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Site preference and alloying effect on elastic properties of ternary B2 RuAl-based alloys
Intermetallics 51 (2014) 24e29
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Site preference and alloying effect on elastic properties of ternary B2 RuAl-based alloys Shuo Huang*, Chuan-Hui Zhang, Rui-Zi Li, Jiang Shen, Nan-Xian Chen Department of Physics, University of Science and Technology Beijing, Beijing 100083, China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 17 August 2013 Received in revised form 25 February 2014 Accepted 26 February 2014 Available online
The structural and elastic properties of ternary B2 RuAl-based alloys are studied using first-principles calculations. Single-crystal elastic constants, atomic volumes, transfer energies, and electronic densities for RuAl-TM are computed, considering all possible transition-metal solute species TM. Calculated elastic constants are used to compute values of some commonly considered elasticity parameters, such as bulk modulus, shear modulus, Yong’s modulus, Pugh ratio, and Cauchy pressure. The present results suggest that the bulk modulus of RuAl-TM increase approximately linearly with increasing electron density. Calculated elastic properties are in favorable accord with available experimental and theoretical data. Ó 2014 Elsevier Ltd. All rights reserved.
Keywords: A. Intermetallics B. Elastic properties D. Site occupancy E. Ab-initio calculations
1. Introduction The B2 compound RuAl possesses a rather high melting point (about 2323 K), strong high-temperature strength, reasonable oxidation and corrosion resistance, and even good roomtemperature toughness [1,2]. These characteristics provoked intense research activity [3,4], and much effort such as microstructural control through processing [5e9], second-phase reinforcement [10e12] and solid-solution hardening [13,14], as well as mechanical alloying [15e18] devoted to RuAl were performed to make it more suitable for applications in technological important area. In particular, RuAl when alloyed with transition metal strongly affects its mechanical properties. For example, hafnium hardened a b-RuAl-based solid solution and decreased the cold-deformation ability, while titanium and chromium in alloys with 5e15 vol% 3-Ru increased the microplastic-deformation ability [19]. Additions up to approximately 40% of cobalt and titanium resulted in the formation of a continuous interfacial “necklace structure”, which contributed to significant plasticity during room temperature compression test [20]. Studies for RuAl-based refractory alloys have shown that RuAl alloyed with iridium improved its compressive strength, while iron, cobalt, and nickel exhibited opposite effects [21]. Moreover, elastic constants were reported to decrease with the increase of the scandium content in a series of ternary RuAl alloys with approximately 50% ruthenium and 2e25% scandium [22], and * Corresponding author. Tel./fax: þ86 62322872. E-mail address: [email protected] (S. Huang). http://dx.doi.org/10.1016/j.intermet.2014.02.020 0966-9795/Ó 2014 Elsevier Ltd. All rights reserved.
shear modulus were also found to decrease with the additions of cobalt, iron, and titanium [23]. On the contrary, a strong solid solution strengthening effect was observed in multiphase RuAl alloys with nickel [24,25], iridium [26], molybdenum [27] and tantalum [28] additions. Besides, the effects of ternary alloying additions such as copper [29], yttrium [30,31], niobium [32], rhodium [33,34], and palladium [35,36] were studied with emphasis on structure and strengthening. Although numerous of studies devoted to RuAl-based alloys have already been published, a systematic understanding of the site preference of transition metal impurities in B2 RuAl and their effect on its elastic properties is still scarce. Notice that the site preference of alloying addition is of importance in understanding and controlling the physical performance of materials and is critical to alloy design [37]. Moreover, the elastic properties provide implications for a vast number of material-related behaviors such as structural stability, hardness, stiffness, brittleness/ductility, anisotropy, melting temperature, etc [38e40]. Hence, to understand, and especially to guide the development of RuAl-based alloys, it is desirable to perform such calculations. In this work the site occupancy, electronic, and elastic properties of RuAl-TM alloys (TM ¼ transition metal) are studied using firstprinciples calculations. We calculate the elastic constants C11, C12 and C44, bulk modulus B, shear modulus G, Young’s modulus E, Pugh ratio B/G and Cauchy pressure (C12-C44) of the RuAl-TM alloys. Analysis of the electronic characteristic suggests that there exist a direct connection between the bonding strength and the elastic properties of alloys.
S. Huang et al. / Intermetallics 51 (2014) 24e29
25
Table 2 Calculated equilibrium lattice parameter (a, in A), elastic constants (Cij, in GPa), aggregate elastic modulus (B, G and Y, in GPa), Poisson’s ratio (n) and Zener anisotropy parameters (AZ) of RuAl compared with the previous experimental and other theoretical data. Present study Previous calculations a C11 C12 C44 B
309.9 148.3 125.8 202.2
G Y
105.3 269.3 0.278 1.56
AZ a b c d e
2. Details of calculations
f g
In this study, structural optimizations and properties calculations are performed using the Cambridge Serial Total Energy Package (CASTEP) [41] plane-wave basis set density functional theory (DFT) code. The effects of the exchange-correlation energy are treated within the generalized gradient approximation (GGA) [42] in the form of PredeweBurkeeErnzerhof (PBE) [43]. Planewave basis set cutoffs for the smooth part of the wave functions is set to 450 eV which ensured convergence in the above test calculations. The Brillouin zone is sampled using 6 6 6 grid of k-points for 16-atom supercells following the MonkhorstePack [44] scheme. Self-consistent field [45] calculations are implemented with a convergence criterion of 106 eV/atom on the total energy. The 16-atom supercell used in this calculation is shown in Fig. 1, which constructed from 8 unit cells of RuAl. The alloying atom TM is placed at the centre of the supercell, replacing a ruthenium or an aluminum atom. 3. Results and discussion 3.1. Site preference To evaluate the site substitution behavior of a TM atom in the RuAl, we calculated the so-called transfer energy ETM(Al / Ru), which corresponds to the energy required to transfer a TM atom from a Al site to a Ru site [46]
Table 1 Transfer energy (ETM, in eV) and site preference of the ternary additions in B2 RuAl alloy. The formation energy of the exchange antisite defect EAntisite is 1.96 eV. Element
ETM(Al / Ru)
Site preference
Element
ETM(Al / Ru)
Site preference
Sc Ti V Cr Mn Fe Co Ni Cu Zn Y Zr Nb Mo
1.75 2.92 3.05 2.51 1.30 0.55 0.01 0.10 0.20 0.55 0.55 2.71 3.26 2.85
Ru, Al Al Al Ru, Ru, Ru, Ru Ru Ru, Ru, Al Al Al
Tc Rh Pd Ag Cd La Hf Ta W Re Os Ir Pt Au
1.39 1.25 1.07 0.95 0.25 0.70 2.34 2.85 3.62 2.43 0.68 1.00 1.15 0.93
Ru, Al Ru Ru Ru Ru Ru, Al Al Al Al Al Ru, Al Ru Ru Ru
Al
Al Al Al
Al Al
h i j k l m n o p q r
Ref. Ref. Ref. Ref. Ref. Ref. Ref. Ref. Ref. Ref. Ref. Ref. Ref. Ref. Ref. Ref. Ref. Ref.
b
c
Experiments
2.99 , 3.002 , 2.969 , 3.006 , 3.005 , 3.03k, 2.95l, 2.992m, 3.02f, 2.96g,3.064h, 3.009i, 3.046j 2.993n, 2.988o, 2.95p 308a, 320b, 346c, 341d 144a, 143b, 162c, 132d 122a, 125b, 141c, 121d 199a, 202b, 223c, 202d, 203e, 207q, 208r 220f, 230g, 208h, 199i, 217j 104a, 112d 107m, 104q, 108r 266a, 232b 267q, 275r 0.28a, 0.308b 0.286q, 0.279r 1.49a
3.005
n Fig. 1. Original model used in the calculation. The alloying atom TM is placed at a Al site at the centre of the supercell, which is supercell Ru8Al7TM; we also use a similar supercell where the alloying atom is placed at a Ru site, which is supercell Ru7Al8TM.
a
d
e
[64]. [65]. [66]. [67]. [51]. [68]. [69]. [70]. [71]. [72]. [59]. [60]. [23]. [1]. [61]. [62]. [22]. [63].
ETM ðAl/RuÞ ¼ EðRu7 Al8 TMÞ EðRu8 Al7 TMÞ þ EðRuAl Þ
(1)
where E(Ru7Al8TM) and E(Ru8Al7TM) are energies of substitution alloys with ternary addition TM on the Ru and Al sublattices, respectively, and E(RuAl) ¼ E(Ru9Al7) E(Ru8Al8) is the energy of the partial antisite defects on the Al sublattice. Besides, the summation of the two transfer energies is equal to the formation energy of the exchange antisite defect in RuAl: ETM(Al / Ru) þ ETM(Ru / Al) ¼ E(RuAl) þ E(AlRu) ¼ EAntisite. According the methodology proposed by Ruban and Skriver [46], if ETM(Al / Ru)＜0, TM exhibits strong Ru site preference; if ETM(Al / Ru)＞EAntisite, TM exhibits strong Al site preference; and Table 3 Calculated elastic constants (Cij, in GPa) of RuAl-based alloys.
Ru8Al7Sc Ru8Al7Ti Ru8Al7V Ru8Al7Cr Ru8Al7Mn Ru8Al7Fe Ru8Al7Co Ru8Al7Zn Ru8Al7Y Ru8Al7Zr Ru8Al7Nb Ru8Al7Mo Ru8Al7Tc Ru8Al7La Ru8Al7Hf Ru8Al7Ta Ru8Al7W Ru8Al7Re Ru8Al7Os
C11
C12
C44
297.0 316.0 325.7 328.2 318.1 311.0 312.7 299.5 277.0 305.4 326.3 334.8 327.9 314.7 307.2 326.3 342.7 351.5 336.7
140.5 149.3 151.8 152.3 148.9 175.0 162.6 147.5 140.2 142.7 148.3 151.9 153.6 124.8 149.0 154.8 153.9 154.2 160.3
114.8 116.1 113.7 110.2 114.4 121.3 128.8 128.5 104.8 110.2 110.7 110.0 100.8 90.3 114.6 116.2 113.4 107.5 113.5
Ru7Al8Sc Ru7Al8Mn Ru7Al8Fe Ru7Al8Co Ru7Al8Ni Ru7Al8Cu Ru7Al8Zn Ru7Al8Y Ru7Al8Tc Ru7Al8Rh Ru7Al8Pd Ru7Al8Ag Ru7Al8Cd Ru7Al8La Ru7Al8Os Ru7Al8Ir Ru7Al8Pt Ru7Al8Au
C11
C12
C44
277.9 310.2 305.5 296.5 292.4 288.9 288.8 266.3 313.8 303.1 296.9 285.7 295.9 251.1 317.0 312.1 307.1 293.0
127.9 138.7 145.0 150.2 145.2 144.4 134.7 116.4 143.9 149.8 146.2 140.3 128.4 108.9 151.6 153.4 148.5 144.6
113.5 127.4 128.6 123.4 120.8 116.5 115.4 104.2 126.1 119.5 117.4 111.4 110.3 93.9 130.2 123.0 118.2 110.5
26
S. Huang et al. / Intermetallics 51 (2014) 24e29
Fig. 2. Calculated lattice parameter and bulk modulus for (a) Ru8Al7TM and (b) Ru7Al8TM.
if 0 ＜ ETM(Al / Ru)＜EAntisite, TM does not show any site preference and randomly occupy Ru or Al sublattice. Previous studies also demonstrate the validity of evaluating the site substitution behavior in alloy systems based on the above criteria [47e50]. The calculated transfer energy and the site preferences of TM in B2 RuAl are presented in Table 1. As can be seen, the metals Ti, V, Cr, Zr, Nb, Mo, Hf, Ta, W, and Re show strong preference for the Al sublattice, whereas Ni, Cu, Rh, Pd, Ag, Cd, Ir, Pt, and Au show strong preference for the Ru sublattice. The metals Sc, Mn, Fe, Co, Zn, Y, Tc, La, and Os are found to have no site preference by randomly substitute both Al and Ru sites. It was reported that Ti predominantly occupies the Al site [20], while Ni and Ir have a very strong preference for the Ru sublattice in RuAl [51e53]. The site substitution behaviors of TM in RuAl predicted in this work are in good agreement with the available experimental and other theoretical results, and provide reference data for the future experimental work on this notable material.
Fig. 3. Correlation between electron density and bulk modulus for the RuAl-based alloys.
Fig. 4. (a) Correlation between the shear modulus G and the Young’s modulus E for the RuAl-based alloys. (b) Correlation in (a) is further extended to a large scale data collected for 29 pure transition metals.
S. Huang et al. / Intermetallics 51 (2014) 24e29
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Fig. 5. Total and site projected density of states for (a) RuAl and (b) Ru8Al7W. Difference charge density contour plots in (1 1 0) plane for (c) RuAl and (d) Ru8Al7W. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
3.2. Elastic properties Elastic properties crystals with cubic symmetry are fully characterized by the three principle elastic tensor elements: C11, C12 and C44. The bulk modulus B and VoigteReusseHill averaged shear modulus G can be obtained by means of the relations that B ¼ (C11 þ 2C12)/3 and G ¼ (GV þ GR)/2, where the effective Voigt shear modulus GV and Reuss shear modulus GR are expressed as [54e56]
GV ¼
1 ðC C12 þ 3C44 Þ; 5 11
GR ¼
5ðC11 C12 ÞC44 4C44 þ 3ðC11 C12 Þ
(2)
and the polycrystalline Young’s modulus E, Poisson’s ratio n, and Zener anisotropy parameters AZ can be obtained from the following equations [57,58]
E ¼
9 BG ; 3BþG
y¼
3 B 2G ; 2ð3 B þ GÞ
AZ ¼
2 C44 C11 C12
(3)
To verify the accuracy of our calculations, we first presented the equilibrium lattice parameter a, elastic constants C11, C12 and C44, bulk modulus B, shear modulus G, Young’s modulus E, Poisson’s ratio n and Zener anisotropy parameters AZ of pure RuAl in Table 2. As can be seen, our measurements are in good agreement with experiments [1,22,23,59e63] and other theoretical calculations [51,64e72]. Therefore, according to the site substitution behavior of a TM atom in the RuAl discussed in section 3.1, we further investigated the effect of TM on the elastic properties of B2 RuAl. Selected results for RuAl-TM alloys are summarized in Table 3. For cubic crystals, the mechanical stability criteria are given by C11 ＞ 0, C44 ＞ 0, C11-C12 ＞ 0 and C11 þ 2C12 ＞ 0 [73]. In our case, the elastic constants of RuAl-TM alloys satisfy the above restrictions equations, indicating that these compounds are mechanically stable. Since the elastic property is usually inversely proportional to the atomic volume for pure elements [74], in Fig. 2 we presented the calculated bulk modulus and lattice parameters of RuAl-TM alloys. Notice that our results agree well with the available theoretical
28
S. Huang et al. / Intermetallics 51 (2014) 24e29
Fig. 6. (a) Correlation between the Cauchy pressure (C12-C44) and the Pugh ratio G/B for the RuAl-based alloys. (b) Renormalized hyperbolic correlation derived by dividing Young’s modulus E from (C12-C44) (as shown in Ref. 82) for the summarized data of (a), along with typical ductile and hard materials for comparison depicted by red balls. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
values [21], and the bulk modulus of RuAl-TM decrease approximately with increasing equilibrium volume. To further illustrate the effect of alloying elements on the variation of bulk modulus of RuAl, in Fig. 3 we presented the relationship between electron density and bulk modulus of RuAl-TM alloys. According to Rose and Shore [75e77], the electron density n is the quotient of the bonding valence ZB and the volume per atom VM in metal. Then we use the following equations to evaluate the electron density n for Ru8Al7TM
nðRu8 Al7 TMÞ ¼ ZB ðRu8 Al7 TMÞ=VM ðRu8 Al7 TMÞ
(4)
where VM (Ru8Al7TM) is the volume (in cm3/mol) of Ru8Al7TM, and ZB (Ru8Al7TM) is the bonding valence (in el/atom) of Ru8Al7TM, which can be evaluated by Vegard’s law, ZB (Ru8Al7TM) ¼ [8Z(Ru) þ 7Z(Al) þ Z(TM)]/16. The bonding valence for pure metal is acquired from Ref. [76]. Similar formulation is used to evaluate the value of n(Ru7Al8TM). With this approach, a nearly linear relationship in B vs. n is observed as shown in Fig. 3, which allows us to anticipate the behavior of bulk modulus of RuAl-TM with respect to the electron density.
The shear modulus G represents the resistance to reversible deformations upon shear stress, while the Young’s modulus E is defined as the ratio of the tensile stress to the corresponding tensile strain [78]. In Fig. 4(a) we presented E against G for the studied RuAl-TM, and a nearly linear relationship is observed. Such a trend was also observed in pure transition-metals as shown in Fig. 4(b) [79]. Notice that the larger the parameters are, the harder the material is. The current calculations suggest that the largest increase of these two parameters of RuAl is due to elements W and Os that prefer Ru and Al site, respectively. It is known that the chemical bonding and corresponding ground state properties of materials are closely associated with its electronic states. In Fig. 5(a) and (b) we presented the total densities of states (DOS), as well as the partial DOS projected to different atoms for Ru8Al8 and Ru8Al7W, respectively. The Fermi energy (EF) is set at zero and marked by the vertical lines. It is immediately clear that the studied materials exhibit metallic characteristics because the value of total DOS at Fermi level is nonzero. Moreover, the current calculations suggest that the main characteristic of the electronic DOS of RuAl is governed by Ru atom. The addition of W that prefers Al site in RuAl does not produce significant effect on the DOS of RuAl. Further insight can be gained by considering the calculated electron density difference maps, which are defined as the electron density difference between the isolated atoms and their bonding states. The results for Ru8Al8 and Ru8Al7W are presented in Fig. 5(c) and (d), respectively, where the red color correspond to higher density region and blue correspond to lower density region. The main part of the charge density distributions surrounding each atom suggests symmetry and does not show evident directionality. Further analysis shows that apparent covalent RueW bonds can be confirmed due to the strong charge accumulations. Namely, the addition of W that prefers Al site in RuAl induced covalent strengthening mechanism. The results are consistent with the observed trend of shear modulus and Yong’s modulus as shown in Fig. 4. In the following we discussed the brittle/ductile behavior of RuAl-TM alloys, which can be described by Pugh ratio G/B [80] and Cauchy pressure (C12-C44) [81]. The relationship between these two parameters for the RuAl-TM alloys is presented in Fig. 6(a). Interestingly, all the studied materials distribute in a ductile region according to the Niu’s reported [82], and a nearly linear relationship between the (C12-C44) and G/B is also built. This unified criterion also unveils the substantial evidence that hardness of materials is correlated with Pugh ratio [83,84]. In order to clarify how good the ductile of RuAl-TM alloys, following the same treatment of Ref. 82, we renormalized the Cauchy pressure (C12-C44) by dividing the Yong’s modulus E and re-plotted the relationship of (C12-C44)/E and G/B based on the data in Fig. 6(a). The data of some typical ductile materials (Au, Ag), brittle materials (Si, Ge), and super-hard materials (cubic BN, diamond) are also presented for comparison [82]. From Fig. 6(b), it is found that the typical ductile and hard materials, as well as the RuAl-TM alloys, can be fitted by a hyperbola curve. Moreover, the closer to the upper left corner, the more ductile and stronger the metallic bonding (e.g. Au), and vice versa, the closer to the bottom right corner, the more brittle and stronger the covalent bonding (e.g. diamond). The general distribution can be used to assess the brittle/ductile behavior of RuAl-TM, which may be helpful for their application in engineering. 4. Conclusion The present work has involved a computational study of the site preference and alloying effect on elastic properties of ternary B2 RuAl-based alloys. It is found that the metals Ti, V, Cr, Zr, Nb, Mo, Hf, Ta, W, and Re have a strong preference for the Al sublattice, whereas
S. Huang et al. / Intermetallics 51 (2014) 24e29
Ni, Cu, Rh, Pd, Ag, Cd, Ir, Pt, and Au have a strong preference for the Ru sublattice. The metals Sc, Mn, Fe, Co, Zn, Y, Tc, La, and Os are predicted to have no site preference. Considering these impurities at their preferential sites, we created a database of the elastic constants C11, C12 and C44, bulk modulus B, shear modulus G, Young’s modulus E, Pugh ratio B/G and Cauchy pressure (C12-C44). The database was then used to analyze the mechanical behavior of these alloys using phenomenological correlations. Furthermore, the electron density, density of states and charge density contour for selected materials are also presented in this study, and the results suggest that the bulk modulus of RuAl-TM increase as a function of electron density. The theoretical predictions on the considered properties of RuAl-based alloys may be helpful for their application in engineering. Acknowledgments This work is supported by the supported by the National Basic Research Program of China (Grant No. 2011CB606401). References [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]
Fleischer R. Metall Trans A 1993;24:227. Wolff IM. JOM 1997;49:34. Mücklich F, Ili c N, Woll K. Intermetallics 2008;16:593. Mücklich F, Ili c N. Intermetallics 2005;13:5. Guitar MA, Woll K, Ramos-Moore E, Mücklich F. Thin Solid Films 2013;527:1. Bax B, Pauly C, Leibenguth P, Woll K, Mücklich F. Surf Coat Technol 2012;206: 3931. Woll K, Holzapfel C, Mücklich F. Intermetallics 2010;18:553. Soldera F, Ili c N, Manent Conesa N, Barrientos I, Mücklich F. Intermetallics 2005;13:101. Ili c N, Soldera F, Mücklich F. Intermetallics 2005;13:444. Zotov N, Woll K, Mücklich F. Intermetallics 2010;18:1507. Acosta M, Reynolds TD, Johnson DR, Kvam EP. Intermetallics 2010;18:2281. Zhu J, Zhang C, Cao W, Yang Y, Zhang F, Chen S, et al. Acta Mater 2009;57:202. Povarova KB, Morozov AE, Padalko AG, Drozdov AA. Russ Metall 2008;2008: 155. Fleischer RL. Acta Metall Mater 1993;41:1197. Liu KW, Mücklich F. Intermetallics 2005;13:373. de Medeiros SN, Machado FLA, Zampiere RB, Santos IA, Paesano Jr A. J Noncryst Solids 2006;352:3718. Cao F, Pollock TM. Metall Mater Trans A 2008;39A:39. Cao F, Nandy TK, Stobbe D, Pollock TM. Intermetallics 2007;15:34. Povarova KB, Morozov AE, Drozdov AA, Kazanskaya NK. Russ Metall 2011;2011:865. Wolff IM, Sauthoff G. Acta Mater 1997;45:2949. Bleskov ID, Isaev EI, Vekilov YK. Phys Solid State 2010;52:1803. Fleischer RL, Field RD, Briant CL. Metall Trans A 1991;22:403. Fleischer RL. Acta Metall Mater 1993;41:863. Wolff IM, Sauthoff G. Metall Mater Trans A 1996;27:1395. Borah A, Robi PS, Mujumdar A, Srinivasan A. Met Mater-Int 2008;14:123. Hill PJ, Cornish LA, Witcomb MJ. J Alloy Compd 1999;291:130. Reynolds TD, Johnson DR. Intermetallics 2004;12:157. Feng Q, Nandy TK, Tryon B, Pollock TM. Intermetallics 2004;12:755. ski B. J Low Temp Phys 2007;149: Rosenbaum R, Mi S, Grushko B, Przepiórzyn 314. Eow K, Lu D, Pollock TM. Scr Mater 1998;38:1065. Fleischer RL, McKee DW. Metall Trans A 1993;24:759.
29
[32] Benarchid MY, David N, Fiorani JM, Vilasi M, Benlaharche T. Thermochim Acta 2009;482:39. [33] Meshi L, Samuha S, Kapush D, Pavlyuchkov D, Grushko B. J Alloy Compd 2011;509:6551. [34] Grushko B, Kapush D, Velikanova TY, Samuha S, Meshi L. J Alloy Compd 2011;509:8018. [35] Pavlyuchkov D, Grushko B, Velikanova TY. J Alloy Compd 2009;469:146. [36] Pavlyuchkov D, Grushko B, Velikanova TY. J Alloy Compd 2008;464:101. [37] Bozzolo GH, Noebe RD, Amador C. Intermetallics 2002;10:149. [38] Clerc DG, Ledbetter HM. J Phys Chem Solids 1998;59:1071. [39] Jhi S, Ihm J, Louie SG, Cohen ML. Nature 1999;399:132. [40] Ravindran P, Fast L, Korzhavyi PA, Johansson B, Wills J, Eriksson O. J Appl Phys 1998;84:4891. [41] Clark SJ, Segall MD, Pickard CJ, Hasnip PJ, Probert MIJ, Refson K, et al. Z Krist 2005;220:567. [42] Perdew JP, Wang Y. Phys Rev B 1992;45:13244. [43] Perdew JP, Burke K, Ernzerhof M. Phys Rev Lett 1996;77:3865. [44] Monkhorst HJ, Pack JD. Phys Rev B 1976;13:5188. [45] Hohenberg P, Kohn W. Phys Rev 1964;136:B864. [46] Ruban AV, Skriver HL. Phys Rev B 1997;55:856. [47] Ponomareva AV, Isaev EI, Vekilov YK, Abrikosov IA. Phys Rev B 2012;85: 144117. [48] Jiang C. Acta Mater 2007;55:4799. [49] Jiang C. Acta Mater 2008;56:6224. [50] Jiang C, Gleeson B. Scr Mater 2006;55:433. [51] Gargano P, Mosca H, Bozzolo G, Noebe RD. Scr Mater 2003;48:695. [52] Prins S, Arroyave R, Liu Z-K. Acta Mater 2007;55:4781. [53] Fleischer RL. J Mater Sci Lett 1988;7:525. [54] Voigt W. Ann Phys-berlin 1889;274:573. [55] Reuss AZ. Angew Math Mech 1929;9:49. [56] Hill R. Proc Phys Soc Sect A 1952;65:349. [57] Schreiber E, Anderson OL, Soga N. Elastic constants and their measurements. New York: McGraw-Hill; 1973. [58] Zener C. Elasticity and anelasticity of metals. Chicago: University of Chicago Press; 1948. [59] Obrowski W. Naturwissenschaften 1960;47:14. [60] Edshammar LE. Acta Chem Scand 1966;20:427. [61] Gobran HA, Liu KW, Heger D, Mücklich F. Scr Mater 2003;49:1097. [62] Mi S, Grushko B. Intermetallics 2004;12:425. [63] Fleischer RL. Isij Int 1991;31:1186. [64] Mehl MJ, Klein BM, Papaconstantopoulos DA. Intermetallic compounds: principles and applications. London: Wiley; 1994. [65] Wen B, Zhao J, Bai F, Li T. Intermetallics 2008;16:333. [66] Nguyen-Manh D, Pettifor DG. Intermetallics 1999;7:1095. [67] Bleskov ID, Smirnova EA, Vekilov YK, Korzhavyi PA, Johansson B, Katsnelson M, et al. Appl Phys Lett 2009;94. [68] Lin W, Xu J, Freeman AJ. J Mater Res 1992;7:592. [69] Mehl MJ, Singh DJ, Papaconstantopoulos DA. Mater Sci Eng A 1993;170:49. [70] Li C, Wu P. Chem Mater 2001;13:4642. [71] Ghosh G, Olson GB. Acta Mater 2007;55:3281. [72] Mosca HO, Bozzolo G, Garcés JE. Scr Mater 2008;58:1025. [73] Wallace DC. Thermodynamics of crystals. New York: Wiley; 1972. [74] Shang SL, Saengdeejing A, Mei ZG, Kim DE, Zhang H, Ganeshan S, et al. Comput Mater Sci 2010;48:813. [75] Rose JH, Shore HB. Phys Rev B 1991;43:11605. [76] Rose JH, Shore HB. Phys Rev B 1993;48:18254. [77] Rose JH, Shore HB. Phys Rev B 1994;49:11588. [78] Nye JF. Physical properties of crystals: their representation by tensors and matrices. New York: Oxford University Press; 1985. [79] Barbalace K. http://environmentalchemistry.com/yogi/periodic/. [80] Pugh SF. Philos Mag 1954;45:823. [81] Pettifor DG. Mater Sci Technol 1992;8:345. [82] Niu HY, Chen XQ, Liu PT, Xing WW, Cheng XY, Li DZ, et al. Sci Rep 2012;2:718. [83] Chen XQ, Niu HY, Li DZ, Li YY. Intermetallics 2011;19:1275. [84] Chen XQ, Niu HY, Franchini C, Li DZ, Li YY. Phys Rev B 2011;84:121405.
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Intermetallics journal homepage: www.elsevier.com/locate/intermet
Site preference and alloying effect on elastic properties of ternary B2 RuAl-based alloys Shuo Huang*, Chuan-Hui Zhang, Rui-Zi Li, Jiang Shen, Nan-Xian Chen Department of Physics, University of Science and Technology Beijing, Beijing 100083, China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 17 August 2013 Received in revised form 25 February 2014 Accepted 26 February 2014 Available online
The structural and elastic properties of ternary B2 RuAl-based alloys are studied using first-principles calculations. Single-crystal elastic constants, atomic volumes, transfer energies, and electronic densities for RuAl-TM are computed, considering all possible transition-metal solute species TM. Calculated elastic constants are used to compute values of some commonly considered elasticity parameters, such as bulk modulus, shear modulus, Yong’s modulus, Pugh ratio, and Cauchy pressure. The present results suggest that the bulk modulus of RuAl-TM increase approximately linearly with increasing electron density. Calculated elastic properties are in favorable accord with available experimental and theoretical data. Ó 2014 Elsevier Ltd. All rights reserved.
Keywords: A. Intermetallics B. Elastic properties D. Site occupancy E. Ab-initio calculations
1. Introduction The B2 compound RuAl possesses a rather high melting point (about 2323 K), strong high-temperature strength, reasonable oxidation and corrosion resistance, and even good roomtemperature toughness [1,2]. These characteristics provoked intense research activity [3,4], and much effort such as microstructural control through processing [5e9], second-phase reinforcement [10e12] and solid-solution hardening [13,14], as well as mechanical alloying [15e18] devoted to RuAl were performed to make it more suitable for applications in technological important area. In particular, RuAl when alloyed with transition metal strongly affects its mechanical properties. For example, hafnium hardened a b-RuAl-based solid solution and decreased the cold-deformation ability, while titanium and chromium in alloys with 5e15 vol% 3-Ru increased the microplastic-deformation ability [19]. Additions up to approximately 40% of cobalt and titanium resulted in the formation of a continuous interfacial “necklace structure”, which contributed to significant plasticity during room temperature compression test [20]. Studies for RuAl-based refractory alloys have shown that RuAl alloyed with iridium improved its compressive strength, while iron, cobalt, and nickel exhibited opposite effects [21]. Moreover, elastic constants were reported to decrease with the increase of the scandium content in a series of ternary RuAl alloys with approximately 50% ruthenium and 2e25% scandium [22], and * Corresponding author. Tel./fax: þ86 62322872. E-mail address: [email protected] (S. Huang). http://dx.doi.org/10.1016/j.intermet.2014.02.020 0966-9795/Ó 2014 Elsevier Ltd. All rights reserved.
shear modulus were also found to decrease with the additions of cobalt, iron, and titanium [23]. On the contrary, a strong solid solution strengthening effect was observed in multiphase RuAl alloys with nickel [24,25], iridium [26], molybdenum [27] and tantalum [28] additions. Besides, the effects of ternary alloying additions such as copper [29], yttrium [30,31], niobium [32], rhodium [33,34], and palladium [35,36] were studied with emphasis on structure and strengthening. Although numerous of studies devoted to RuAl-based alloys have already been published, a systematic understanding of the site preference of transition metal impurities in B2 RuAl and their effect on its elastic properties is still scarce. Notice that the site preference of alloying addition is of importance in understanding and controlling the physical performance of materials and is critical to alloy design [37]. Moreover, the elastic properties provide implications for a vast number of material-related behaviors such as structural stability, hardness, stiffness, brittleness/ductility, anisotropy, melting temperature, etc [38e40]. Hence, to understand, and especially to guide the development of RuAl-based alloys, it is desirable to perform such calculations. In this work the site occupancy, electronic, and elastic properties of RuAl-TM alloys (TM ¼ transition metal) are studied using firstprinciples calculations. We calculate the elastic constants C11, C12 and C44, bulk modulus B, shear modulus G, Young’s modulus E, Pugh ratio B/G and Cauchy pressure (C12-C44) of the RuAl-TM alloys. Analysis of the electronic characteristic suggests that there exist a direct connection between the bonding strength and the elastic properties of alloys.
S. Huang et al. / Intermetallics 51 (2014) 24e29
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Table 2 Calculated equilibrium lattice parameter (a, in A), elastic constants (Cij, in GPa), aggregate elastic modulus (B, G and Y, in GPa), Poisson’s ratio (n) and Zener anisotropy parameters (AZ) of RuAl compared with the previous experimental and other theoretical data. Present study Previous calculations a C11 C12 C44 B
309.9 148.3 125.8 202.2
G Y
105.3 269.3 0.278 1.56
AZ a b c d e
2. Details of calculations
f g
In this study, structural optimizations and properties calculations are performed using the Cambridge Serial Total Energy Package (CASTEP) [41] plane-wave basis set density functional theory (DFT) code. The effects of the exchange-correlation energy are treated within the generalized gradient approximation (GGA) [42] in the form of PredeweBurkeeErnzerhof (PBE) [43]. Planewave basis set cutoffs for the smooth part of the wave functions is set to 450 eV which ensured convergence in the above test calculations. The Brillouin zone is sampled using 6 6 6 grid of k-points for 16-atom supercells following the MonkhorstePack [44] scheme. Self-consistent field [45] calculations are implemented with a convergence criterion of 106 eV/atom on the total energy. The 16-atom supercell used in this calculation is shown in Fig. 1, which constructed from 8 unit cells of RuAl. The alloying atom TM is placed at the centre of the supercell, replacing a ruthenium or an aluminum atom. 3. Results and discussion 3.1. Site preference To evaluate the site substitution behavior of a TM atom in the RuAl, we calculated the so-called transfer energy ETM(Al / Ru), which corresponds to the energy required to transfer a TM atom from a Al site to a Ru site [46]
Table 1 Transfer energy (ETM, in eV) and site preference of the ternary additions in B2 RuAl alloy. The formation energy of the exchange antisite defect EAntisite is 1.96 eV. Element
ETM(Al / Ru)
Site preference
Element
ETM(Al / Ru)
Site preference
Sc Ti V Cr Mn Fe Co Ni Cu Zn Y Zr Nb Mo
1.75 2.92 3.05 2.51 1.30 0.55 0.01 0.10 0.20 0.55 0.55 2.71 3.26 2.85
Ru, Al Al Al Ru, Ru, Ru, Ru Ru Ru, Ru, Al Al Al
Tc Rh Pd Ag Cd La Hf Ta W Re Os Ir Pt Au
1.39 1.25 1.07 0.95 0.25 0.70 2.34 2.85 3.62 2.43 0.68 1.00 1.15 0.93
Ru, Al Ru Ru Ru Ru Ru, Al Al Al Al Al Ru, Al Ru Ru Ru
Al
Al Al Al
Al Al
h i j k l m n o p q r
Ref. Ref. Ref. Ref. Ref. Ref. Ref. Ref. Ref. Ref. Ref. Ref. Ref. Ref. Ref. Ref. Ref. Ref.
b
c
Experiments
2.99 , 3.002 , 2.969 , 3.006 , 3.005 , 3.03k, 2.95l, 2.992m, 3.02f, 2.96g,3.064h, 3.009i, 3.046j 2.993n, 2.988o, 2.95p 308a, 320b, 346c, 341d 144a, 143b, 162c, 132d 122a, 125b, 141c, 121d 199a, 202b, 223c, 202d, 203e, 207q, 208r 220f, 230g, 208h, 199i, 217j 104a, 112d 107m, 104q, 108r 266a, 232b 267q, 275r 0.28a, 0.308b 0.286q, 0.279r 1.49a
3.005
n Fig. 1. Original model used in the calculation. The alloying atom TM is placed at a Al site at the centre of the supercell, which is supercell Ru8Al7TM; we also use a similar supercell where the alloying atom is placed at a Ru site, which is supercell Ru7Al8TM.
a
d
e
[64]. [65]. [66]. [67]. [51]. [68]. [69]. [70]. [71]. [72]. [59]. [60]. [23]. [1]. [61]. [62]. [22]. [63].
ETM ðAl/RuÞ ¼ EðRu7 Al8 TMÞ EðRu8 Al7 TMÞ þ EðRuAl Þ
(1)
where E(Ru7Al8TM) and E(Ru8Al7TM) are energies of substitution alloys with ternary addition TM on the Ru and Al sublattices, respectively, and E(RuAl) ¼ E(Ru9Al7) E(Ru8Al8) is the energy of the partial antisite defects on the Al sublattice. Besides, the summation of the two transfer energies is equal to the formation energy of the exchange antisite defect in RuAl: ETM(Al / Ru) þ ETM(Ru / Al) ¼ E(RuAl) þ E(AlRu) ¼ EAntisite. According the methodology proposed by Ruban and Skriver [46], if ETM(Al / Ru)＜0, TM exhibits strong Ru site preference; if ETM(Al / Ru)＞EAntisite, TM exhibits strong Al site preference; and Table 3 Calculated elastic constants (Cij, in GPa) of RuAl-based alloys.
Ru8Al7Sc Ru8Al7Ti Ru8Al7V Ru8Al7Cr Ru8Al7Mn Ru8Al7Fe Ru8Al7Co Ru8Al7Zn Ru8Al7Y Ru8Al7Zr Ru8Al7Nb Ru8Al7Mo Ru8Al7Tc Ru8Al7La Ru8Al7Hf Ru8Al7Ta Ru8Al7W Ru8Al7Re Ru8Al7Os
C11
C12
C44
297.0 316.0 325.7 328.2 318.1 311.0 312.7 299.5 277.0 305.4 326.3 334.8 327.9 314.7 307.2 326.3 342.7 351.5 336.7
140.5 149.3 151.8 152.3 148.9 175.0 162.6 147.5 140.2 142.7 148.3 151.9 153.6 124.8 149.0 154.8 153.9 154.2 160.3
114.8 116.1 113.7 110.2 114.4 121.3 128.8 128.5 104.8 110.2 110.7 110.0 100.8 90.3 114.6 116.2 113.4 107.5 113.5
Ru7Al8Sc Ru7Al8Mn Ru7Al8Fe Ru7Al8Co Ru7Al8Ni Ru7Al8Cu Ru7Al8Zn Ru7Al8Y Ru7Al8Tc Ru7Al8Rh Ru7Al8Pd Ru7Al8Ag Ru7Al8Cd Ru7Al8La Ru7Al8Os Ru7Al8Ir Ru7Al8Pt Ru7Al8Au
C11
C12
C44
277.9 310.2 305.5 296.5 292.4 288.9 288.8 266.3 313.8 303.1 296.9 285.7 295.9 251.1 317.0 312.1 307.1 293.0
127.9 138.7 145.0 150.2 145.2 144.4 134.7 116.4 143.9 149.8 146.2 140.3 128.4 108.9 151.6 153.4 148.5 144.6
113.5 127.4 128.6 123.4 120.8 116.5 115.4 104.2 126.1 119.5 117.4 111.4 110.3 93.9 130.2 123.0 118.2 110.5
26
S. Huang et al. / Intermetallics 51 (2014) 24e29
Fig. 2. Calculated lattice parameter and bulk modulus for (a) Ru8Al7TM and (b) Ru7Al8TM.
if 0 ＜ ETM(Al / Ru)＜EAntisite, TM does not show any site preference and randomly occupy Ru or Al sublattice. Previous studies also demonstrate the validity of evaluating the site substitution behavior in alloy systems based on the above criteria [47e50]. The calculated transfer energy and the site preferences of TM in B2 RuAl are presented in Table 1. As can be seen, the metals Ti, V, Cr, Zr, Nb, Mo, Hf, Ta, W, and Re show strong preference for the Al sublattice, whereas Ni, Cu, Rh, Pd, Ag, Cd, Ir, Pt, and Au show strong preference for the Ru sublattice. The metals Sc, Mn, Fe, Co, Zn, Y, Tc, La, and Os are found to have no site preference by randomly substitute both Al and Ru sites. It was reported that Ti predominantly occupies the Al site [20], while Ni and Ir have a very strong preference for the Ru sublattice in RuAl [51e53]. The site substitution behaviors of TM in RuAl predicted in this work are in good agreement with the available experimental and other theoretical results, and provide reference data for the future experimental work on this notable material.
Fig. 3. Correlation between electron density and bulk modulus for the RuAl-based alloys.
Fig. 4. (a) Correlation between the shear modulus G and the Young’s modulus E for the RuAl-based alloys. (b) Correlation in (a) is further extended to a large scale data collected for 29 pure transition metals.
S. Huang et al. / Intermetallics 51 (2014) 24e29
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Fig. 5. Total and site projected density of states for (a) RuAl and (b) Ru8Al7W. Difference charge density contour plots in (1 1 0) plane for (c) RuAl and (d) Ru8Al7W. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
3.2. Elastic properties Elastic properties crystals with cubic symmetry are fully characterized by the three principle elastic tensor elements: C11, C12 and C44. The bulk modulus B and VoigteReusseHill averaged shear modulus G can be obtained by means of the relations that B ¼ (C11 þ 2C12)/3 and G ¼ (GV þ GR)/2, where the effective Voigt shear modulus GV and Reuss shear modulus GR are expressed as [54e56]
GV ¼
1 ðC C12 þ 3C44 Þ; 5 11
GR ¼
5ðC11 C12 ÞC44 4C44 þ 3ðC11 C12 Þ
(2)
and the polycrystalline Young’s modulus E, Poisson’s ratio n, and Zener anisotropy parameters AZ can be obtained from the following equations [57,58]
E ¼
9 BG ; 3BþG
y¼
3 B 2G ; 2ð3 B þ GÞ
AZ ¼
2 C44 C11 C12
(3)
To verify the accuracy of our calculations, we first presented the equilibrium lattice parameter a, elastic constants C11, C12 and C44, bulk modulus B, shear modulus G, Young’s modulus E, Poisson’s ratio n and Zener anisotropy parameters AZ of pure RuAl in Table 2. As can be seen, our measurements are in good agreement with experiments [1,22,23,59e63] and other theoretical calculations [51,64e72]. Therefore, according to the site substitution behavior of a TM atom in the RuAl discussed in section 3.1, we further investigated the effect of TM on the elastic properties of B2 RuAl. Selected results for RuAl-TM alloys are summarized in Table 3. For cubic crystals, the mechanical stability criteria are given by C11 ＞ 0, C44 ＞ 0, C11-C12 ＞ 0 and C11 þ 2C12 ＞ 0 [73]. In our case, the elastic constants of RuAl-TM alloys satisfy the above restrictions equations, indicating that these compounds are mechanically stable. Since the elastic property is usually inversely proportional to the atomic volume for pure elements [74], in Fig. 2 we presented the calculated bulk modulus and lattice parameters of RuAl-TM alloys. Notice that our results agree well with the available theoretical
28
S. Huang et al. / Intermetallics 51 (2014) 24e29
Fig. 6. (a) Correlation between the Cauchy pressure (C12-C44) and the Pugh ratio G/B for the RuAl-based alloys. (b) Renormalized hyperbolic correlation derived by dividing Young’s modulus E from (C12-C44) (as shown in Ref. 82) for the summarized data of (a), along with typical ductile and hard materials for comparison depicted by red balls. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
values [21], and the bulk modulus of RuAl-TM decrease approximately with increasing equilibrium volume. To further illustrate the effect of alloying elements on the variation of bulk modulus of RuAl, in Fig. 3 we presented the relationship between electron density and bulk modulus of RuAl-TM alloys. According to Rose and Shore [75e77], the electron density n is the quotient of the bonding valence ZB and the volume per atom VM in metal. Then we use the following equations to evaluate the electron density n for Ru8Al7TM
nðRu8 Al7 TMÞ ¼ ZB ðRu8 Al7 TMÞ=VM ðRu8 Al7 TMÞ
(4)
where VM (Ru8Al7TM) is the volume (in cm3/mol) of Ru8Al7TM, and ZB (Ru8Al7TM) is the bonding valence (in el/atom) of Ru8Al7TM, which can be evaluated by Vegard’s law, ZB (Ru8Al7TM) ¼ [8Z(Ru) þ 7Z(Al) þ Z(TM)]/16. The bonding valence for pure metal is acquired from Ref. [76]. Similar formulation is used to evaluate the value of n(Ru7Al8TM). With this approach, a nearly linear relationship in B vs. n is observed as shown in Fig. 3, which allows us to anticipate the behavior of bulk modulus of RuAl-TM with respect to the electron density.
The shear modulus G represents the resistance to reversible deformations upon shear stress, while the Young’s modulus E is defined as the ratio of the tensile stress to the corresponding tensile strain [78]. In Fig. 4(a) we presented E against G for the studied RuAl-TM, and a nearly linear relationship is observed. Such a trend was also observed in pure transition-metals as shown in Fig. 4(b) [79]. Notice that the larger the parameters are, the harder the material is. The current calculations suggest that the largest increase of these two parameters of RuAl is due to elements W and Os that prefer Ru and Al site, respectively. It is known that the chemical bonding and corresponding ground state properties of materials are closely associated with its electronic states. In Fig. 5(a) and (b) we presented the total densities of states (DOS), as well as the partial DOS projected to different atoms for Ru8Al8 and Ru8Al7W, respectively. The Fermi energy (EF) is set at zero and marked by the vertical lines. It is immediately clear that the studied materials exhibit metallic characteristics because the value of total DOS at Fermi level is nonzero. Moreover, the current calculations suggest that the main characteristic of the electronic DOS of RuAl is governed by Ru atom. The addition of W that prefers Al site in RuAl does not produce significant effect on the DOS of RuAl. Further insight can be gained by considering the calculated electron density difference maps, which are defined as the electron density difference between the isolated atoms and their bonding states. The results for Ru8Al8 and Ru8Al7W are presented in Fig. 5(c) and (d), respectively, where the red color correspond to higher density region and blue correspond to lower density region. The main part of the charge density distributions surrounding each atom suggests symmetry and does not show evident directionality. Further analysis shows that apparent covalent RueW bonds can be confirmed due to the strong charge accumulations. Namely, the addition of W that prefers Al site in RuAl induced covalent strengthening mechanism. The results are consistent with the observed trend of shear modulus and Yong’s modulus as shown in Fig. 4. In the following we discussed the brittle/ductile behavior of RuAl-TM alloys, which can be described by Pugh ratio G/B [80] and Cauchy pressure (C12-C44) [81]. The relationship between these two parameters for the RuAl-TM alloys is presented in Fig. 6(a). Interestingly, all the studied materials distribute in a ductile region according to the Niu’s reported [82], and a nearly linear relationship between the (C12-C44) and G/B is also built. This unified criterion also unveils the substantial evidence that hardness of materials is correlated with Pugh ratio [83,84]. In order to clarify how good the ductile of RuAl-TM alloys, following the same treatment of Ref. 82, we renormalized the Cauchy pressure (C12-C44) by dividing the Yong’s modulus E and re-plotted the relationship of (C12-C44)/E and G/B based on the data in Fig. 6(a). The data of some typical ductile materials (Au, Ag), brittle materials (Si, Ge), and super-hard materials (cubic BN, diamond) are also presented for comparison [82]. From Fig. 6(b), it is found that the typical ductile and hard materials, as well as the RuAl-TM alloys, can be fitted by a hyperbola curve. Moreover, the closer to the upper left corner, the more ductile and stronger the metallic bonding (e.g. Au), and vice versa, the closer to the bottom right corner, the more brittle and stronger the covalent bonding (e.g. diamond). The general distribution can be used to assess the brittle/ductile behavior of RuAl-TM, which may be helpful for their application in engineering. 4. Conclusion The present work has involved a computational study of the site preference and alloying effect on elastic properties of ternary B2 RuAl-based alloys. It is found that the metals Ti, V, Cr, Zr, Nb, Mo, Hf, Ta, W, and Re have a strong preference for the Al sublattice, whereas
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Ni, Cu, Rh, Pd, Ag, Cd, Ir, Pt, and Au have a strong preference for the Ru sublattice. The metals Sc, Mn, Fe, Co, Zn, Y, Tc, La, and Os are predicted to have no site preference. Considering these impurities at their preferential sites, we created a database of the elastic constants C11, C12 and C44, bulk modulus B, shear modulus G, Young’s modulus E, Pugh ratio B/G and Cauchy pressure (C12-C44). The database was then used to analyze the mechanical behavior of these alloys using phenomenological correlations. Furthermore, the electron density, density of states and charge density contour for selected materials are also presented in this study, and the results suggest that the bulk modulus of RuAl-TM increase as a function of electron density. The theoretical predictions on the considered properties of RuAl-based alloys may be helpful for their application in engineering. Acknowledgments This work is supported by the supported by the National Basic Research Program of China (Grant No. 2011CB606401). References [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]
Fleischer R. Metall Trans A 1993;24:227. Wolff IM. JOM 1997;49:34. Mücklich F, Ili c N, Woll K. Intermetallics 2008;16:593. Mücklich F, Ili c N. Intermetallics 2005;13:5. Guitar MA, Woll K, Ramos-Moore E, Mücklich F. Thin Solid Films 2013;527:1. Bax B, Pauly C, Leibenguth P, Woll K, Mücklich F. Surf Coat Technol 2012;206: 3931. Woll K, Holzapfel C, Mücklich F. Intermetallics 2010;18:553. Soldera F, Ili c N, Manent Conesa N, Barrientos I, Mücklich F. Intermetallics 2005;13:101. Ili c N, Soldera F, Mücklich F. Intermetallics 2005;13:444. Zotov N, Woll K, Mücklich F. Intermetallics 2010;18:1507. Acosta M, Reynolds TD, Johnson DR, Kvam EP. Intermetallics 2010;18:2281. Zhu J, Zhang C, Cao W, Yang Y, Zhang F, Chen S, et al. Acta Mater 2009;57:202. Povarova KB, Morozov AE, Padalko AG, Drozdov AA. Russ Metall 2008;2008: 155. Fleischer RL. Acta Metall Mater 1993;41:1197. Liu KW, Mücklich F. Intermetallics 2005;13:373. de Medeiros SN, Machado FLA, Zampiere RB, Santos IA, Paesano Jr A. J Noncryst Solids 2006;352:3718. Cao F, Pollock TM. Metall Mater Trans A 2008;39A:39. Cao F, Nandy TK, Stobbe D, Pollock TM. Intermetallics 2007;15:34. Povarova KB, Morozov AE, Drozdov AA, Kazanskaya NK. Russ Metall 2011;2011:865. Wolff IM, Sauthoff G. Acta Mater 1997;45:2949. Bleskov ID, Isaev EI, Vekilov YK. Phys Solid State 2010;52:1803. Fleischer RL, Field RD, Briant CL. Metall Trans A 1991;22:403. Fleischer RL. Acta Metall Mater 1993;41:863. Wolff IM, Sauthoff G. Metall Mater Trans A 1996;27:1395. Borah A, Robi PS, Mujumdar A, Srinivasan A. Met Mater-Int 2008;14:123. Hill PJ, Cornish LA, Witcomb MJ. J Alloy Compd 1999;291:130. Reynolds TD, Johnson DR. Intermetallics 2004;12:157. Feng Q, Nandy TK, Tryon B, Pollock TM. Intermetallics 2004;12:755. ski B. J Low Temp Phys 2007;149: Rosenbaum R, Mi S, Grushko B, Przepiórzyn 314. Eow K, Lu D, Pollock TM. Scr Mater 1998;38:1065. Fleischer RL, McKee DW. Metall Trans A 1993;24:759.
29
[32] Benarchid MY, David N, Fiorani JM, Vilasi M, Benlaharche T. Thermochim Acta 2009;482:39. [33] Meshi L, Samuha S, Kapush D, Pavlyuchkov D, Grushko B. J Alloy Compd 2011;509:6551. [34] Grushko B, Kapush D, Velikanova TY, Samuha S, Meshi L. J Alloy Compd 2011;509:8018. [35] Pavlyuchkov D, Grushko B, Velikanova TY. J Alloy Compd 2009;469:146. [36] Pavlyuchkov D, Grushko B, Velikanova TY. J Alloy Compd 2008;464:101. [37] Bozzolo GH, Noebe RD, Amador C. Intermetallics 2002;10:149. [38] Clerc DG, Ledbetter HM. J Phys Chem Solids 1998;59:1071. [39] Jhi S, Ihm J, Louie SG, Cohen ML. Nature 1999;399:132. [40] Ravindran P, Fast L, Korzhavyi PA, Johansson B, Wills J, Eriksson O. J Appl Phys 1998;84:4891. [41] Clark SJ, Segall MD, Pickard CJ, Hasnip PJ, Probert MIJ, Refson K, et al. Z Krist 2005;220:567. [42] Perdew JP, Wang Y. Phys Rev B 1992;45:13244. [43] Perdew JP, Burke K, Ernzerhof M. Phys Rev Lett 1996;77:3865. [44] Monkhorst HJ, Pack JD. Phys Rev B 1976;13:5188. [45] Hohenberg P, Kohn W. Phys Rev 1964;136:B864. [46] Ruban AV, Skriver HL. Phys Rev B 1997;55:856. [47] Ponomareva AV, Isaev EI, Vekilov YK, Abrikosov IA. Phys Rev B 2012;85: 144117. [48] Jiang C. Acta Mater 2007;55:4799. [49] Jiang C. Acta Mater 2008;56:6224. [50] Jiang C, Gleeson B. Scr Mater 2006;55:433. [51] Gargano P, Mosca H, Bozzolo G, Noebe RD. Scr Mater 2003;48:695. [52] Prins S, Arroyave R, Liu Z-K. Acta Mater 2007;55:4781. [53] Fleischer RL. J Mater Sci Lett 1988;7:525. [54] Voigt W. Ann Phys-berlin 1889;274:573. [55] Reuss AZ. Angew Math Mech 1929;9:49. [56] Hill R. Proc Phys Soc Sect A 1952;65:349. [57] Schreiber E, Anderson OL, Soga N. Elastic constants and their measurements. New York: McGraw-Hill; 1973. [58] Zener C. Elasticity and anelasticity of metals. Chicago: University of Chicago Press; 1948. [59] Obrowski W. Naturwissenschaften 1960;47:14. [60] Edshammar LE. Acta Chem Scand 1966;20:427. [61] Gobran HA, Liu KW, Heger D, Mücklich F. Scr Mater 2003;49:1097. [62] Mi S, Grushko B. Intermetallics 2004;12:425. [63] Fleischer RL. Isij Int 1991;31:1186. [64] Mehl MJ, Klein BM, Papaconstantopoulos DA. Intermetallic compounds: principles and applications. London: Wiley; 1994. [65] Wen B, Zhao J, Bai F, Li T. Intermetallics 2008;16:333. [66] Nguyen-Manh D, Pettifor DG. Intermetallics 1999;7:1095. [67] Bleskov ID, Smirnova EA, Vekilov YK, Korzhavyi PA, Johansson B, Katsnelson M, et al. Appl Phys Lett 2009;94. [68] Lin W, Xu J, Freeman AJ. J Mater Res 1992;7:592. [69] Mehl MJ, Singh DJ, Papaconstantopoulos DA. Mater Sci Eng A 1993;170:49. [70] Li C, Wu P. Chem Mater 2001;13:4642. [71] Ghosh G, Olson GB. Acta Mater 2007;55:3281. [72] Mosca HO, Bozzolo G, Garcés JE. Scr Mater 2008;58:1025. [73] Wallace DC. Thermodynamics of crystals. New York: Wiley; 1972. [74] Shang SL, Saengdeejing A, Mei ZG, Kim DE, Zhang H, Ganeshan S, et al. Comput Mater Sci 2010;48:813. [75] Rose JH, Shore HB. Phys Rev B 1991;43:11605. [76] Rose JH, Shore HB. Phys Rev B 1993;48:18254. [77] Rose JH, Shore HB. Phys Rev B 1994;49:11588. [78] Nye JF. Physical properties of crystals: their representation by tensors and matrices. New York: Oxford University Press; 1985. [79] Barbalace K. http://environmentalchemistry.com/yogi/periodic/. [80] Pugh SF. Philos Mag 1954;45:823. [81] Pettifor DG. Mater Sci Technol 1992;8:345. [82] Niu HY, Chen XQ, Liu PT, Xing WW, Cheng XY, Li DZ, et al. Sci Rep 2012;2:718. [83] Chen XQ, Niu HY, Li DZ, Li YY. Intermetallics 2011;19:1275. [84] Chen XQ, Niu HY, Franchini C, Li DZ, Li YY. Phys Rev B 2011;84:121405.