from well-described chemical bonding by the bond valence model, which makes it powerful for exploring novel superhard materials.
1. INTRODUCTION Superhard materials are of great interest because of their extensive variety of applications in modern science and technology, including abrasives, cutting, polishing tools, and wear-resistant coatings. Diamond is the hardest known material and has been the most widely used. However, diamond1 has limited ability in cutting ferrous metals because of a chemical reaction. Another well-known ultrahard material, cubic boron nitride (c-BN),2,3 can partly replace diamond and be used to effectively cut ferrous metals, but this material is quite expensive because of the synthesis conditions with extreme pressure and temperature. Over the last 2 decades, searching for a new class of superhard materials has attracted great interest, in which the focus is on selecting intrinsic superhard materials with a high valence-electron concentration and the presence of covalent bonding.1,4 B−C−N systems such as C3N4,5 BC2N,6 and some transition-metal borides,7 carbides,8 and nitrides9 were predicted to be new potential superhard materials, but these materials with bulk quantities have rarely been obtained. To purposely search for new superhard materials, we must understand the origin of the superhardness of some materials and explore a practical method to estimate the hardness of materials. Experimentally, the hardness value is obtained by indentation hardness in which a diamond with a standard shape is impressed into a specimen’s surface. Hardness measures a © XXXX American Chemical Society
material’s ability to resist deformation and is determined by elastic stiffness and plastic resistance. Because elastic stiffness and plastic deformation are all closely related to the resistant force of the chemical bond, it is suggested that the hardness of an ideal solid is mainly determined by the nature of its chemical bonding. Additionally, it is suggested that the stiffness of the solids is also related to the delocalized bonding picture of molecular orbitals in the electronic structure.10 Some results1,7 have revealed some design parameters for the development of superhard materials: high valence electron density and the incorporation of bond covalency. Thus, appropriate methods for relating the hardness to the nature of chemical bonding are essential to estimate the hardness. At the early stages, the hardness values have mostly been compared with the values of cohesive energies,11 bulk moduli,5 or shear moduli.1 Recently, some empirical methods12−14 have been proposed to calculate the hardness based on three factors of the nature of its chemical bonding: bond density, ionicity, and resistant force of the chemical bond. In particular, it is important to quantitatively characterize the resistant force of the chemical bond. Gao et al.12 correlated the resistance of the bond with the energy gap of the microscopic electronic structure. Šimůnek and Vackar13 introduced the reference energy accessible by the firstReceived: July 19, 2016
A
DOI: 10.1021/acs.inorgchem.6b01729 Inorg. Chem. XXXX, XXX, XXX−XXX
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Inorganic Chemistry
Table 1. Hardness and Parameters Related to the Hardness Calculation of Typical Binary Covalent and Ionic Crystals, Where Hk,calc and Hexp Are the Calculated and Experimental Knoop Hardness Values, Respectivelya crystal
Rij (Å)
R0 (Å)
Sij (vu)
f ij
Nv (Å−3)
Hk,calc (GPa)
Hexp (GPa)
diamond Si Ge Sn c-BN InN AlP GaP InP AlAs GaAs InAs AlN GaN ZrO2 SnO2 BeO HfO2 Y2O3 RuO2 TiN ZrN NbN HfN TaN NaCl KCl
1.544 2.341 2.450 2.810 1.567 2.136 2.347 2.307 2.532 2.438 2.385 2.623 1.890 1.934 2.224 2.162 1.898 2.215 2.280 1.989 2.135 2.269 2.221 2.255 2.165 2.875 3.112
1.54 2.34 2.45 2.81 1.482 2.03 2.24 2.26 2.43 2.32 2.34 2.51 1.79 1.84 1.937 1.905 1.381 1.923 2.014 1.834 1.93 2.11 2.06 2.090 2.010 2.15 2.52
0.99 1.00 1.00 1.00 0.79 0.75 0.75 0.88 0.76 0.73 0.89 0.74 0.76 0.78 0.46 0.50 0.25 0.45 0.49 0.66 0.57 0.65 0.65 0.64 0.66 0.14 0.20
1.00 1.00 1.00 1.00 0.40 0.39 0.34 0.40 0.40 0.32 0.40 0.38 0.35 0.33 0.19 0.21 0.05 0.14 0.20 0.32 0.21 0.31 0.31 0.24 0.25 0.02 0.04
0.35 0.10 0.09 0.06 0.34 0.13 0.10 0.11 0.08 0.09 0.10 0.07 0.19 0.18 0.24 0.26 0.44 0.24 0.22 0.21 0.31 0.26 0.27 0.26 0.30 0.13 0.10
100.0 12.2 10.4 5.3 50.9 10.6 6.1 7.5 4.4 4.9 6.8 3.8 18.1 15.9 11.4 14.2 12.0 9.8 10.5 17.1 16.9 15.9 16.5 14.2 18.4 0.9 0.7
90−100b 14 11.3 4.5 48 9.0 9.4 9.5 5.4 5 7.5 3.8 18c 15.1c 11.6 13.8 12.5 9.9 7.5 20 17.7 15.1 17 16.3d 22d 0.3 0.2
a Unless noted, all experimental Knoop hardness values are taken from refs 14, 25, and 26. bKnoop hardness from ref 27. cVicker hardness from ref 12. dVicker hardness from ref 28.
elastically resist volume compression and generally possesses high hardness. Furthermore, it is expected that the bond valence model would provide us with the primary clue to relate to the material’s hardness, by which the nature of the chemical bonding can be well characterized. In the present work, for the first time to our knowledge, we present a new empirical method to calculate the hardness of inorganic crystals based on the bond valence model. The resistant force of the chemical bond related to the hardness can be evaluated empirically by the bond covalent components− bond valence correlation. The empirical method between the hardness and resistant force of the chemical bond is established for binary systems, and this method has been developed to estimate the hardness of multibond crystal systems by an average of the hardness of all constituted binary systems.
principles technique to characterize the resistant force of the bond. Li et al.14 defined a new scale of bond electronegativity to describe the electron-holding energy of two bonded atoms. However, the resistant force of the chemical bond mentioned above is a complex and indistinct quantity, thus simpler and more accurate characterization of the nature of the chemical bonding is needed for an empirical hardness method. In recent years, the bond valence model,15,16 derived from Pauling’s second rule, has evolved into a simple but highly predictive model to provide a quantitative picture of the chemical structure. This model has been validated by a correlation between the theoretically determined bond valences and the experimentally observed bond lengths in a large number of well-determined structures, which makes it an advantage of great authenticity to describe many phenomena of inorganic chemical bonding. The presentation of the chemical bonding by the bond valence model has been complementary to the quantum-mechanical approaches.15 The bulk modulus measures the material’s resistance to uniform compression and is highly related to the nature of the chemical bonding.5 It is revealed that the bond valence is an effective parameter to estimate the material’s bulk modulus. For example, Brown et al.17 have deduced an expression by bond valence to calculate the bulk modulus of a chemical bond in binary crystals with high-symmetry structures. Most recently, we also proposed a new empirical method18 based on bond valences to estimate the compressibility of complex spinel-type crystals. The hardness is a complex property related to the elasticity and plasticity. A material having a high bulk modulus tends to
2. THEORETICAL METHODS AND RESULTS 2.1. Calculation of the Hardness for Binary Crystals. According to static indentation tests,19 the hardness can be measured by the load per unit area of the impression, when a pyramid is forced into the material’s surface. The hardness can be defined as the resistance of a material to permanent indentation, which is mainly related to the resistant force per unit area.12,20 In our opinion, the resistant force per unit area of chemical bonds mainly determines the fundamental hardness of a material on the atomic scale. In solids, simple bonding models assume that the resistant force of the chemical bond is mainly composed of the attractive Coulomb force and repulsive force. The attractive Coulomb force is usually calculated using the B
DOI: 10.1021/acs.inorgchem.6b01729 Inorg. Chem. XXXX, XXX, XXX−XXX
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Inorganic Chemistry
structure is taken from the Inorganic Crystal Structure Database (see the Supporting Information).24 In Table 1, the calculated hardness values are compared with the experimental data, and the good agreement demonstrates the predictive power of eq 5. It is suggested that the higher hardness values of the ultrahard material diamond and c-BN are related to their shorter bond lengths, higher bond valences, and bond densities. Additionally, lower hardness values of the ionic NaCl and KCl are also in reasonable agreement with the experiment data. The comparison between the calculated hardness and experimental values is shown in Figure 1. The predictability of our calculation
classical electrostatics, but the repulsive force requires a complex quantum treatment and is hard to accurately estimate. In the bond valence model,15 the bond valence (Sij) between the two atoms i and j obeys ⎛ R 0 − R ij ⎞ Sij = exp⎜ ⎟ b ⎝ ⎠
(1) 21
where R0 is the bond valence parameter, Rij is the length of the bond between the two atoms i and j, and b is a universal constant equal to 0.37 Å. The bond valence is the same as the electrostatic flux that links the two bonded atoms, which is a complete description of the Coulomb field of a structure. Therefore, the attractive Coulomb force of the chemical bond can be described by its bond valence.15 Brown’s work also indicates that the repulsive force of the chemical bond can be evaluated by bond covalent components ( f ij) from the exponential equation related to the bond valence as follows: fij = TSij M
(2)
T and M are fitted constants, and f ij is assigned as 1 for pure covalent bonds (see the Supporting Information). Furthermore, some hardness results12 of ionic materials have shown that the attractive Coulomb force does not have a direct relationship with the macroscopic hardness. As a result, the resistant force of the chemical bond related to the hardness is mainly determined by the repulsive force. Thus, the resistant force per unit area of the chemical bond can be evaluated empirically by covalent components per unit area of a bond (Pij) by the equation 22
⎛ f ⎞2/3 ⎛ TS M ⎞2/3 ij ij ⎟ Pij = ⎜⎜ 3 ⎟⎟ = ⎜⎜ 3 ⎟ ⎝ cR ij ⎠ ⎝ cR ij ⎠
Figure 1. Calculated hardness from different models versus the experimental values for typical binary covalent and ionic crystals. The inset shows a magnified image between 0 and 40 GPa.
could be comparable with other popular empirical methods. Unlike the other three models, the main input parameters in our model can be directly deduced from the bonding properties with simple arithmetic calculations based on the bond valence model and directly correlate the hardness of the crystal with the bonding properties. 2.2. Calculation of the Hardness for Multibond Crystals. For a more complex multibond crystal, its hardness is considered as a geometric average of the hardness of all binary systems in the solid.12,13 For a multicomponent compound system with n types of binary systems, its hardness can be calculated as
(3)
where the bond volume is similar to Rij3 23 and c is a proportionality constant. The hardness is expected to be proportional to Pij. Moreover, the hardness of an ideal single crystal is also proportional to the bond number per unit cell of the crystal, i.e., bond density Nv = N/V,14 where N is the number of bonds per unit cell and V is the volume of the unit cell in cubic angstroms. Thus, the hardness of an ideal single crystal can be assumed by using the empirical expression ⎛ TS M ⎞2/3 ij ⎟ Nv H (GPa) = BPijNv = D⎜⎜ 3 ⎟ ⎝ R ij ⎠
n
Hk (GPa) = (∏ Hk μ)1/ n
(4)
μ=1
where B and D are constants. Because the bond valence model has been shown to be valid in all types of localized bonds including covalent and ionic bonds, eq 4 is expected to be suitable for both covalent and ionic crystals. In order to determine the relationship of eq 4, the nominal Knoop hardness Hk of a natural single-crystal diamond (100 GPa) is adopted,20 and D is found to be about 680.3 GPa·Å5. Therefore, the Knoop hardness of the ideal single crystal can be expressed as ⎛ TS M ⎞2/3 ij ⎟ Nv Hk (GPa) = 680.3⎜⎜ 3 ⎟ ⎝ R ij ⎠
1/ n ⎧ n ⎡ ⎛ TS M ⎞2/3 ⎤⎫ ⎪ ⎪ ij , μ ⎥ ⎢ ⎟ Nv μ ⎬ = ⎨∏ ⎢680.3⎜⎜ ⎥⎪ R ij , μ3 ⎟⎠ ⎪ μ=1 ⎣ ⎝ ⎦⎭ ⎩
(6)
Hkμ
where is the hardness of the binary system composed of μtype bonds, Rij,μ and Sij,μ are the bond length and bond valence of μ-type bonds, respectively. The chemical bond density of μtype bond Nvμ can be expressed as Nvμ = Nμ/Vμ,29 where Nμ is the bond multiplicity of μ-type bonds in the unit cell, Vμ is the volume of the binary system composed of μ-type bonds per cubic angstroms and can be calculated as Vμ = VNμRij,μ3/ (∑nμ=1NμRij,μ3). Here we calculate the hardness of α-Al2O3 to show how our hardness model works for multibond crystals. In α-Al2O3, the hexagonal unit cell contains 12 six-coordinated Al atoms and 18 four-coordinated O atoms, while there are two
(5)
To confirm the practicability of eq 5, the Knoop hardness values of typical binary covalent and ionic crystals are also calculated. Throughout our work, information on the chemical C
DOI: 10.1021/acs.inorgchem.6b01729 Inorg. Chem. XXXX, XXX, XXX−XXX
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Inorganic Chemistry Table 2. Hardness and Parameters Related to the Hardness Calculation of Typical Multicomponent Compoundsa crystal α-Al2O3 Stishovite TiO2 c-Zr3N4 β-Si3N4
a
Rij,μ (Å)
R0 (Å)
Sij,μ (vu)
f ij
Nvμ (Å−3)
Hkμ (GPa)
Hk,calc (GPa)
Hexp (GPa)
1.813 1.902 1.724 1.738 1.961 1.978 2.192 2.488 1.633 1.730 1.764 1.844
1.651 1.651 1.64 1.64 1.815 1.815 2.11 2.11 1.724 1.724 1.724 1.724
0.65 0.51 0.80 0.77 0.67 0.64 0.80 0.36 1.28 0.98 0.90 0.72
0.26 0.18 0.37 0.35 0.26 0.25 0.43 0.13 0.81 0.53 0.45 0.32
0.33 0.29 0.28 0.28 0.19 0.19 0.39 0.26 0.20 0.17 0.16 0.14
26.9 17.8 32.4 30.5 14.2 12.9 31.8 7.1 44.9 25.4 20.7 13.3
21.9
21
31.7
32
13.8
12b
15.0
17.5
23.7
21
Unless noted, all experimental Knoop hardness values are taken from refs 14, 25, and 26. bVicker hardness from ref 30.
Figure 2. Bonding environments of the tetrahedral and octahedral C atoms as observed in γ-C3N4. The bond valences (vu) are shown in the lower part of each figure and bond lengths (Å) in the upper part.
types of Al−O bonds with different bond lengths. The bond multiplicities of the Al1−O and Al2−O bonds are 36 and 36, respectively. Hkμ for the Al1−O and Al2−O components determined from eq 5 are 26.9 and 17.8 GPa, respectively. Using eq 6, the hardness of α-Al2O3 is calculated to be 21.9 GPa, in good agreement with the experimental value of 21 GPa. For illustration in Table 2, eq 6 is applied to calculate the hardness values for more typical multicomponent compounds such as Stishovite, TiO2, c-Zr3N4 and β-Si3N4. The calculated values agree well with the experimental data. The nitride spinel materials γ-A3N4 (A = C, Si, Ge) are widely studied as considerable superhard materials. The spineltype structure consists of an almost ideal cubic-close-packed arrangement of anions, in which 8 tetrahedrally coordinated cations and 16 octahedrally coordinated cations are distributed in the unit cell.31 Therefore, the nitride spinel system can be divided into two binary systems (n = 2). As illustrated in Figure 2, the sketches of the local tetrahedral and octahedral environments around C atoms in the normal spinel structures for γ-C3N4 are shown. The coordination environments all adopt highly symmetrical structures. The tetrahedral and octahedral C atoms have observed bond valence sums of 3.56 and 3.84 vu,
respectively, which are close to the expected value of 4.00 vu. From Table 3, it is shown that the calculated hardness values for γ-C3N4, γ-Si3N4, and γ-Ge3N4 are in reasonable agreement with Gao’s, Šimůnek’s, and Li’s calculations and also close to the available experimental data. It is revealed that the hardness of the tetrahedral binary system is much higher than that of the octahedral binary system for the nitride spinel materials. The octahedral binary system has higher coordination number of the cations, which can contribute to an increase in the packing efficiency of the crystal structures. However, simultaneously it results in an increase in the average bond length and a decrease in both the bond valence and bond density, which finally leads to a decrease in the hardness. Consequently, the origin of the superhardness of γ-A3N4 is mainly related to its tetrahedral chemical bonding. In addition, the hardness of typical oxide spinel materials is also predicted using our empirical method, as shown in Table 3. The calculated hardness values for γMgAl2O4 and γ-ZnAl2O4 are compared favorably with the experimental ones. As can see from the calculated data, the hardness values of both γ-MgX2O4 and γ-ZnX2O4 (X = Al, Ga, In) decrease in the order of Al, Ga, and In, partly because of the D
DOI: 10.1021/acs.inorgchem.6b01729 Inorg. Chem. XXXX, XXX, XXX−XXX
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Table 3. Hardness and Parameters Related to the Hardness Calculation of Predicted Nitride Spinels and Typical Oxide Spinels crystal γ-C3N4 γ-C3N4g γ-Si3N4 γ-Si3N4g γ-Ge3N4 γ-Ge3N4g γ-MgAl2O4 γ-MgGa2O4 γ-MgIn2O4 γ-ZnAl2O4 γ-ZnGa2O4 γ-ZnIn2O4h
Rij,μ (Å)
R0 (Å)
Sij,μ (vu)
f ij
Nvμ (Å−3)
Hkμ (GPa)
Hk,calc (GPa)
1.512 1.636 1.510 1.630 1.809 1.885 1.750 1.860 1.919 2.012 1.850 1.950 1.917 1.930 1.954 1.978 2.067 2.221 1.880 1.955 1.973 1.990 2.002 2.161
1.47 1.47 1.47 1.47 1.77 1.77 1.77 1.77 1.88 1.88 1.88 1.88 1.693 1.651 1.693 1.73 1.693 1.902 1.704 1.651 1.704 1.73 1.704 1.902
0.89 0.64 0.90 0.65 0.90 0.73 1.06 0.78 0.90 0.70 1.08 0.83 0.55 0.47 0.49 0.51 0.36 0.42 0.62 0.44 0.48 0.50 0.45 0.50
0.49 0.28 0.50 0.28 0.45 0.32 0.59 0.36 0.41 0.28 0.56 0.36 0.20 0.16 0.17 0.17 0.10 0.16 0.23 0.14 0.16 0.16 0.14 0.21
0.51 0.41 0.51 0.41 0.30 0.26 0.33 0.28 0.25 0.22 0.28 0.24 0.25 0.24 0.23 0.22 0.20 0.16 0.26 0.23 0.23 0.22 0.22 0.18
93.7 44.6 97.1 44.6 36.7 23.0 51.6 28.6 25.5 16.5 38.1 21.2 15.3 13.1 12.5 12.0 6.8 6.5 19.5 11.0 11.0 12.0 10.5 9.8
53.7 54.2 25.9 33.1 18.4 24.6 13.6
Hexp (GPa) 56.7a, 70.1b, 71.1c 30d, 30.9a, 27.4b, 29.1c 24.3a, 19.1b, 22.2c 12e
12.1 6.6 12.6
10.5f
11.7 10.0
a Vicker hardness from ref 12. bVicker hardness from ref 13. cKnoop hardness from ref 14. dVicker hardness from ref 32. eKnoop hardness from ref 33. fKnoop hardness from ref 34. gCalculated crystal structure data are taken from ref 35. hCalculated crystal structure data are taken from ref 36.
Table 4. Hardness and Parameters Related to the Hardness Calculation of a Typical Ternary B−C−N System crystal e
c-BC2N-1
c-BC2N-2e
β-BC2Nf
bond type
Rij,μ (Å)
R0 (Å)
Sij,μ (vu)
f ij
Nvμ (Å−3)
Hkμ (GPa)
Hk,calc (GPa)
C−C B−N B−C C−N C−C B−N B−C C−N C−C B−N B−C C−N
1.577 1.577 1.577 1.577 1.577 1.577 1.577 1.577 1.515 1.562 1.573 1.564
1.54 1.482 1.58 1.47 1.54 1.482 1.58 1.47 1.54 1.482 1.58 1.47
0.90 0.77 1.01 0.75 0.90 0.77 1.01 0.75 1.07 0.81 1.02 0.78
0.50 0.38 0.61 0.36 0.50 0.38 0.61 0.36 0.67 0.41 0.62 0.39
0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.37 0.34 0.33 0.34
56.1 47.1 65.1 44.9 56.1 47.1 65.1 44.9 85.6 53.2 65.1 50.9
52.7
52.1
Hexp (GPa) 55a, 71.9b, 63.0c, 54.1c
62.3 78d, 76.5b
a Knoop hardness from ref 37. bVicker hardness from ref 13. cKnoop hardness from ref 14. dVicker hardness from ref 12. eCalculated crystal structure data are taken from ref 37. fCalculated crystal structure data are taken from ref 12.
different ratios of chemical bonds have been proposed. In cBC2N-1 with a zinc blende structure, the ratio of the chemical bonds N(C−C)/N(B−N)/N(B−C)/N(C−N) is 1:1:1:1, and the other c-BC2N-2 with superlattices is 3:3:1:1.37,38 As shown in Table 4, the hardness values have been calculated for these two structures of c-BC2N, which are all very close to the experimental Knoop hardness values. These results also demonstrate that B−C bonds are the strongest, while C−N bonds are the weakest among all constituted bonds of c-BC2N. In the theoretical studies of ternary B−C−N systems, a β-phase crystal of the BC2N ternary compound derived from the optimization is also considered reasonable.12 As shown in Figure 3, in the tetragonal unit cell, there are four types of chemical bonds, C−C, B−N, B−C, and C−N, with different bond lengths, and the number of each type of chemical bond is
increase of the bond length and the decrease of the bond density. The ternary B−C−N system is considered as a potential ultrahard material with thermal and chemical stability, which is due to its structural similarity to those of diamond and c-BN. Recently, much effort has been devoted to the theoretical and experimental studies of these kinds of superhard ternary compounds. c-BC2N has been synthesized by Solozhenko et al.,37 which has the experimental Knoop hardness value of 55 GPa. In c-BC2N, there are eight atoms in the tetragonal unit cell, and all atoms are tetrahedral bonded. It is suggested that four types of chemical bonds exist, including the C−C, B−N, B−C, and C−N bonds, and all interatomic distances are the same. However, there have been controversial atom positions for c-BC2N available so far. Two possible structures with E
DOI: 10.1021/acs.inorgchem.6b01729 Inorg. Chem. XXXX, XXX, XXX−XXX
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all higher than 40 GPa. On the other hand, heavy transition metals such as Os, Re, and Ir have a relatively high number of unpaired valence d electrons,15 which can distribute higher bond valences to the chemical bonds and allow them to make directional bonds with stronger resistant force. Furthermore, transition metals are able to form coordination compounds with relatively more coordination numbers, which contributes to higher bond density. It is generally believed that a new family of transition-metal compounds formed by heavy transition metals and light elements are potential superhard materials. Light-element-constituting bonds have shorter bond lengths and higher bond valences, which can play a significant role in higher hardness. Meanwhile, it is expected to reduce the cost of production by substituting some less-expensive light elements for more expensive transition metals. Recently, many of these compounds such as OsB27 and ReB242 have been successfully synthesized, and their hardness values have been proven to be comparable with those of traditional superhard materials. Because crystal structures and the nature of the chemical bonding can be well described in a simple mathematical form based on the bond valence model, numerous superhard materials with more complex structure and composition are expected to be purposely designed using the empirical hardness method.
Figure 3. Crystal structure of β-BC2N and bonding environments of the B, C, and N atoms.
four. The hardness of β-BC2N is calculated and also listed in Table 4. From the above-mentioned calculations, it can be found that the calculated hardness value for β-BC2N is higher than that of c-BC2N. In all chemical bonds, the C−C bonds in β-BC2N are the strongest because of their shortest bond length and highest bond valence and bond density.
4. CONCLUSIONS In summary, a new empirical hardness method is presented to estimate the hardness of covalent and ionic crystals by introducing the resistant force of the chemical bond based on the bond valence model. The calculated hardness values of single and multibond crystals are in good agreement with their available experiment results. It is revealed that short bonds with a high degree of bond valence and bond density are critical to superhard materials. On the basis of the advantages of the bond valence model, it is demonstrated that the empirical method can give simple quantitative predictions of the hardness for crystals with complex structure. Further, combining the bond valence model with its insight into crystal structures and the nature of the chemical bonding, our present empirical hardness method could show a powerful predictive ability to design novel superhard materials.
3. DISCUSSION In our method, input variables can all be directly deduced from the crystal structure by the bond valence model, which can provide a general and convenient tool for hardness estimation. The resistant force of the chemical bond related to the hardness is evaluated empirically in terms of the bond valence based on the bond valence model. The bond valence can provide a description of both covalent and ionic bonding,15 in which the bonding electron pairs are directly distributed within the bonds by a covalent view, and the bonding electrons reside on the anion by an ionic view. Because the bond valence model has been widely validated in a large number of well-determined structures,39 it can make a more quantitative prediction of the nature of the chemical bonding. As a result, our empirical hardness method on the basis of the bond valence model can fit a wide range of covalent and ionic crystals and is expected to be accurate because of the incorporation of the well-described chemical bonding. While more physical models provide pictures to describe the chemical bonding of crystals, the bond valence model has an advantage of reducing the rules of chemistry to the simplest mathematical form. In this form, the model can provide a simple and robust account of the chemical bonding of complex systems; thus, our empirical hardness method has the ability to give more simple predictions of the hardness for crystals with complex structure. According to our calculation from eq 5, it is noted that the hardness of a material is mainly determined by the bond length, bond valence, and bond density. Further, our empirical hardness method can give useful clues for finding new superhard materials. This indicates that short bonds with a high degree of bond valence and bond density are crucial for high hardness. On the one hand, light elements such as B, C, N, and O have the ability to form short and strong bonds with a high degree of bond valence. Thus, a number of superhard materials have been found in the B/C/N/O system. In particular, new superhard materials such as c-BC2N,37 c-BC5,40 and B6O41 have been synthesized, and their hardness values are
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ASSOCIATED CONTENT
S Supporting Information *
The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.inorgchem.6b01729. Details of bond covalent component calculation and ICSD number of taken crystal structure and parameter values related to the hardness calculation (PDF)
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS This work was supported by the National Natural Science Foundation of China (Grants 51472195, 51502219, and 51521001) and the Ministry of Science and Technology of F
DOI: 10.1021/acs.inorgchem.6b01729 Inorg. Chem. XXXX, XXX, XXX−XXX
Article
Inorganic Chemistry
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China (Grant 2015DFR50650). The authors thank Prof. Ian David Brown for his assistance with this research.
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DOI: 10.1021/acs.inorgchem.6b01729 Inorg. Chem. XXXX, XXX, XXX−XXX
