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Journal of Alloys and Compounds 720 (2017) 466e472

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Journal of Alloys and Compounds

journal homepage: http://www.elsevier.com/locate/jalcom

Compositional effects on ideal shear strength in Fe-Cr alloys

* Luis Casillas-Trujillo, Liubin Xu, Haixuan Xu

University of Tennessee, Department of and Engineering, Knoxville, TN 37996, USA article info abstract

Article history: The ideal shear strength is the minimum stress needed to plastically deform a defect free crystal; it is of Received 8 August 2016 engineering interest since it sets the upper bound of the strength of a real crystal and connects to the Received in revised form nucleation of . In this study, we have employed spin-polarized functional theory to 25 March 2017 calculate the ideal shear strength, elastic constants and various moduli of body centered cubic Fe-Cr Accepted 15 May 2017 alloys. We have determined the magnetic ground state of the Fe-Cr solid solutions, and noticed that Available online 17 May 2017 calculations without the correct magnetic ground state would to incorrect results of the lattice and elastic constants. We have determined the ideal shear strength along the 〈111〉{110} and 〈111〉{112} slip Keywords: Compositional effects systems and established the relationship between alloy composition and mechanical properties. We 〈 〉 Ideal observe strengthening in the 111 {110} system as a function of composition, while there is no Density functional theory change in strength in the 〈111〉{112} system. The observed differences can be explained by the response Magnetism of the magnetic moments as a function of applied strain. This study provides insights on how electronic FeCr and magnetic interaction of constituent alloying elements may influence the properties of the resulting alloys and their dependence on alloy compositions. © 2017 Elsevier B.V. All rights reserved.

1. Introduction sample made the experimental determination of the ISS difficult. In fact, experimental values of the ISS have been obtained only Fe-Cr alloys, among other based alloys, are currently being recently by nanoindentation measurements [15]. In these experi- used as critical structural materials for various energy applications ments an indenter with a very sharp tip, usually between 50 nm [1e4]. In addition, they are candidates for structural components and 1 mm, is pressed against a material with a low defect density. for next-generation fission and fusion energy systems [5]. These The volume under the indenter can be considered defect-free. The applications require alloys to work in extremely challenging con- value of the stress required to initiate is either the ditions, demanding improved mechanical properties and radiation ideal shear strength or the stress required to nucleate dislocations resistance [6,7]. To achieve optimized performance for these ap- homogeneously [16]. Similar deformation processes also happen plications, it is essential to fundamentally understand the compo- during the measurement, except the latter is at a much sitional effects on the mechanical properties of these alloys. larger length scale. The ideal shear strength (ISS) is defined as the stress that ex- The requirements of a periodic defect-free crystal make first ceeds the limit of elastic stability in a defect-free crystal. It is the principles calculations a suitable way to estimate the ISS [17]. Thus, required stress to move layers of atoms on top of each other and is the ideal strength is one of the mechanical properties that are relevant to the nucleation process. Therefore the ISS is predictable by computational means [16]. Ab initio computations of an important parameter in plasticity and fracture [8,9]. Moreover, the ISS of several materials have been performed, mostly on single during extremely high strain rate events, the ISS represents the element systems: pure bcc [18e20], pure fcc metals, and lattice resistance to dislocation motion [10,11]. In practice, it is of semiconductors [21]. For instance, studies on ferromagnetic bcc engineering interest since the ISS provides an upper limit of the iron have been done by Clatterbuck et al. [22]. Ogata et al. [23] strength of a material. Studies of the ISS have been performed since explored ferromagnetic and iron. In addition, calculations the 1920's [12e14]. However, the requirement of a defect-free have been done in ceramics, carbides, nitrides [24,25], and recently in TiVNbMo high entropy alloys [26]. However, to our knowledge no studies of magnetic alloy systems have been performed, partially because calculations of the ISS in magnetic systems are * Corresponding author. E-mail address: [email protected] (H. Xu). challenging. First, the magnetic ground state of the alloy must be http://dx.doi.org/10.1016/j.jallcom.2017.05.167 0925-8388/© 2017 Elsevier B.V. All rights reserved. L. Casillas-Trujillo et al. / Journal of Alloys and Compounds 720 (2017) 466e472 467 determined, and secondly, the magnetic moments will experience In this work, we have calculated the ideal shear strength by frustration; the magnetic moments of chromium atoms would like applying incremental affine shears to the crystal along the bcc to point in the opposite direction to the iron magnetic moments, common slip systems, 〈111〉{110} and 〈111〉{112}. The ISS t is ob- and at the same time in the opposite direction to other chromium tained by calculating the energy and volume as a function of the atoms, as described by Soderlind et al. [27]. strain, and taking the derivative of the energy E with respect to the In this work, the compositional effects on ISS in bcc Fe-Cr system strain g, shown in Equation (1). The ideal shear strength is the are investigated using density functional theory calculations. The maximum of the stress-strain curve [28]. challenge associated with theoretical investigation of Fe-Cr has To apply the shear strain it is convenient to use a coordinate been widely recognized, due to the subtle but critical magnetic system defined by the slip system. This Cartesian coordinate system interactions in the alloy systems. Specifically, we have determined has unit vectors ei(i ¼ 1,2,3), where e1 is pointing in the slip di- the ISS of magnetic Fe-Cr up 12.9% Cr along the 〈111〉{110} and 〈111〉 rection, and e2 and e3 are perpendicular to e1. This transformation {112} slip systems. To determine the electronic and magnetic origin converts the cubic unit cell to an orthorhombic supercell, with the of the observed trend, the cohesive energy, elastic constants, and lattice vectors aa(a ¼ 1,2,3) pointing in the direction of the unit several elastic moduli of Fe-Cr are also obtained and compared with vectors ei. To apply the shear strain in the slip plane perpendicular previous studies. to a3, the supercell is deformed by applying a deformation matrix D with a non-zero shear element. We have chosen an alternative way to perform the ISS calcula- 2. Methodology tions in order to preserve cubic symmetry. Instead of transforming the coordinate system to the slip system reference, we transform 2.1. Ideal shear stress the deformation matrix to the cubic crystal system. Fig. 1a shows the usual transformation used for ISS calculations, a cubic system Shear deformation can be applied by two different ways: affine with basis vectors ð100Þ, ð010Þ, and ð001Þ is transformed to a co- and alias [7]. In the affine shear deformation, all atoms are shifted ordinate system with basis vectors ð111Þ, ð110Þ, and ð112Þ, with parallel to the direction of shearing. Practically, only lattice vectors ð111Þ pointing in the slip direction, and with vectorsð110Þ, and are changed, while the fractional coordinates of the atoms within ð112Þ perpendicular to the slip planes. The deformation is applied the supercell remain the same. In the alias deformation only the top to the orthogonal supercell. Fig. 1b is the transformation used in layer of the cell is initially displaced in the shear direction, while the this work. The process is similar to that described above, but an atoms in all other layers remain in their original positions. The inverse transformation is applied to the deformation matrix to displacement of the top layer initially only influences the layer next convert it to the cubic crystal system. to it, but atomistic relaxation to the propagation of the displacement from top to bottom. Comparing these two methods, fi the af ne deformation gives a homogeneous deformation of the 1 vE lattice, while the alias deformation depends on the shear plane. We t ¼ (1) V vg have chosen to use the affine shear, since it offers a homogeneous deformation, which is well suited to the alloy systems.

Fig. 1. Transformations used to apply the shear deformation. A is the basis vector matrix, D is the deformation matrix, and S is the transformation matrix to a Cartesian system with ¡ the slip system as basis vectors. S 1 is the inverse transformation, used to convert the system back to the cubic crystal system. (a) The coordinate system commonly used for ISS calculations, where the basis vectors are associated with the slip system. (b) The coordinate system used in this work, where the deformation matrix transformed to the to the cubic crystal system reference. 468 L. Casillas-Trujillo et al. / Journal of Alloys and Compounds 720 (2017) 466e472

2.2. Elastic properties [35]. The search of the SQS was restricted to the 54 atom 3 3 3 cubic supercell using the mcsqs [36] program of the Alloy Theoretic The elastic properties of cubic single crystals are described by Automated Toolkit (ATAT) [37]. the three independent elastic constants C11, C12, and C44.Wehave calculated the polycrystalline elastic moduli from their relationship with the elastic constants. For cubic solids, the bulk-modulus (B)is 3. Results and discussion equivalent to the polycrystalline bulk-modulus and its relationship with the elastic constants is given by Equation (2). 3.1. Magnetic ground state, lattice constant and cohesive energies of Fe-Cr alloys ðC þ 2C Þ B ¼ 11 12 (2) 3 Before we determine the mechanical properties of the alloys, the fi The polycrystalline shear modulus (G) has been obtained by the ground state properties are examined rst. The magnetic state of fl Hill averaging method [29], the binary alloy will be in uenced by the magnetic states of the parent elements, and the resulting state is a function of the ðG þ G Þ composition of the alloy. In solid solution, it is known that chro- G ¼ V R (3) 2 mium atoms will try to align their magnetic moment in the opposite direction to the iron atoms, but at the same time they will where the Voigt (GV) and Reuss (GR) bounds are given by Ref. [30], also want to have it opposite to other chromium atoms, leading to magnetic frustration in the lattice as the chromium concentration ð Þ ð þ Þ ¼ 5 C11 C12 C44 ¼ C11 C12 3C44 increases. Olsson et al. showed that for concentrations 〈20% chro- GR ð þ Þ GV (4) 4C44 3C11 3C12 5 mium, the system is well described with a bcc structure in the e Finally the Young's modulus (E), is connected to B and G by ferromagnetic state [38 40], with a Curie temperature around e Ref. [30], 900 1500 K [41]. The spinodal decomposition clustering occurs at high chromium concentrations [42,43]. We focus on the region of 9BG the phase diagram where Fe-Cr forms single phase solid solutions. E ¼ (5) ð3B þ GÞ At 300 C bcc iron forms a solid solution up to 9% chromium; at higher temperatures chromium is completely soluble in iron for all compositions [44]. Previous theoretical studies have performed calculations for the whole compositional range of Fe-Cr alloys. We 2.3. Computational details have performed calculations up to 12.9% chromium. We initially calculated the magnetic ground state for the parent The DFT calculations were carried out using the generalized elements, iron and chromium, using collinear spin calculations and gradient approximation (GGA) with the PerdeweBurkeeErnzerhof considering ferromagnetic, antiferromagnetic, and nonmagnetic functional [31]. The electroneion interaction is described by the configurations. Fig. 2 shows the energy versus lattice constant for projector augmented wave method (PAW) [32], and the plane wave different magnetic states of iron (Fig. 2a), and chromium (Fig. 2b). basis energy cutoff was set to 500 eV. The Vienna ab initio package Our results are in agreement with previous experimental and VASP [33,34] was employed to perform the simulations. For the Fe- theoretical results. The magnetic state with the lowest energy for Cr simulations, we employed a 3 3 3 supercell with a 2 2 2 bcc iron is ferromagnetic, which is consistent with [45,46]. Chro- k-point sampling grid. The force on each atom is relaxed to less than mium possesses a spin density wave antiferromagnetic ordering 0.01 eV/Å. To assess magnetism, spin polarized DFT was employed. [47]. We found that the antiferromagnetic configuration possesses The effect of local configuration becomes more pronounced in the lowest energy for chromium. In addition, we performed non- magnetic systems due to the frustration of the magnetic moments. collinear spin calculations for both iron and chromium, as can be We have averaged the results of three different configurations seen in Fig. 2, the difference in energy between the lowest collinear generated using the special quasirandom structures approach (SQS) and non-collinear calculations is very small, ~0.003 eV, which is in

Fig. 2. Energy versus lattice constant for non-magnetic, antiferromagnetic, ferromagnetic and non-collinear calculations for (a) bcc iron, (b) chromium. L. Casillas-Trujillo et al. / Journal of Alloys and Compounds 720 (2017) 466e472 469

Table 1 Lattice constant of iron and chromium for different magnetic states.

Antiferromagnetic Ferromagnetic Non-magnetic Non-collinear (Å) (Å) (Å) (Å)

Fe 2.8030 2.8310 2.7501 2.83076 Cr 2.8625 2.8376 2.8374 2.8502

agreement with the values reported by Hafner et al. [48]. However, the non-collinear feature did not change the overall magnetic configurations. ISS calculations are more sensitive to the energy difference rather than absolute values of energies. Therefore, collinear spin calculations are used in this work, due to the small energy difference and computational costs associated with the non-collinear calculations. Table 1 tabulates the value of the lattice constant for the different magnetic states from Fig. 2. Vegard's law establishes a linear trend between the lattice constants of solid solutions and the concentration of the alloying elements. This behavior is not observed in the Fe-Cr solid solutions. Experimental measurements of the lattice constant as a function of composition do not follow Vegard's law, and a clear linear trend Fig. 4. Mixing energy of Fe-Cr. We compare the results obtained in this work with cannot be established. This behavior can be appreciated in Fig. 3a, previous theoretical studies. but it becomes more evident for the whole compositional range as in Ref. [49]. The non-linear behavior is accentuated in the theo- amount, but all of them exhibit the same trend. retical results shown in Fig. 3a with this study's DFT results, and the The good agreement obtained in lattice constant and mixing results obtained with exact muffin- orbitals (EMTO) [50]. The energy between the results of this work and previous studies in- dashed line in Fig. 3 represents Vegard's law for the DFT data. The dicates our simulations are able to reproduce the ground state lattice constant obtained from DFT is an underestimation for all properties of the Fe-Cr alloys. The obtained relaxed structures compositions compared with EMTO. However, both methods pre- represent a good starting point for the ISS and elastic constant dict qualitatively the same trend. A maximum in the lattice con- calculations. stant is obtained between 7.5 and 10% chromium in both approaches. We have identified that the alloy with the largest lat- tice constant corresponds to the configuration with the highest 3.2. ISS and elastic constants of Fe-Cr magnetic moment of iron. We found there is no correlation be- tween the magnetic moment of chromium and the lattice constant. Fig. 5a shows the Fe-Cr energy versus strain curve for different Fig. 3b plots the lattice constant and the magnetic moment of iron chromium compositions of the 〈111〉{112} slip system. The ISS re- as a function of chromium concentration, the maximum value for sults for both 〈111〉{112} and 〈111〉{110} slip systems are summa- both parameters occurs at 9.25% chromium content. A similar rized in Fig. 5b. The ISS shows different behaviors between the two behavior has been reported for Fe-Ni alloys [51] where the slip systems. First, it can be noted that the magnitude of the ISS in maximum in lattice constant corresponds to the composition the 〈111〉{110} slip system is greater than that of the 〈111〉{112} slip where the magnetic moments reach their maximum value. system for all concentrations. Secondly, they exhibit a different Based on the obtained ground states, the mixing energy as a behavior as a function of concentration. An increase in magnitude function of chromium composition is calculated and shown in can be observed in the 〈111〉{110} system as more chromium atoms Fig. 4, and compared with previous results from theoretical calcu- are incorporated. The magnitude of the ISS for the 〈111〉{112} slip lations for the low chromium concentration regime [38,39,52]. system remains almost constant as a function of chromium Quantitatively the value for each of these studies differs by a small composition.

Fig. 3. (a) Lattice constant as a function chromium concentration, from experimental results of Pearson [49], EMTO calculations of Zhang [50], and the DFT results from this work. The dashed line indicates Vegard's law for the DFT calculations. (b) The lattice constant as a function of concentration for the current work is plotted in the left y axis, and the average magnetic moment for iron atoms in the right y axis. 470 L. Casillas-Trujillo et al. / Journal of Alloys and Compounds 720 (2017) 466e472

Fig. 5. (a) Energy versus strain curve of Fe-Cr for different compositions for the 〈111〉{112} slip system. The overall energy of the system decreases as the concentration of chromium rises. The ideal shear strength will be given by the maximum of the derivative of this curve. (b) Ideal shear stress as a function of chromium concentration for the 〈111〉{110} slip system and the 〈111〉{112} slip system.

In order to determine the origin of the nonlinear behavior of the with the experimental values. The magnitude of the moduli in- ISS as a function of strain in the 〈111〉{110} system, we carried out creases almost linearly as a function of chromium composition. We calculations to determine the elastic properties of the alloy systems therefore conclude that elastic properties and ISS show different to compare the elastic and plastic behavior of the alloys. The elastic compositional dependence and only the plastic deformation constants were calculated by performing six finite distortions of the involving 〈111〉{110} slip system exhibits non-linear behavior. lattice and deriving the elastic constants from the strain-stress We subsequently performed magnetic structure analysis and relationship [53]. The value of the elastic constants increases as found that the difference in behavior between the 〈111〉{112} and the concentration of chromium increases, and consequently the the 〈111〉{110} can be rationalized by magnetic response of the magnitude of the elastic moduli also increases, which means the system. Fig. 7 plots the change in magnetic moment for chromium, elastic properties of iron are improved as chromium is added to the iron, and the total moment of the system as a function of chromium system up to 12.9%. Fig. 6a shows the compositional effect on the composition. We observe a different behavior in the response of the elastic constants. We have compared the DFT results from this chromium magnetic moments in these two slip systems. In the study with elastic constants obtained by EMTO [50]. In both cases, 〈111〉{110} slip system (Fig. 7a) the chromium magnetic moment we observe an increase in an almost linear fashion. In the DFT case, presents a maximum, while for the 〈111〉{112} slip system (Fig. 7b) results show an increase in C12 and C44, while a subtle increase is the opposite behavior is observed; the magnetic moment of chro- observed in C11. The increase in value is more prominent for the mium possesses a minimum. More importantly, in the 〈111〉{110} EMTO results. The shear modulus and Young's modulus are shown system, the magnetic moment of chromium changes with chro- as a function of composition in Fig. 6b. We again compare with the mium composition; whereas in the 〈111〉{112} system the relative EMTO calculations, and with experimental results [54]. The results change of chromium magnetic moment are the same for all com- obtained in this study by DFT are in better agreement with the positions. For the iron atoms, in the 〈111〉{110} slip system (Fig. 7c), experimental results than those reported by EMTO. The magni- similar behaviors are observed for different chromium concentra- tudes of the theoretical values are overestimated in comparison tions, while in the 〈111〉{112} slip system (Fig. 7d) it can be observed

Fig. 6. (a) Elastic constants of Fe-Cr as a function of Cr composition from this work calculated using DFT, and previous EMTO calculations [50]. (b) Elastic and shear moduli comparison between experimental results [54], EMTO and DFT values from this work. L. Casillas-Trujillo et al. / Journal of Alloys and Compounds 720 (2017) 466e472 471

Fig. 7. Change in magnetic moment per atom as a function of strain, for chromium in the (a) 〈111〉{110} slip system, (b) 〈111〉{112} slip system; iron (c) 〈111〉{110} slip system, (d) 〈111〉{112} slip system; and total magnetic moment of the system (e) 〈111〉{110} slip system, (f) 〈111〉{112} slip system. the presence of a maximum for low chromium concentrations and change consistently with composition. transitions to two local maximums and a local minimum as the Based on the above results, the addition of chromium improves chromium concentration increases. The total magnetic moment the strength of the alloy, which is consistent with previous studies shows similar behavior as iron (Fig. 7e and f). In addition, the [50]. It should be noted that solutes in bcc metals could lead to chromium magnetic moment is sensitive to the local configuration, softening at very low temperatures [55]. One aspect that has not as it can be seen from the large error bars in the figure. Based on been considered in this study is that chromium atoms could alter these observations, we believe the difference in behavior of ISS can the structure of the dislocation core, giving rise to the ductile-to- be attributed to the magnetic response of chromium, since both ISS brittle transition instead of hardening. Since we are treating and change in chromium magnetic moment depend on alloy dislocation free crystals, the hardening observed in this study is due composition in the 〈111〉{110} slip system and are independent of to the changes in electronic and magnetic structure and the strain chromium concentration in the 〈111〉{112} system. In comparison, introduced by substitutional impurities. both the magnetic moment of iron and total magnetic moment 472 L. Casillas-Trujillo et al. / Journal of Alloys and Compounds 720 (2017) 466e472

4. Conclusions Philos. Mag. A 81 (2001) 1725e1747. [20] W. Xu, J.A. Moriarty, Atomistic simulation of ideal shear strength, point de- fects, and screw dislocations in bcc transition metals: Mo as a prototype, Phys. In this work, we have established the relationship between Rev. B 54 (1996) 6941. mechanical properties and composition in Fe-Cr alloys, and [21] D. Roundy, M.L. Cohen, Ideal strength of , Si, and Ge, Phys. Rev. B 64 assessed the impact of magnetism in these properties. We have (2001) 212103. [22] D. Clatterbuck, D. Chrzan, J. Morris, The ideal strength of iron in tension and calculated the lattice constant, ideal shear stress, elastic constants shear, Acta Mater. 51 (2003) 2271e2283. and cohesive energies for Fe-Cr (up to 12.9%) solid solutions as a [23] S. Ogata, J. Li, N. Hirosaki, Y. Shibutani, S. Yip, Ideal shear strain of metals and function of composition. The calculations of the lattice constant as a ceramics, Phys. Rev. B 70 (2004) 104104. [24] S.-H. Jhi, S.G. Louie, M.L. Cohen, J. Morris Jr., Mechanical instability and ideal function of composition present a maximum at 9.25% chromium, shear strength of transition carbides and nitrides, Phys. Rev. Lett. 87 which corresponds to the composition with the maximum value of (2001) 075503. the iron magnetic moment. The addition of chromium to the Fe-Cr [25] Y.-J. Wang, C.-Y. Wang, A comparison of the ideal strength between L1 2 Co 3 (Al, W) and Ni 3 Al under tension and shear from first-principles calculations, system improves its elastic properties within the compositional Appl. Phys. Lett. 94 (2009) 261909. range considered in this study. The magnitude of the shear and [26] F. Tian, D. Wang, J. Shen, Y. Wang, An ab initio investigation of ideal tensile Young's modulus also increase as more chromium is added to the and shear strength of TiVNbMo high-entropy alloy, Mater. Lett. 166 (2016) e system. Finally, the ISS for the two slip systems in question show 271 275. [27] P. Soderlind,€ J.A. Moriarty, J.M. Wills, First-principles theory of iron up to different behaviors. The ISS of the 〈111〉{112} slip system is inde- earth-core : structural, vibrational, and elastic properties, Phys. Rev. pendent of composition, whereas the ISS of the 〈111〉{110} slip B 53 (1996) 14063. system increases with chromium concentration. This difference in [28] D. Roundy, C. Krenn, M.L. Cohen, J. Morris Jr., Ideal shear strengths of fcc aluminum and , Phys. Rev. Lett. 82 (1999) 2713. behavior can be attributed to the response of the magnetic moment [29] R. Hill, The elastic behaviour of a crystalline aggregate, Proc. Phys. Soc. Sect. A of chromium to the strain conditions in each of the systems. 65 (1952) 349. [30] L. Vitos, Computational Quantum Mechanics for Materials Engineers: the EMTO Method and Applications, Springer Science & Business Media, 2007. Acknowledgements [31] J.P. Perdew, J. Chevary, S. Vosko, K.A. Jackson, M.R. Pederson, D. Singh, C. Fiolhais, Erratum: atoms, molecules, solids, and surfaces: applications of the We are thankful to Par Olsson for providing some of the input generalized gradient approximation for exchange and correlation, Phys. Rev. B fi 48 (1993) 4978. les for comparing with previous DFT calculations. The research is [32] P.E. Blochl,€ Projector augmented-wave method, Phys. Rev. B 50 (1994) 17953. sponsored by the U.S. Department of Energy Nuclear Energy Uni- [33] G. Kresse, J. Furthmüller, Efficiency of ab-initio total energy calculations for versity Program (NEUP) under project number 14-6346. This metals and semiconductors using a plane-wave basis set, Comput. Mater. Sci. 6 (1996) 15e50. research used resources of The National Institute for Computational [34] G. Kresse, J. Hafner, Ab initio molecular dynamics for metals, Phys. Rev. Sciences at UT under contract UT-TENN0112. B 47 (1993) 558. [35] A. Zunger, S.-H. Wei, L. Ferreira, J.E. Bernard, Special quasirandom structures, References Phys. Rev. Lett. 65 (1990) 353. [36] A. Van de Walle, P. Tiwary, M. De Jong, D. Olmsted, M. Asta, A. Dick, D. Shin, Y. Wang, L.-Q. Chen, Z.-K. Liu, Efficient stochastic generation of special qua- [1] L.K. Beland, Y.N. Osetsky, R.E. Stoller, H. Xu, Interstitial loop transformations in sirandom structures, Calphad 42 (2013) 13e18. FeCr, J. Alloys Compd. 640 (2015) 219e225. [37] A. van de Walle, Alloy Theoretic Automated Toolkit (ATAT), Cal Tech, Pasa- [2] C. Heintze, F. Bergner, A. Ulbricht, H. Eckerlebe, The microstructure of dena, CA, 2008. neutron-irradiated FeeCr alloys: a small-angle neutron scattering study, [38] P. Olsson, I.A. Abrikosov, L. Vitos, J. Wallenius, Ab initio formation energies of J. Nucl. Mater. 409 (2011) 106e111. FeeCr alloys, J. Nucl. Mater. 321 (2003) 84e90. [3] Z. Jiao, G. Was, Novel features of radiation-induced segregation and radiation- [39] P. Olsson, C. Domain, J. Wallenius, Ab initio study of Cr interactions with point induced precipitation in austenitic stainless , Acta Mater. 59 (2011) defects in bcc Fe, Phys. Rev. B 75 (2007) 014110. 1220e1238. [40] C. Jiang, C. Wolverton, J. Sofo, L.-Q. Chen, Z.-K. Liu, First-principles study of [4] H. Xu, R.E. Stoller, Y.N. Osetsky, D. Terentyev, Solving the puzzle of ˂100˃ binary bcc alloys using special quasirandom structures, Phys. Rev. B 69 (2004) interstitial loop formation in bcc iron, Phys. Rev. Lett. 110 (2013) 265503. 214202. [5] A. Hishinuma, A. Kohyama, R. Klueh, D. Gelles, W. Dietz, K. Ehrlich, Current [41] R. Hultgren, P.D. Desai, D.T. Hawkins, M. Gleiser, K.K. Kelley, Selected Values of status and future R&D for reduced-activation ferritic/martensitic steels, the Thermodynamic Properties of Binary Alloys, 1973. DTIC Document. J. Nucl. Mater. 258 (1998) 193e204. [42] J. Cieslak, S. Dubiel, B. Sepiol, Mossbauer-effect€ study of the phase separation [6] F. Garner, M. Toloczko, B. Sencer, Comparison of swelling and irradiation creep in the Fe-Cr system, J. Phys. Condens. Matter 12 (2000) 6709. behavior of fcc-austenitic and bcc-ferritic/martensitic alloys at high neutron [43] S. Tavares, R. De Noronha, M. Da Silva, J. Neto, S. Pairis, 475 C embrittlement in exposure, J. Nucl. Mater. 276 (2000) 123e142. a duplex stainless UNS S31803, Mater. Res. 4 (2001) 237e240. [7] M. Jahnatek, J. Hafner, M. Krajcí, Shear deformation, ideal strength, and [44] A. Handbook, Alloy Phase Diagrams, vol. 3, ASM International, Materials Park, stacking fault formation of fcc metals: a density-functional study of Al and Cu, OH, USA, 1992, 2.48. Phys. Rev. B 79 (2009) 224103. [45] J. Kübler, Magnetic moments of ferromagnetic and antiferromagnetic bcc and [8] Y. Umeno, M. Cerný, Effect of normal stress on the ideal shear strength in fcc iron, Phys. Lett. A 81 (1981) 81e83. covalent crystals, Phys. Rev. B 77 (2008) 100101. [46] V. Moruzzi, P. Marcus, K. Schwarz, P. Mohn, Ferromagnetic phases of bcc and [9] K.J. Van Vliet, J. Li, T. Zhu, S. Yip, S. Suresh, Quantifying the early stages of fcc Fe, Co, and Ni, Phys. Rev. B 34 (1986) 1784. plasticity through nanoscale experiments and simulations, Phys. Rev. B 67 [47] L. Corliss, J. Hastings, R. Weiss, Antiphase antiferromagnetic structure of (2003) 104105. chromium, Phys. Rev. Lett. 3 (1959) 211. [10] E.W. Hart, Lattice resistance to dislocation motion at high velocity, Phys. Rev. [48] R. Hafner, D. Spisak, R. Lorenz, J. Hafner, Magnetic ground state of Cr in 98 (1955) 1775e1776. density-functional theory, Phys. Rev. B 65 (2002) 184432. [11] W. Johnston, J.J. Gilman, Dislocation velocities, dislocation , and [49] W. Pearson, WB Pearson a Handbook of Lattice Spacings and Structures of plastic flow in fluoride crystals, J. Appl. Phys. 30 (1959) 129e144. Metals and Alloys, vol. 1, Pergamon Press, London, 1958. € [12] J. Frenkel, Zur theorie der elastizitatsgrenze und der festigkeit kristallinischer [50] H. Zhang, B. Johansson, L. Vitos, Ab initio calculations of elastic properties of € korper, Z. Phys. 37 (1926) 572e609. bcc Fe-Mg and Fe-Cr random alloys, Phys. Rev. B 79 (2009) 224201. [13] A. Kelly, N.H. Macmillan, Strong Solids, Oxford University Press, Walton Street, [51] G. Cacciamani, A. Dinsdale, M. Palumbo, A. Pasturel, The FeeNi system: Oxford OX 2 6 DP, UK, 1986 (1986). thermodynamic modelling assisted by atomistic calculations, Intermetallics [14] M. Polanyi, The X-ray fiber diagram, Z Phys. 7 (1921) 149e180. 18 (2010) 1148e1162. [15] A. Gouldstone, H.-J. Koh, K.-Y. Zeng, A. Giannakopoulos, S. Suresh, Discrete and [52] G. Bonny, D. Terentyev, L. Malerba, The hardening of ironechromium alloys continuous deformation during nanoindentation of thin films, Acta Mater. 48 under thermal ageing: an atomistic study, J. Nucl. Mater. 385 (2009) 278e283. (2000) 2277e2295. [53] Y. Le Page, P. Saxe, Symmetry-general least-squares extraction of elastic data [16] C. Krenn, D. Roundy, M.L. Cohen, D. Chrzan, J. Morris Jr., Connecting atomistic for strained materials from ab initio calculations of stress, Phys. Rev. B 65 and experimental estimates of ideal strength, Phys. Rev. B 65 (2002) 134111. (2002) 104104. [17] W. Luo, D. Roundy, M.L. Cohen, J. Morris Jr., Ideal strength of bcc [54] G. Speich, A. Schwoeble, W.C. Leslie, Elastic constants of binary iron-base al- and , Phys. Rev. B 66 (2002) 094110. loys, Metall. Trans. 3 (1972) 2031e2037. [18] C. Krenn, D. Roundy, J. Morris, M.L. Cohen, Ideal strengths of bcc metals, [55] D.R. Trinkle, C. Woodward, The chemistry of deformation: how solutes soften Mater. Sci. Eng. A 319 (2001) 111e114. pure metals, Science 310 (2005) 1665e1667. [19] D. Roundy, C. Krenn, M.L. Cohen, J. Morris Jr., The ideal strength of ,