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The prediction of stable state of high-entropy alloys (HEAs) is crucial to obtain fundamental insight to the excellent properties of HEAs. Taking a FeCuCrMnMo alloy as a case study, we combined Monte Carlo (MC) method with the density functional thoery (DFT) calculations (MC/DFT) to predict the equilibrium structure of high-entropy alloys in a finite unit cell. Instead of approaching the ideal random state obtained from special quasi-random approximation (SQS) method, physical factors such as atomic size, mixing enthalpy of atomic pairs, and interatomic interactions in the alloy are fully considered and implemented in our simulation by MC/DFT calculations. MC codes ensure the energy convergence of the system to the equilibrium state through the atom exchange process. The equilibrium structures exhibit Cu-rich short-range orders (SRO), which is consistent with the observation in experiments. Comparing with ideal random state structure, SRO structure is more stable in energy, and more closely packed in atomic arrangement. Moreover, the analyses of order parameters and radial distribution functions (RDFs) are performed to character the structure of high-entropy alloy. The order parameter of Cu-Cu atomic pair reaches to –0.53 in the SRO equilibrium structure, which indicates that Cu-rich regions appear in the alloy. The RDFs show that the atomic distance distribution of the SRO structure is between 2.25 Å to 2.7 Å, which is smaller than the range of 2.16 Å to 2.84 Å in the SQS structure, indicating that the lattice distortions is relatively small in the SRO structure after structural optimization. The appearing of SRO phenomena is attributed to the inherent characteristics of atoms, including (i) atomic size, (ii) interatomic mixing enthalpy and (iii) the interaction of atoms. Atomic sizes in the FeCuCrMnMo alloy are in the order of Fe (11.78) < Cu (11.81) < Cr (11.97) < Mn (14.38) < Mo (15.58), in unit of Å3/atom. The relatively large sizes of Mn and Mo atoms should disadvantage the pairing of Mo-Mo and Mn-Mn. The mixing enthalpy of Cu with other atoms are all positive values, indicating that Cu is not favor of pairing other elements and precipitate itself. The analyses of density of state (DOS) and Crystal Orbital Hamilton Population (COHP) also support the results. The reason is exactly attributed to the inactive valence electrons of Cu. Furthermore, the effect of SRO on the magnetic and mechanical properties are investigated. The existence of SRO decreases the mean value of magnetic moment, and results in an increase of elastic moduli (B, G and E) and a decrease in the ductility and anisotropy properties.
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Keywords:
- first-principles calculations /
- Monte Carlo method /
- short range order /
- high entropy alloys
[1] Yeh J W, Chen S K, Lin S J, Gan J Y, Chin T S, Shun T T, Tsau C H, Chang S Y 2004 Adv. Eng. Mater. 6 299
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Google Scholar
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Google Scholar
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[13] Shang S L, Wang Y, Kim D E, Zacherl C L, Du Y, Liu Z K 2011 Phys. Rev. B 83 144204
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Google Scholar
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Google Scholar
[18] Feng W Q, Qi Y, Wang S Q 2017 Metals 7 482
Google Scholar
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[21] Cowley J M 1950 Phys. Rev. 77 669
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图 1 温度分别在800, 1000, 1200和1500 K下, FeCuCrMnMo合金结构总能和有序参数随MC步数的演化 (a), (c), (e)和(g)为MC步数-结构总能关系, (b), (d), (f)和(h)为MC步数-有序参数关系图
Figure 1. System evolution vs MC steps at 800, 1000, 1200 and 1500 K. (a), (c), (e) and (g) System energy; (b), (d), (f), and (h) SRO parameters for atomic pairs.
图 2 FeCuCrMnMo合金平衡态晶格结构, 其中Cu原子(蓝色原子球)的SRO现象用黑色虚线框标出
Figure 2. The lattice structure of FeCuCrMnMo alloy, in which Cu-rich short range order (SRO) is framed by a dotted box. Blue spheres represent Cu atoms.
图 4 FeCuCrMnMo合金的6个SQS结构 (a)有序参数; (b)结构总能(单个原子)
Figure 4. Six SQS structures of FeCuCrMnMo alloy: (a) Order parameters of each atomic pairs; (b) total energy per atom.
图 5 高熵合金理想BBC结构, 结构优化后的SRO结构和SQS结构的径向分布函数计算结果
Figure 5. Average radial distribution functions for BCC structure (unrelaxed), SRO structure (relaxed) and SQS structure (relaxed) of FeCuCrMnMo alloy.
图 6 高熵合金结构优化后原子对间径向分布函数图 (a), (b) SQS结构; (c), (d) SRO结构. 虚线标明无畸变晶格中第一近邻(1NN)和第二近邻(2NN)的距离值
Figure 6. Partial pair distribution functions of FeCuCrMnMo alloy: (a), (b) SQS structure; (c), (d) SRO structure. Dotted lines show the distances of first nearest neighbor (1NN) and second nearest neighbor in unrelaxed lattice.
图 7 FeCuCrMnMo合金中原子的电子(自旋向上)分波态密度, 虚线处代表费米能级
Figure 7. Partial density of states of FeCuCrMnMo alloy (spin-up). Dotted line shows the Fermi level.
图 8 FeCuCrMnMo合金中Fe-Mo, Cu-Mo, Cr-Mo和Mn-Mo原子对间的COHP分析
Figure 8. COHP for FeCuCrMnMo alloy describing Fe-Mo, Cu-Mo, Cr-Mo and Mn-Mo interactions.
图 9 FeCuCrMnMo合金SRO结构和SQS结构电子态密度
Figure 9. Density of state of FeCuCrMnMo alloy in SRO lattice and SQS lattice.
图 10 FeCuCrMnMo合金和纯金属W的{100}方向电荷密度图 (a) SRO结构; (b)SQS结构, 其中蓝色区域附近的“圆形状原子”为Cu原子; (c)纯金属W
Figure 10. Electron density of {100} atomic plane: (a) SRO structure; (b) SQS structure; (c) W lattice.
图 11 FeCuCrMnMo合金SRO和SQS结构中各原子的磁矩
Figure 11. Magnetic moments of individual atoms in SRO structure and SQS structure of FeCuCrMnMo alloy.
表 1 FeCuCrMnMo合金的SRO结构、SQS结构及相关纯金属的平均单个原子能、平均单个原子体积和晶格常数值. 其中FM, NM和AFM分别代表铁磁性、顺磁性和反铁磁性
Table 1. Cohesive energy per atom, structural volume per atom, and lattice parameters for SRO and SQS structures, and Fe, Cu, Cr, Mn, Mo, and W metals. FM, AFM, and NM denote ferromagnetism, antiferromagnetism, and nonmagnetic, respectively.
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表 2 FeCuCrMnMo合金中不同原子对间的混合焓[24] (单位: KJ/mol)
Table 2. Enthalpy of mixing of binary systems containing the elements in FeCuCrMnMo alloy(Unit: KJ/mol).
混合焓 Cr Mn Mo Fe Cu 12 4 19 13 Cr 2 0 –1 Mn 5 0 Mo –2
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表 3 FeCuCrMnMo合金中不同原子对的ICOHP平均值
Table 3. Mean values of ICOHPs for each atomic pair in FeCuCrMnMo alloy.
原子对 Cu-Fe Cu-Mn Cu-Mo Cu-Cr Cu-Cu ICOHP –0.54 –0.59 –0.88 –0.66 –0.39 原子对 Fe-Cr Fe-Mo Mn-Mo Cr-Mo Cr-Mn ICOHP –1.45 –1.68 –1.72 –1.78 –1.61
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表 4 FeCuCrMnMo合金SRO结构和SQS结构的力学性质, 其中弹性模量的单位为GPa
Table 4. Calculated mechanical properties for SRO structure and SQS structure of FeCuCrMnMo alloy. The unit for the elastic moduli is GPa.
结构 C11 C12 C44 B BEOS G E ν G/B AZ SRO 263.4 187.5 100.5 212.8 188.7 88.2 232.5 0.318 2.415 2.65 SQS 156.1 139.3 85.1 144.9 154.4 43.4 118.4 0.364 3.323 10.1
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[1] Yeh J W, Chen S K, Lin S J, Gan J Y, Chin T S, Shun T T, Tsau C H, Chang S Y 2004 Adv. Eng. Mater. 6 299
Google Scholar
[2] Zhang W, Liaw P K, Zhang Y 2018 Sci. China. Mater. 61 2
Google Scholar
[3] Cantor B, Chang I T H, Knight P, Vincent A J B 2004 Mater. Sci. Eng. A 213 375
[4] Granberg F, Nordlund K, Ullah M W, Jin K, Lu C, Bei H, Wang L M, Djurabekova F, Weber W J, Zhang Y 2016 Phys. Rev. Lett. 116 135504
Google Scholar
[5] Lu C, Niu L, Chen N, Jin K, Yang T, Xiu P, Zhang Y, Gao F, Bei H, Shi S, He M R, Robertson I M, Weber W J, Wang L 2016 Nat. Commun. 7 13564
Google Scholar
[6] Kumar N A P K, Li C, Leonard K J, Bei H, Zinkle S J 2016 Acta Mater. 113 230
Google Scholar
[7] Zhong Z H, Xu Q, Mori K, Tokitani M 2019 Philos. Mag. 99 1515
Google Scholar
[8] Tian F 2017 Front. Mater. 4
Google Scholar
[9] Soven P 1967 Phys. Rev. 156 809
Google Scholar
[10] Abrikosov I A, Simak S I, Johansson B, Ruban A V, Skriver H L 1997 Phys. Rev. B 56 9319
Google Scholar
[11] Zunger A, Wei S, Ferreira L G, Bernard J E 1990 Phys. Rev. Lett. 65 353
Google Scholar
[12] van de Walle A, Tiwary P, de Jong M, Olmsted D L, Asta M, Dick A, Shin D, Wang Y, Chen L Q, Liu Z K 2013 Calphad 42 13
Google Scholar
[13] Shang S L, Wang Y, Kim D E, Zacherl C L, Du Y, Liu Z K 2011 Phys. Rev. B 83 144204
Google Scholar
[14] Pickering E J, Muñoz-Moreno R, Stone H J, Jones N G 2016 Scripta Mater. 113 106
Google Scholar
[15] Santodonato L J, Zhang Y, Feygenson M, Parish C M, Gao M C, Weber R J, Neuefeind J C, Tang Z, Liaw P K 2015 Nat. Commun. 6 5964
Google Scholar
[16] Singh S, Wanderka N, Murty B S, Glatzel U, Banhart J 2011 Acta Mater. 59 182
Google Scholar
[17] Kresse G, Furthmüller J 1996 Phys. Rev. B 54 11169
Google Scholar
[18] Feng W Q, Qi Y, Wang S Q 2017 Metals 7 482
Google Scholar
[19] Niu C, Windl W, Ghazisaeidi M 2017 Scripta Mater. 132 9
Google Scholar
[20] Ren X L, Shi P H, Zhang W W, Wu X Y, Xu Q, Wang Y X 2019 Acta Mater. 180 189
Google Scholar
[21] Cowley J M 1950 Phys. Rev. 77 669
Google Scholar
[22] Jiang D E, Carter E A 2004 Phys. Rev. B 70 064102
Google Scholar
[23] Brandes E A, Brook G B 1992 Smithells Metals Reference Book (7th Ed.) (Oxford: Reed Educational and Publishing Ltd.) pp129−136
[24] Takeuchi A, Inoue A 2005 Mater. Trans. 46 2817
Google Scholar
[25] Richard D 1993 J. Phys. Chem. 97 8617
Google Scholar
[26] 姚宝殿, 胡桂青, 于治水, 张慧芬, 施立群, 沈浩, 王月霞 2016 65 026202
Google Scholar
Yao B D, Hu G Q, Yu Z S, Zhang H F, Shi L Q, Shen H, Wang Y X 2016 Acta Phys. Sin. 65 026202
Google Scholar
[27] Ding Y C, Chen M, Gao X Y, Jiang M H 2012 Chin. Phys. B 21 067101
Google Scholar
[28] Feng J 2014 APL Mater. 2 081801
Google Scholar
[29] Ding Y, Chen M, Wu W 2014 Phys. B: Condens. Matter 433 48
Google Scholar
[30] 王浩玉, 农智学, 王继杰, 朱景川 2019 68 136401
Wang H Y, Nong Z S, Wang J J, Zhu J C 2019 Acta Phys. Sin. 68 136401
[31] Miao N, Sa B, Zhou J, Sun Z 2011 Comput. Mater. Sci. 50 1559
Google Scholar
[32] Gueler E, Gueler M 2015 Chin. J. Phys. 53 040807
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