Anisotropic hardness prediction of crystalline hard materials from the electronegativity

Available online at www.sciencedirect.com

Acta Materialia 60 (2012) 35–42 www.elsevier.com/locate/actamat

Anisotropic hardness prediction of crystalline hard materials from the electronegativity Keyan Li a, Peng Yang a, Dongfeng Xue a,b,⇑ a

State Key Laboratory of Fine Chemicals, School of Chemical Engineering, Dalian University of Technology, Dalian 116024, People’s Republic of China b State Key Laboratory of Rare Earth Resource Utilization, Changchun Institute of Applied Chemistry, Chinese Academy of Sciences, Changchun 130022, People’s Republic of China Received 12 August 2011; received in revised form 6 September 2011; accepted 7 September 2011 Available online 26 October 2011

Abstract We have presented an efficient method to predict the anisotropic hardness of crystalline materials along different crystallographic directions or on different crystallographic planes in terms of electronegativity. Bond stretching and bending strengths, respectively, are proposed to characterize the ability of a chemical bond to resist stretching and bending deformation, which are the main microscopic deformations in single crystals when measuring indentation hardness. Good agreement between the calculated and experimental values of anisotropic hardness for a large range of crystalline materials has been achieved, including sphalerite, wurtzite and rocksalt structured materials, as well as oxides (e.g. a-SiO2 and LaGaO3) and graphite. The anisotropic hardness values of other important materials, such as B12 analogs, group IVA nitrides, tungsten carbide structured materials, and transition metal di- and tetra-borides, were quantitatively predicted. We found that materials with the same crystal structure have the same or similar hardness anisotropy. Furthermore, the more orderly bond arrangement in single crystals and the greater bond ionicity often result in greater hardness anisotropy. This work shines a light on the nature of hardness and on studies of the anisotropy of other macroscopic properties of crystalline materials. Ó 2011 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Hardness; Anisotropy; Crystalline materials; Electronegativity

1. Introduction Hardness is a complex property related to the extent to which solids resist both elastic and plastic deformation [1]. The force of the indenter is applied perpendicularly to the surface of a sample when measuring hardness, and will then be diverted sideways, thus the sample is subjected to a combination of compression, shear, and tension [2,3]. Much work [1–9] has been carried out to determine the nature of hardness, motivated by the industrial application of hard materials. Liu et al. [1] and Sung et al. [2] used the bulk modulus as a surrogate of hardness. Teter [4] and ⇑ Corresponding author at: State Key Laboratory of Rare Earth Resource Utilization, Changchun Institute of Applied Chemistry, Chinese Academy of Sciences, Changchun 130022, People’s Republic of China. Tel.: +86 431 8526 2294; fax: +86 431 8526 2294. E-mail address: [email protected] (D. Xue).

Brazhkin et al. [5] pointed out that the shear modulus can better reflect hardness. Recently several empirical and semi-empirical models have been proposed to estimate the intrinsic hardness in terms of different parameters, including the reference potential [3], energy gap [6], electronegativity [7], and atomization energy [8], using which good agreement was reached between the calculated and experimental hardness values. Technologically relevant materials often possess anisotropic physical properties, such as optical properties, elastic properties, and hardness. Hardness anisotropy is often observed in various single crystals when indentation tests are conducted along different crystallographic directions or on different crystallographic planes [9–12], which is closely related to the practical application of general materials. For single crystal growth and scratch-resistant coating preparation hardness anisotropy prediction can

1359-6454/$36.00 Ó 2011 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2011.09.011

K. Li et al. / Acta Materialia 60 (2012) 35–42

provide useful guidance as to the crystallization process to form the hardest planes and to prevent the formation of soft planes. It can also supply important clues or directions to cut and polish gemstones/jewels. However, anisotropic hardness has not yet been well understood due to the indentation size effect [12,13] and the dependence of hardness on the applied load, and the nature of the hardness anisotropy of materials needs further investigation. Recently Simunek [9] proposed an expression for hardness anisotropy, and calculated the hardnesses of ReB2 and OsB2 along the c-axis, which agree well with experimental measurements. Consequently, a general theoretical model for anisotropic hardness is highly desirable. Electronegativity is an important parameter in chemistry, physics and materials science which provides primary information about the microscopic bonding between atoms. In the last decade we have proposed two sets of electronegativity scales for ionic [14] and covalent [15] crystals, which have been used for correlation with various physical properties such as electronic polarizability [16], bulk modulus [17], and hardness [7,15]. In this work we established a model to predict the anisotropic hardness of crystalline materials from the viewpoint of electronegativity. Bond stretching and bending strength were defined to describe the total strength of a bond, which is correlated with the anisotropic hardness combined with the bond density and the angles between the bond axes and crystallographic directions. The anisotropic hardness values of materials with various compositions and structures were quantitatively calculated and the relationships of hardness anisotropy to crystal structure and bond characteristics revealed. 2. Methods From the viewpoint of microscopic chemical bonds the hardness of crystals is determined by the nature of their constituent chemical bonds. When pressed by an indenter the chemical bonds in crystals suffer from both stretching and bending deformation [18]. The hardness of a material H is directly proportional to the density and strength of chemical bonds [3,6,15]. The total strength of a chemical bond dt can be described by both the stretching strength ds and bending strength db. vA vB ; d 0:5 v v db ¼ 33:5 A 2 B e9:7f w ; d

ds ¼ 26:9

jvA  vB j ; z2 ðvA þ vB Þ

dt ¼

db ds : db cos2 u þ ds sin2 u

ð2Þ

ð3Þ

ð4Þ

From Eq. (4) we can find dt = ds when u = 0, which indicates that bond A–B undergoes only stretching deformation. Analogously, dt = db when u = 90°, which indicates that bond A–B undergoes only bending deformation. To determine the specific relationship between H, dt and q, the average values of cosu and sinu are deduced (equal to 1/2 and p/4 respectively, see Appendix B) and introduced into Eq. (4), which gives the average dt. By plotting the “isotropic” hardness of materials Hiso against dt and q (Fig. 1), we obtain an expression for hardness, H ¼ adbt qc ;

ð5Þ

where constants a, b and c are determined to be 1.5, 1 and 0.5 for Vickers hardness and 1.3, 1 and 0.5 for Knoop hardness. Further, the anisotropic hardness Hani can be calculated using Eq. (5) by substituting individual values for the average cosu and sinu. For all crystalline materials, including simple substances, as well as binary and multi-composition compounds, the bonds in the crystallographic frame can be classified into different subsystems, each of which contains only one type of bond (the same constituent atoms, same bond length and same u). For a crystal with j subsystems the anisotropic hardness can be obtained using the following equations

ð1Þ

where vA and vB are the electronegativities of atoms A and B, respectively (the electronegativity values, if unavailable in Li et al. [15], were calculated using Eq. (2) in Li et al. [15] using the radius data from Pauling [19,20]), d is the bond length, and fw is the strength weakening factor, which is proportional to the ionicity of the bond, fw ¼

where z is the charge number transferred between bonded atoms A and B. Eq. (3) indicates that a positive (or negative) effective charge attracts (or repulses) the bonding electrons, which can decrease the actual ionicity and, hence, decrease the weakening factor fw. All constants in Eqs. (1) and (2) are determined by plotting ds and db against the bond stretching force constant a and bending force constant b [21,22], respectively. By analyzing the interaction between a bond and the load along a direction at an angle of u between them (Appendix A), dt is deduced as

Hardness (GPa)

36

2

100

Hv

R =0.993

80

Hk

R =0.991

2

60 40 20 0 0

10

20

30

40

50

60

70

b c

δ ρ

Fig. 1. Plot of ‘isotropic’ hardness of materials against the total strength dt and density of bonds q. The dashed and solid lines correspond to the Knoop and Vickers hardness, respectively.

K. Li et al. / Acta Materialia 60 (2012) 35–42

" H ani ¼

j Y

 j #1 P ni H ni i

i¼1

ð6Þ

;

i¼1

H i ¼ adbtðiÞ qci ;

ð7Þ

qi ¼ ni =V i ;

ð8Þ

V i ¼ ni d 3i V

, j X

nk d 3k ;

ð9Þ

k¼1

where ni is the bond number in the ith subsystem, Vi the volume of the ith subsystem, and di the bond length of the ith subsystem. Taking the [0 0 1] direction of BC2N as an example, the bonds are classified into four subsystems, ˚ and an ani.e. the C–N bond with a bond length of 1.552 A gle of 54.54° between its axis and the [0 0 1] direction, the ˚ and angle of 56.03°, C–C bond with a length of 1.509 A ˚ and angle of the C–B bond with a length of 1.567 A ˚ and an53.79° and the B–N bond with a length of 1.552 A gle of 53.72°. The hardness of each subsystem Hi can be

37

calculated using Eqs. (7)–(9), then the hardness of BC2N along the [0 0 1] direction can be calculated as the geometric average of the hardness of the above four subsystems using Eq. (6). 3. Results and discussion Using Eqs. (6)–(9) we calculated the anisotropic hardness of materials with different compositions and structures, and the results are listed in Tables 1–5. Table 1 shows the calculated anisotropic hardness of tetrahedrally coordinated crystalline materials. The calculated results are consistent with the available experimental data for bSiC, ZnS, a-SiC and BeO [2,10]. Experiments by Zhang et al. [24] showed that the hardness values of the (1 1 0) and (1 1 1) planes are 1.02 and 1.21 GPa for a Zn0.04 Cd0.96Te crystal, which agree well with our calculated hardness values of 0.97 and 1.26 GPa, respectively. It is known that the (1 1 1) face is harder than the (0 0 1) face for diamond [2], which agrees with our calculations of 89.7 GPa

Table 1 Calculated anisotropic hardness values of tetrahedrally coordinated crystalline materials. Crystala

Hv

[1 0 0]

[1 1 0]

[1 1 1]

Diamond Si Ge b-SiC c-BN BP BAs AlN GaN ZnS Zn0.04Cd0.96Te

88.5 9.9 8.2 24.2b (26.7c) 56.6 20.6 17.0 19.0 17.4 2.1 (1.5d) 0.87

89.3 10.4 8.5 24.9b 58.2 21.4 17.6 20.0 18.3 2.3 (1.8d) 0.97 (1.02e)

89.7 10.7 8.9 25.4b (28.2c) 59.4 22.0 18.1 21.1 19.4 2.9 (2.0d) 1.26 (1.21e)

½2 1 1 0

½0 1 1 0

[0 0 0 1]

Lonsdaleite w-BN a-SiC BeO

89.4 59.9 28.7 (27.4d) 12.4

89.5 60.1 28.9 (27.3d) 13 (9.15d)

89.9 61.1 29.4 (32.7d) 14.5 (13.7d)

BC2N

[1 0 0] 73.8

[0 1 0] 73.1

[0 0 1] 72.6

cal

(different directions)

fw

Hv

0 0 0 0.013 0.039 0.016 0.012 0.056 0.056 0.145

85f 12g 9.7h 34g 47f, 63i 26h 19.9h 18j 15.1k 2.7h

0 0.039 0.013 0.164

60–70l, 97m 50–78i 21–29l 10–15l

13o

76n

55n

bulk

Hk

bulk

88.2c 14c 8.8c 24.8c 48c 32c 12.25o 1.7c

Hv cal is, unless labeled otherwise, the calculated Vickers hardness. The available experimental anisotropic hardness values are listed in parentheses. Hv bulk and Hk bulk are the “isotropic” Vickers and Knoop hardness of bulk materials measured experimentally or calculated by other researchers, respectively. a The structural data are from FIZ Karlsruhe [23]. b Knoop scale. c Sung and Sung [2]. d Westbrook and Jorgensen [10]. e Zhang et al. [24]. f Zhao et al. [25]. g Gao et al. [6]. h Calculated value, Simunek and Vackar [3]. i Andrievski [26]. j Yonenaga et al. [27]. k Drory et al. [28]. l Brazhkin et al. [5]. m Calculated value, Guo et al. [29]. n Solozhenko et al. [30]. o Lide [31].

38

K. Li et al. / Acta Materialia 60 (2012) 35–42

Table 2 Calculated anisotropic compounds. Crystala

MgO TiC ZrC HfC VC NbC TaC RuCb TiN ZrN HfN VN NbN TaN PbS

RuC WC ReC OsC IrC OsBc RuBc a

b c d e f g h i j k l m n

Hv

cal

hardness

values

(different directions)

[1 0 0]

[1 1 0]

[1 1 1]

6.6 (7.3d) 23.7 17.1 17.5 28.9 21.1 21.3 25.8 18.6 13.2 14.0 20.7 17.0 17.7 1.44

4.2 (6.9d) 21.6 15.2 15.5 26.6 19.0 19.2 23.5 15.8 10.8 11.6 17.8 14.3 15.0 0.81

4 (6.5d) 21.2 14.9 15.2 26.1 18.7 18.8 23.1 15.4 10.5 11.3 17.3 13.9 14.6 0.78

½2  1 1 0

½0 1  1 0

[0 0 0 1]

21.5 21.3 22.1 21.2 21.4 9.5 10.0

21.5 21.4 22.1 21.3 21.5 9.5 10.0

23.6 23.5 24.3 23.3 22.0 10.6 11.1

of fw 0.177 0.016 0.018 0.018 0.014 0.016 0.016 0.013 0.056 0.060 0.059 0.055 0.055 0.055 0.164

0.013 0.018 0.017 0.014 0.013 0.014 0.013

six-fold Hv

coordinated

bulk

18–32e 20.9f

Hk

bulk

3.7n 24.7n 17.9f

29g 18g 18f 25.2h 18e 15i 15.9i 15i 14i, 17g 19j

23.6k 24f 29.4l 24.3k, 22.7l 15.9l 10.6m 8.0m

18n n

17.7 15.1n

Table 3 Calculated anisotropic hardness values of main group borides and nitrides. Crystala

Hv cal (different directions) ½2 1 1 0 ½0 1 1 0 [0 0 0 1]

Hv

a-Boron B6O B4C B13C2 B6P B6As a-Si3N4 b-Si3N4 a-C3N4 b-C3N4

32.2 33.6 27.6 33.3 34.0 29.0 27.5 24.1 62.2 62.3

32.1 34.3 27.6 33.4 34.1 29.0 27.6 24.2 62.2 62.4

31.3 32.9 29.4 33.3 34.4 28.9 27.5 24.6 62.5 62.8

34c 35d 27.8–39.5e 41.5c 32.9f 28.6f 32.2g 21.3g

[1 0 0]

[1 1 0]

[1 1 1]

55.8 26.4 20.8 13.2 52.7

54.4 25.5 19.9 12.4 51.4

54.2 25.3 19.8 12.3 51.1

c-C3N4 c-Si3N4 c-Ge3N4 c-Sn3N4 CN2b a

18.8n

The structure data, unless labeled otherwise, are from FIZ Karlsruhe [23]. Structural data are from Abidri et al. [34]. Structural data are from Gu et al. [35]. Khan et al. [36]. Brazhkin et al. [5]. Shackelford et al. [37]. Simunek and Vackar [3]. Calculated value, Abidri et al. [34]. Gou et al. [38]. Yang et al. [39]. Calculated value, Simunek. [40]. Calculated value, Gou et al. [38]. Gu et al. [35]. Lide [31].

for the [1 1 1] direction and 88.5 GPa for the [1 0 0] direction. Both Westbrook et al. [10] and Shaffer [32] showed that the hardness of the (0 0 0 1) face of a-SiC is greater than those of both the ð1 1  2 0Þ and ð1 0  1 0Þ faces, which is consistent with our calculations. According to Brookes [33] materials with the same crystal structure and with common slip systems possess similar anisotropic properties. Our calculations show that materials with the same structure possess the same or similar hardness anisotropy. As shown in Table 1, the [1 1 1] direction is harder than the [1 1 0] direction, which is harder than the [1 0 0] direction for sphalerite or diamond structured materials; the [0 0 0 1] direction is harder than the ½0 1  1 0 direction, which is appreciably harder than the ½2  1 1 0 direction for wurtzite structured materials. Materials with the same crystal structure have the same (or a similar) atomic coordinates and the same (or approximate) bond angle and, therefore, the same (or a similar) u value along the same direction, which

b c d e f g h i j k l

bulk

Hk

bulk

28l

21g

60.4h

56.7i 30–43j 28k 11k 58i

The structural data, unless labeled otherwise, are from FIZ Karlsruhe [23]. Structural data are from Weihrich et al. [42]. Gao et al. [43]. Brazhkin et al. [5]. Nino et al. [44]. Calculated value, Gao et al. [43]. Sung and Sung [2]. Calculated value, Gao [45]. Calculated values, Gao et al. [46]. Zerr et al. [47]. Shemkunas et al. [48]. Lide [31].

is the decisive factor for anisotropy of hardness. Besides, our calculations show that orthorhombic BC2N has almost equivalent hardnesses along the a-, b- and c-axes. The anisotropy degree of hardness increases with increasing fw, which varies synchronously with the ionicity of bonds. For example, for diamond fw is 0 and the ratio of H[111] to H[100] is 1.01, while for ZnS fw is 0.145 and the corresponding ratio is 1.38. Table 2 shows the calculated anisotropic hardness of sixfold coordinated materials, including rocksalt and tungsten carbide structures. Khan et al. [36] showed that the (1 0 0) face has a greater hardness than the (1 1 0) face which is harder than the (1 1 1) face for MgO, which is consistent with our calculations. Young et al. showed an experimental hardness order of (1 0 0) > (1 1 0) > (1 1 1) for PbS (galena) [10,41], which is in good agreement with our present calculations. According to Table 2, the [1 0 0] direction is harder than the [1 1 0] direction which is harder than the [1 1 1] direction for rocksalt structured materials; the [0 0 0 1] direction is harder than the ½2 1 1 0 direction, which has almost the same hardness as the ½0 1 1 0 direction for tungsten carbide structured materials. For the rocksalt structure the ratio of H[100] to H[111] increases from 1.12 for RuC to 1.21 for TiN to 1.65 for MgO as fw increases from 0.013 to 0.056 to 0.177, respectively. However, such a phenomenon

K. Li et al. / Acta Materialia 60 (2012) 35–42

39

Table 4 Calculated anisotropic hardness values of transition metal di- and tetra-borides. Crystala

RuB2 OsB2

WB4 ReB2 TcB2 MnB2 WB2 TiB2 ZrB2 HfB2 VB2 NbB2 TaB2 CrB2 MgB2 MoB2 a b c d e f g h i

Hv

cal

(different directions)

Hv

[1 0 0]

[0 1 0]

[0 0 1]

26.3 25.8

25.1 24.6

30.6 30.0

½2 1 1 0

½0 1 1 0

[0 0 0 1]

30.9 24.5 (\c:27b) 25.2 30.5 25.2 19.2 14.2 14.9 27.0 22.3 22.8 15.8 6.7 24.6

30.8 24.5 (\c:27b) 25.2 30.6 25.3 19.2 14.3 15.0 27.1 22.4 22.9 15.9 7.6 24.7

32.2 31.2(31b) 32.1 38.1 29.1 20.6 15.8 16.5 28.3 24.0 24.1 16.9 7.8 28.3

bulk

Hk

bulk

19.9c 23.5c

31.8–46.2c 20.7–31.1d 37e 43.9e 27.7c 27f 20f 20f 26.7f 24.4f 29.5f, 17g 20.7f 4.3–10.3h 21.7–24.7f

28.5i 15.6i

26i

The structure data are from FIZ Karlsruhe [23]. Chung et al. [13]. Gu et al. [35]. Locci et al. [49]. Calculated value, Aydin and Simsek [50]. Okada et al. [51]. Microhardness, Shackelford and Alexander [37]. Kolemen [52]. Lide [31].

is not significant for tungsten carbide structured materials due to their almost equivalent fw values. The bonds in materials with a rocksalt structure are highly regularly arranged in three directions, namely along the a-, b- and c-axes, which results in a relatively large difference between dt in different directions, leading to a large difference of hardness in different directions. We applied our model to main group nitrides and borides, and the results are tabulated in Table 3. Our calculations show a greater hardness along the c-axis than perpendicular to the c-axis for both a- and b-Si3N4, which agrees with the experimental results [2]. As shown in Table 3, the hardness anisotropies of all materials are low, which may be caused by the disorderly arrangement of chemical bonds in the crystallographic frame. The bonds in the a and b phases of group IVA nitrides and B12 cage-like icosahedrons are variously directional, which results in a wide range of u values and, hence, approximate dt and approximate hardness values. However, the c phase of group IVA nitrides has relatively orderly arranged bonds compared with the a and b phases, and thus relatively high anisotropies of hardness were observed. The atomic coordination and arrangement of the a and b phases of group IVA nitrides are the same or similar, and their bond lengths are almost equivalent, which means they have comparable dt values in the same direction and, hence, approximate hardness values in the same direction. In the search for superhard materials much attention has been paid to transition metal di- and tetra-borides.

The calculated results of anisotropic hardness of these materials are presented in Table 4. Chung et al. [13] showed that the average hardness along the c-axis of ReB2 (31 GPa) is greater than that of directions perpendicular to the c-axis (27 GPa), which agrees well with our calculations of 31.2 GPa for the c-axis and 24.5 GPa for both the ½2 1 1 0 and ½0 1 1 0 directions perpendicular to the c-axis. Hebbache et al. [53] indicated that the hardness of OsB2 is extremely high, in particular along the c-axis. Herein, the greatest hardness from our calculations was observed along the c-axis in OsB2. Other borides also have the greatest hardness along the c-axis. To further validate our present model we investigated the hardness anisotropy of some oxides and graphite, and the calculated results are listed in Table 5. According to Table 5 good agreements between the calculated and experimental values of anisotropic hardness for Al2O3, a-SiO2, YPO4 and LaGaO3 [10,33,54,55] were obtained. Our calculated results for stishovite show that the hardness value along the c-axis is higher than those along both the [1 1 0] and [1 0 0] directions, which is consistent with the experimental measurements [11]. Graphite, which has the highest hardness along the c-axis [9], is composed of monolayer graphenes through weak Van der Waals interactions. We consider that such interactions involve p electrons of C atoms (Fig. 2). One-third of the p electrons of each C2 atom bond to C1 atom plus one p electron of the central C1 atom as a singleton (two bonding electrons) bonds to another one of the neighboring monolayer. The bonding

40

K. Li et al. / Acta Materialia 60 (2012) 35–42

Table 5 Calculated anisotropic hardness values of some oxides and graphite. Crystala

Hv cal (different directions or planes) ½2 1 1 0 ½0 1 1 0

[0 0 0 1]

Al2O3 Graphite

11.1b 0.19b

10.9b(16c) 0.19b

10.5b(13.5c) 0.95b

a-SiO2

ð1 0 1 0Þ 12.6(11.4d)

ð1 0 1 1Þ 12.7(12.4d)

ð0 1 1 1Þ 12.9(12.5d)

c-SiO2 TiO2 SnO2 RuO2 ZrO2 YPO4 LaGaO3

(1 0 0) 22.7(\c:26.2e) 13.7 10.3 15.9 11.1 9.8(6.8–7.9f) 6.2(8g)

(1 1 0) 25.9(\c:26.2e) 15.3 11.6 17.4 10.0 8.5 5.8

(0 0 1) 26.2(31.8e) 15.4 12.0 18.6 9.8 10.2(8–9.8f) 5.8(7g)

HfO2

(1 0 0) 10.3

(0 1 0) 10.3

(0 0 1) 10.4

9.9n

Y2O3

(1 0 0) 4.2

(1 1 0) 4.7

(1 1 1) 5.6

7.5j

a b c d e f g h i j k l m n

Hv

bulk

15h

Hk

bulk

11.67k 0.12l 8.2m

17–33i 12j 11.1j 20j 12m

The structure data are from FIZ Karlsruhe [23]. Knoop values. Brookes et al. [33]. Westbrook and Jorgensen [10]. Brazhkin et al. [11]. Mogilevsky and Parthasarathy [54]. Pathak et al. [55]. Krell [56]. Brazhkin et al. [5]. Gao [45]. Load independent hardness, Kaji et al. [57]. Sung and Sung [2]. Lide [31]. Okutomi et al. [58].

Fig. 2. Scheme of bonding between monolayer graphenes in graphite.

radius of the singleton is substituted by the sum of the radius of the C2 atom and the distance between the C1 and C2 atoms. Consequently, we obtain a hardness of 0.95 GPa for the c-axis and 0.19 GPa for the ½2  1 1 0 and  ½0 1 1 0 directions, which agrees well with the experimental

“isotropic” value of 0.12 GPa [2]. In this work our calculated results on the sequencing of different facets in a specified material consist with the work of other researchers for a large range of crystalline materials. However, for some compounds, such as stishovite and corundum, the calculated values still deviated from the work of other researchers. For example, although the hardness of stishovite along the c-axis (26.2 GPa) is in the range of experimental “isotropic” hardness values 17–33 GPa [5], it is underestimated compared with the experimental value of 31.8 GPa along the c-axis. Our method is applicable to compounds with a relatively high covalent character which are readily identified as hard or superhard materials, while the anisotropic hardness of compounds which contain highly ionic bonds, such as NaCl, CaF2 and LiNbO3, cannot yet be accurately estimated. Therefore, the nature of hardness still needs further investigation and related work is necessary to solve these problems. 4. Conclusions On the basis of electronegativity we established theoretical equations to quantitatively calculate the anisotropic hardness of hard materials with different compositions

K. Li et al. / Acta Materialia 60 (2012) 35–42

and structures. Our calculations agree well with the available experimental results. Taking into consideration the experimental errors due to the effects of the diagonal direction of Vickers and Knoop indenters, the indentation load and duration, and impurities in the samples, etc., our model is a powerful tool to predict the anisotropic hardness of crystalline materials. Some conclusions have been drawn. (i) The hardness of materials is dominated by both the stretching and bending strength of the constituent bonds, which correspond to the bulk modulus and shear modulus, respectively. (ii) Materials with the same crystal structure have the same or similar hardness anisotropy. The corresponding bond angles are the same or similar in the same structures, which can lead to the same or similar orders of bond strength in different directions and, hence, the same or similar orders of hardness. (iii) The more orderly the bond arrangement in crystals such as rocksalt and sphalerite structures, in which the bonds are arranged in just three and four directions, respectively, e.g. along the a-, b- and c-axes for the rocksalt structure and four different body diagonal directions for the sphalerite structure, the greater the anisotropy of hardness, and vice versa. (iv) The larger the ionicity of a bond, which varies synchronously with fw, the greater the anisotropy of hardness, and vice versa. The current work provides an important insight into the nature of hardness and the mechanism of anisotropy of various physical properties of crystalline materials. Acknowledgement We gratefully acknowledge financial support from the National Natural Science Foundation of China (Grant Nos. 50872016 and 20973033). Appendix A. Deduction of total bond strength dt

41

Therefore, we can calculate the displacements of atom B occurring under load p along B ! D, Dd, which is equal to the absolute value of BF in magnitude, and along B ! E, Dl, which is equal to the absolute value of BG in magnitude (suppose atom A is fixed and consider ds and db as approximately constant under subtle deformation of bond A–B), p cos u Dd ¼ ; ðA1Þ ds p sin u Dl ¼ : ðA2Þ db According to vector relationships the resultant displacement of atom B, Ds, should be B ! B0 (because ds is always larger than db for any bond the direction of Ds is not superimposed on the direction of load p but deviates from p in a direction perpendicular to bond A–B). The component displacement of Ds along load p should be Dscosc, where c is the angle between Ds and load p, equaling the difference between h and u, where h is the angle between bond A–B and Ds. pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Ds ¼ Dd 2 þ Dl2 ðA3Þ h ¼ arccos

Dd Ds

c ¼ h  u; Ds cos c ¼

ðA4Þ ðA5Þ

p cos2 u p sin2 u þ : ds db

ðA6Þ

So dt, the total strength of bond A–B along p, should be dt ¼

p db ds ¼ : 2 Ds cos c db cos u þ ds sin2 u

ðA7Þ

Appendix B. Mean projection of a bond into all indentation planes and in all indenting directions

The angle between the axis of bond A–B (see Fig. A1) and the direction of indentation load p (N) is denoted u. Then we can obtain the component forces of load p along the bond axis A–B and along B ! E, perpendicular to bond axis A–B and coplanar with bond A–B and load p, i.e. pcosu and psinu, which are equal to the absolute values of BD and BE, respectively, in magnitude. The stretching strength along B ! D of bond A–B and bending strength perpendicular to bond A–B are ds and db, respectively.

The length of bond A–B and the angle between its axis and the normal of a plane of indentation n are denoted as unit 1 and u, respectively. Then we obtain the projection of bond A–B into plane n and in the indenting direction, namely sinu and cosu, respectively. M is a mobile dot on a spherical surface with equation r = 1 in a spherical coordinate system (r, h, u) (Fig. B1). The projection of bond A–B into tangent plane n (the plane

Fig. A1. Force analysis of bond A–B under load p.

Fig. B1. Scheme of mean projection of bond A–B in all planes (or all directions) of indentation.

42

K. Li et al. / Acta Materialia 60 (2012) 35–42

of indentation) at dot M (1, h, u) is sinu, which is regarded as the density of dot M. Then the spherical shell will become uneven in density when the projections of bond A–B at all M dots of the spherical surface are regarded as their corresponding densities. Thus we obtain the mass of the spherical shell as Z p msin ¼ 2p sin2 udu ¼ p2 : ðB1Þ 0

The area of the spherical surface is 4p, so the mean value of the projections of bond A–B into all planes of indentation should be sin u ¼

p2 p ¼ : 4p 4

ðB2Þ

Analogously, the mass of the spherical shell formed by cosu, the projections of bond A–B in all indenting directions, should be Z p mcos ¼ 2p sin uj cos ujdu ¼ 2p: ðB3Þ 0

So the mean value of the projections of bond A–B in all indenting directions should be cos u ¼

2p 1 ¼ : 4p 2

[7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19]

[24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37]

ðB4Þ

References [1] [2] [3] [4] [5] [6]

[20] [21] [22] [23]

Liu AY, Cohen ML. Science 1989;245:841. Sung CM, Sung M. Mater Chem Phys 1996;43:1. Simunek A, Vackar J. Phys Rev Lett 2006;96:085501. Teter DM. Mater Res Soc Bull 1998;23:22. Brazhkin VV, Lyapin AG, Hemley RJ. Philos Mag A 2002;82:231. Gao F, He J, Wu E, Liu S, Yu D, Li D, et al. Phys Rev Lett 2003;91:015502. Li K, Wang X, Zhang F, Xue D. Phys Rev Lett 2008;100:235504. Mukhanov VA, Kurakevych OO, Solozhenko VL. J Superhard Mater 2008;30:368. Simunek A. Phys Rev B 2009;80:060103. Westbrook JH, Jorgensen PJ. Am Mineral 1968;53:1899. Brazhkin VV, Grimsditch M, Guedes I, Bendeliani NA, Dyuzheva TI, Lityagina LM. Phys Usp 2002;45:447. Aguilar-Santillan J. Acta Mater 2008;56:2476. Chung HY, Weinberger MB, Levine JB, Kavner A, Yang JM, Tolbert SH, et al. Science 2007;316:436. Li K, Xue D. J Phys Chem A 2006;110:11332. Li K, Wang X, Xue D. J Phys Chem A 2008;112:7894. Li K, Xue D. Mater Res Bull 2010;45:288. Li K, Ding Z, Xue D. Phys Status Solidi B 2011;248:1227. Gilman JJ, Cumberland RW, Kaner RB. Int J Refract Met Hard Mater 2006;24:1. Pauling L. The nature of the chemical bond. New York: Cornell University Press; 1960.

[38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50] [51] [52] [53] [54] [55] [56] [57] [58]

Pauling L. J Am Chem Soc 1947;69:542. Martin RM. Phys Rev B 1970;1:4005. Wang SQ, Ye HQ. Phys Status Solidi B 2003;240:45. Karlsruhe FIZ. Inorganic crystal structure database. Karlsruhe: Leibniz Institute for Information Infrastructure; 2008. Zhang Z, Gao H, Jie W, Guo D, Kang R, Li Y. Semicond Sci Technol 2008;23:105023. Zhao Y, He DW, Daemen LL, Shen TD, Schwarz RB, Zhu Y, et al. J Mater Res 2002;17:3139. Andrievski RA. Int J Refract Met Hard Mater 2001;19:447. Yonenaga I, Shima T, Sluiter MHF. Jpn J Appl Phys 2002;41:4620. Drory MD, Ager JW, Suski T, Grzegory I, Porowski S. Appl Phys Lett 1996;69:4044. Guo X, Xu B, Liu Z, Yu D, He J, Guo L. Chin Phys Lett 2008;25:2158. Solozhenko VL, Andrault D, Fiquet G, Mezouar M, Rubie DC. Appl Phys Lett 2001;78:1385. Lide DR. CRC handbook of chemistry and physics. 87th ed. Boca Raton, FL: CRC Press; 2007. Shaffer PTB. J Am Ceram Soc 1964;47:466. Brookes CA, O’Neill JB, Redfern BAW. Proc R Soc Lond A 1971;322:73. Abidri B, Rabah M, Rached D, Baltache H, Rached H, Merzoug I, et al. J Phys Chem Solids 2010;71:1780. Gu Q, Krauss G, Steurer W. Adv Mater 2008;20:3620. Khan MY, Brown LM, Chaudhri MM. J Phys D Appl Phys 1992;25:A257. Shackelford JF, Alexander W. CRC materials science and engineering handbook. 3rd ed. Baca Raton, FL: CRC Press; 2001. Gou H, Hou L, Zhang J, Gao F. Appl Phys Lett 2008;92:241901. Yang JF, Yuan ZG, Wang XP, Fang QF. Nanosci Nanotechnol Lett 2011;3:280. Simunek A. Phys Rev B 2007;75:172108. Yong BB, Millman AP. Trans Inst Min Met (London) 1964;7:437. Weihrich R, Eyert V, Matar SF. Chem Phys Lett 2003;373:636. Gao F, Hou L, He Y. J Phys Chem B 2004;108:13069. Nino A, Tanaka A, Sugiyama S, Taimatsu H. Mater Trans 2010;51:1621. Gao F. Phys Rev B 2006;73:132104. Gao F, Xu R, Liu K. Phys Rev B 2005;71:052103. Zerr A, Kempf M, Schwarz M, Kroke E, Goken M, Riedel R. J Am Ceram Soc 2002;85:86. Shemkunas MP, Petuskey WT, Chizmeshya AVG, Leinenweber K, Wolf GH. J Mater Res 2004;19:1392. Locci AM, Licheri R, Orru R, Cao G. Ceram Int 2009;35:397. Aydin S, Simsek M. Phys Rev B 2009;80:134107. Okada S, Atoda T, Higashi I, Takahashi Y. J Mater Sci 1987;22:2993. Kolemen U. J Alloys Compd 2006;425:429. Hebbache M, Stuparevic L, Zivkovic D. Solid State Commun 2006;139:227. Mogilevsky P, Parthasarathy TA. Mater Sci Eng A 2007;454–455:227. Pathak S, Kalidindi SR, Moser B, Klemenz C, Orlovskaya N. J Eur Ceram Soc 2008;28:2039. Krell A. Cryst Res Technol 1980;15:1467. Kaji M, Stevenson ME, Bradt RC. J Am Ceram Soc 2002;85:415. Okutomi M, Kasamatsu M, Tsukamoto K, Shiratori S, Uchiyama F. Appl Phys Lett 1984;44:1132.