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Mechanical properties of structural materials from first-principles
Current Opinion in Solid State and Materials Science 10 (2006) 19–25
Mechanical properties of structural materials from first-principles Qing Miao Hu *, Rui Yang Shenyang National Laboratory for Materials Science, Institute of Metal Research, Chinese Academy of Sciences, 72 Wenhua Road, Shenyang 110016, China Received 3 December 2005
Abstract First-principles method based on the electronic structure theory is one of the most promising approaches of computational materials design. Although only a few of mechanical properties (e.g., ideal strength and elastic constants) are accessible directly by first-principles calculations, such methods may predict the complex mechanical properties by extracting appropriate calculable parameters (e.g., the ratio of bulk modulus to shear modulus, the formation energies of and interaction energies between lattice defects) and adopting proper models (e.g., Peierls–Nabarro model for dislocation core). In this paper, we briefly review recent first-principles investigations of mechanical properties of structural materials, covering topics of ideal strength, elastic constants, and lattice defects. Some of the major recent advances, such as the application of coherent potential approximation coupled first-principles methods (accurate enough for the calculating of the elastic constants of random alloys with complex compositions), the appreciation of the importance of low C11–C12 to the ‘super properties’ of BCC-Ti based alloys, and the relationship between solute–vacancy interaction and creep resistance, etc., are highlighted. Ó 2006 Elsevier Ltd. All rights reserved. Keywords: First-principles methods; Mechanical properties; Ideal strength; Elastic modulus; Lattice defects
1. Introduction Mechanical properties are the main concerns for structural materials. The classical way of obtaining materials with desired properties is so called ‘trial and error’, i.e., trying to find the ‘formula’ of compositions of the materials more or less randomly from combinations of hundreds of chemical species as shown in the periodic table. Being costly both in time and economics, this kind of ‘materials design’ is obviously not the optimal. Materials scientists have long expected that they could identify the appropriate ‘formula’ for the desired properties more efficiently, i.e., select the chemical species intentionally according to the target mechanical properties. With the development of materials modeling methods and computer technologies, the dream of materials scientists is closer to reality. Among diverse *
Corresponding author. Tel.: +86 24 2397 1946; fax: +86 24 2397 2021. E-mail addresses: [email protected] (Q.M. Hu), [email protected] (R. Yang). 1359-0286/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.cossms.2006.02.002
materials modeling methods, first-principles (or ab initio) methods based on electronic structure theory [1] are one of the most promising. By solving self-consistently the Schro¨dinger equation of a system with given chemical composition and lattice structure (no experimental or empirical parameters are needed), these methods generate the electronic wave-functions and related physical quantities, such as total energy, interatomic force, etc., of the system. Most properties of solids depend on the behavior of electrons since electrons are the ‘glue’ which binds the atomic nuclei to form solids, and therefore can be understood on the basis of the electronic structures of the solids. However, understanding the mechanical properties through first-principles electronic structure calculations is sometimes not straightforward because many factors, from electronic to atomic, microstructural, and continuum medium level, are involved. Only a few simple mechanical properties such as ideal strength and elastic constants are accessible directly from first-principles calculations. Nevertheless, most mechanical properties such as yield strength are lattice structure-sensitive and
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highly dependent on the lattice defects such as vacancies, interstitial or substitutional atoms, dislocations, grain boundaries, stacking faults, etc. Therefore, first-principles investigations of lattice defects may provide an indirect but efficient way of prediction of mechanical properties. In this paper, we attempt to give a brief review of the status of the investigation of mechanical properties by the use of first-principles methods during mainly the past two years, covering topics of ideal strength, elastic constants, and lattice defects. Limited by the length of this review, we should point out that we cannot mention all of the publications in this very active area, and cannot go into much detail of each topic. Due to the same reason, this review does not include some other topics (such as phase stability and phase transition, phase diagram, lattice dynamics, etc.) that may be also of interest to the design of structural materials. 2. Ideal strength Ideal strength represents the upper strength limit of a crystal, provides insights into how the crystal is bonded and, therefore, is of both engineering and theoretical interests. It is one of the properties accessible directly from firstprinciples calculations. Applying a series of incremental strains to an unstrained crystal (stretching or shearing the crystal in the desired direction), one may obtain the resulted stresses using first-principles method. The ideal strength is defined as the maximum stress needed to render the materials elastically unstable. During the past two years, first-principles calculations of the ideal strengths of materials such as pure metals (e.g., Fe [*2], Al [3–5], Si [4], and Cu [5]), intermetallic compounds (e.g., transition metal aluminides [6,7]), and ceramics (e.g., Si3N4 [8,9]), etc., have been extensively reported in literature. Besides the ideal strength, it was recently proposed by Yip and collaborators that, the maximum shear strain where the ideal shear strength occurs, i.e., the shearability as they defined it, is also an intrinsic property of materials which shows quantitatively the electronic and atomic response of solid at the point of bond breaking [**10]. Yip and collaborators have checked the stress–strain responses of 22 simple metals and ceramics. They found that there exists a gap between the shearabilities of metals and ceramics [**10]. The concept of shearability may be used to explain why Al has larger ideal strength than Cu though Al has a smaller modulus in f1 1 1gh1 1 2i shear [5], i.e., Al has a larger shearability (more extended deformation range before elastic instability) than Cu. First-principles calculations of ideal strength have been mostly focused on perfect crystals because a larger system is needed to represent crystal defects, and consequently more computational resource is required. With the development of electronic structure theory (e.g., first-principles pseudopotential method) and computer technology, theoreticians have also paid some attention to the ideal strength of defective crystals. The ideal tensile strength of an Al R9 grain boundary has been studied by the use of first-princi-
ples pseudopotential method [*11]. It was shown that the ideal strength of the grain boundary area does not decrease very much compared to the perfect crystal due to the reconstruction of the grain boundary. The ideal strength of defective crystals presents some information on intrinsic local properties of the defected region in real materials. 3. Elastic properties Elastic constant is another mechanical property that can be evaluated directly by first-principles methods. While ideal strength represents the large-strain nonlinear response of the materials, elastic constant reflects the linear response of the lattice to small strain around equilibrium. By applying a series of strain pattern on a crystal in equilibrium, a first-principles method gives the resulted stress and total energies. Fitting the stress–strain or total energies–strain curves according to their elastic relationships, one gets the elastic constants of the crystal (see Ref. [12] for review). In recent years, the Laves phase [12–15], transition/ noble metal aluminides/nitrides/carbides [16–19], and platinum–metal based compounds [20], possessing relatively high strength and melting temperature, have drawn much attention due to their potential high-temperature applications (e.g., in aerospace or airspace engine). The elastic constants of these intermetallic compounds have been extensively studied by first-principles calculations [12–20]. The occupations of the atoms on the lattice of the abovementioned materials are well defined. However, most real structural materials are of very complex composition and may contain alloying atoms distributing randomly in the crystal. In principle, the site on the crystal lattice can only be occupied by one species in the framework of first-principles methods, which makes the direct calculations of random alloys, especially those with complex multiple components, very inconvenient. This difficulty is now resolved by the implementation of the coherent potential approximation (CPA) [21] in first-principles methods by Vitos et al. [22]. By the use of full charge-density scheme and extended muffin–tin orbital for the basis sets [22,23], the first-principles CPA method is accurate enough to calculate the elastic constants of random alloys. This method has been applied successfully in the calculations of elastic constants of austenitic stainless steels (Fe–Cr–Ni and Fe– Cr–Ni–Mo) [**24] and random Al–Li alloys [25]. Results in good agreement with experimental values were obtained. The first-principles CPA method provides a feasible way for the first-principles calculation of the properties of real alloys. From the first-principles calculated elastic constants, one may get the isotropic elastic moduli, bulk modulus B, shear modulus G, and Young’s modulus E using schemes such as Heuss, Voigt, or Hill averaging [26]. Some other elastic properties such as Possion’s ratio, anisotropy value, etc., can also be derived. These quantities are closely related to the more complex mechanical properties of materials such as ductility and plasticity, etc.
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In 2003, Gschneidner et al. found a family of ductile rareearth intermetallic compounds with B2 structure [**27]. Their calculations showed that the ductile compounds have Poisson ratios and anisotropy values of about 0.3 and 1.0, respectively, close to those of BCC transition metals, indicating that the compounds are more isotropic in nature than the brittle non-rare-earth B2 phases [**27,*28]. This may account for the absence of brittleness of rare-earth compounds. The exploration of the physics behind the mechanical properties of the rare-earth may shed new light on the improvement of the ductility of other intermetallic compounds. In as early as 1954, on the basis of an analysis of the ratio of bulk modulus to shear modulus, B/G, of pure metals, Pugh concluded that B/G reflects the ductility of the metals [29]: the larger is the B/G, the more ductile the metals. Similar trend was also found for other materials such as intermetallic compounds. B/G is now widely accepted as a predictor in first-principles materials design. Vitos et al. have calculated the elastic constants and B/G value as well as shear modulus as functions of compositions using first-principles CPA methods [**24]. From the chemical composition-B/G and composition–shear modulus relations, they predicted two new basic compositions for austenitic stainless steels: One (Fe–13Cr–8Ni) with excellent hardness and the other (Fe–18Cr–24Ni) with remarkable resistance to various forms of localized corrosion and intermediate hardness. Beside B/G, it was recently found that the C11–C12 is also very significant on the mechanical properties of materials. Ikehata et al. have calculated the elastic constants of binary Ti alloys with alloying elements of V, Nb, Ta, Mo, or W using first-principles pseudopotential supercell method [*30]. The results showed that, when the alloy falls to a range of compositions with valence electron number of 4.20–4.24, the C11–C12 approaches zero, which was believed to contribute to the low Young’s modulus, superplasiticity, and superelasticity of the so called ‘gum metal’ [31]. Similarly, the calculation of Souvatzis et al. showed that some other binary alloys such as W–Re, W–Tc, Mo–Tc and Mo–Re with valence electron number of 6.6–6.9 also have vanishingly low C11–C12, and therefore may be candidates for ‘gum metal’ [32]. According to these investigations, C11–C12 could be treated as another useful index in the theoretical search of new materials. Elastic moduli also enter into some classical models for mechanical properties as parameters, which makes it possible to predict more complicated mechanical properties on the basis of first-principles calculations. For example, the theoretical stress to create a crack is given by the Orowan criterion, rmax ¼ ðEcs =a0 Þ
1=2
;
with E being the Young’s modulus, cs the surface energy, and a0 the inter-planar spacing. The elastic driving force on an equilibrium crack is C ¼ ð1 mÞK 2I =2G; with G being the shear modulus, m the Poisson’s ratio, and KI the stress intensity factor. These first-principle combined
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models have recently been used to study the correlation between hardness and fracture toughness of brittle materials B6O, cBN, 3C-SiC, and Si by Ding et al. [33]. 4. Lattice defect Unlike the ideal strength and elastic properties that are directly dominated by the nature of the bonds, some other mechanical properties, such as yield strength, fracture strength, creep resistance, etc., are sensitive to the lattice structures. A well-known example of the structure-sensitive properties is that, due to the defects in the lattice, the yield strength of a real crystal is only about 1/1000 of its theoretical value with perfect lattice. Therefore, it is critical to understand the behavior of the lattice defects in order to predict the mechanical properties of the structural materials. According to their spatial pattern, there are three kinds of lattice defects: zero-dimensional point defects such as vacancy, interstitial atoms, and alloying atoms; one-dimensional dislocation; two-dimensional defects such as stacking fault, grain boundary, etc. We will discuss recent first-principles investigation of these lattice defects accordingly. 4.1. Zero-dimensional defects: point defects Point defects mostly correlate to properties related to atomic diffusion, for example, creep resistance. First-principles investigations of point defects in metals e.g., [34], intermetallic compounds e.g., [35–37], random alloys e.g., [38], and so on, are abundant in literature. However, how on earth the point defects affect the mechanical properties is seldom discussed in these papers. We have calculated the interaction energy between a series of alloying atoms and vacancy in a-Ti alloys using linear muffin–tin orbital method within atomic sphere approximation (LMTOASA) [*39]. Comparing the interaction energies with available experimental creep resistance, we found that those alloying atoms attractive to vacancy increase the creep resistance whereas those repulsive to vacancy do not (see Fig. 1 and Tables 1 and 2). This is understandable because the interaction between alloying atoms and vacancy facilitates the alloying atoms to diffuse to dislocations and form Cottrell atmosphere near the dislocation, therefore inhibiting the glide of dislocation when creeping. We have also investigated the interaction between alloying atoms in aTi alloys, which gives the clustering/ordering tendency of the alloy [40]. Using the Flinn model, the interaction energy can be used to predict the contribution of the short-range order to the critical shear stress. Correct trend was found for the strengthening effect of Al, Si, Ga, Ge on a-Ti alloys. Recently, Mayer et al. have studied the effect of interstitial H on the bulk modulus and C44 of Nb using fullpotential variant LMTO method [16]. It was shown that, with the increase in the content of H, the bulk modulus of Nb increases but C44 decreases, both in agreement with experiments. In the same paper, they also reported the
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Y Zr Nb Mo Tc Ru Rh Pd Ag Cd In Sn 0.1
ΔE (eV)
0.0 -0.1 -0.2 -0.3
Y to Sn Al, Si, and Sc to Ge Mn to Cu, spin polorized
-0.4 Al Si
Sc Ti V Cr Mn Fe Co Ni Cu Zn Ga Ge
Alloying Atoms Fig. 1. Interaction energy between vacancy and alloying atom in a-Ti, defined as DE = [E(V + S, N) + E(N)] [E(V, N) + E(S + N)]. E(V + S, N), E(V, N), and E(S, N) are the total energy of the N-site supercell containing one vacancy and one alloying atom that are the nearest neighbor of each other, one vacancy only, and one alloying atom only, respectively; E(N) is the total energy of the perfect supercell. Negative value refers to attractive interaction between the vacancy and the alloying atom, whereas positive to repulsive.
Table 1 Experimental 100-h creep rupture stress of unalloyed a titanium and a titanium based alloys at different temperature (see [*39, and reference therein]) Alloy (wt.%)
Rupture stress (MPa) 425 °C
500 °C
550 °C
600 °C
Ti Ti–2Al Ti–3Al Ti–4Al Ti–7Ga Ti–11In Ti–10Zr Ti–10Sn
157.8 274.4 – 382.2 – – – –
– – 151.9 – 311.6 121.5 – –
46.6 82.3 – – – – – –
– – – 73.5 – – 73.5 49.0a
a
dislocation can be described successfully by using continuum elasticity methods. However, the dislocation core region that sometimes affects the plasticity of materials dramatically is out of the capability of continuum elasticity methods. First-principles investigations of dislocation core region have drawn much attention in recent years. Direct modeling of an isolated dislocation by using supercell (with periodic boundary condition) based first-principles methods is not practical since: (1) the stress field induced by a dislocation is long ranged, therefore extremely large supercell must be used to avoid the interaction between the dislocations in nearby supercells, which is out of the scope of current first-principles methods; (2) the geometry of an isolated dislocation breaks periodic symmetry. Several approaches have been developed to tackle the above problems: (1) simulation of dislocation dipole arrays by non-isolated dislocation (e.g., [41]); (2) direct simulation of isolated dislocation using flexible boundary conditions (e.g., [42,43]); (3) conjunction of first-principles derived generalized staking fault energy (so called c surface) to twodimensional Peierls–Nabarro description of the dislocation core (see [*44] for review). For details of these approaches, we refer the readers to a recent review paper by Woodward [*45]. A most recent development in the area of first-principles investigation of dislocation, not covered in Ref. [*45], is the incorporation of c-surface to phase field model, proposed by Shen and Wang [*46]. Using this approach, they could realistically treat the dissociation of dislocation with arbitrary configuration, interaction among extended dislocations, and creation and annihilation of various planar faults. 4.3. Two-dimensional defects: stacking fault and twin
150-h creep rupture stress.
Table 2 Creep deformation of a based Ti–5Al–5Sn–2Zr–0.8Mo–0.5Si alloyed with some b stabilizers at 538 °C and 379 MPa after 96 h (see [35] and references therein) Additions
0
1.2Mo
2.0V
4.2Nb
0.9Cr
0.9Ni
Creep deformation (%)
0.58
0.27
0.54
0.45
0.71
>10.0
elastic constants of NbCr2xVx as a function of composition, where C44 remains almost constant and the bulk modulus decreases with increasing x [16]. 4.2. One-dimensional defects: dislocations Plastic deformation is mainly controlled by the glide of dislocations. In simulation, the long-range stress field of
When the glide of dislocation is blocked for some reasons, crystals have to resort to the generation of stacking fault or twin (planar lattice defects) to deform. Higher tendency of this type of plastic deformation (defined as twinnability in Ref. [47]) correlates to a lower stacking fault or twin formation energy. These formation energies can be easily obtained by comparing the total energies of the defective and perfect crystals from first-principles calculations. In Ref. [47], the intrinsic stacking fault energy, unstable stacking fault energy, and unstable twinning energy, as well as extrinsic stacking fault energy and twin boundary energy have been reported for eight FCC metals. The ‘twinnability’ of the metals predicted from these energies was found to be in agreement with the experimental results. 4.4. Interaction between defects of different dimension Since most structural materials are alloys and impurities (e.g., H, C, P, S, etc.) are introduced inevitably into the alloys during the process of smelting or through the chemical reaction between the materials and the atmosphere, a knowledge of the interactions between these point defects (alloying atoms/impurities) and other lattices defects,
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such as dislocations, stacking fault, and grain boundaries, sometimes is crucial to understanding the mechanical properties of materials. In a recent paper of Medvedeva, Gornostyrev, and Freeman [**48], interactions between alloying atoms and generalized stacking fault energy (GSF, i.e., c surface) in BCC-Mo alloys have been investigated by using fullpotential LMTO method, and the electronic origin of the solid solution softening in these alloys was revealed. The authors demonstrated that additions with more 5d-electrons (Re, Os, Ir, and Pt) than Mo lead to a decrease in the GSF energy and those with less 5d-electrons (Hf and Ta) to its sharp increase. Using the generalized Peierls– Nabarro model for a nonplanar dislocation core, they associate the local reduction of the GSF energy with an enhancement of double kink nucleation and an increase of the dislocation mobility, which contributes to the solid solution softening of the BCC-Mo alloys. Grain boundary can also be considered as a kind of planar defects. The relative strength of grain boundaries to the bulk crystal controls the manner of fracture: if the grain boundaries are weaker/stronger than the bulk crystal, the metal is most likely to exhibit intergranular/transgranular fracture. The segregation of alloying elements or impurities can affect the strength of the grain boundary significantly. This topic has been extensively investigated in literature for metals (e.g., Al [49], Fe [50–53], Ni [53–56], Cu [*57,58]), intermetallic compounds (e.g., Ni3Al [*59]), and ceramics (e.g., Al2O3 [60,61]). For the interstitial impurities, H, P, S are mostly identified by first-principles calculations as embrittlement elements due to their decohesive effect on the grain boundaries [50,54,*59], in agreement with experimental findings. The segregation of these elements to grain boundary is the main reason for the brittle intergranular fracture according to the theory of Rice and Wang [62]. The first-principles investigation of H-induced embrittlement is recently reviewed by Geng et al. [63]. Contrary to H, P, S, etc., the segregation of B generally leads to cohesive enhancement of the grain boundary [51,*59]. There were generally two arguments concerning the origin of the cohesive strengthening/weakening effect of the alloying atoms or impurities on the grain boundary: (1) electronic effect, for example, B is believed to form strong covalent bond with Ni atoms in the grain boundary of Ni and Ni3Al [*59, and reference therein] which enhances the binding of the grain boundary; (2) atomic size effect, for example, it was argued that the large Bi atoms weaken the interatomic bonding by pushing apart the host Cu atoms at the grain boundary, and therefore make the grain boundary brittle [*57]. 5. Conclusion and remark In conclusion, we presented a brief review of the first-principles investigations of mechanical properties of structural materials in recent years. The results of these investigations demonstrated that first-principles methods
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can be practical tools for the design of structural materials. Since first-principles calculations are in principle at electronic level and performed at zero temperature, while the mechanical properties of structural materials are generally macroscopic and related to finite temperature, further efforts to identify or define more calculable parameters and models so as to bridge the size and temperature gaps between first-principles method and complex mechanical properties will be important for the design of new materials. Acknowledgements The authors are very grateful for the support from Chinese Academy of Sciences (Grant INF105-SCE-02-04), the Chinese MoST (Grant TG2000067105), and NSFC (Grant 50301014). References The papers of particular interest have been highlighted as: * of special interest; ** of very special interest. [1] Dreizler RM, Gross EKU. Density functional theory. Berlin: Springer; 1998. [*2] Clatterbuck DM, Chrzan DC, Morris Jr JW. The ideal strength of iron in tension and shear. Acta Mater 2003;51:2271–83. [3] Clatterbuck DM, Krenn CR, Cohen ML, Morris Jr JW. Phonon instabilities and the ideal strength of aluminum. Phys Rev Lett 2003;91:135501. [4] Yashiro K, Oho M, Tomita Y. Ab initio study of the lattice instability of silicon and aluminum under [0 0 1] tension. Comput Mater Sci 2004;29:397–406. [5] Ogata S, Li J, Yip S. Ideal pure shear strength of aluminum and copper. Science 2002;298:807–11. [6] Jahnatek M, Krajci M, Hafner J. Interatomic bonding, elastic properties, and ideal strength of transition metal aluminides: a case study for Al3(V, Ti). Phys Rev B 2005;71:024101. [7] Li T, Morris Jr JW, Chrzan DC. Ideal strength of B2 transitionmetal aluminides. Phys Rev B 2004;70:054107. [8] Ogata S, Hirosaki N, Kocer C, Shibutani Y. A comparative ab initio study of the ideal strength of single crystal a- and b-Si3N4. Acta Mater 2004;52:233–8. [9] Kiefer B, Shieh SR, Duffy TS, Sekine T. Strength, elasticity, and equation of state of the nanocrystalline cubic silicon nitride c-Si3N4 to 68 GPa. Phys Rev B 2005;72:014102. [**10] Ogata S, Li J, Hirosaki N, Shibutani Y, Yip S. Ideal shear strain of metals and ceramics. Phys Rev B 2004;70:104104. [*11] Lu GH, Deng SH, Wang TM, Kohyama M, Yamamoto R. Theoretical tensile strength of an Al grain boundary. Phys Rev B 2004;69:134106. [12] Mehl MJ, Klein BM, Papaconstantopoulos DA. First principles calculations of elastic properties of metals. In: Westbrook JH, Fleischer RL, editors. Intermetallic compounds: principles and practice. London: John Wiley and Sons; 1994. p. 195–210. [13] Sun J, Jiang B. Ab initio calculation of the phase stability, mechanical properties and electronic structure of ZrCr2 Laves phase compounds. Philos Mag 2004;84:3133–44. [14] Kellou A, Grosdidier T, Coddet C, Aourag H. Theoretical study of structural, electronic, and thermal properties of Cr2(Zr, Nb) Laves alloys. Acta Mater 2005;53:1459–66.
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[*39] Hu QM, Xu DS, Li D. First-principles investigations of solute– vacancy interaction energy and its effect on the creep properties of a-titanium. Philos Mag A 2001;81:2809–21. [40] Hu QM, Xu DS, Yang R, Li D. First-principles investigations of ordering in binary a-Ti solid solutions. Philos Mag 2003;83:217–29. [41] Frederiksen SL, Jacobsen K. Density functional theory of screw dislocation core structure in bcc metals. Philos Mag 2003;83: 365–75. [42] Yan JA, Wang CY, Wang SY. Generalized-stacking-fault energy and dislocation properties in bcc Fe: a first-principles study. Phys Rev B 2004;70:174105. [43] Woodward C, Rao SI. Ab-initio simulation of (a/2)h1 1 0i screw dislocation in c-TiAl. Philos Mag 2004;84:401–13. [*44] Schoeck G. The Peierls model: progress and limitations. Mater Sci Eng A 2005;400–401:7–17. [*45] Woodward C. First-principles simulations of dislocation cores. Mater Sci Eng A 2005;400–401:59–67. A review. [*46] Shen C, Wang Y. Incorporation of c-surface to phase field model of dislocations: simulating dislocation dissociation in fcc crystals. Acta Mater 2004;52:683–91. [47] Bernstein N, Tadmor EB. Tight-binding calculations of stacking energies and twinnability in fcc metals. Phys Rev B 2004;69:094116; Tadmor EB, Bernstein N. A first-principles measure for the twinnability of FCC metals. J Mech Phys Solids 2004;52:2507–19. [**48] Medvedeva NI, Gornostyrev YN, Freeman AJ. Electronic origin of solid solution softening in bcc molybdenum alloys. Phys Rev Lett 2005;94:136402. [49] Liu XG, Wang XW, Wang JY, Zhang HY. First-principles investigation of Mg segregation at R11(1 1 3) grain boundaries in Al. J Phys: Condens Matter 2005;17:4301–8. [50] Braithwaite JS, Rez P. Grain boundary impurities in iron. Acta Mater 2005;53:2715–26. [51] Wang SY, Wang CY, Zhao DL. Effect of double boron impurities on the alpha-iron grain boundary cohesion. J Alloys Compd 2004;368:308–11. [52] Kim M, Geller CB, Freeman AJ. The effect of interstitial N on grain boundary cohesive strength in Fe. Scripta Mater 2004;50:1341–3. [53] Yang R, Huang RZ, Wang YM, Ye HQ, Wang CY. The effects of 3d alloying elements on grain boundary cohesion in gamma-iron: a first principles study on interface embrittlement due to the segregation. J Phys: Condens Matter 2003;15:8339–49. [54] Yamaguchi M, Shiga M, Kaburaki H. Grain boundary decohesion by impurity segregation in a nickel–sulfur system. Science 2005;307:393–7; Yamaguchi M, Shiga M, Kaburaki H. Response to comment on ‘Grain boundary decohesion by impurity segregation in a nickel– sulfur system. Science 2005;309:1677d; Geng WT, Wang JS, Olson GB. Comment on ‘Grain boundary decohesion by impurity segregation in a nickel–sulfur system. Science 2005;309:1677. [55] Yamaguchi M, Shiga M, Kaburaki H. Energetics of segregation and embrittling potency for non-transition elements in the Ni R5(0 1 2) symmetrical tilt grain boundary: a first-principles study. J Phys: Condens Matter 2004;16:3933–56. [56] Siegel DJ, Hamilton JC. Computational study of carbon segregation and diffusion within a nickel grain boundary. Acta Mater 2005;53:87–96. [*57] Schweinfest R, Paxton AT, Finnis MW. Bismuth embrittlement of copper is an atomic size effect. Nature 2004;432:1008–11. [58] Duscher G, Chisholm MF, Alber U, Ruhle M. Bismuth-induced embrittlement of copper grain boundaries. Nature Mater 2004;3: 621–6. [*59] Hu QM, Yang R, Xu DS, Hao YL, Li D. Energetics and electronic structure of grain boundaries and surfaces of B- and H-doped Ni3Al. Phys Rev B 2003;67:224203. [60] Elsasser C, Elsasser T. Codoping and grain-boundary cosegregation of substitutional cations in alpha-Al2O3: a density-functionaltheory study. J Am Ceram Soc 2005;88:1–14.
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Mechanical properties of structural materials from first-principles Qing Miao Hu *, Rui Yang Shenyang National Laboratory for Materials Science, Institute of Metal Research, Chinese Academy of Sciences, 72 Wenhua Road, Shenyang 110016, China Received 3 December 2005
Abstract First-principles method based on the electronic structure theory is one of the most promising approaches of computational materials design. Although only a few of mechanical properties (e.g., ideal strength and elastic constants) are accessible directly by first-principles calculations, such methods may predict the complex mechanical properties by extracting appropriate calculable parameters (e.g., the ratio of bulk modulus to shear modulus, the formation energies of and interaction energies between lattice defects) and adopting proper models (e.g., Peierls–Nabarro model for dislocation core). In this paper, we briefly review recent first-principles investigations of mechanical properties of structural materials, covering topics of ideal strength, elastic constants, and lattice defects. Some of the major recent advances, such as the application of coherent potential approximation coupled first-principles methods (accurate enough for the calculating of the elastic constants of random alloys with complex compositions), the appreciation of the importance of low C11–C12 to the ‘super properties’ of BCC-Ti based alloys, and the relationship between solute–vacancy interaction and creep resistance, etc., are highlighted. Ó 2006 Elsevier Ltd. All rights reserved. Keywords: First-principles methods; Mechanical properties; Ideal strength; Elastic modulus; Lattice defects
1. Introduction Mechanical properties are the main concerns for structural materials. The classical way of obtaining materials with desired properties is so called ‘trial and error’, i.e., trying to find the ‘formula’ of compositions of the materials more or less randomly from combinations of hundreds of chemical species as shown in the periodic table. Being costly both in time and economics, this kind of ‘materials design’ is obviously not the optimal. Materials scientists have long expected that they could identify the appropriate ‘formula’ for the desired properties more efficiently, i.e., select the chemical species intentionally according to the target mechanical properties. With the development of materials modeling methods and computer technologies, the dream of materials scientists is closer to reality. Among diverse *
Corresponding author. Tel.: +86 24 2397 1946; fax: +86 24 2397 2021. E-mail addresses: [email protected] (Q.M. Hu), [email protected] (R. Yang). 1359-0286/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.cossms.2006.02.002
materials modeling methods, first-principles (or ab initio) methods based on electronic structure theory [1] are one of the most promising. By solving self-consistently the Schro¨dinger equation of a system with given chemical composition and lattice structure (no experimental or empirical parameters are needed), these methods generate the electronic wave-functions and related physical quantities, such as total energy, interatomic force, etc., of the system. Most properties of solids depend on the behavior of electrons since electrons are the ‘glue’ which binds the atomic nuclei to form solids, and therefore can be understood on the basis of the electronic structures of the solids. However, understanding the mechanical properties through first-principles electronic structure calculations is sometimes not straightforward because many factors, from electronic to atomic, microstructural, and continuum medium level, are involved. Only a few simple mechanical properties such as ideal strength and elastic constants are accessible directly from first-principles calculations. Nevertheless, most mechanical properties such as yield strength are lattice structure-sensitive and
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highly dependent on the lattice defects such as vacancies, interstitial or substitutional atoms, dislocations, grain boundaries, stacking faults, etc. Therefore, first-principles investigations of lattice defects may provide an indirect but efficient way of prediction of mechanical properties. In this paper, we attempt to give a brief review of the status of the investigation of mechanical properties by the use of first-principles methods during mainly the past two years, covering topics of ideal strength, elastic constants, and lattice defects. Limited by the length of this review, we should point out that we cannot mention all of the publications in this very active area, and cannot go into much detail of each topic. Due to the same reason, this review does not include some other topics (such as phase stability and phase transition, phase diagram, lattice dynamics, etc.) that may be also of interest to the design of structural materials. 2. Ideal strength Ideal strength represents the upper strength limit of a crystal, provides insights into how the crystal is bonded and, therefore, is of both engineering and theoretical interests. It is one of the properties accessible directly from firstprinciples calculations. Applying a series of incremental strains to an unstrained crystal (stretching or shearing the crystal in the desired direction), one may obtain the resulted stresses using first-principles method. The ideal strength is defined as the maximum stress needed to render the materials elastically unstable. During the past two years, first-principles calculations of the ideal strengths of materials such as pure metals (e.g., Fe [*2], Al [3–5], Si [4], and Cu [5]), intermetallic compounds (e.g., transition metal aluminides [6,7]), and ceramics (e.g., Si3N4 [8,9]), etc., have been extensively reported in literature. Besides the ideal strength, it was recently proposed by Yip and collaborators that, the maximum shear strain where the ideal shear strength occurs, i.e., the shearability as they defined it, is also an intrinsic property of materials which shows quantitatively the electronic and atomic response of solid at the point of bond breaking [**10]. Yip and collaborators have checked the stress–strain responses of 22 simple metals and ceramics. They found that there exists a gap between the shearabilities of metals and ceramics [**10]. The concept of shearability may be used to explain why Al has larger ideal strength than Cu though Al has a smaller modulus in f1 1 1gh1 1 2i shear [5], i.e., Al has a larger shearability (more extended deformation range before elastic instability) than Cu. First-principles calculations of ideal strength have been mostly focused on perfect crystals because a larger system is needed to represent crystal defects, and consequently more computational resource is required. With the development of electronic structure theory (e.g., first-principles pseudopotential method) and computer technology, theoreticians have also paid some attention to the ideal strength of defective crystals. The ideal tensile strength of an Al R9 grain boundary has been studied by the use of first-princi-
ples pseudopotential method [*11]. It was shown that the ideal strength of the grain boundary area does not decrease very much compared to the perfect crystal due to the reconstruction of the grain boundary. The ideal strength of defective crystals presents some information on intrinsic local properties of the defected region in real materials. 3. Elastic properties Elastic constant is another mechanical property that can be evaluated directly by first-principles methods. While ideal strength represents the large-strain nonlinear response of the materials, elastic constant reflects the linear response of the lattice to small strain around equilibrium. By applying a series of strain pattern on a crystal in equilibrium, a first-principles method gives the resulted stress and total energies. Fitting the stress–strain or total energies–strain curves according to their elastic relationships, one gets the elastic constants of the crystal (see Ref. [12] for review). In recent years, the Laves phase [12–15], transition/ noble metal aluminides/nitrides/carbides [16–19], and platinum–metal based compounds [20], possessing relatively high strength and melting temperature, have drawn much attention due to their potential high-temperature applications (e.g., in aerospace or airspace engine). The elastic constants of these intermetallic compounds have been extensively studied by first-principles calculations [12–20]. The occupations of the atoms on the lattice of the abovementioned materials are well defined. However, most real structural materials are of very complex composition and may contain alloying atoms distributing randomly in the crystal. In principle, the site on the crystal lattice can only be occupied by one species in the framework of first-principles methods, which makes the direct calculations of random alloys, especially those with complex multiple components, very inconvenient. This difficulty is now resolved by the implementation of the coherent potential approximation (CPA) [21] in first-principles methods by Vitos et al. [22]. By the use of full charge-density scheme and extended muffin–tin orbital for the basis sets [22,23], the first-principles CPA method is accurate enough to calculate the elastic constants of random alloys. This method has been applied successfully in the calculations of elastic constants of austenitic stainless steels (Fe–Cr–Ni and Fe– Cr–Ni–Mo) [**24] and random Al–Li alloys [25]. Results in good agreement with experimental values were obtained. The first-principles CPA method provides a feasible way for the first-principles calculation of the properties of real alloys. From the first-principles calculated elastic constants, one may get the isotropic elastic moduli, bulk modulus B, shear modulus G, and Young’s modulus E using schemes such as Heuss, Voigt, or Hill averaging [26]. Some other elastic properties such as Possion’s ratio, anisotropy value, etc., can also be derived. These quantities are closely related to the more complex mechanical properties of materials such as ductility and plasticity, etc.
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In 2003, Gschneidner et al. found a family of ductile rareearth intermetallic compounds with B2 structure [**27]. Their calculations showed that the ductile compounds have Poisson ratios and anisotropy values of about 0.3 and 1.0, respectively, close to those of BCC transition metals, indicating that the compounds are more isotropic in nature than the brittle non-rare-earth B2 phases [**27,*28]. This may account for the absence of brittleness of rare-earth compounds. The exploration of the physics behind the mechanical properties of the rare-earth may shed new light on the improvement of the ductility of other intermetallic compounds. In as early as 1954, on the basis of an analysis of the ratio of bulk modulus to shear modulus, B/G, of pure metals, Pugh concluded that B/G reflects the ductility of the metals [29]: the larger is the B/G, the more ductile the metals. Similar trend was also found for other materials such as intermetallic compounds. B/G is now widely accepted as a predictor in first-principles materials design. Vitos et al. have calculated the elastic constants and B/G value as well as shear modulus as functions of compositions using first-principles CPA methods [**24]. From the chemical composition-B/G and composition–shear modulus relations, they predicted two new basic compositions for austenitic stainless steels: One (Fe–13Cr–8Ni) with excellent hardness and the other (Fe–18Cr–24Ni) with remarkable resistance to various forms of localized corrosion and intermediate hardness. Beside B/G, it was recently found that the C11–C12 is also very significant on the mechanical properties of materials. Ikehata et al. have calculated the elastic constants of binary Ti alloys with alloying elements of V, Nb, Ta, Mo, or W using first-principles pseudopotential supercell method [*30]. The results showed that, when the alloy falls to a range of compositions with valence electron number of 4.20–4.24, the C11–C12 approaches zero, which was believed to contribute to the low Young’s modulus, superplasiticity, and superelasticity of the so called ‘gum metal’ [31]. Similarly, the calculation of Souvatzis et al. showed that some other binary alloys such as W–Re, W–Tc, Mo–Tc and Mo–Re with valence electron number of 6.6–6.9 also have vanishingly low C11–C12, and therefore may be candidates for ‘gum metal’ [32]. According to these investigations, C11–C12 could be treated as another useful index in the theoretical search of new materials. Elastic moduli also enter into some classical models for mechanical properties as parameters, which makes it possible to predict more complicated mechanical properties on the basis of first-principles calculations. For example, the theoretical stress to create a crack is given by the Orowan criterion, rmax ¼ ðEcs =a0 Þ
1=2
;
with E being the Young’s modulus, cs the surface energy, and a0 the inter-planar spacing. The elastic driving force on an equilibrium crack is C ¼ ð1 mÞK 2I =2G; with G being the shear modulus, m the Poisson’s ratio, and KI the stress intensity factor. These first-principle combined
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models have recently been used to study the correlation between hardness and fracture toughness of brittle materials B6O, cBN, 3C-SiC, and Si by Ding et al. [33]. 4. Lattice defect Unlike the ideal strength and elastic properties that are directly dominated by the nature of the bonds, some other mechanical properties, such as yield strength, fracture strength, creep resistance, etc., are sensitive to the lattice structures. A well-known example of the structure-sensitive properties is that, due to the defects in the lattice, the yield strength of a real crystal is only about 1/1000 of its theoretical value with perfect lattice. Therefore, it is critical to understand the behavior of the lattice defects in order to predict the mechanical properties of the structural materials. According to their spatial pattern, there are three kinds of lattice defects: zero-dimensional point defects such as vacancy, interstitial atoms, and alloying atoms; one-dimensional dislocation; two-dimensional defects such as stacking fault, grain boundary, etc. We will discuss recent first-principles investigation of these lattice defects accordingly. 4.1. Zero-dimensional defects: point defects Point defects mostly correlate to properties related to atomic diffusion, for example, creep resistance. First-principles investigations of point defects in metals e.g., [34], intermetallic compounds e.g., [35–37], random alloys e.g., [38], and so on, are abundant in literature. However, how on earth the point defects affect the mechanical properties is seldom discussed in these papers. We have calculated the interaction energy between a series of alloying atoms and vacancy in a-Ti alloys using linear muffin–tin orbital method within atomic sphere approximation (LMTOASA) [*39]. Comparing the interaction energies with available experimental creep resistance, we found that those alloying atoms attractive to vacancy increase the creep resistance whereas those repulsive to vacancy do not (see Fig. 1 and Tables 1 and 2). This is understandable because the interaction between alloying atoms and vacancy facilitates the alloying atoms to diffuse to dislocations and form Cottrell atmosphere near the dislocation, therefore inhibiting the glide of dislocation when creeping. We have also investigated the interaction between alloying atoms in aTi alloys, which gives the clustering/ordering tendency of the alloy [40]. Using the Flinn model, the interaction energy can be used to predict the contribution of the short-range order to the critical shear stress. Correct trend was found for the strengthening effect of Al, Si, Ga, Ge on a-Ti alloys. Recently, Mayer et al. have studied the effect of interstitial H on the bulk modulus and C44 of Nb using fullpotential variant LMTO method [16]. It was shown that, with the increase in the content of H, the bulk modulus of Nb increases but C44 decreases, both in agreement with experiments. In the same paper, they also reported the
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Y Zr Nb Mo Tc Ru Rh Pd Ag Cd In Sn 0.1
ΔE (eV)
0.0 -0.1 -0.2 -0.3
Y to Sn Al, Si, and Sc to Ge Mn to Cu, spin polorized
-0.4 Al Si
Sc Ti V Cr Mn Fe Co Ni Cu Zn Ga Ge
Alloying Atoms Fig. 1. Interaction energy between vacancy and alloying atom in a-Ti, defined as DE = [E(V + S, N) + E(N)] [E(V, N) + E(S + N)]. E(V + S, N), E(V, N), and E(S, N) are the total energy of the N-site supercell containing one vacancy and one alloying atom that are the nearest neighbor of each other, one vacancy only, and one alloying atom only, respectively; E(N) is the total energy of the perfect supercell. Negative value refers to attractive interaction between the vacancy and the alloying atom, whereas positive to repulsive.
Table 1 Experimental 100-h creep rupture stress of unalloyed a titanium and a titanium based alloys at different temperature (see [*39, and reference therein]) Alloy (wt.%)
Rupture stress (MPa) 425 °C
500 °C
550 °C
600 °C
Ti Ti–2Al Ti–3Al Ti–4Al Ti–7Ga Ti–11In Ti–10Zr Ti–10Sn
157.8 274.4 – 382.2 – – – –
– – 151.9 – 311.6 121.5 – –
46.6 82.3 – – – – – –
– – – 73.5 – – 73.5 49.0a
a
dislocation can be described successfully by using continuum elasticity methods. However, the dislocation core region that sometimes affects the plasticity of materials dramatically is out of the capability of continuum elasticity methods. First-principles investigations of dislocation core region have drawn much attention in recent years. Direct modeling of an isolated dislocation by using supercell (with periodic boundary condition) based first-principles methods is not practical since: (1) the stress field induced by a dislocation is long ranged, therefore extremely large supercell must be used to avoid the interaction between the dislocations in nearby supercells, which is out of the scope of current first-principles methods; (2) the geometry of an isolated dislocation breaks periodic symmetry. Several approaches have been developed to tackle the above problems: (1) simulation of dislocation dipole arrays by non-isolated dislocation (e.g., [41]); (2) direct simulation of isolated dislocation using flexible boundary conditions (e.g., [42,43]); (3) conjunction of first-principles derived generalized staking fault energy (so called c surface) to twodimensional Peierls–Nabarro description of the dislocation core (see [*44] for review). For details of these approaches, we refer the readers to a recent review paper by Woodward [*45]. A most recent development in the area of first-principles investigation of dislocation, not covered in Ref. [*45], is the incorporation of c-surface to phase field model, proposed by Shen and Wang [*46]. Using this approach, they could realistically treat the dissociation of dislocation with arbitrary configuration, interaction among extended dislocations, and creation and annihilation of various planar faults. 4.3. Two-dimensional defects: stacking fault and twin
150-h creep rupture stress.
Table 2 Creep deformation of a based Ti–5Al–5Sn–2Zr–0.8Mo–0.5Si alloyed with some b stabilizers at 538 °C and 379 MPa after 96 h (see [35] and references therein) Additions
0
1.2Mo
2.0V
4.2Nb
0.9Cr
0.9Ni
Creep deformation (%)
0.58
0.27
0.54
0.45
0.71
>10.0
elastic constants of NbCr2xVx as a function of composition, where C44 remains almost constant and the bulk modulus decreases with increasing x [16]. 4.2. One-dimensional defects: dislocations Plastic deformation is mainly controlled by the glide of dislocations. In simulation, the long-range stress field of
When the glide of dislocation is blocked for some reasons, crystals have to resort to the generation of stacking fault or twin (planar lattice defects) to deform. Higher tendency of this type of plastic deformation (defined as twinnability in Ref. [47]) correlates to a lower stacking fault or twin formation energy. These formation energies can be easily obtained by comparing the total energies of the defective and perfect crystals from first-principles calculations. In Ref. [47], the intrinsic stacking fault energy, unstable stacking fault energy, and unstable twinning energy, as well as extrinsic stacking fault energy and twin boundary energy have been reported for eight FCC metals. The ‘twinnability’ of the metals predicted from these energies was found to be in agreement with the experimental results. 4.4. Interaction between defects of different dimension Since most structural materials are alloys and impurities (e.g., H, C, P, S, etc.) are introduced inevitably into the alloys during the process of smelting or through the chemical reaction between the materials and the atmosphere, a knowledge of the interactions between these point defects (alloying atoms/impurities) and other lattices defects,
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such as dislocations, stacking fault, and grain boundaries, sometimes is crucial to understanding the mechanical properties of materials. In a recent paper of Medvedeva, Gornostyrev, and Freeman [**48], interactions between alloying atoms and generalized stacking fault energy (GSF, i.e., c surface) in BCC-Mo alloys have been investigated by using fullpotential LMTO method, and the electronic origin of the solid solution softening in these alloys was revealed. The authors demonstrated that additions with more 5d-electrons (Re, Os, Ir, and Pt) than Mo lead to a decrease in the GSF energy and those with less 5d-electrons (Hf and Ta) to its sharp increase. Using the generalized Peierls– Nabarro model for a nonplanar dislocation core, they associate the local reduction of the GSF energy with an enhancement of double kink nucleation and an increase of the dislocation mobility, which contributes to the solid solution softening of the BCC-Mo alloys. Grain boundary can also be considered as a kind of planar defects. The relative strength of grain boundaries to the bulk crystal controls the manner of fracture: if the grain boundaries are weaker/stronger than the bulk crystal, the metal is most likely to exhibit intergranular/transgranular fracture. The segregation of alloying elements or impurities can affect the strength of the grain boundary significantly. This topic has been extensively investigated in literature for metals (e.g., Al [49], Fe [50–53], Ni [53–56], Cu [*57,58]), intermetallic compounds (e.g., Ni3Al [*59]), and ceramics (e.g., Al2O3 [60,61]). For the interstitial impurities, H, P, S are mostly identified by first-principles calculations as embrittlement elements due to their decohesive effect on the grain boundaries [50,54,*59], in agreement with experimental findings. The segregation of these elements to grain boundary is the main reason for the brittle intergranular fracture according to the theory of Rice and Wang [62]. The first-principles investigation of H-induced embrittlement is recently reviewed by Geng et al. [63]. Contrary to H, P, S, etc., the segregation of B generally leads to cohesive enhancement of the grain boundary [51,*59]. There were generally two arguments concerning the origin of the cohesive strengthening/weakening effect of the alloying atoms or impurities on the grain boundary: (1) electronic effect, for example, B is believed to form strong covalent bond with Ni atoms in the grain boundary of Ni and Ni3Al [*59, and reference therein] which enhances the binding of the grain boundary; (2) atomic size effect, for example, it was argued that the large Bi atoms weaken the interatomic bonding by pushing apart the host Cu atoms at the grain boundary, and therefore make the grain boundary brittle [*57]. 5. Conclusion and remark In conclusion, we presented a brief review of the first-principles investigations of mechanical properties of structural materials in recent years. The results of these investigations demonstrated that first-principles methods
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can be practical tools for the design of structural materials. Since first-principles calculations are in principle at electronic level and performed at zero temperature, while the mechanical properties of structural materials are generally macroscopic and related to finite temperature, further efforts to identify or define more calculable parameters and models so as to bridge the size and temperature gaps between first-principles method and complex mechanical properties will be important for the design of new materials. Acknowledgements The authors are very grateful for the support from Chinese Academy of Sciences (Grant INF105-SCE-02-04), the Chinese MoST (Grant TG2000067105), and NSFC (Grant 50301014). References The papers of particular interest have been highlighted as: * of special interest; ** of very special interest. [1] Dreizler RM, Gross EKU. Density functional theory. Berlin: Springer; 1998. [*2] Clatterbuck DM, Chrzan DC, Morris Jr JW. The ideal strength of iron in tension and shear. Acta Mater 2003;51:2271–83. [3] Clatterbuck DM, Krenn CR, Cohen ML, Morris Jr JW. Phonon instabilities and the ideal strength of aluminum. Phys Rev Lett 2003;91:135501. [4] Yashiro K, Oho M, Tomita Y. Ab initio study of the lattice instability of silicon and aluminum under [0 0 1] tension. Comput Mater Sci 2004;29:397–406. [5] Ogata S, Li J, Yip S. Ideal pure shear strength of aluminum and copper. Science 2002;298:807–11. [6] Jahnatek M, Krajci M, Hafner J. Interatomic bonding, elastic properties, and ideal strength of transition metal aluminides: a case study for Al3(V, Ti). Phys Rev B 2005;71:024101. [7] Li T, Morris Jr JW, Chrzan DC. Ideal strength of B2 transitionmetal aluminides. Phys Rev B 2004;70:054107. [8] Ogata S, Hirosaki N, Kocer C, Shibutani Y. A comparative ab initio study of the ideal strength of single crystal a- and b-Si3N4. Acta Mater 2004;52:233–8. [9] Kiefer B, Shieh SR, Duffy TS, Sekine T. Strength, elasticity, and equation of state of the nanocrystalline cubic silicon nitride c-Si3N4 to 68 GPa. Phys Rev B 2005;72:014102. [**10] Ogata S, Li J, Hirosaki N, Shibutani Y, Yip S. Ideal shear strain of metals and ceramics. Phys Rev B 2004;70:104104. [*11] Lu GH, Deng SH, Wang TM, Kohyama M, Yamamoto R. Theoretical tensile strength of an Al grain boundary. Phys Rev B 2004;69:134106. [12] Mehl MJ, Klein BM, Papaconstantopoulos DA. First principles calculations of elastic properties of metals. In: Westbrook JH, Fleischer RL, editors. Intermetallic compounds: principles and practice. London: John Wiley and Sons; 1994. p. 195–210. [13] Sun J, Jiang B. Ab initio calculation of the phase stability, mechanical properties and electronic structure of ZrCr2 Laves phase compounds. Philos Mag 2004;84:3133–44. [14] Kellou A, Grosdidier T, Coddet C, Aourag H. Theoretical study of structural, electronic, and thermal properties of Cr2(Zr, Nb) Laves alloys. Acta Mater 2005;53:1459–66.
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