Mechanical Properties of Random Alloys from Quantum Mechanical ...
Mechanical Properties of Random Alloys from Quantum Mechanical Simulations Levente Vitos1,2,3 and B¨ orje Johansson1,2 1
Applied Materials Physics, Department of Materials Science and Engineering, Royal Institute of Technology, SE-10044 Stockholm, Sweden 2 Condensed Matter Theory Group, Physics Department, Uppsala University, SE-75121 Uppsala, Box 530, Sweden [email protected] 3 Research Institute for Solid State Physics and Optics, H-1525 Budapest, P.O. Box 49, Hungary
Abstract. Today, a direct determination of the mechanical properties of complex alloys from first-principles theory is not feasible. On the other hand, well established phenomenological models exist, which are suitable for an accurate description of materials behavior under various mechanical loads. These models involve a large set of atomic-level physical parameters. Unfortunately, in many cases the available parameters have unacceptably large experimental error bars. Here we demonstrate that computational modeling based on modern first-principles alloy theory can yield fundamental physical parameters with high accuracy. We illustrate this in the case of aluminum and transition metal alloys and austenitic stainless steels by computing the size and elastic misfit parameters, and the surface and stacking fault energies as functions of chemical composition.
1
Introduction
The mechanical properties represent the behavior of materials under applied forces. They are of vital importance in fabrication processes and use. Materials behavior are usually described in terms of stress or force per unit area and strain or displacement per unit distance. On the basis of stress and strain relations, one can distinguish elastic and plastic regimes. At small stress, the displacement and applied force obey Hook’s law and the specimen returns to its original shape on uploading. Beyond the so called elastic limit, upon strain release the material is left with a permanent shape. Several models of elastic and plastic phenomena in solids have been established. For a detailed discussion of these models we refer to [1,2,3,4,5]. Within the elastic regime, the single crystal elastic constants and polycrystalline elastic moduli play the principal role in describing the stress-strain relation. Within the plastic regime, the importance of lattice defects in influencing the mechanical behavior of crystalline solids was recognized long time ago. Plastic deformations are primarily facilitated by dislocation motion and can occur B. K˚ agstr¨ om et al. (Eds.): PARA 2006, LNCS 4699, pp. 510–519, 2007. c Springer-Verlag Berlin Heidelberg 2007
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at stress levels far below those required for dislocation-free crystals. Dislocation theory is a widely studied field within material science. Most recently, the mechanism of dislocation motion was also confirmed in complex materials [6]. The mechanical hardness represents the resistance of material to plastic deformation. It may be related to the yield stress separating the elastic and plastic regions, above which a substantial dislocation activity develops. In an ideal crystal dislocations can move easily because they experience only the weak periodic lattice potential. In real crystals, however, the motion of dislocations is impeded by obstacles, leading to an elevation of the yield strength. According to this scenario, the yield stress is decomposed into the Peierls stress, needed to move a dislocation in the crystal potential, and the solid-solution strengthening contribution, due to dislocation pinning by the randomly distributed solute atoms. The Peierls stress of pure metals is found to be approximately proportional to the shear modulus [5]. Dislocation pinning by random obstacles has been studied by classical theories [2,3,4] and it was found to be mostly determined by the size misfit and elastic misfit parameters. The concentration (c) dependence of the Peierls term is governed by that of the elastic constants, whereas the solidsolution strengthening contribution depends on concentration as c2/3 [2,3,4]. Besides the above described bulk parameters, the formation energies of twodimensional defects are also important in describing the mechanical characteristics of solids. The surface energy, defined as the excess free energy of a free surface, is a key parameter in brittle fracture. According to Griffith theory [5], the fracture stress is proportional to the square root of the surface energy, that is, the larger the surface energy is, the larger the load could be before the solid starts to break apart. Another important planar defect is the stacking fault in close-packed lattices, such as the face-centered cubic (f cc) or hexagonal closepacked (hcp) lattice. In these structures, the dislocations may split into energetically more favorable partial dislocations having Burgers vectors smaller than a unit lattice translation [1]. The partial dislocations are bound together and move as a unit across the slip plane. In the ribbon connecting the partials the original ideal stacking of close-packed lattice is faulted. The energy associated with this miss-packing is the stacking-fault energy (SFE). The balance between the SFE and the energy gain by splitting the dislocation determines the size of the stacking fault ribbon. The width of the stacking fault ribbon is of importance in many aspects of plasticity, as in the case of dislocation intersection or cross-slip. In both cases, the two partial dislocations have to be brought together to form an unextended dislocation before intersection or cross-slip can occur. By changing the SFE or the dislocation strain energy, wider or narrower dislocations can be produced and the mechanical properties can be altered accordingly. For instance, materials with high SFE permit dislocations to cross slip easily. In materials with low SFE, cross slip is difficult and dislocations are constrained to move in a more planar fashion. In this case, the constriction process becomes more difficult and hindered plastic deformation ensues. Designing for low SFE, in order to restrict dislocation movement and enhance hardness was adopted, e.g., in transition metal carbides [7].
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The principal problem related to modeling the mechanical properties of complex solid solutions is the lack of reliable experimental data of the alloying effects on the fundamental bulk and surface parameters. While the volume misfit parameters are available for almost all the solid solutions, experimental values of the elastic misfit parameters are scarce. There are experimental techniques to establish the polar dependence of the surface energy, but a direct measurement of its magnitude is not yet feasible [8]. In contrast to the surface energy, the stacking fault energy can be determined from experiments. For instance, one can find a large number of measurements on the stacking fault energy of austenitic stainless steels [9,10]. However, different sets of experimental data published on similar steel compositions differ significantly, indicating large error bars in these measurements. On the theoretical side, the number of accurate calculations on solid solutions is also very limited. In fact, the complexity of the problem connected with the presence of disorder impeded any former attempts to calculate the above parameters from ab initio methods. Our ability to determine the physical parameters of solid solutions from first-principles has become possible with the Exact MuffinTin Orbitals (EMTO) method [11,12] based on the density functional theory [13] and efficient alloy theories [14]. Within this approach, we could reach a level of accuracy where many fundamental physical quantities of random alloys could be determined with an accuracy equal to or in many cases even better than experiments. The EMTO method has proved an accurate tool in the theoretical description of the simple and transition metal alloys [14,15,16,17,18,19] and, in particular, Fe-based random alloys [20,21,22,23,24]. In this work, we illustrate the possible impact of such calculations on modeling the mechanical properties of simple and transition metal binary alloys and austenitic stainless steels.
2
Theory
In this section, we briefly review the theory of the elastic constants, surface energy and stacking fault energy and give the most important numerical details used in the ab initio determination of these physical parameters. 2.1
Physical Parameters
Volume and elastic misfit parameters: The equilibrium volume V (c) of a pseudo-binary alloy A1−c Bc is obtained from a Morse [25] or a Murnaghan [26] type of function fitted to the free energies F (V ) calculated for different volumes. The elastic constants are the second order derivatives of the free energy with respect to the strain tensor ekl (k, l = 1, 2, 3), viz. cijkl =
1 ∂F , V ∂eij ∂ekl
(1)
where the derivatives are calculated at the equilibrium volume and at constant e’s other than eij and ekl . In a cubic system there are 3 independent elastic
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constants. Employing the Voigt notations, these are c11 , c12 and c44 . In a hexagonal crystal, there are 5 different elastic constants: c11 , c12 , c13 , c33 and c44 . On a large scale, a polycrystalline material can be considered to be isotropic in a statistical sense. Such system is completely described by the bulk modulus B and the shear modulus G1 . The only way to establish the ab initio polycrystalline B and G is to average the single crystal elastic constants cij by suitable methods based on statistical mechanics. A large variety of averaging methods has been proposed. According to Hershey’s averaging method [27], for a cubic system the average shear modulus G is a solution of equation G3 + αG2 + βG + γ = 0,
(2)
where α = (5c11 + 4c12 )/8, β = −c44 (7c11 − 4c12 )/8, γ = −c44 (c11 − c12 )(c11 + 2c12 )/8. This approach turned out to give the most accurate relation between cubic single-crystal and polycrystalline data in the case of Fe-Cr-Ni alloys [28]. For cubic crystals, the polycrystalline bulk modulus coincides with the singlecrystal B = (c11 + 2c12 )/3. The size (εb ) and elastic (εG ) misfit parameters are calculated from the concentration dependent Burgers vector b(c) or lattice parameter, and shear modulus G(c) as 1 ∂b(c) 1 ∂G(c) , and εG = . (3) εb = b(c) ∂c G(c) ∂c According to Labusch-Nabarro model [3,4], solid solution hardening is propor2/3 4/3 tional to c εL , where εL ≡ εG 2 + (αεb )2 is the Fleischer parameter, εG ≡ εG /(1 + 0.5|εG |) and α = 9 − 16. Surface energy: The surface free energy (γS ) represent the excess free energy per unit area associated with an infinitely large surface. For a random alloy, this can be calculated from the free energy of the surface region F S ({cα }) as γS =
c) ES FS ({cα }) − FB (˜ = , A2D A2D
(4)
where {cα } denotes the equilibrium surface concentration profile, α = 1, 2, ... is the layer index, FB (˜ c) is the bulk free energy referring to the same number of atoms as FS ({cα }), and c˜ is set to be equal with the average concentration from the surface region. A2D is the area of the surface. The surface concentration profile is determined for each bulk concentration c by minimizing the free energy of the surface and bulk subsystems as described, e.g., in [19]. Stacking fault energy: A perfect f cc crystal has the ideal ABCABCAB... stacking sequence, where the letters denote adjacent (111) atomic layers. The intrinsic stacking fault is the most commonly found fault in experiments on f cc metals. This fault is produced by a shearing operation described by the 1
The Young modulus E and Poisson ratio ν are connected to B and G by the relations E = 9BG/(3B + G) and ν = (3B − 2G)/(6B + 2G).
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transformation ABC → BCA to the right-hand side of an (111) atomic layer, and it corresponds to the ABCAC˙ A˙ B˙ C˙ stacking sequence, where the translated layers are marked by dot. The formation energy of an extended stacking fault is defined as the excess free energy per unit area, i.e. γSF =
FSF − FB , A2D
(5)
where FSF and FB are the free energies of the system with and without the stacking fault, respectively. Since the intrinsic stacking fault creates a negligible stress near the fault core, the faulted lattice approximately preserves the close packing of the atoms, and can be modeled by an ideal close-packed lattice. Within the third order axial interaction model [29], for the excess free energy we find FSF − FB ≈ Fhcp + 2Fdhcp − 3Ff cc , where Ff cc , Fhcp and Fdhcp are the energies of f cc, hcp and double-hcp (dhcp) structures, respectively. 2.2
The ab initio Calculations
In the present application, the elastic constants have been derived by calculating the total energy2 as a function of small strains δ applied on the parent lattice. For a cubic lattice, the two cubic shear constants, c = (c11 −c12 )/2 and c44 , have been obtained from volume-conserving orthorhombic and monoclinic distortions, respectively. Details about these distortions can be found in [15,16]. The bulk modulus B has been determined from the equation of state fitted to the total energies of undistorted cubic structure (δ = 0). For a hexagonal lattice, at each volume V , the theoretical hexagonal axial ratio (c/a)0 has been determined by minimizing the total energy E(V, c/a) calculated for different c/a ratios close to the energy minimum. The hexagonal bulk modulus has been obtained from the equation of state fitted to the energy minima E(V, (c/a)0 ). The five hexagonal elastic constants have been obtained from the bulk modulus, the logarithmic volume derivative of (c/a)0 (V ), and three isochoric strains, as described in [30]. In the EMTO total energy calculations, the one-electron equations were solved within the scalar-relativistic and frozen-core approximations. To obtain the accuracy needed for the calculation of the elastic constants, we used about ∼ 105 uniformly distributed k−points in the irreducible wedge of the Brillouin zone of the ideal and distorted lattices. In surface calculations, the close-packed f cc (111) surface was modeled by 8 atomic layers separated by 4 layers of empty sites simulating the vacuum region. The 2D Brillouin zone was sampled by ∼ 102 uniformly distributed k−points in the irreducible wedge. The EMTO basis set included sp and d orbitals in the case of simple metal alloys, and spd and f orbitals for Ag-Zn, Pd-Ag and Fe-based alloys. The exchange-correlation was treated within the generalized gradient approximation (GGA) [31].
2
For the temperature dependence of the elastic constants of random alloys the reader is referred to [18].
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Table 1. Theoretical (present results) and experimental [32] equilibrium volume V (units of Bohr3 ), hexagonal axial ratio (c/a)0 , and elastic constants (units of GPa) of the hcp Ag0.3 Zn0.7 random alloy Ag0.3 Zn0.7 V (c/a)0 theory 110.8 1.579 experiment 104.3 1.582 percent error 6 0.2
3 3.1
c11 110 130 15
c12 56 65 14
c13 63 64 2
c33 129 158 18
c44 27 41 34
Results Misfit Parameters
Hexagonal Ag-Zn: We illustrate the accuracy of the present ab initio approach by comparing in Table 1 the theoretical results obtained for the Ag0.3 Zn0.7 random alloy with experimental data [32]. The deviation between the theoretical and experimental equilibrium volume and equilibrium hexagonal axial ratio (c/a)0 are 6 % and 0.2 %, respectively. The calculated elastic constants are somewhat small when compared with the measured values, but the relative magnitudes are well reproduced by the theory. Actually, the difference between the two sets of data is typical for what has been obtained for simple and transition metals in connection with the GGA for the exchange-correlation functional [31]. Therefore, the overall agreement between theory and experiment can be considered to be very satisfactory. Aluminium alloys: On the left panel of Figure 1, the theoretical single-crystal elastic constants for Al-Mg are compared to the available experimental data [33]. We observe that the experimental value is slightly overestimated for c44 and c12 and underestimated for c11 . Such deviations are typically obtained for elemental metals. Nevertheless, for all three elastic constants we find that the variations of the theoretical values with concentration are in perfect agreement with the experimental data. In particular, we point out that both the experimental and theoretical c11 and c12 decrease whereas c44 slightly increases with Mg addition. The calculated size misfit parameters (εb ) for five Al-based solid solutions are compared to the experimental values on the right panel of Figure 1. An excellent agreement between the computed and experimental values is observed. This figure also shows the calculated elastic misfit parameter (εG ) as a function of theoretical εb . According to these data, we can see that the elastic misfit 4/3 for Mn, Cu and Mg contributes by ∼ 25% to εL , i.e. to the solid solution hardening, whereas this effect is below 3% in the case of Si and Zn. Austenitic stainless steels: The volume and shear modulus of Fe100−c−n Crc Nin alloys have been determined as functions of chemical composition for 13.5 < c < 25.5 and 8 < n < 24. The calculated composition-shear modulus map is presented in Figure 2 (left panel). Alloys with large shear modulus correspond to low and intermediate Cr (< 20%) and low Ni (< 15%) concentrations. Within this group of al-
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Fig. 1. Left panel: Composition dependence of the theoretical (present results) and experimental [33] single-crystal elastic constants of Al-Mg random alloys. Right panel: Misfit parameters for selected Al-based alloys (alloying elements shown at the bottom of the figure). Left axis: theoretical εb versus experimental εb (Ref. [34]: circles, Ref. [35]: squares); right axis: theoretical εG versus theoretical εb .
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10xεb
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Fig. 2. Left panel: Calculated shear modulus of FeCrNi alloys as a function of the chemical composition. Right panel: Theoretical misfit parameters for Fe58 Cr18 Ni24 alloy comprising a few percentage of Al, Si, V, Cu, Nb, Mo, Re, Os or Ir. Note that the size misfit parameters have been multiplied by a factor of 10.
loys G decreases monotonically with both Cr and Ni from a pronounced maximum of 81 GPa (near Fe78 Cr14 Ni8 ) to approximately 77 GPa. The high Cr content alloys define the second family of austenites possessing the lowest shear moduli (< 75 GPa) with a minimum around Fe55 Cr25 Ni20 . The third family of austenites, with intermediate G values, is located at moderate Cr (< 20%) and high Ni (> 15%) concentrations, where G shows no significant chemical composition dependence.
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The effect of alloying additions on the volume and shear modulus of alloy with composition Fe58 Cr18 Ni24 is demonstrated on the right panel of Figure 2. It is found that Nb and Mo give the largest elastic misfit parameters (|εG | = 3.9 and 2.4, respectively). The size misfit is negligible for Al, Si, V and Cu, but it has a sizable value (between 0.21 and 0.33) for the 4d and 5d dopants. The Fleischer parameter is 5.4 for Nb, 4.1 for Mo, and ∼ 3.5 for the 5d elements. All the other dopants give εL < 1.5. Hence, assuming that the Labusch-Nabarro model is valid in the case of Fe-Cr-Ni alloys encompassing a few percent of additional elements, Nb and Mo are expected to yield the largest solid solution hardening. However, one should also take into account that the 4d metals, in contrast to the 5d metals, significantly decrease G and thus the Peierls stress [20]. Therefore, the overall hardening effect might be different from the one expressed merely via the Fleischer parameter. 3.2
Surface Energy of Pd-Ag Alloys
The surface energy and the top layer Ag concentration (c1 ) of the f cc (111) surface of the Ag-Pd random alloy3 , calculated as a function of temperature and bulk Ag concentration, are plotted on the left panel of Figure 3. At 0 K, the surface energy is mainly determined by the pure Ag surface layer, which is reflected by an almost flat ES (0K, c) ≈ ES,Ag line for c 0.1. With increasing temperature, ES (T, c) converges towards the value estimated using a linear interpolation between end members. Note that the temperature dependence of ES is very similar to that of c1 . Although, at intermediate bulk concentrations, the subsurface Ag concentration (c2 ) shows strong temperature dependence [19], this effect is imperceptible in the surface energy. Therefore, the variation of the surface energy for a close-packed facet with temperature and bulk composition is, to a large extent, governed by the surface layer, and the subsurface layers play only a secondary role. 3.3
Stacking Fault Energy of Austenitic Stainless Steels
The calculated room-temperature SFE of Fe-Cr-Ni alloys is shown on the right panel of Figure 3 as a function of chemical composition. We can observe a strongly nonlinear composition dependence. This behavior is a consequence of the persisting local moments in austenitic steels, which are the take-off for many basic properties that the austenitic stainless steels exhibit [23]. The local magnetic moments disappear near the stacking faults, except in the high-Ni–low-Cr alloys, where they are comparable to those from the bulk. This gives a significant reduction of the magnetic fluctuation contribution to the SFE in the high-Ni– low-Cr alloys, but for the rest of the alloys, it is found that the magnetic part of γSF has the same order of magnitude as the total SFE . We have found that the most common austenitic steels, i.e. those with low-Ni and intermediate and 3
For the effect of structural relaxation on the surface energy and equilibrium concentration profile the reader is referred to [36].
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Fig. 3. Left panel: Surface energy (ES ) and top layer Ag concentration (c1 ) for the f cc (111) surface of the Ag-Pd random alloy as functions of temperature and bulk Ag concentration. Lines are added to guide to the eye. Right panel: Stacking fault energy of Fe-Cr-Ni random alloys. The calculated SFE is shown for T = 300 K as a function of Cr and Ni content.
high-Cr content are in fact stabilized by the magnetic entropy. They possess γSF 0, and have enhanced hardness. The high-Ni–low-Cr alloys are more ductile compared to the rest of the compositions, since they have the largest γSF . The role of the additional alloying elements has been investigated in Fe-CrNi-M alloys encompassing a few percent of Nb or Mn. While the effect of Mn is found to be similar to that of Cr [9], the relative effect of Nb can be as large as ∼ 30% [24]. However, the absolute effect of Nb on the SFE depends strongly on the initial composition. For instance, in alloys close to the hcp magnetic transition, Nb decreases the SFE [24]. Therefore, in a steel design process, both the alloying element and the composition of the host material are key parameters for predicting the role of alloying. This finding is in contrast to the widely employed models for compositional dependence of SFE, and it clearly demonstrates that no universal composition equations for the SFE can be established.
4
Summary
We have demonstrated that computational methods based on modern ab initio alloy theory can yield essential physical parameters for random alloys with an accuracy comparable to the experiment. These parameters can be used in phenomenological models to trace the variation of the mechanical properties with alloying additions. Acknowledgments. The Swedish Research Council, the Swedish Foundation for Strategic Research and the Hungarian Scientific Research Fund (OTKA T046773 and T048827) are acknowledged for financial support.
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References 1. Argon, A.S., Backer, S., McClintock, F.A., Reichenbach, G.S., Orowan, E., Shaw, M.C., Rabinowicz, E.: Metallurgy and Materials. Addison-Wesley Series. AddisonWesley Publishing Company, Ontario (1966) 2. Fleischer, R.L.: Acta Metal 11, 203–209 (1963) 3. Labusch, R.: Acta Met. 20, 917–927 (1972) 4. Nabarro, F.R.N.: Philosophical magazine 35, 613–622 (1977) 5. Lung, C.W., March, N.H.: World Scientific Publishing Co. Pte. Ltd (1999) 6. Chisholm, M.F., Kumar, S., Hazzledine, P.: Science 307, 701–703 (2005) 7. Hugosson, H., Jansson, U., Johansson, B., Eriksson, O.: Science 293, 2423–2437 (2001) 8. Sander, D.: Current Opinion in Solid State and Materials Science 7, 51 (2003) 9. Schramm, R.E., Reed, R.P.: Met. Trans. A 6A, 1345–1351 (and references therein) (1975) 10. Miodownik, A.P.: Calphad 2, 207–226 (and references therein) (1978) 11. Andersen, O. K., Jepsen, O., Krier, G.: In: Kumar, V., Andersen, O. K., Mookerjee, A. (eds.) Lectures on Methods of Electronic Structure Calculations, World Scientific Publishing, Singapore, (1994) pp. 63–124 12. Vitos, L.: Phys. Rev. B 64, 014107(11) (2001) 13. Hohenberg, P., Kohn, W.: Phys. Rev. 136, B864–B871 (1964) 14. Vitos, L., Abrikosov, I.A., Johansson, B.: Phys. Rev. Lett. 87, 156401(4) (2001) 15. Taga, A., Vitos, L., Johansson, B., Grimvall, G.: Phys. Rev. B 71, 014201(9) (2005) 16. Magyari-K¨ ope, B., Grimvall, G., Vitos, L.: Phys. Rev. B 66, 064210, (2002) ibid 66, 179902 (2002) 17. Magyari-K¨ ope, B., Vitos, L., Grimvall, G.: Phys. Rev. B 70, 052102(4) (2004) 18. Huang, L., Vitos, L., Kwon, S.K., Johansson, B., Ahuja, R.: Phys. Rev. B 73, 104203 (2006) 19. Ropo, M., Kokko, K., Vitos, L., Koll´ ar, J., Johansson, B.: Surf. Sci. 600, 904–913 (2006) 20. Vitos, L., Korzhavyi, P.A., Johansson, B.: Nature Materials 2, 25–28 (2003) 21. Dubrovinsky, L., et al.: Nature 422, 58–61 (2003) 22. Dubrovinsky, N., et al.: Phys. Rev. Lett. 95, 245502(4) (2005) 23. Vitos, L., Korzhavyi, P.A., Johansson, B.: Phys. Rev. Lett. 96, 117210(4) (2006) 24. Vitos, L., Nilsson, J.-O., Johansson, B.: Acta Materialia 54, 3821–3826 (2006) 25. Moruzzi, V.L., Janak, J.F., Schwarz, K.: Phys. Rev. B. 37, 790–799 (1988) 26. Murnaghan, F.D.: Proc. Nat. Acad. Sci. 30, 244–247 (1944) 27. Ledbetter, H.M.: J. Appl. Phys. 44, 1451–1454 (1973) 28. Ledbetter, H.M.: In: Levy, M., Bass, H.E., Stern, R.R. (eds.) Handbook of Elastic Properties of Solids, Liquids, Gases. Academic, San Diego, vol. III, p. 313 (2001) 29. Denteneer, P.J.H., van Haeringen, W.: J. Phys. C: Solid State Phys. 20, L883–L887 (1987) 30. Steinle-Neumann, G., Stixrude, L., Cohen, R.E.: Phys. Rev. B 60, 791–799 (1999) 31. Perdew, J.P., Burke, K., Ernzerhof, M.: Phys. Rev. Lett. 77, 3865–3868 (1996) 32. Matsuo, Y.: J. Phys. Soc. Japan 53, 1360–1365 (1984) 33. Gault, C., Dauger, A., Boch, P.: Physica Status Solidi (a) 43, 625–632 (1977) 34. King, H.W.: Journal of Materials Science 1, 79–90 (1966) 35. Pearson, W.B.: In: Raynor, G.V. (ed.) Handbook of lattice spacings and structures of metals and alloys, Pergamon Press Ltd., New York (1958) 36. Ropo, M.: Phys. Rev. B 74, 195401(4) (2006)
Applied Materials Physics, Department of Materials Science and Engineering, Royal Institute of Technology, SE-10044 Stockholm, Sweden 2 Condensed Matter Theory Group, Physics Department, Uppsala University, SE-75121 Uppsala, Box 530, Sweden [email protected] 3 Research Institute for Solid State Physics and Optics, H-1525 Budapest, P.O. Box 49, Hungary
Abstract. Today, a direct determination of the mechanical properties of complex alloys from first-principles theory is not feasible. On the other hand, well established phenomenological models exist, which are suitable for an accurate description of materials behavior under various mechanical loads. These models involve a large set of atomic-level physical parameters. Unfortunately, in many cases the available parameters have unacceptably large experimental error bars. Here we demonstrate that computational modeling based on modern first-principles alloy theory can yield fundamental physical parameters with high accuracy. We illustrate this in the case of aluminum and transition metal alloys and austenitic stainless steels by computing the size and elastic misfit parameters, and the surface and stacking fault energies as functions of chemical composition.
1
Introduction
The mechanical properties represent the behavior of materials under applied forces. They are of vital importance in fabrication processes and use. Materials behavior are usually described in terms of stress or force per unit area and strain or displacement per unit distance. On the basis of stress and strain relations, one can distinguish elastic and plastic regimes. At small stress, the displacement and applied force obey Hook’s law and the specimen returns to its original shape on uploading. Beyond the so called elastic limit, upon strain release the material is left with a permanent shape. Several models of elastic and plastic phenomena in solids have been established. For a detailed discussion of these models we refer to [1,2,3,4,5]. Within the elastic regime, the single crystal elastic constants and polycrystalline elastic moduli play the principal role in describing the stress-strain relation. Within the plastic regime, the importance of lattice defects in influencing the mechanical behavior of crystalline solids was recognized long time ago. Plastic deformations are primarily facilitated by dislocation motion and can occur B. K˚ agstr¨ om et al. (Eds.): PARA 2006, LNCS 4699, pp. 510–519, 2007. c Springer-Verlag Berlin Heidelberg 2007
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at stress levels far below those required for dislocation-free crystals. Dislocation theory is a widely studied field within material science. Most recently, the mechanism of dislocation motion was also confirmed in complex materials [6]. The mechanical hardness represents the resistance of material to plastic deformation. It may be related to the yield stress separating the elastic and plastic regions, above which a substantial dislocation activity develops. In an ideal crystal dislocations can move easily because they experience only the weak periodic lattice potential. In real crystals, however, the motion of dislocations is impeded by obstacles, leading to an elevation of the yield strength. According to this scenario, the yield stress is decomposed into the Peierls stress, needed to move a dislocation in the crystal potential, and the solid-solution strengthening contribution, due to dislocation pinning by the randomly distributed solute atoms. The Peierls stress of pure metals is found to be approximately proportional to the shear modulus [5]. Dislocation pinning by random obstacles has been studied by classical theories [2,3,4] and it was found to be mostly determined by the size misfit and elastic misfit parameters. The concentration (c) dependence of the Peierls term is governed by that of the elastic constants, whereas the solidsolution strengthening contribution depends on concentration as c2/3 [2,3,4]. Besides the above described bulk parameters, the formation energies of twodimensional defects are also important in describing the mechanical characteristics of solids. The surface energy, defined as the excess free energy of a free surface, is a key parameter in brittle fracture. According to Griffith theory [5], the fracture stress is proportional to the square root of the surface energy, that is, the larger the surface energy is, the larger the load could be before the solid starts to break apart. Another important planar defect is the stacking fault in close-packed lattices, such as the face-centered cubic (f cc) or hexagonal closepacked (hcp) lattice. In these structures, the dislocations may split into energetically more favorable partial dislocations having Burgers vectors smaller than a unit lattice translation [1]. The partial dislocations are bound together and move as a unit across the slip plane. In the ribbon connecting the partials the original ideal stacking of close-packed lattice is faulted. The energy associated with this miss-packing is the stacking-fault energy (SFE). The balance between the SFE and the energy gain by splitting the dislocation determines the size of the stacking fault ribbon. The width of the stacking fault ribbon is of importance in many aspects of plasticity, as in the case of dislocation intersection or cross-slip. In both cases, the two partial dislocations have to be brought together to form an unextended dislocation before intersection or cross-slip can occur. By changing the SFE or the dislocation strain energy, wider or narrower dislocations can be produced and the mechanical properties can be altered accordingly. For instance, materials with high SFE permit dislocations to cross slip easily. In materials with low SFE, cross slip is difficult and dislocations are constrained to move in a more planar fashion. In this case, the constriction process becomes more difficult and hindered plastic deformation ensues. Designing for low SFE, in order to restrict dislocation movement and enhance hardness was adopted, e.g., in transition metal carbides [7].
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The principal problem related to modeling the mechanical properties of complex solid solutions is the lack of reliable experimental data of the alloying effects on the fundamental bulk and surface parameters. While the volume misfit parameters are available for almost all the solid solutions, experimental values of the elastic misfit parameters are scarce. There are experimental techniques to establish the polar dependence of the surface energy, but a direct measurement of its magnitude is not yet feasible [8]. In contrast to the surface energy, the stacking fault energy can be determined from experiments. For instance, one can find a large number of measurements on the stacking fault energy of austenitic stainless steels [9,10]. However, different sets of experimental data published on similar steel compositions differ significantly, indicating large error bars in these measurements. On the theoretical side, the number of accurate calculations on solid solutions is also very limited. In fact, the complexity of the problem connected with the presence of disorder impeded any former attempts to calculate the above parameters from ab initio methods. Our ability to determine the physical parameters of solid solutions from first-principles has become possible with the Exact MuffinTin Orbitals (EMTO) method [11,12] based on the density functional theory [13] and efficient alloy theories [14]. Within this approach, we could reach a level of accuracy where many fundamental physical quantities of random alloys could be determined with an accuracy equal to or in many cases even better than experiments. The EMTO method has proved an accurate tool in the theoretical description of the simple and transition metal alloys [14,15,16,17,18,19] and, in particular, Fe-based random alloys [20,21,22,23,24]. In this work, we illustrate the possible impact of such calculations on modeling the mechanical properties of simple and transition metal binary alloys and austenitic stainless steels.
2
Theory
In this section, we briefly review the theory of the elastic constants, surface energy and stacking fault energy and give the most important numerical details used in the ab initio determination of these physical parameters. 2.1
Physical Parameters
Volume and elastic misfit parameters: The equilibrium volume V (c) of a pseudo-binary alloy A1−c Bc is obtained from a Morse [25] or a Murnaghan [26] type of function fitted to the free energies F (V ) calculated for different volumes. The elastic constants are the second order derivatives of the free energy with respect to the strain tensor ekl (k, l = 1, 2, 3), viz. cijkl =
1 ∂F , V ∂eij ∂ekl
(1)
where the derivatives are calculated at the equilibrium volume and at constant e’s other than eij and ekl . In a cubic system there are 3 independent elastic
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constants. Employing the Voigt notations, these are c11 , c12 and c44 . In a hexagonal crystal, there are 5 different elastic constants: c11 , c12 , c13 , c33 and c44 . On a large scale, a polycrystalline material can be considered to be isotropic in a statistical sense. Such system is completely described by the bulk modulus B and the shear modulus G1 . The only way to establish the ab initio polycrystalline B and G is to average the single crystal elastic constants cij by suitable methods based on statistical mechanics. A large variety of averaging methods has been proposed. According to Hershey’s averaging method [27], for a cubic system the average shear modulus G is a solution of equation G3 + αG2 + βG + γ = 0,
(2)
where α = (5c11 + 4c12 )/8, β = −c44 (7c11 − 4c12 )/8, γ = −c44 (c11 − c12 )(c11 + 2c12 )/8. This approach turned out to give the most accurate relation between cubic single-crystal and polycrystalline data in the case of Fe-Cr-Ni alloys [28]. For cubic crystals, the polycrystalline bulk modulus coincides with the singlecrystal B = (c11 + 2c12 )/3. The size (εb ) and elastic (εG ) misfit parameters are calculated from the concentration dependent Burgers vector b(c) or lattice parameter, and shear modulus G(c) as 1 ∂b(c) 1 ∂G(c) , and εG = . (3) εb = b(c) ∂c G(c) ∂c According to Labusch-Nabarro model [3,4], solid solution hardening is propor2/3 4/3 tional to c εL , where εL ≡ εG 2 + (αεb )2 is the Fleischer parameter, εG ≡ εG /(1 + 0.5|εG |) and α = 9 − 16. Surface energy: The surface free energy (γS ) represent the excess free energy per unit area associated with an infinitely large surface. For a random alloy, this can be calculated from the free energy of the surface region F S ({cα }) as γS =
c) ES FS ({cα }) − FB (˜ = , A2D A2D
(4)
where {cα } denotes the equilibrium surface concentration profile, α = 1, 2, ... is the layer index, FB (˜ c) is the bulk free energy referring to the same number of atoms as FS ({cα }), and c˜ is set to be equal with the average concentration from the surface region. A2D is the area of the surface. The surface concentration profile is determined for each bulk concentration c by minimizing the free energy of the surface and bulk subsystems as described, e.g., in [19]. Stacking fault energy: A perfect f cc crystal has the ideal ABCABCAB... stacking sequence, where the letters denote adjacent (111) atomic layers. The intrinsic stacking fault is the most commonly found fault in experiments on f cc metals. This fault is produced by a shearing operation described by the 1
The Young modulus E and Poisson ratio ν are connected to B and G by the relations E = 9BG/(3B + G) and ν = (3B − 2G)/(6B + 2G).
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transformation ABC → BCA to the right-hand side of an (111) atomic layer, and it corresponds to the ABCAC˙ A˙ B˙ C˙ stacking sequence, where the translated layers are marked by dot. The formation energy of an extended stacking fault is defined as the excess free energy per unit area, i.e. γSF =
FSF − FB , A2D
(5)
where FSF and FB are the free energies of the system with and without the stacking fault, respectively. Since the intrinsic stacking fault creates a negligible stress near the fault core, the faulted lattice approximately preserves the close packing of the atoms, and can be modeled by an ideal close-packed lattice. Within the third order axial interaction model [29], for the excess free energy we find FSF − FB ≈ Fhcp + 2Fdhcp − 3Ff cc , where Ff cc , Fhcp and Fdhcp are the energies of f cc, hcp and double-hcp (dhcp) structures, respectively. 2.2
The ab initio Calculations
In the present application, the elastic constants have been derived by calculating the total energy2 as a function of small strains δ applied on the parent lattice. For a cubic lattice, the two cubic shear constants, c = (c11 −c12 )/2 and c44 , have been obtained from volume-conserving orthorhombic and monoclinic distortions, respectively. Details about these distortions can be found in [15,16]. The bulk modulus B has been determined from the equation of state fitted to the total energies of undistorted cubic structure (δ = 0). For a hexagonal lattice, at each volume V , the theoretical hexagonal axial ratio (c/a)0 has been determined by minimizing the total energy E(V, c/a) calculated for different c/a ratios close to the energy minimum. The hexagonal bulk modulus has been obtained from the equation of state fitted to the energy minima E(V, (c/a)0 ). The five hexagonal elastic constants have been obtained from the bulk modulus, the logarithmic volume derivative of (c/a)0 (V ), and three isochoric strains, as described in [30]. In the EMTO total energy calculations, the one-electron equations were solved within the scalar-relativistic and frozen-core approximations. To obtain the accuracy needed for the calculation of the elastic constants, we used about ∼ 105 uniformly distributed k−points in the irreducible wedge of the Brillouin zone of the ideal and distorted lattices. In surface calculations, the close-packed f cc (111) surface was modeled by 8 atomic layers separated by 4 layers of empty sites simulating the vacuum region. The 2D Brillouin zone was sampled by ∼ 102 uniformly distributed k−points in the irreducible wedge. The EMTO basis set included sp and d orbitals in the case of simple metal alloys, and spd and f orbitals for Ag-Zn, Pd-Ag and Fe-based alloys. The exchange-correlation was treated within the generalized gradient approximation (GGA) [31].
2
For the temperature dependence of the elastic constants of random alloys the reader is referred to [18].
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Table 1. Theoretical (present results) and experimental [32] equilibrium volume V (units of Bohr3 ), hexagonal axial ratio (c/a)0 , and elastic constants (units of GPa) of the hcp Ag0.3 Zn0.7 random alloy Ag0.3 Zn0.7 V (c/a)0 theory 110.8 1.579 experiment 104.3 1.582 percent error 6 0.2
3 3.1
c11 110 130 15
c12 56 65 14
c13 63 64 2
c33 129 158 18
c44 27 41 34
Results Misfit Parameters
Hexagonal Ag-Zn: We illustrate the accuracy of the present ab initio approach by comparing in Table 1 the theoretical results obtained for the Ag0.3 Zn0.7 random alloy with experimental data [32]. The deviation between the theoretical and experimental equilibrium volume and equilibrium hexagonal axial ratio (c/a)0 are 6 % and 0.2 %, respectively. The calculated elastic constants are somewhat small when compared with the measured values, but the relative magnitudes are well reproduced by the theory. Actually, the difference between the two sets of data is typical for what has been obtained for simple and transition metals in connection with the GGA for the exchange-correlation functional [31]. Therefore, the overall agreement between theory and experiment can be considered to be very satisfactory. Aluminium alloys: On the left panel of Figure 1, the theoretical single-crystal elastic constants for Al-Mg are compared to the available experimental data [33]. We observe that the experimental value is slightly overestimated for c44 and c12 and underestimated for c11 . Such deviations are typically obtained for elemental metals. Nevertheless, for all three elastic constants we find that the variations of the theoretical values with concentration are in perfect agreement with the experimental data. In particular, we point out that both the experimental and theoretical c11 and c12 decrease whereas c44 slightly increases with Mg addition. The calculated size misfit parameters (εb ) for five Al-based solid solutions are compared to the experimental values on the right panel of Figure 1. An excellent agreement between the computed and experimental values is observed. This figure also shows the calculated elastic misfit parameter (εG ) as a function of theoretical εb . According to these data, we can see that the elastic misfit 4/3 for Mn, Cu and Mg contributes by ∼ 25% to εL , i.e. to the solid solution hardening, whereas this effect is below 3% in the case of Si and Zn. Austenitic stainless steels: The volume and shear modulus of Fe100−c−n Crc Nin alloys have been determined as functions of chemical composition for 13.5 < c < 25.5 and 8 < n < 24. The calculated composition-shear modulus map is presented in Figure 2 (left panel). Alloys with large shear modulus correspond to low and intermediate Cr (< 20%) and low Ni (< 15%) concentrations. Within this group of al-
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Fig. 1. Left panel: Composition dependence of the theoretical (present results) and experimental [33] single-crystal elastic constants of Al-Mg random alloys. Right panel: Misfit parameters for selected Al-based alloys (alloying elements shown at the bottom of the figure). Left axis: theoretical εb versus experimental εb (Ref. [34]: circles, Ref. [35]: squares); right axis: theoretical εG versus theoretical εb .
5
misfit parameter
10xεb
Fe0.58−xCr0.18Ni0.24Mx
2
0
−2
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Si
V Cu Nb Mo Re Os
Ir
Fig. 2. Left panel: Calculated shear modulus of FeCrNi alloys as a function of the chemical composition. Right panel: Theoretical misfit parameters for Fe58 Cr18 Ni24 alloy comprising a few percentage of Al, Si, V, Cu, Nb, Mo, Re, Os or Ir. Note that the size misfit parameters have been multiplied by a factor of 10.
loys G decreases monotonically with both Cr and Ni from a pronounced maximum of 81 GPa (near Fe78 Cr14 Ni8 ) to approximately 77 GPa. The high Cr content alloys define the second family of austenites possessing the lowest shear moduli (< 75 GPa) with a minimum around Fe55 Cr25 Ni20 . The third family of austenites, with intermediate G values, is located at moderate Cr (< 20%) and high Ni (> 15%) concentrations, where G shows no significant chemical composition dependence.
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The effect of alloying additions on the volume and shear modulus of alloy with composition Fe58 Cr18 Ni24 is demonstrated on the right panel of Figure 2. It is found that Nb and Mo give the largest elastic misfit parameters (|εG | = 3.9 and 2.4, respectively). The size misfit is negligible for Al, Si, V and Cu, but it has a sizable value (between 0.21 and 0.33) for the 4d and 5d dopants. The Fleischer parameter is 5.4 for Nb, 4.1 for Mo, and ∼ 3.5 for the 5d elements. All the other dopants give εL < 1.5. Hence, assuming that the Labusch-Nabarro model is valid in the case of Fe-Cr-Ni alloys encompassing a few percent of additional elements, Nb and Mo are expected to yield the largest solid solution hardening. However, one should also take into account that the 4d metals, in contrast to the 5d metals, significantly decrease G and thus the Peierls stress [20]. Therefore, the overall hardening effect might be different from the one expressed merely via the Fleischer parameter. 3.2
Surface Energy of Pd-Ag Alloys
The surface energy and the top layer Ag concentration (c1 ) of the f cc (111) surface of the Ag-Pd random alloy3 , calculated as a function of temperature and bulk Ag concentration, are plotted on the left panel of Figure 3. At 0 K, the surface energy is mainly determined by the pure Ag surface layer, which is reflected by an almost flat ES (0K, c) ≈ ES,Ag line for c 0.1. With increasing temperature, ES (T, c) converges towards the value estimated using a linear interpolation between end members. Note that the temperature dependence of ES is very similar to that of c1 . Although, at intermediate bulk concentrations, the subsurface Ag concentration (c2 ) shows strong temperature dependence [19], this effect is imperceptible in the surface energy. Therefore, the variation of the surface energy for a close-packed facet with temperature and bulk composition is, to a large extent, governed by the surface layer, and the subsurface layers play only a secondary role. 3.3
Stacking Fault Energy of Austenitic Stainless Steels
The calculated room-temperature SFE of Fe-Cr-Ni alloys is shown on the right panel of Figure 3 as a function of chemical composition. We can observe a strongly nonlinear composition dependence. This behavior is a consequence of the persisting local moments in austenitic steels, which are the take-off for many basic properties that the austenitic stainless steels exhibit [23]. The local magnetic moments disappear near the stacking faults, except in the high-Ni–low-Cr alloys, where they are comparable to those from the bulk. This gives a significant reduction of the magnetic fluctuation contribution to the SFE in the high-Ni– low-Cr alloys, but for the rest of the alloys, it is found that the magnetic part of γSF has the same order of magnitude as the total SFE . We have found that the most common austenitic steels, i.e. those with low-Ni and intermediate and 3
For the effect of structural relaxation on the surface energy and equilibrium concentration profile the reader is referred to [36].
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Fig. 3. Left panel: Surface energy (ES ) and top layer Ag concentration (c1 ) for the f cc (111) surface of the Ag-Pd random alloy as functions of temperature and bulk Ag concentration. Lines are added to guide to the eye. Right panel: Stacking fault energy of Fe-Cr-Ni random alloys. The calculated SFE is shown for T = 300 K as a function of Cr and Ni content.
high-Cr content are in fact stabilized by the magnetic entropy. They possess γSF 0, and have enhanced hardness. The high-Ni–low-Cr alloys are more ductile compared to the rest of the compositions, since they have the largest γSF . The role of the additional alloying elements has been investigated in Fe-CrNi-M alloys encompassing a few percent of Nb or Mn. While the effect of Mn is found to be similar to that of Cr [9], the relative effect of Nb can be as large as ∼ 30% [24]. However, the absolute effect of Nb on the SFE depends strongly on the initial composition. For instance, in alloys close to the hcp magnetic transition, Nb decreases the SFE [24]. Therefore, in a steel design process, both the alloying element and the composition of the host material are key parameters for predicting the role of alloying. This finding is in contrast to the widely employed models for compositional dependence of SFE, and it clearly demonstrates that no universal composition equations for the SFE can be established.
4
Summary
We have demonstrated that computational methods based on modern ab initio alloy theory can yield essential physical parameters for random alloys with an accuracy comparable to the experiment. These parameters can be used in phenomenological models to trace the variation of the mechanical properties with alloying additions. Acknowledgments. The Swedish Research Council, the Swedish Foundation for Strategic Research and the Hungarian Scientific Research Fund (OTKA T046773 and T048827) are acknowledged for financial support.
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References 1. Argon, A.S., Backer, S., McClintock, F.A., Reichenbach, G.S., Orowan, E., Shaw, M.C., Rabinowicz, E.: Metallurgy and Materials. Addison-Wesley Series. AddisonWesley Publishing Company, Ontario (1966) 2. Fleischer, R.L.: Acta Metal 11, 203–209 (1963) 3. Labusch, R.: Acta Met. 20, 917–927 (1972) 4. Nabarro, F.R.N.: Philosophical magazine 35, 613–622 (1977) 5. Lung, C.W., March, N.H.: World Scientific Publishing Co. Pte. Ltd (1999) 6. Chisholm, M.F., Kumar, S., Hazzledine, P.: Science 307, 701–703 (2005) 7. Hugosson, H., Jansson, U., Johansson, B., Eriksson, O.: Science 293, 2423–2437 (2001) 8. Sander, D.: Current Opinion in Solid State and Materials Science 7, 51 (2003) 9. Schramm, R.E., Reed, R.P.: Met. Trans. A 6A, 1345–1351 (and references therein) (1975) 10. Miodownik, A.P.: Calphad 2, 207–226 (and references therein) (1978) 11. Andersen, O. K., Jepsen, O., Krier, G.: In: Kumar, V., Andersen, O. K., Mookerjee, A. (eds.) Lectures on Methods of Electronic Structure Calculations, World Scientific Publishing, Singapore, (1994) pp. 63–124 12. Vitos, L.: Phys. Rev. B 64, 014107(11) (2001) 13. Hohenberg, P., Kohn, W.: Phys. Rev. 136, B864–B871 (1964) 14. Vitos, L., Abrikosov, I.A., Johansson, B.: Phys. Rev. Lett. 87, 156401(4) (2001) 15. Taga, A., Vitos, L., Johansson, B., Grimvall, G.: Phys. Rev. B 71, 014201(9) (2005) 16. Magyari-K¨ ope, B., Grimvall, G., Vitos, L.: Phys. Rev. B 66, 064210, (2002) ibid 66, 179902 (2002) 17. Magyari-K¨ ope, B., Vitos, L., Grimvall, G.: Phys. Rev. B 70, 052102(4) (2004) 18. Huang, L., Vitos, L., Kwon, S.K., Johansson, B., Ahuja, R.: Phys. Rev. B 73, 104203 (2006) 19. Ropo, M., Kokko, K., Vitos, L., Koll´ ar, J., Johansson, B.: Surf. Sci. 600, 904–913 (2006) 20. Vitos, L., Korzhavyi, P.A., Johansson, B.: Nature Materials 2, 25–28 (2003) 21. Dubrovinsky, L., et al.: Nature 422, 58–61 (2003) 22. Dubrovinsky, N., et al.: Phys. Rev. Lett. 95, 245502(4) (2005) 23. Vitos, L., Korzhavyi, P.A., Johansson, B.: Phys. Rev. Lett. 96, 117210(4) (2006) 24. Vitos, L., Nilsson, J.-O., Johansson, B.: Acta Materialia 54, 3821–3826 (2006) 25. Moruzzi, V.L., Janak, J.F., Schwarz, K.: Phys. Rev. B. 37, 790–799 (1988) 26. Murnaghan, F.D.: Proc. Nat. Acad. Sci. 30, 244–247 (1944) 27. Ledbetter, H.M.: J. Appl. Phys. 44, 1451–1454 (1973) 28. Ledbetter, H.M.: In: Levy, M., Bass, H.E., Stern, R.R. (eds.) Handbook of Elastic Properties of Solids, Liquids, Gases. Academic, San Diego, vol. III, p. 313 (2001) 29. Denteneer, P.J.H., van Haeringen, W.: J. Phys. C: Solid State Phys. 20, L883–L887 (1987) 30. Steinle-Neumann, G., Stixrude, L., Cohen, R.E.: Phys. Rev. B 60, 791–799 (1999) 31. Perdew, J.P., Burke, K., Ernzerhof, M.: Phys. Rev. Lett. 77, 3865–3868 (1996) 32. Matsuo, Y.: J. Phys. Soc. Japan 53, 1360–1365 (1984) 33. Gault, C., Dauger, A., Boch, P.: Physica Status Solidi (a) 43, 625–632 (1977) 34. King, H.W.: Journal of Materials Science 1, 79–90 (1966) 35. Pearson, W.B.: In: Raynor, G.V. (ed.) Handbook of lattice spacings and structures of metals and alloys, Pergamon Press Ltd., New York (1958) 36. Ropo, M.: Phys. Rev. B 74, 195401(4) (2006)
