Temperature dependent stacking fault energy of FeCrCoNiMn high entropy alloy

Temperature dependent stacking fault energy of FeCrCoNiMn high entropy alloy

SMM 10688 No. of Pages 4, Model 5G 15 June 2015 Scripta Materialia xxx (2015) xxx–xxx 1 Contents lists available at ScienceDirect Scripta Material...

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SMM 10688

No. of Pages 4, Model 5G

15 June 2015 Scripta Materialia xxx (2015) xxx–xxx 1

Contents lists available at ScienceDirect

Scripta Materialia journal homepage: www.elsevier.com/locate/scriptamat 5 6

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Temperature dependent stacking fault energy of FeCrCoNiMn high entropy alloy

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Shuo Huang a,⇑, Wei Li a, Song Lu a, Fuyang Tian b, Jiang Shen b, Erik Holmström c, Levente Vitos a,d,e,⇑

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Applied Materials Physics, Department of Materials Science and Engineering, Royal Institute of Technology, Stockholm SE-100 44, Sweden Department of Physics, University of Science and Technology Beijing, Beijing 100083, China c Sandvik Coromant R&D, 126 80 Stockholm, Sweden d Department of Physics and Astronomy, Division of Materials Theory, Uppsala University, Box 516, SE-75120 Uppsala, Sweden e Institute for Solid State Physics and Optics, Wigner Research Centre for Physics, P.O. Box 49, H-1525 Budapest, Hungary b

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Article history: Received 27 April 2015 Accepted 17 May 2015 Available online xxxx Keywords: High-entropy alloy Stacking fault energy Twinning First-principles calculation

a b s t r a c t The stacking fault energy (SFE) of paramagnetic FeCrCoNiMn high entropy alloy is investigated as a function of temperature via ab initio calculations. We divide the SFE into three major contributions: chemical, magnetic and strain parts. Structural energies, local magnetic moments and elastic moduli are used to estimate the effect of temperature on each term. The present results explain the recently reported twinning observed below room-temperature and predict the occurrence of the hexagonal phase at cryogenic conditions. Ó 2015 Published by Elsevier Ltd. on behalf of Acta Materialia Inc.

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High entropy alloys (HEAs) are multi-component alloys with elements having equal or nearly equal concentrations. Since 2004, HEAs have been one of the most promising fields of metallic materials and opened a new exciting research area in condensed matter physics [1–3]. Due to the sluggish diffusion, severe lattice distortion and cocktail effects [4,5], HEAs often exhibit many excellent properties, such as high strength [6], outstanding thermal stability [7,8], as well as good resistances to wear, corrosion and oxidation [9–11]. As potential engineering materials, FeCrCoNiMn HEA triggered intense research activity [2,12–31]. It was found that equiatomic FeCrCoNiMn remains single face-centered cubic (fcc) disordered solid solution even after elevated temperature exposures of several days [12,13]. The yield strength, ultimate tensile strength and elongation to fracture of this particular alloy increase simultaneously with decreasing temperature [14,15]. Recent microstructural observation indicated that this phenomenon is related to the change in the deformation mechanism from conventional dislocation glide at room temperature to nano-twinning at relatively low temperatures [15,16]. It is known that stacking fault energy (SFE) affects the propensity to form deformation twins, and low-SFE materials are more

⇑ Corresponding authors at: Applied Materials Physics, Department of Materials Science and Engineering, Royal Institute of Technology, Stockholm SE-100 44, Sweden (L. Vitos). E-mail addresses: [email protected] (S. Huang), [email protected] (L. Vitos).

likely to deform by twinning, with increased strain hardening rate, dislocation storage capacity, and ductility [32–35]. Recent experiment indicated that FeCrCoNiMn HEA is a strong brass type deformation texture typical of low-SFE materials [30]. The SFE in this alloy has been quantified at room temperature using experimental measurements in combination with elastic moduli obtained from ab initio calculations [31]. This pioneering work by Zaddach et al. employed a well-established semi-empirical relationship connecting the SFE to the elastic moduli and stacking fault probability [36] but unfortunately provided no information about the temperature dependence of the SFE. In fact, to our best knowledge, no ab initio investigations of the SFE of the above or any other HEAs have been published so far, which may be attributed to the complexity of the problem related to the chemical and magnetic disorder present in many HEAs. In this letter, using first-principles alloy theory we put forward a comprehensive study of the chemical, magnetic and strain effects responsible for the temperature dependence of the SFE for the FeCrCoNiMn HEA. In ab initio calculations, the SFE is often computed as the excess free energy associated with the formation of an intrinsic stacking fault in an otherwise perfect fcc lattice. In the present application, the total energies were calculated using the exact muffin-tin orbitals (EMTO) method [37,38], in combination with the coherent potential approximation (CPA) [39,40]. The one-electron Kohn–Sham equations were solved within the scalar-relativistic approximation and the soft-core scheme. The self-consistent

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Temperature (K) Fig. 1. Theoretical stacking fault energy of FeCrCoNiMn high entropy alloy. Panel (a) show the total SFE cSFE = (cchem + cmag + cstrain), and panel (b) the individual contribution: chemical part cchem, magnetic part cmag and strain part cstrain. The quoted semi-empirical data is taken from Ref. [31]. In panel (a) we highlight the observed deformation regimes (SLIP: dislocation glide; TWIP: twinning) as discussed in Ref. [16] and indicate a possible transition from TWIP + SLIP towards TRIP (phase transformation) regime with decreasing temperature.

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the hcp–fcc structural energy difference increases with increasing volume (not shown), which explains the obtained positive temperature slope of cchem. The magnetic part is defined as cmag = T(Sisf  S0)/A, where Sisf and S0 denote the magnetic entropy in the faulted and perfect lattice, respectively. The magnetic entropy is estimated using the mean-field expression S = Ri kBciln (1 + li), where ci and li are the concentration and the local magnetic moments for atom i, respectively (kB is the Boltzmann constant). This expression corresponds to a completely disordered paramagnetic state [46]. As indicated in Fig. 1b, cmag increases rapidly at low temperature and decreases slightly at high temperature. To identify the magnetic contribution, in Fig. 2a we present the magnetic moment difference between perfect and faulted lattices for each element with respect to the number of layers N (N represents the Nth-nearest layer around stacking fault plane) at room temperature. It is found that the Fe and Mn atoms in layer 1 contribute a major part to the magnetic moment difference, and starting from layer 3 the difference can be neglected. The inset shows the temperature dependence of the magnetic moment for each element in the perfect phase. It is immediately clear from the inset that temperature enhances the magnetic moments of Fe and Mn atoms. We notice that the magnetic moment increase from Fig. 2a is due merely to the thermal lattice expansion (i.e., the longitudinal spin fluctuations have been neglected). In this particular alloy, the average room temperature moments per Fe and Mn atoms are 1.64 lB and 1.40 lB, respectively, while for Cr, Co and Ni atoms, the static DLM magnetic moments are nearly zero. Compared to the bulk values the Fe and Mn magnetic moments in the first layer next to the stacking fault decrease by 16% and 49%, respectively. The large difference in the magnetic moments results in a large magnetic entropy difference, and hence a large cmag (14.8 mJ/m2) at room temperature. As the effect of temperature is considerable, in Fig. 2b we also display the magnetic moment difference for each element in layer 1 as a function of temperature. Notice that the magnetic moment differences for Fe and Mn decrease monotonously with increasing temperature, indicating a continuously decreasing magnetic entropy difference. This result explains the slightly decreasing trend of cmag at high temperatures. In experiments, the SFE is determined by observing stacking faults terminated by partial dislocations inside the grains. Since the staking fault leads to additional structural relaxations, the

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calculations were performed within the generalized gradient approximation proposed by Perdew, Burke and Ernzerhof (PBE) [41]. The paramagnetic state observed in the present HEA was simulated by the disordered local moments (DLM) model [42]. The lattice parameters corresponding to temperature T were derived from the room-temperature experimental value taking into account the thermal expansion coefficient [25]. To minimize the errors induced by k-point sampling and structure relaxation, for both SFE and bulk energy calculations we adopted a supercell consisting of nine fcc (1 1 1) layers with and without one intrinsic stacking fault [43]. The effect of temperature on SFE was accounted for via the quasi-harmonic approximation and including the magnetic entropy term. The electronic entropy and the explicit lattice vibrational free energy were neglected as their contributions to SFE were verified to be relatively insignificant [44,45]. In Fig. 1a, we present the temperature dependence of the SFE for FeCrCoNiMn alloy. The calculated SFE at room temperature is 21 mJ/m2, which agrees well with the X-ray diffraction measurement (18.3–27.3 mJ/m2) [31]. The obtained results suggest that temperature has a remarkable effect on SFE, especially at relatively low temperatures. With increasing temperature, the slope of SFE with respect to temperature slightly decreases, showing a tendency to saturate at high temperatures. The general trend of SFE, as well as the surprisingly low SFE at low temperatures (e.g. 3.4 mJ/m2 near 0 K), demonstrate that FeCrCoNiMn is more likely to deform by twinning with decreasing temperature, which is consistent with the experiment observation [16]. To obtain an insight into the atomic-level mechanism behind the trend of SFE, in Fig. 1b we present and discuss the three main contributions: the chemical part cchem, the magnetic part cmag and the strain part cstrain of the total SFE. The chemical contribution to SFE represents the free energy change due to the stacking fault in an otherwise perfect fcc crystal. It is calculated as cchem = (Eisf  E0)/A, where Eisf and E0 are the free energies in the faulted and perfect lattice, respectively, and A is the area of the stacking fault. It can be seen from Fig. 1b that cchem increases almost linearly with increasing temperature. Since the atomic packing near the stacking fault follows a hexagonal close-packed (hcp) pattern, the chemical term is often approximated by the free energy difference between the hcp and fcc lattices [44,45]. According to the present total energy calculations,

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Fig. 2. Magnetic moment difference between the perfect and faulted lattice in paramagnetic FeCrCoNiMn high entropy alloy for each alloy component plotted as a function of (a) number of layers N at room temperature, and (b) temperature for atoms in layer 1. The inset shows the temperature dependence of the magnetic moments in the perfect fcc lattice.

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termination may yield a strain energy which adds a positive contribution to the measured SFE [47,48]. For parallel partial dislocations, the strain part of the SFE can be written as cstrain = 0.5sGe2/(1  m), where s is the inter-planar spacing between the close-packed planes parallel to the fault plane, G is the shear modulus, m is Poisson’s ratio, and e is the strain normal to the fault plane [47,48]. Relaxing the interlayer distance near the ideal (infinitely large) stacking fault, we estimated e to be approximately 1.6% for the present HEA. This value compares reasonably well with the experimental result found for austenitic stainless steels (2%) [49]. In the present application, we assumed that e is independent of temperature. In order to estimate the shear modulus and Poisson’s ratio, we calculated the single-crystal elastic constants using the conventional energy-strain approach [38], and the quasi-harmonic approximation. The polycrystalline elastic moduli were computed via the Voigt–Reuss–Hill averaging method [50]. In Fig. 3, we plot the temperature dependence of the shear constant C44 and the tetragonal shear constant C0 = (C11  C12)/2, as well as the bulk modulus B, shear modulus G and Young’s modulus E for the present HEA. The values of B, G and E at room temperature are 140 GPa, 80 GPa and 201 GPa, respectively, which compare reasonably well with the corresponding experimental data [25–27]. The calculated elastic modulus decrease with temperature (due to the lattice expansion) following a normal softening behavior [51,52]. Using the above parameters, we find that at room temperature the estimated value of cstrain for the present HEA is 2.8 mJ/m2, and decreases slightly with increasing temperature as seen in Fig. 1b. The reported excellent combinations of strength and ductility in FeCrCoNiMn HEA at cryogenic conditions was explained by an additional plasticity mechanism (twinning) developing as the temperature is lowered below the room-temperature [16]. Here we use our calculated SFE values to provide a first-principles background for the above observation. According to phenomenological models [53], the competition between the three deformation mechanisms during plastic deformation of an fcc alloy, namely planar slip (dislocation glide), twinning, and phase transformation to hcp lattice is controlled by the size of the SFE. By lowering the SFE, deformation twinning becomes significant, which is commonly referred to as the Twinning Induced Plasticity (TWIP) effect. Using the present SFE results and the experimental observations [16], we conclude that the SFE in FeCrCoNiMn HEA should be less than 21 mJ/m2 for the activation of the TWIP effect. On the other

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hand, very small or negative SFE has been considered as indicator of the Transformation Induced Plasticity (TRIP) phenomena. Since no TRIP effect was reported in FeCrCoNiMn for temperatures around and above 77 K [16], we tend to place the TWIP–TRIP boundary in the present HEA somewhere close to 8 mJ/m2. Nevertheless, considering that in austenitic stainless steels SFE less than 10–18 mJ/m2 resulted in hexagonal close packed (e) phase formation [54–56], one cannot rule out that in the present HEA the TRIP effect might also occur at low homologous temperatures and contribute to the observed fracture toughness. In summary, we have determined the temperature dependence of the SFE for the FeCrCoNiMn high entropy alloy using quantum mechanical first-principles methods. The chemical, magnetic and strain effects responsible for the temperature dependence of the SFE have been analyzed. The present ab initio results are in line with the available room-temperature experimental data. Theory predicts a large positive temperature factor for the SFE, which explains the observed TWIP effect at sub-zero temperatures in FeCrCoNiMn. On the other hand, the very low SFE values at cryogenic conditions suggest the presence of the TRIP effect, which might also contribute to the outstanding combinations of strength and ductility observed in FeCrCoNiMn HEA.

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We acknowledge helpful discussion with P.J. Ferreira. Work supported by the Swedish Research Council, the Swedish Foundation for Strategic Research, Sweden’s Innovation Agency (VINNOVA Grant No. 2014-03374) and the China Scholarship Council. The National 973 Project of China (Grant No. 2011CB606401) and the Hungarian Scientific Research Fund (OTKA 84078 and 109570) are also acknowledged for financial support.

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References

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[1] J.W. Yeh, S.K. Chen, S.J. Lin, J.Y. Gan, T.S. Chin, T.T. Shun, C.H. Tsau, S.Y. Chang, Adv. Eng. Mater. 6 (2004) 299–303. [2] B. Cantor, I.T.H. Chang, P. Knight, A.J.B. Vincent, Mater. Sci. Eng., A 375–377 (2004) 213–218. [3] Y. Zhang, T.T. Zuo, Z. Tang, M.C. Gao, K.A. Dahmen, P.K. Liaw, Z.P. Lu, Prog. Mater Sci. 61 (2014) 1–93. [4] J.W. Yeh, Ann. Chim. Sci. Mater. 31 (2006) 633–648. [5] M.H. Tsai, J.W. Yeh, Mater. Res. Lett. 2 (2014) 107–123. [6] C.J. Tong, M.R. Chen, J.W. Yeh, S.J. Lin, S.K. Chen, T.T. Shun, S.Y. Chang, Metall. Mater. Trans. A 36 (2005) 1263–1271. [7] V. Dolique, A.L. Thomann, P. Brault, Y. Tessier, P. Gillon, Surf. Coat. Technol. 204 (2010) 1989–1992. [8] M.H. Tsai, C.W. Wang, C.W. Tsai, W.J. Shen, J.W. Yeh, J.Y. Gan, W.W. Wu, J. Electrochem. Soc. 158 (2011) H1161–H1165. [9] Y.L. Chou, Y.C. Wang, J.W. Yeh, H.C. Shih, Corros. Sci. 52 (2010) 3481–3491. [10] Y.F. Kao, T.D. Lee, S.K. Chen, Y.S. Chang, Corros. Sci. 52 (2010) 1026–1034. [11] M.H. Chuang, M.H. Tsai, W.R. Wang, S.J. Lin, J.W. Yeh, Acta Mater. 59 (2011) 6308–6317. [12] F. Otto, Y. Yang, H. Bei, E.P. George, Acta Mater. 61 (2013) 2628–2638. [13] F. Otto, N.L. Hanold, E.P. George, Intermetallics 54 (2014) 39–48. [14] A. Gali, E.P. George, Intermetallics 39 (2013) 74–78. [15] F. Otto, A. Dlouhy´, C. Somsen, H. Bei, G. Eggeler, E.P. George, Acta Mater. 61 (2013) 5743–5755. [16] B. Gludovatz, A. Hohenwarter, D. Catoor, E.H. Chang, E.P. George, R.O. Ritchie, Science 345 (2014) 1153–1158. [17] C. Zhu, Z.P. Lu, T.G. Nieh, Acta Mater. 61 (2013) 2993–3001. [18] Y. Wu, W.H. Liu, X.L. Wang, D. Ma, A.D. Stoica, T.G. Nieh, Z.B. He, Z.P. Lu, Appl. Phys. Lett. 104 (2014) 051910. [19] P.P. Bhattacharjee, G.D. Sathiaraj, M. Zaid, J.R. Gatti, C. Lee, C.W. Tsai, J.W. Yeh, J. Alloys Compd. 587 (2014) 544–552. [20] K.Y. Tsai, M.H. Tsai, J.W. Yeh, Acta Mater. 61 (2013) 4887–4897. [21] J.Y. He, C. Zhu, D.Q. Zhou, W.H. Liu, T.G. Nieh, Z.P. Lu, Intermetallics 55 (2014) 9–14. [22] C.C. Tasan, Y. Deng, K.G. Pradeep, M.J. Yao, H. Springer, D. Raabe, JOM 66 (2014) 1993–2001. [23] M.J. Yao, K.G. Pradeep, C.C. Tasan, D. Raabe, Scripta Mater. 72–73 (2014) 5–8. [24] E.W. Huang, D. Yu, J.W. Yeh, C. Lee, K. An, S.Y. Tu, Scripta Mater. 101 (2015) 32– 35. [25] G. Laplanche, P. Gadaud, O. Horst, F. Otto, G. Eggeler, E.P. George, J. Alloys Compd. 623 (2015) 348–353.

241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280

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No. of Pages 4, Model 5G

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S. Huang et al. / Scripta Materialia xxx (2015) xxx–xxx

[26] A. Haglund, M. Koehler, D. Catoor, E.P. George, V. Keppens, Intermetallics 58 (2015) 62–64. [27] Z. Wu, H. Bei, G.M. Pharr, E.P. George, Acta Mater. 81 (2014) 428–441. [28] M. Laurent-Brocq, A. Akhatova, L. Perrière, S. Chebini, X. Sauvage, E. Leroy, Y. Champion, Acta Mater. 88 (2015) 355–365. [29] N. Stepanov, M. Tikhonovsky, N. Yurchenko, D. Zyabkin, M. Klimova, S. Zherebtsov, A. Efimov, G. Salishchev, Intermetallics 59 (2015) 8–17. [30] G.D. Sathiaraj, P.P. Bhattacharjee, J. Alloys Compd. 637 (2015) 267–276. [31] A.J. Zaddach, C. Niu, C.C. Koch, D.L. Irving, JOM 65 (2013) 1780–1789. [32] P.L. Sun, Y.H. Zhao, J.C. Cooley, M.E. Kassner, Z. Horita, T.G. Langdon, E.J. Lavernia, Y.T. Zhu, Mater. Sci. Eng., A 525 (2009) 83–86. [33] H. Bahmanpour, A. Kauffmann, M.S. Khoshkhoo, K.M. Youssef, S. Mula, J. Freudenberger, J. Eckert, R.O. Scattergood, C.C. Koch, Mater. Sci. Eng., A 529 (2011) 230–236. [34] K. Youssef, M. Sakaliyska, H. Bahmanpour, R. Scattergood, C. Koch, Acta Mater. 59 (2011) 5758–5764. [35] Y.L. Gong, C.E. Wen, Y.C. Li, X.X. Wu, L.P. Cheng, X.C. Han, X.K. Zhu, Mater. Sci. Eng., A 569 (2013) 144–149. [36] R.P. Reed, R.E. Schramm, J. Appl. Phys. 45 (1974) 4705–4711. [37] L. Vitos, Phys. Rev. B 64 (2001) 014107. [38] L. Vitos, Computational Quantum Mechanics for Materials Engineers, Springer, London, 2007. [39] P. Soven, Phys. Rev. 156 (1967) 809–813.

[40] L. Vitos, I.A. Abrikosov, B. Johansson, Phys. Rev. Lett. 87 (2001) 156401. [41] J.P. Perdew, K. Burke, M. Ernzerhof, Phys. Rev. Lett. 77 (1996) 3865–3868. [42] B.L. Gyorffy, A.J. Pindor, J. Staunton, G.M. Stocks, H. Winter, J. Phys. F: Met. Phys. 15 (1985) 1337–1386. [43] S. Kibey, J.B. Liu, D.D. Johnson, H. Sehitoglu, Acta Mater. 55 (2007) 6843–6851. [44] L. Vitos, P.A. Korzhavyi, B. Johansson, Phys. Rev. Lett. 96 (2006) 117210. [45] L. Vitos, J.O. Nilsson, B. Johansson, Acta Mater. 54 (2006) 3821–3826. [46] G. Grimvall, Phys. Rev. B 39 (1989) 12300–12301. [47] P.J. Ferreira, P. Müllner, Acta Mater. 46 (1998) 4479–4484. [48] P. Müllner, P.J. Ferreira, Philos. Mag. Lett. 73 (1996) 289–298. [49] J.W. Brooks, M.H. Loretto, R.E. Smallman, Acta Metall. 27 (1979) 1839–1847. [50] R. Hill, Proc. Phys. Soc. A 65 (1952) 349–354. [51] R.F.S. Hearmon, Rev. Mod. Phys. 18 (1946) 409–440. [52] C. Kittel, Introduction to Solid State Physics, Wiley, New York, 1976. [53] B.C. De Cooman, K.G. Chin, J.K. Kim, in: M. Chiaberge (Ed.), New Trends and Developments in Automotive System Engineering, InTech, Rijeka, 2011, pp. 101–128. [54] S. Allain, J.P. Chateau, O. Bouaziz, S. Migot, N. Guelton, Mater. Sci. Eng., A 387– 389 (2004) 158–162. [55] G. Frommeyer, U. Brüx, P. Neumann, ISIJ Int. 43 (2003) 438–446. [56] A. Dumay, J.P. Chateau, S. Allain, S. Migot, O. Bouaziz, Mater. Sci. Eng., A 483– 484 (2008) 184–187.

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