4642
Chem. Mater. 2001, 13, 4642-4648
Correlation of Bulk Modulus and the Constituent Element Properties of Binary Intermetallic Compounds Chonghe Li and Ping Wu* Institute of High Performance Computing of Singapore, 89C Science Park Drive, #02-11/12 The Rutherford Singapore Science Park 1, Singapore 118261, Singapore Received April 24, 2001
The correlation between the bulk modulus B and the atomic properties of binary intermetallics (element molar volume and element valence electron density) is studied using literature data of 240 binary intermetallic compounds and alloys. It is found that the bulk modulus of binary systems may be predicted (with an average error limit of (11%) by B ) nws2V, where nws and V are estimates from elemental valence electron density and molar volume, respectively.
1. Introduction Intermetallic compounds are of interest to scientists and engineers because of their superior structural, electrical, chemical, and mechanical stability at high temperature.1-5 They are candidate materials not only for the next generation of efficient turbines and engines,1,2,6 but also for new microelectronic systems.4,5 The elastic constant (bulk modulus or shear modulus) is an important mechanical property in the design and selection of materials. The bulk modulus of intermetallics is less sensitive to thermophysical processing histories or microstructural properties. Because it is a measure of the intrinsic interatomic force, it is often used to estimate interatomic potentials used in computer simulations. Although elastic constants for a number of binary intermetallic compounds and alloys7-14 have been published, many have not, especially for those compounds with high melting temperature. Thus, it is important to predict or estimate the elastic constants for either theoretical study or industrial applications. Grammatickake15-17 and Varotsos18,19 used the thermodynamical model developed by Varotsos and Alexopoulos20 to explain the elastic properties of substitutional alloys:
BdBA[1+x(VB/VA - 1)]/[1 + x(BAVB/BB/VA - 1)]
(1)
* To whom correspondence should be addressed. (1) Taub, A. I.; Fleischer R. L. Science 1989, 243, 616. (2) Tanaka, K.; Koiwa, M. High Temp. Mater. Processes 1999, 18, 323. (3) Fleischer, R. L. ISIJ Int. 1991, 31, 1186. (4) Wuu, D. S.; Chan, C. C.; Horng, R. H. J. Vac. Sci. Technol. 1999, A17 (6), 3327. (5) Wuu, D. S.; Lee, M. L.; Lin, T. Y.; Horng, R. H. Mater. Chem. Phys. 1996, 45, 163. (6) Fleischer, R. L. J. Mater. Sci. 1987, 22, 2281. (7) Makamura, M. In Intermetallic Compounds: Vol. 1, Principles; Westbrook, J. H., Fleischer, R. L., Eds.; John Wiley and Sons Ltd., 1994, p 873. (8) Fine, M. E.; Brown, L. D.; Marcus, H. L. Scr. Metall. 1984, 18, 951. (9) Fleischer, R. L.; Gilmore, R. S.; Zabala, R. J. Acta Metall. 1989, 37 (10), 2801. (10) Nakamura, M. Metall. Trans. A 1994, 25A, 331. (11) Yamanaka, S.; Yamada, K.; Tsuzuki, T.; Iguchi, T.; Katsura, M.; Hoshino, Y.; Saiki, W. J. Alloys Compd. 1998, 271-273, 549. (12) Tanaka, K.; Koiwa, M. Intermetallics 1996, 4, s29.
where B, BA, and BB are the bulk moduli of alloy A1-xBx, consisting of metal A and metal B, respectively; VA and VB are the molar volumes of metal A and metal B. Normally, good agreement with experiment is reached in dilute substitutional alloy systems where the volume change after mixing is proportional linearly to the atomic fraction of the alloy. By the well-known Vegard’s law, Giri obtained the following expression:21
B)[(1 - x)aABA + xaBBB]/a
(2)
where B, BA, and BB have the same definitions as above; a, aA, and aB are lattice constants for alloy A1-xBx, metal A, and metal B, respectively. Reasonable agreement can be obtained when alloy A1-xBx has the same crystal structure with the constituent metal A and metal B. In another study, Cohen22 proposed a simple empirical model which can accurately predict bulk modulus for covalent solids; using this model, a new superhard material (β-C3N4) is predicted and synthesized. However, investigations focused on the bulk modulus of engineering materials have scarcely been performed with the empirical approach, especially for intermetallic compounds. In this paper, starting from either Miedema’s model23 of alloys or Cho’s model24 with regards to bulk electron concentration, we derive a simple empirical model by (13) Chu, F.; Mitchell, T. E.; Majumdar, B.; Miracle, D.; Nandy, T. K.; Banerjee, D. Intermetallics 1997, 5, 147. (14) Davidson, D. L. Mater. Sci. Eng., A 2000, 293, 281. (15) Grammatikakis, J.; Manolopoulos, M.; Katsika, V.; Kostopoulos, D. Phys. Status Solidi 1994, 144A, 317. (16) Grammatikakis, J.; Papathanassiou, A. N. Phys. Status Solidi 1996, 155A, 365. (17) Grammatikakis, J.; Katsika, V. J. Phys.: Condens. Matter 1990, 2, 3903. (18) Varotsos, P. Phys. Status Solidi 1980, 99B, k93. (19) Varotsos, P.; Alexopoulos, K. Phys. Status Solidi 1980, 102B, k67. (20) Varotsos, P.; Alexopoulos, K. In Thermodynamics of Point Defects and Their Relation with Bulk Properties Series, Defects in Solids, Amelinckx, S., Gevers, R., Nihoul, J., Eds.; North-Holland Publ. Co.: Amsterdam, 1986; pp 325-347 and pp 156-158. (21) Giri, A. K. Mater. Lett. 1993, 17, 353. (22) Liu, A. Y.; Cohen, M. L. Science 1989, 245, 841. (23) Miedema, A. R. Physica B 1992, 182, 1. (24) Cho, S. A. Trans. Jpn. Inst. Met. 1981, 22 (9), 643.
10.1021/cm0104203 CCC: $20.00 © 2001 American Chemical Society Published on Web 11/27/2001
Binary Intermetallic Compounds
Chem. Mater., Vol. 13, No. 12, 2001 4643 Table 1. The Value of B, V, and nws of 73 Elements
atom
B (Gpa)
nws (e/au3)
volume (10-6 m3)
(B/V)1/2
atom
B (Gpa)
nws (e/au3)
volume (10-6 m3)
(B/V)1/2
Li Na K Rb Cs H Be Zn Cd Hg Mg Ca Sr Ba B Al Ga In As Sb Bi P Tl N C Si Ge Sn Pb Pu Sc Th Lu Tm Er Y Dy Ho Tb Gd Sm Pm Sc
11.6 6.8 3.2 3.1 1.57 0.2 110 69.4 51 38.2 35.6 17.2 12 10.2 178 75.2 56.9 41.1 38.3 38.3 31.5 30.4 28.5 1.2 443 98 77.2 58.2 45.8 76 56.6 54 47.6 44.5 44.4 41.2 40.5 40.2 38.7 37.9 37.8 33 56.6
0.94 0.55 0.27 0.22 0.17 3.38 4.66 2.30 1.91 1.91 1.60 0.75 0.59 0.53 5.36 2.70 2.25 1.60 3.00 2.00 1.56 4.49 1.40 4.49 5.55 3.38 2.57 1.90 1.52 2.99 2.05 2.10 1.91 1.86 1.86 1.77 1.82 1.82 1.82 1.77 1.77 1.77 2.05
13.00 23.78 45.63 56.07 69.23 1.70 4.90 9.17 13.00 14.08 14.00 26.2 33.93 38.1 4.70 10.00 11.82 15.75 11.85 16.95 19.32 8.60 17.23 4.10 3.25 8.60 9.87 16.30 18.28 12.06 15.03 19.8 17.77 18.12 18.45 19.90 19.00 18.76 19.32 19.90 20.01 20.25 15.03
0.94 0.53 0.26 0.24 0.15 0.34 4.74 2.75 1.98 1.65 1.59 0.81 0.59 0.52 6.15 2.74 2.19 1.62 1.80 1.50 1.28 1.88 1.29 0.54 11.68 3.38 2.80 1.89 1.58 2.51 1.94 1.65 1.64 1.57 1.55 1.44 1.46 1.46 1.42 1.38 1.37 1.28 1.94
Th Lu Tm Er Y Dy Ho Tb Gd Sm Pm Nd Yb Pr La Ce Eu Hf Ti Zr Au Cu Ag Ta Nb V U Pt Pd Ni Ir W Rh Mo Co Cr Os Ru Fe Re Mn Tc
54 47.6 44.5 44.4 41.2 40.5 40.2 38.7 37.9 37.8 33 31.8 30.5 28.8 27.9 21.5 14.7 109 108 89.8 171 138 104 196 170 158 97.9 276 187 177 371 311 276 261 182 160 373 286 166 334 92.6 281
2.10 1.91 1.86 1.86 1.77 1.82 1.82 1.82 1.77 1.77 1.77 1.73 1.86 1.73 1.64 1.69 1.77 3.05 3.51 2.8 3.87 3.18 2.52 4.33 4.41 4.41 3.44 5.64 4.66 5.36 6.13 5.93 5.45 5.55 5.36 5.18 6.33 6.13 5.55 6.33 4.17 5.93
19.80 17.77 18.12 18.45 19.90 19.00 18.76 19.32 19.90 20.01 20.25 20.58 17.97 20.79 22.55 21.62 19.97 13.45 10.58 14.00 10.20 7.12 10.25 10.81 10.80 8.36 13.15 9.10 8.90 6.60 8.52 9.55 8.30 9.40 6.70 7.23 8.45 8.20 7.09 8.85 7.35 8.64
1.65 1.64 1.57 1.55 1.44 1.46 1.46 1.42 1.38 1.37 1.28 1.24 1.30 1.18 1.11 1.00 0.86 2.85 3.20 2.53 4.09 4.40 3.18 4.26 3.97 4.35 2.73 5.51 4.58 5.18 6.60 5.71 5.77 5.27 5.20 4.71 6.64 5.91 4.84 6.14 3.55 5.70
using literature data. This model may be used to predict or estimate the bulk modulus of binary intermetallic compounds and alloys. 2. Bulk Modulus for Pure Elements Miedema23 proposed an empirical model for the prediction of alloy formation enthalpy. In this model, two semiempirical physical parameters, the work function φ, and the electron density nws are used for each element. The work function φ has about the same meaning as the electronegativity. The second parameter, nws, is defined as the electron density at the boundary of Wigner-Seitz cell. For nontransition metals, it is obtained by summation of electron densities of free atoms, while for transition metals, it is derived from bulk modulus data. A nice linear relationship between nws and (B/V)1/2 was presented by Miedema for Al, Cu, Fe, and five alkali metals,23 where B and V are the bulk modulus and the molar volume of the elements, respectively. To extend this correlation beyond the above 8 elements, another 73 metal and semimetal elements are selected for testing. Their values of B, V, and nws are listed in Table 1. Figure 1 illustrates a very good linear relationship between nws and (B/V)1/2 for all except H, N, P, and C.
Figure 1. The linear relationship between nws and (B/V)1/2 of 73 elements, nws in arbitrary density units [electron/(au)3], B in Gpa, and V in 10-6 m3 per mole.
This relationship can be expressed:
nws ) (B/V)1/2
(3)
The abnormity for H, N, P, and C elements may result from their typical covalent features. The elements As, Cu, Fe, P, and U also deviate to some extent from the empirical regularity. The effect of this deviation to the
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Li and Wu
bulk modulus of binary alloys containing these elements will be discussed later. 3. Bulk Modulus for Binary Systems As mentioned above, a good linear relationship between nws and (B/V)1/2 exists for metal or semimetal elements. Can this rule be extended to binary intermetallic compounds and alloys? If so, how are the electronic density nws and molar volume V estimated? An answer may be found by applying Miedema’s semiempirical or macroscopic atom model,23 where atoms are considered “blocks” of the elements and compounds or alloys are considered “piles” of the blocks. As an example, consider an intermetallic compound A1-xBx that is formed by atom A (with the electronic density nwsA and the molar volume VA) and atom B (with the electronic density nwsB and the molar volume VB). The total valence electron numbers N contained in (1-x) mole element A and x mole element B can be obtained by Vegard’s Law:
N ) nwsAVA(1 - x) + nwsBVBx
(4)
The molar volume V of compound A1-xBx can be predicted by Miedema’s model.23 Suppose that the total valence electron numbers N remain unchanged after compound A1-xBx is formed. Then the “electron density” of compound nws can be obtained by the following equation:
nws ) N/V ) [nwsAVA(1 - x) + nwsBVBx)]/V
(5)
Table 2 lists the value of bulk modulus B, calculated molar volume V, total valence electron numbers N, calculated bulk modulus Bcal, and their prediction errors for 240 binary compounds and alloys. The values of ∆H, E, and Bcal′ are also listed in this table; their definitions will be given later. The values of bulk modulus for these compounds and alloys are obtained from the literature.7-14 Figure 2 illustrates the relationship between nws and (B/V)1/2 of these compounds and alloys. From this figure, a good linear relationship can be obtained for binary metal systems. A linear fit between nws and (B/V)1/2 of 69 metallic or semimetallic elements is also drawn in this figure. It is astonishing that the linear relationship for binary compounds is the same as for pure elements. The relationship can be expressed in eq 3, or it can be rewritten in eq 6; bulk modulus can also be estimated using this equation:
B ) nws2V
(6)
In Table 2 and Figure 2, the prediction using eq 6 is close to the reported value with an average error limit of (11%. Of special interest are the intermetallic compounds or semiconductor compounds whose chemical constitution is near 50at. %; for example, No7 (Al51Co49), No19 (Al2Zr), No60 (NiAl), No177 (ZnS), and No180 (CdS) in Table 2. Their prediction errors are about 30% or more. Elastic constants are determined by the derivative of free energy with respect to strain. All factors that contribute to the total energy may add on to the elastic constant. Compounds whose atomic ratio of constituent elements is close to 1 show a large enthalpy of formation
(or called heat of formation). Therefore, the volume change after mixing may not be linearly proportional to the chemical constituent. To predict the bulk modulus more accurately, the enthalpy change (or short-range ordering) should be included. According to Miedema’s model, cohesive enthalpies of intermetallic compounds E, consist of two parts: the average value of cohesive energy of constituent elements EAB and the additional formation enthalpy ∆H. For compound A1-xBx, this cohesive enthalpy is defined as
E ) ∆H + EAB ) ∆H + (1 - x)EA + xEB
(7)
where EA, EB, and EAB are cohesive energy of metal A, metal B, and their linear summation, respectively. The formation enthalpy ∆H can be predicted by Miedema’s model.23 In eq 6, the contribution of the formation enthalpy to bulk modulus is not considered; this may result in a larger prediction error for compounds whose formation enthalpy can no longer be neglected. Therefore, the contribution of formation enthalpy ∆H to bulk modulus has to be calculated. Total bulk modulus can be expressed as
B ) B 1 + B2
(8)
where B, B1, and B2 are the total bulk modulus, the contribution by the linear summation energy of constituent elements, and the modification by the formation enthalpy, respectively. B1 is obtained by eq 6, and B2 is calculated by eq 9:
B2 ) B1∆H/(∆H + EAB)
(9)
Equation 9 shows that the modification with regards to the formation enthalpy is proportional to the fraction of the formation enthalpy in the total cohesive energy. From eqs 8 and 9, the bulk modulus may be predicted by
B ) B1[1 + ∆H/(EAB + ∆H)]
(10)
Table 2 lists the value of formation enthalpy ∆H, total cohesive enthalpy E, revised calculated bulk modulus Bcal′, and their predicted errors. It is seen from Table 2 that the new model may provide better predictions for the intermetallics of equal elemental ratio. 4. Discussion According to Martin’s one-gap model,25 bulk moduli for tetrahedrally coordinated compounds are proportional to -3.5 power of the interatomic distance. For the diamond and the zinc blende solid, Cohen26 found good analytic relation of the bulk modulus B)1761d-3.5, where d is the bond length. For elemental substances, the bulk modulus B can be determined by the effective pseudopotential radius rps through an empirical relation:27 BdCrpsm, where C and m are constants related to the bonding character; the powers m range from -3.1 to -3.4 (for the highly sp-bonded elemental substance). Consider our empirical model with regards to this argument. Equation 6 can be rewritten as BdCz2d-3 (25) Martin, R. M. Phys. Rev. B 1970, 1, 4005. (26) Cohen, M. L. Phys. Rev. B 1985, 32, 7988. (27) Makino, Y. J. Alloys Compd. 1996, 242, 122.
Binary Intermetallic Compounds
Chem. Mater., Vol. 13, No. 12, 2001 4645
Table 2. The Value of nws and (B/V)1/2 for Binary Systems no
compound
B (GPa)
N (M e)a
V (10- 6m3)
∆H (KJ)
E (KJ)
Bcal (GPa)
error (%)
Bcal′ (GPa)
error (%)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76
Al42.5Co57.5 Al51Co49 Al52.5Co47.5 Aal48Co52 Al49Co51 AlCo Al51Co49 Al46Co54 Al38Co62 Al2Gd Al52Ru48 Al48Ru52 Al50Ru50 Al40Ru60 Al42Ru58 Al2U Al2Y Al3Zr Al2Zr AuCd B6La Be17Nb2 Be12Nb Be12Ti Be13Zr CaAl2 Co2Hf Co2Zr Cr2Nb CoSi2 CrSi2 TiSi2 Cu2Mg CuZn ErFe2 GdAl2 IrNb LaAl2 MgAg Mg2Si Mg2Sn !MnS2 Mo48.7Nb Mo0.25Nb0.75 Mo0.31Nb0.69 Mo24.8Nb Mo0.37Nb0.63 Mo0.44Nb0.56 Mo0.53Nb0.47 Mo0.72Nb0.28 Mo16.1Re Mo27Re Mo35Re MoSi2 Nb33Cr77 NbCr2 Nb23.3Mo Nb33.9Mo NbSi2 NiAl Ni3Al Ni3Fe Ni3Ga Ni3Ge Ni3Mn NiTi PdTi ReSi2 RuSc Ru53Ta47 Ra11Ta9 Ru49Ta51 Ru11Ta9 Ru51Ta49 SbY Si2V
136.0 138.0 153.0 156.0 158.0 162.0 173.0 180.0 189.0 78.0 199.0 206.0 208.0 222.0 222.0 83.0 89.2 95.4 117.0 85.0 90.0 141.0 177.0 117.0 122.0 49.0 167.0 153.0 200.0 201.1 172.0 148.9 88.0 116.0 103.0 80.0 301.0 71.0 66.0 55.0 41.0 64.0 215.5 188.8 194.5 243.5 194.5 206.6 217.1 235.8 282.0 292.5 293.0 209.7 192.7 200.0 187.7 197.4 191.5 166.0 170.9 180.6 146.2 182.6 193.1 140.3 143.7 165.0 148.0 251.0 255.0 255.0 255.0 259.0 66.0 166.0
31.88 31.16 31.03 31.41 31.33 31.24 31.16 31.58 32.26 29.74 37.99 38.91 38.45 40.74 40.28 29.39 29.74 30.06 31.09 32.13 24.79 25.38 24.67 23.84 23.92 24.47 37.29 36.74 40.90 39.00 39.66 39.52 28.25 26.14 34.31 29.74 50.17 30.35 24.12 28.50 25.20 27.92 50.02 48.88 49.14 51.05 49.40 49.70 50.09 50.91 52.75 53.19 53.50 44.55 40.56 40.90 48.81 49.27 43.11 31.16 33.24 35.07 33.13 35.26 34.19 36.18 39.14 45.87 40.37 48.57 48.62 48.45 48.62 48.51 35.80 39.44
7.81 8.10 8.16 8.00 8.03 8.07 8.10 7.93 7.67 12.94 8.71 8.63 8.66 8.49 8.52 10.85 12.94 11.02 11.36 11.03 6.65 5.50 5.33 5.28 5.44 12.54 8.55 8.64 8.40 10.19 10.46 11.57 8.76 8.02 9.94 12.95 9.53 13.73 11.76 12.73 14.61 13.74 10.04 10.45 10.35 9.72 10.26 10.15 10.02 9.76 9.30 9.23 9.19 11.18 8.29 8.40 10.47 10.31 11.73 8.01 7.23 6.71 7.53 8.16 6.75 8.35 9.50 10.89 10.31 9.26 9.21 9.37 9.21 9.32 18.86 10.86
43.1 42.4 41.9 43.1 42.9 42.7 42.4 43.3 41.8 65.2 48.0 48.6 48.4 46.7 47.6 55.7 65.2 58.5 73.2 19.8 28.2 14.0 10.2 12.1 17.4 13.7 48.8 56.9 9.7 37.6 38.0 81.4 -25.6 -11.1 33.1 65.2 78.5 66.0 15.8 12.2 11.1 104.6 8.4 5.1 6.3 5.6 7.3 8.1 8.4 6.2 4.1 6.9 8.6 39.9 9.0 9.7 4.8 6.8 69.2 48.0 35.5 0.9 28.8 28.8 8.1 50.2 95.4 33.3 62.9 58.5 58.4 57.8 58.4 58.3 65.8 53.4
425.9 416.9 414.9 420.6 419.4 418.2 416.9 422.7 428.9 416.5 530.0 543.5 536.9 567.5 561.9 452.3 423.9 454.5 492.2 259.8 364.1 377.2 361.8 343.5 357.6 291.0 538.5 540.6 516.3 476.3 467.0 534.8 246.7 221.9 414.1 416.5 778.5 427.7 230.3 257.6 208.8 381.9 701.4 717.1 714.0 681.4 710.6 706.4 700.2 684.4 680.9 696.5 707.6 556.5 504.5 516.3 718.0 712.4 609.8 425.5 438.3 425.1 417.5 442.8 399.6 498.2 517.4 589.0 575.9 770.6 767.8 775.1 767.8 773.0 409.3 521.4
130.1 119.8 118.0 123.3 122.2 121.0 119.8 125.8 135.7 68.3 165.8 175.5 170.6 195.4 190.4 79.6 68.3 82.0 85.1 93.6 92.4 117.1 114.1 107.6 105.1 47.7 162.6 156.3 199.1 149.2 150.4 135.0 91.2 85.2 118.4 68.3 264.0 67.1 49.5 63.8 43.5 56.7 249.0 228.7 233.3 268.2 237.9 243.3 250.4 265.6 299.4 306.4 311.6 177.5 198.5 199.1 227.5 235.5 158.5 121.3 152.8 183.3 145.9 152.3 173.1 156.6 161.2 193.2 158.1 254.7 256.8 250.5 256.8 252.6 67.9 143.2
4.3 13.2 22.8 20.9 22.6 25.3 30.7 30.1 28.1 12.3 16.6 14.8 17.9 11.9 14.2 4.1 23.3 14.0 27.2 10.0 2.6 16.9 35.5 8.0 13.8 2.6 2.6 2.1 0.4 25.7 12.5 9.3 3.5 26.5 14.9 14.6 12.2 5.5 25.0 16.0 5.9 11.3 15.5 21.1 19.9 10.1 22.3 17.7 15.3 12.6 6.1 4.7 6.3 15.3 3.0 0.4 21.2 19.2 17.2 26.9 10.5 1.4 0.2 16.6 10.3 11.6 12.2 17.1 6.8 1.4 0.6 1.7 0.6 2.4 2.9 13.7
143.2 132.0 129.9 136.0 134.7 133.3 132.0 138.7 148.9 79.0 180.8 191.2 186.0 211.5 206.5 89.4 78.9 92.6 97.7 100.7 99.6 121.4 117.3 111.4 110.2 50.0 177.4 172.8 202.8 161.0 162.6 155.5 81.7 81.0 127.9 79.0 290.7 77.4 52.9 66.8 45.8 72.2 252.0 230.4 235.3 270.4 240.3 246.1 253.4 268.0 301.2 309.4 315.4 190.2 202.1 202.8 229.0 237.8 176.5 134.9 165.2 183.7 156.0 162.2 176.6 172.4 191.0 204.2 175.4 274.0 276.3 269.2 276.3 271.6 78.9 157.9
5.3 4.3 15.0 12.8 14.7 17.7 23.7 22.9 21.1 1.3 9.1 7.1 10.5 4.7 6.9 7.6 11.6 2.9 16.4 18.4 10.6 13.8 33.7 4.8 9.6 1.9 6.2 12.9 1.4 19.9 5.4 4.4 7.1 30.1 24.1 1.2 3.4 9.0 19.9 21.5 11.6 12.8 16.9 22.0 21.0 11.0 23.5 19.1 16.7 13.6 6.8 5.7 7.6 9.2 4.8 1.4 22.0 20.4 7.8 18.7 3.3 1.7 6.6 11.1 8.5 22.9 32.9 23.7 18.4 9.1 8.3 5.5 8.3 4.8 19.5 4.8
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Table 2 (Continued) no
compound
B (GPa)
N (M e)a
V (10- 6m3)
∆H (KJ)
E (KJ)
Bcal (GPa)
error (%)
Bcal′ (GPa)
error (%)
77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152
SiV3 Si3V5 Sn3Ce TaSi2 TaV2 TbFe2 TiAl3 Ti3Al Ti28.4Cr Ti50Cr50 Ti46Cr54 Ti44Cr56 Ti42Cr58 Ti40Cr60 Ti48Cr52 TiCr2 Ti13.8Cr Ti9.4Cr Ti7Cr Ti5Ge3 Ti94Ge6 Ti20Re80 Ti5Re24 Ti64Ru36 TiRu Ti87Sn13 Ti3Sn Ufe2 Uni2 Uni5 Upd3 Urh3 Uru3 V3Ge Vsi2 V0.294Ti0.706 V0.385Ti0.615 V0.53Ti0.47 V0.73Ti0.27 Wsi2 W0.096Ta0.904 W0.785Ta0.215 W0.60Ta0.40 W0.36Ta0.64 W0.10Ta0.90 Yal2 Yfe2 Yzn ZrCo2 ZrCr2 TiAl Al-Li alloy Al-Li alloy Al-Li alloy Al-Li alloy Al-Li alloy AlNi Co68Ni32 NbCr2 HfV2 Mo7.0Re Mo16.6Re Mo26.9Re Nb16.8Mo Nb23.3Mo Nb33.9Mo Nb51.6Mo Nb75.2Mo Nb92.1Mo Cr67Zr33 Cr64Cr36 Cr69Zr31 Mo5Si3 Nb83.2Mo16.8 Nb76.7Mo23.3 Nb66.1Mo35.9
175.0 187.0 55.0 192.5 218.0 93.0 106.0 118.0 115.7 147.0 147.0 150.0 153.0 154.0 155.0 159.0 108.6 107.8 108.5 82.0 116.0 309.0 322.0 173.0 247.0 97.0 97.5 134.0 126.0 182.0 167.0 261.0 264.0 168.0 167.2 113.0 116.7 125.9 138.1 222.4 205.1 216.7 237.9 266.1 297.7 82.0 97.0 62.0 156.0 142.0 110.0 72.0 68.5 69.0 67.0 65.5 156.0 183.1 229.4 104.0 270.8 278.5 284.1 179.3 183.4 196.4 216.8 242.8 257.7 147.0 147.0 152.0 264.7 179.3 185.2 196.4
37.78 38.27 30.32 42.86 40.21 34.59 29.51 34.52 37.15 37.24 37.26 37.27 37.27 37.28 37.25 37.31 37.09 37.07 37.06 36.29 36.91 52.27 52.80 41.66 43.46 36.23 35.49 34.26 34.94 35.13 39.47 42.43 45.96 36.35 39.44 36.96 36.93 36.90 36.85 46.01 47.97 49.10 50.85 53.11 55.57 29.74 34.61 28.15 36.74 38.05 32.02 26.26 25.52 25.15 24.78 24.41 31.16 35.43 40.90 38.15 52.39 52.77 54.25 48.53 48.81 49.27 50.03 51.05 51.77 38.05 38.10 38.01 47.86 48.53 48.81 49.27
9.31 9.79 16.34 11.73 9.18 10.18 10.18 10.45 9.53 8.77 8.63 8.57 8.50 8.44 8.70 8.23 10.05 10.21 10.30 11.86 10.76 9.03 9.01 9.40 9.01 11.36 12.10 8.66 8.23 7.38 9.47 9.00 8.85 9.73 10.86 9.87 9.66 9.33 8.90 11.20 10.72 10.54 10.25 9.93 9.64 12.94 10.32 14.32 8.65 9.19 10.33 9.93 9.87 9.83 9.80 9.77 8.03 6.61 8.40 9.93 9.35 9.29 9.08 10.57 10.47 10.31 10.04 9.71 9.49 9.15 9.34 9.03 10.40 10.57 10.47 10.31
50.9 67.6 56.6 69.8 1.3 31.9 39.4 38.1 7.1 10.9 11.0 11.0 10.9 10.8 11.0 9.9 3.5 2.4 1.8 11.5 14.3 20.4 17.6 52.9 63.5 19.9 37.6 7.9 16.9 9.7 46.6 28.2 13.6 37.0 53.4 1.7 2.2 2.5 1.8 34.7 2.6 5.8 10.0 9.8 2.9 65.2 29.6 51.3 56.9 16.8 60.4 0.7 1.4 1.8 2.1 2.5 48.0 0.3 9.7 3.0 1.8 4.2 10.1 3.4 4.8 6.8 8.4 5.6 1.8 16.7 17.3 16.1 48.1 3.4 4.8 6.8
546.4 554.9 388.1 627.8 603.3 437.6 401.6 470.8 454.4 442.4 439.6 438.1 436.6 435.0 441.0 429.2 461.4 463.5 464.7 443.5 476.5 734.0 739.7 586.4 622.5 466.5 464.3 461.9 480.9 455.7 462.6 577.7 635.1 514.0 521.4 482.7 487.1 493.8 501.9 618.4 792.0 804.4 822.8 841.1 854.2 423.9 445.6 327.3 540.6 481.1 457.9 319.3 311.5 307.7 303.8 299.9 425.5 425.5 516.3 551.3 668.0 681.7 731.0 721.3 718.0 712.4 701.2 681.4 665.5 480.3 487.2 475.6 626.6 721.3 718.0 712.4
153.3 149.7 56.3 156.6 176.1 117.6 85.6 114.0 144.9 158.2 160.8 162.1 163.4 164.7 159.5 169.2 136.8 134.6 133.4 111.1 126.6 302.6 309.5 184.6 209.7 115.5 104.0 135.5 148.3 167.1 164.5 200.0 238.7 135.7 143.2 138.4 141.2 145.8 152.5 189.1 214.7 228.8 252.1 283.9 320.3 68.3 116.1 55.3 156.1 157.6 99.2 69.4 66.0 64.3 62.7 61.0 121.0 189.9 199.1 146.7 293.6 299.7 324.0 222.7 227.5 235.5 249.3 268.2 282.4 158.2 155.4 160.1 220.2 222.7 227.5 235.5
12.3 19.9 2.2 18.6 19.2 26.4 19.2 3.3 25.2 7.5 9.3 8.0 6.8 6.9 2.8 6.4 25.9 24.8 22.9 35.4 9.1 2.0 3.8 6.7 15.1 19.1 6.7 1.1 17.7 8.1 1.4 23.3 9.5 19.2 14.3 22.4 21.0 15.8 10.3 14.9 4.6 5.5 5.9 6.7 7.5 16.6 19.7 10.7 0.0 11.0 9.8 3.5 3.6 6.7 6.4 6.8 22.4 3.7 13.2 41.0 8.4 7.6 14.0 24.2 24.0 19.8 14.9 10.4 9.5 7.6 5.7 5.3 16.8 24.2 22.8 19.8
167.6 167.9 64.5 174.0 176.5 126.1 93.9 123.3 147.1 162.1 164.8 166.2 167.5 168.8 163.4 173.1 137.9 135.3 133.9 113.9 130.4 311.0 316.9 201.2 231.1 120.5 112.5 137.8 153.5 170.7 181.1 209.8 243.8 145.5 157.8 138.9 141.8 146.6 153.0 199.7 215.4 230.4 255.2 287.3 321.4 78.8 123.9 64.0 172.5 163.1 112.3 69.6 66.3 64.7 63.1 61.5 134.6 190.0 202.8 147.5 294.4 301.6 328.5 223.8 229.0 237.8 252.3 270.4 283.1 163.7 160.9 165.5 237.1 223.8 229.0 237.8
4.2 10.1 17.2 9.5 19.0 35.6 11.3 4.4 27.1 10.2 12.1 10.7 9.4 9.6 5.4 8.8 26.9 25.4 23.3 38.9 12.4 0.6 1.5 16.3 6.4 24.1 15.3 2.8 21.8 6.2 8.4 19.6 7.6 13.3 5.6 22.9 21.5 16.3 10.7 10.2 4.9 6.3 7.2 7.9 7.9 3.8 27.6 3.2 10.6 14.8 2.0 3.3 3.2 6.2 5.8 6.0 13.7 3.7 11.5 41.7 8.7 8.2 15.6 24.8 24.8 21.0 16.3 11.3 9.8 11.3 9.4 8.8 10.4 24.8 23.6 21.0
Binary Intermetallic Compounds
Chem. Mater., Vol. 13, No. 12, 2001 4647
Table 2 (Continued) no
compound
B (GPa)
N (M e)a
V (10- 6m3)
∆H (KJ)
E (KJ)
Bcal (GPa)
error (%)
Bcal′ (GPa)
error (%)
153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228
Nb48.4Mo51.6 Nb24.8Mo75.2 Nb7.2Mo92.8 Ta90W10 Ta70W30 TaW Ta33W67 Ta17W83 Cu92.7Ni7.3 Cu82.2Ni17.8 Cu80Ni20 Ci77.2Ni22.8 Cu64.5Ni35.5 Cu53.8Ni46.2 Cu31.1Ni68.9 AlP AlAs AlSb GaP GaAs GaSb InP InAs InSb ZnS ZnSe ZnTe CdS CdSe CdTe HgSe HgTe Cu alloy Cu alloy Cu alloy Cu alloy Cu alloy Cu alloy Cu alloy Cu alloy Cu alloy Cu alloy Cu-Ni alloy Cu-Ni alloy Cu-Ni alloy Cu-Ni alloy Cu-Ni alloy Cu-Au alloy Cu-Au alloy Cu-Au alloy Cu-Al alloy Cu-Al alloy Cu-Al alloy Cu-Al alloy Cu-Al alloy Cu-Al alloy Cu-Al alloy Cu-Al alloy Cu-Al alloy Cu-Al alloy Cu-Al alloy Pb-Tl alloy Pb-Tl alloy Pb-Tl alloy Pb-Tl alloy Pb-Tl alloy Pb-Tl alloy Pb-Tl alloy Pb-Tl alloy Zn-Cu alloy Zn-Cu alloy Fe-Ni alloy Au-Ni alloy Au-Ni alloy Au-Ni alloy Au-Ni alloy
216.8 242.8 257.1 201.7 220.7 243.7 268.9 288.9 181.2 175.5 176.2 174.2 169.7 164.7 151.0 86.0 77.0 58.2 88.7 74.8 57.0 71.0 60.0 47.4 77.1 62.4 51.0 62.0 53.0 42.4 50.0 42.3 133.9 136.3 131.6 134.5 135.7 138.7 132.8 139.5 136.2 133.2 137.7 137.7 138.0 138.6 139.2 138.8 139.0 145.5 136.9 135.5 134.3 137.6 136.9 136.6 135.8 135.4 135.2 134.4 133.0 44.8 45.0 44.5 44.7 44.3 43.9 42.8 42.6 133.9 126.6 173.1 172.6 171.4 170.7 170.5
50.03 51.05 51.80 48.02 49.94 51.85 53.48 55.01 31.50 31.93 32.02 32.14 32.66 33.10 34.04 27.22 27.72 29.43 27.00 27.51 29.21 26.29 26.79 28.49 25.62 24.74 24.65 25.66 24.78 24.69 25.81 25.72 30.73 30.99 30.78 31.12 31.00 31.59 31.69 31.93 31.24 31.26 31.29 31.32 31.38 31.45 31.60 31.21 31.43 32.02 31.07 30.89 30.74 31.16 31.05 30.99 30.91 30.88 30.84 30.76 30.64 27.78 27.75 27.72 27.70 27.66 27.56 27.24 27.14 29.97 29.33 36.42 39.31 39.04 38.44 37.69
10.04 9.71 9.48 10.72 10.41 10.12 9.91 9.73 7.05 6.98 6.97 6.95 6.88 6.82 6.71 9.17 10.98 14.18 9.88 11.79 15.09 11.28 13.41 16.89 13.34 13.82 15.91 14.57 15.28 17.58 15.79 18.05 7.17 7.22 7.36 7.16 7.27 7.30 7.35 7.48 7.16 7.20 7.08 7.07 7.06 7.05 7.03 7.10 7.18 7.41 7.18 7.29 7.39 7.11 7.18 7.22 7.28 7.30 7.32 7.37 7.45 18.28 18.27 18.26 18.25 18.24 18.21 18.11 18.08 7.45 7.63 6.73 10.08 9.84 9.32 8.66
8.4 5.6 1.6 2.7 8.0 11.0 9.3 5.0 2.6 6.4 7.2 8.2 12.3 14.2 11.4 44.1 8.8 -8.1 47.1 13.2 -0.4 47.3 15.2 7.4 139.1 58.9 10.6 169.1 83.7 36.0 71.7 26.9 -1.4 0.3 0.5 -0.5 -1.2 2.1 2.6 3.8 0.0 -0.1 0.9 1.1 1.6 2.2 3.5 0.1 1.3 4.5 0.2 0.4 0.6 0.0 0.2 0.3 0.4 0.4 0.4 0.5 0.7 0.0 0.0 0.0 0.1 0.1 0.1 0.4 0.4 -0.4 -0.6 1.6 -0.7 -2.3 -5.8 -9.6
701.2 681.4 664.8 792.4 813.1 831.5 842.9 850.9 345.3 358.8 361.6 365.2 380.9 392.7 410.8 373.1 315.0 287.9 348.1 291.3 267.6 334.3 279.3 261.4 341.6 242.4 181.1 362.6 258.2 197.5 222.7 164.9 325.1 335.8 335.6 334.5 332.0 342.7 344.3 348.3 336.3 336.5 339.0 339.9 341.8 343.7 348.4 336.2 338.2 343.7 335.9 335.7 335.6 336.0 335.9 335.8 335.7 335.7 335.7 335.6 335.5 196.0 195.9 195.8 195.7 195.6 195.3 194.3 194.0 296.5 275.7 425.6 369.1 371.5 376.7 383.8
249.3 268.2 283.0 215.2 239.6 265.7 288.6 310.9 140.8 146.0 147.2 148.6 155.2 160.8 172.7 80.8 70.0 61.1 73.8 64.2 56.6 61.2 53.5 48.1 49.2 44.3 38.2 45.2 40.2 34.7 42.2 36.6 131.7 133.0 128.6 135.3 132.1 136.7 136.6 136.3 136.3 135.7 138.4 138.7 139.4 140.2 142.0 137.3 137.6 138.3 134.5 130.8 127.9 136.6 134.2 133.0 131.2 130.7 129.9 128.4 125.9 42.2 42.1 42.1 42.0 41.9 41.7 41.0 40.7 120.7 112.7 197.1 153.3 154.9 158.6 164.1
14.9 10.4 10.0 6.7 8.5 9.0 7.3 7.6 22.2 16.7 16.4 14.7 8.5 2.4 14.3 6.0 9.0 4.9 16.8 14.2 0.7 13.7 10.8 1.3 36.1 29.0 25.1 27.0 24.2 18.2 15.6 13.3 1.6 2.4 2.2 0.6 2.6 1.4 2.8 2.2 0.0 1.8 0.5 0.7 1.0 1.1 2.0 1.0 1.0 4.9 1.7 3.4 4.7 0.7 1.9 2.6 3.3 3.4 3.8 4.4 5.3 5.6 6.3 5.4 5.9 5.3 4.9 4.2 4.4 9.8 11.0 13.8 11.1 9.6 7.0 3.7
252.3 270.4 283.7 215.9 242.0 269.2 291.8 312.7 141.9 148.6 150.1 151.9 160.1 166.6 177.5 90.3 72.0 59.3 83.8 67.1 56.5 69.9 56.4 49.4 69.3 55.0 40.4 66.3 53.2 41.0 55.7 42.6 131.1 133.1 128.8 135.1 131.6 137.6 137.6 137.8 136.3 135.6 138.7 139.2 140.1 141.1 143.4 137.3 138.1 140.2 134.6 130.9 128.2 136.6 134.3 133.1 131.4 130.9 130.1 128.6 126.2 42.2 42.1 42.1 42.0 42.0 41.7 41.0 40.8 120.5 112.4 197.9 153.0 153.9 156.2 160.0
16.3 11.3 10.3 7.0 9.6 10.4 8.4 8.2 21.6 15.3 14.8 12.8 5.6 1.1 17.5 5.0 6.5 1.9 5.5 10.3 0.9 1.5 5.9 4.2 10.1 11.7 20.7 6.9 0.3 3.3 11.4 0.7 2.0 2.3 2.1 0.4 2.9 0.8 3.6 1.1 0.0 1.8 0.7 1.0 1.5 1.7 3.0 1.0 0.6 3.6 1.7 3.3 4.5 0.7 1.8 2.5 3.2 3.3 3.7 4.3 5.1 5.6 6.3 5.4 5.9 5.2 4.8 4.0 4.2 10.0 11.1 14.3 11.3 10.2 8.5 6.1
4648
Chem. Mater., Vol. 13, No. 12, 2001
Li and Wu
Table 2 (Continued) no
compound
B (GPa)
N (M e)a
V (10- 6m3)
∆H (KJ)
E (KJ)
Bcal (GPa)
error (%)
Bcal′ (GPa)
error (%)
229 230 231 232 233 234 235 236 237 238 239 240
Cu-Ni alloy Cu-Ni alloy Cu-Ni alloy Zn-Cu alloy Mg-Li alloy Mg-Li alloy Mg-Li alloy Mg-Li alloy Mg-Li alloy Mg-Li alloy Mg-Li alloy Mg-Li alloy
181.1 175.9 174.9 133.9 37.1 36.9 36.7 36.2 35.8 35.4 35.2 34.7
35.02 34.59 34.38 30.87 22.21 22.06 21.85 21.66 21.35 21.15 20.93 20.75
6.62 6.66 6.68 7.18 13.95 13.92 13.87 13.83 13.77 13.73 13.68 13.64
2.8 6.7 8.6 -0.1 0.0 0.0 0.1 0.1 0.1 0.1 0.2 0.2
424.0 418.4 415.6 325.6 145.2 145.4 145.7 146.0 146.4 146.7 147.0 147.3
185.3 179.7 177.0 132.7 35.4 35.0 34.4 33.9 33.1 32.6 32.0 31.6
2.3 2.1 1.2 0.9 4.5 5.1 6.1 6.4 7.5 8.0 8.9 9.0
186.5 182.6 180.7 132.6 35.4 35.0 34.4 33.9 33.1 32.6 32.1 31.6
3.0 3.7 3.3 0.9 4.5 5.1 6.0 6.3 7.4 7.9 8.8 8.9
average a
11.0
10.3
M is a constant and equals 6.748 × 1024.
Figure 2. The linear relationship between nws and (B/V)1/2 of pure elements and binary compounds, nws in arbitrary density units [electron/(au)3], B in Gpa, and V in 10-6 m3 per mole. O, Pure elements; ×, binary intermetallics.
because nws is proportional to z/V, where V is roughly proportional to d3, C is a constant, z is the number of valence electrons contained in the unit cell, V is the unit cell volume, and d is the equilibrium interatomic distance. In our model, bulk modulus B is also proportional to dm, and the power m is equal to -3 which is very close to the previous results. For a terminal solid solution, the formation enthalpy (or heat of formation) ∆H is close to zero; eq 10 will reduce to eq 9, which means our model can also be applied to a terminal solid solution. As to more complicated intermetallic compounds and alloys, because their formation enthalpies can no longer be neglected, the additional term ∆H/(E + ∆H) must be considered to predict their bulk modulus. That is, our model can predict the bulk modulus for intermetallic compounds, alloys, and ideal dilute solid solutions. The trends in the elastic constant of the simple and transition elemental cubic metals is investigated by Rose and Shore28 in terms of a uniform electron-gas theory. The elastic properties of metals depend primarily on two parameters: the average electron density (28) Rose J. H.; Shore H. B. Phys. Rev. B 1994, 49, 11588.
at the cell boundary and less importantly the bonding valence. This conclusion is very similar to ours which may be rearranged as B ) nws2V ) nws(nwsV) ) nwsZ. But unlike the model by Rose and Shore, eq 6 may cover binary alloys. As noted above, carbon does not obey this linear relationship, so carbides (e.g., TiC and SiC) are not included in Figure 2, which means that the carbides may not obey this empirical rule. Attention is also needed for copper alloys because copper also shows some abnormality (see Figure 1). If we still use the original value of nws ) 3.18 in Figure 2, all alloys containing Cu will obviously deviate from the ideal line in this figure. Therefore, it is reasonable to use the curve fit best estimate, nws ) 4.40, for copper alloys. The same correction is also applied to element As(2.4), P(3.19), Fe(4.84), and U(2.72). The value of nws for elements S(1.71), Se(1.5), and Te(1.2) is not given in Miedema’s model; the value of nws is supposed to equal the value of (B/V)1/2 to estimate the bulk modulus of compounds containing these elements. 5. Conclusions The correlation between the bulk modulus and the constituent element properties of binary intermetallic compounds and alloys has been constructed on the basis of literature data of 240 compounds and alloys. Using this model, the following results have been obtained: (1) A relationship among bulk modulus B, molar volume V, and valence electron density nws for 69 metal or semimetal elements (except H, N, P, and C elements) exists and can be expressed by eq 6. (2) A relationship among bulk modulus B, molar volume V, and valence electron density nws for binary compounds and alloys exists and can also be expressed by eq 6, with average predicted error limits of (11%. (3) For compounds whose atomic ratio is near to 1 and which have a large formation enthalpy, the contribution of formation enthalpy must be considered to predict their bulk modulus; the bulk modulus may be predicted by eq 10. CM0104203
