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对于计算材料科学的研究者来说, 经常由于找不到合适的原子间势而工作受阻. 本文将在Finnis-Sinclair势的框架下, 通过开发金属Nb的Finnis-Sinclair势而给出较详细的原子间势拟合、检验、修正的过程. 首先建立原子间势与材料宏观性能之间的关系, 然后通过再现金属Nb的结合能、体模量、表面能、空位形成能及平衡点阵常数的实验数据的方法拟合金属Nb的Finnis-Sinclair势. 利用所构建的原子间势计算金属Nb的弹性常数、剪切模量及柯西压力来检验势函数. 讨论势函数曲线形状对间隙形成能的影响, 进而根据间隙能的计算数据修正已构建的原子间势. 讨论截断距离的处理方法. 本文的结果一方面为构建原子间势函数库提供资料, 为构建与Nb相关的合金原子间势奠定基础; 另一方面, 为开发和改善原子间势质量提供方法和依据.
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关键词:
- 原子间势构建方法 /
- 金属Nb /
- Finnis-Sinclair势 /
- 原子间势函数形式
Researchers’ work on computational materials is often hampered by the lack of suitable intera tomic potentials. In this paper, under the framework of Finnis-Sinclair (FS) potentials, the process of fitting, testing and correction of interatomic potential is given in detail by developing the FS potential for metal Nb. First, the relationship between the interatomic potential and the macroscopic properties of the material is established. Then, the Finnis-Sinclair potential of metal Nb is fitted by reproducing the experimental data, such as the cohesive energy, bulk modulus, surface energy, vacancy formation energy and equilibrium lattice constant, and the fitting mean square error is less than 10–7. In order to test the interatomic potential, the elastic constant, shear modulus and Cauchy pressure of metal Nb are calculated by the constructed interatomic potential. In addition, how the form of the interatomic potential function affects the interstitial performance is discussed, and the constructed interatomic potential is modified according to the results of density functional theory (DFT) of the interstitial formation energy. The treatment of cutoff distance is also discussed. In the paper, the results are as follows. 1) The original form of FS potential is not suitable for extending the atomic interaction to the third nearest neighbor. Through analysis and test, it is found that when the modified electron density function is in the form of the fourth power and the form of the pair potential function is in the form of the sixth power polynomial, the interatomic potential can better describe the interatomic interaction; 2) The result of interstitial formation energy is taken as the target value to modify the behavior of the pair potential function in the near distance, and the modified interatomic potential gives the interstitial formation energy close to the result of DFT. When the interstitial energy calculated by the interatomic potential is larger than the target value, the pair potential curve of near distance will be softened by the superposition of a polynomial term, otherwise, the pair potential curve will be stiffened; 3) When the physical quantity under equilibrium state is used as the fitting data, the fitted potential parameters and the elastic constant results will not be affected, while adjusting the curve form of the potential function, as long as none of the function value, the slope and the curvature of the function curve is changed at each neighbor position. The magnitude of interstitial energy will be affected by changing the shape of the curve that is less than the first neighbor range; 4) Under the cutoff strategy in this paper, changing the cutoff distance has almost no influence on the calculated results of potential parameters or crystal properties, but has a slight influence on the mean square error of the fitting results. The results of this paper provide some information for the construction of interatomic potentials database, and lay a foundation for constructing the Nb-related interatomic potential of alloy. And it also provides a method and basis for developing and improving the quality of interatomic potential. -
Keywords:
- construction of interatomic potential /
- metal Nb /
- Finnis-Sinclair potential /
- function form of interatomic potential
[1] Fellinger M R, Park H, Wilkins J W 2010 Phys. Rev. B 81 144119
Google Scholar
[2] Finnis M W, Sinclair J E 1984 Phil. Mag. A 50 45
Google Scholar
[3] Liao X C, Gong H F, Chen Y C, Liu G D, Liu T, Shu R, Liu Z X, Hu W Y, Gao F, Jiang C, Deng H Q 2020 J. Nucl. Mater. 541 152421
Google Scholar
[4] Johnson R A, Oh D J 1989 J. Mater. Res. 4 1195
Google Scholar
[5] Ackland G J 2012 J. Phys. Conf. Ser. 402 012001
Google Scholar
[6] Cheng C, Ma Y L, Bao Q L, Wang X, Sun J X, Zhou G, Wang H, Liu Y X, Xu D S 2020 Comput. Mater. Sci. 173 109432
Google Scholar
[7] Pasianot R., Savino E. J. 1992 Phys. Rev. B 45 12704
Google Scholar
[8] Ackland G J 1992 Phil. Mag. A 66 917
Google Scholar
[9] Kim Y M, Lee B J, Baskes M I 2006 Phys. Rev. B 74 014101
Google Scholar
[10] Mendelev M I, Underwood T L, Ackland G J 2016 J. Chem. Phys. 145 154102
Google Scholar
[11] Farkas D, Jones C 1996 Mater. Sci. Eng. 4 23
[12] Jones C, Farkas D 1996 Comput. Mater. Sci. 6 231
Google Scholar
[13] Li Y J, Wu A P, Li Q, Zhao Y, Zhu R C, Wang G Q 2019 Trans. Nonferrous Met. Soc. China 29 1873
Google Scholar
[14] Smirnova D E, Starikov S V 2017 Comput. Mater. Sci. 129 259
Google Scholar
[15] Ghosh G, Olson G B 2007 Acta Mater. 55 3281
Google Scholar
[16] Daw M S, Baskes M I, 1983 Phys. Rev. Lett. 50 1285
Google Scholar
[17] Daw M S, Baskes M I 1984 Phys. Rev. B Condens. Matter 29 6443
Google Scholar
[18] Yang C M, Qi L 2019 Comput. Mater. Sci. 161 351
Google Scholar
[19] Hu W Y, Zhang B W, Shu X L, Huang 1999 J. Alloys Compd. 289 159
[20] Li J H, Dai X D, Liang S H, Tai K P, Kong Y, Liu B X 2008 Phys. Rep. 455 1
Google Scholar
[21] Ackland G J, Thetford R 1987 Phil. Mag. A 56 15
Google Scholar
[22] Rebonato R, Welch D O, Hatcher R D, Bilello J C 1987 Phil. Mag. A 55 655
Google Scholar
[23] Zope R R, Mishin Y 2003 Phys. Rev. B 68 024102
Google Scholar
[24] Nguyen-Manh D, Horsfield A P, Dudarev S L 2006 Phys. Rev. B 73 020101
Google Scholar
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Google Scholar
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图 2 势函数曲线 (a) 电子密度曲线; (b) 对势曲线; (c) 有效对势曲线
Fig. 2. Potential function curve: (a) Electron density curve; (b) potential curve; (c) effective pair potential curve.
表 1 截断距离内的各间隙原子的距离及等价原子数
Table 1. Distance and equivalent atomic number of each atom withinthe cutoff distance from the interstitial atom.
间隙构型 距离及等价原子数 挤列子 距离 $\dfrac{ {\sqrt 3 } }{ {4} }{a }$ $\dfrac{{\sqrt 11 }}{{4}}{a}$ $\dfrac{{\sqrt {{\rm{19}}} }}{{4}}{\rm{a}}$ $\dfrac{{\sqrt {{\rm{27}}} }}{{4}}{a}$ $\dfrac{{\sqrt {{\rm{35}}} }}{{4}}{a}$ $\dfrac{{\sqrt {{\rm{43}}} }}{{4}}{a}$ 原子数 2 6 6 8 12 6 八面体 距离 $\dfrac{1}{2}{a}$ $\dfrac{{\sqrt 2 }}{2}{a}$ $\dfrac{{\sqrt {\rm{5}} }}{2}{a}$ $\dfrac{{\sqrt 6 }}{2}{a}$ $\dfrac{3}{2}{a}$ 原子数 2 4 8 8 10 四面体 距离 $\dfrac{{\sqrt {\rm{5}} }}{{4}}{a}$ $\dfrac{{\sqrt {{\rm{13}}} }}{{4}}{a}$ $\dfrac{{\sqrt {{\rm{21}}} }}{{4}}{a}$ $\dfrac{{\sqrt {{\rm{29}}} }}{{4}}{a}$ $\dfrac{{\sqrt {{\rm{37}}} }}{{4}}{a}$ 原子数 4 4 8 12 4 
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表 2 拟合用金属Nb的实验数据及计算结果
Table 2. Experimental and calculation data of metal Nb for fitting interatomic potential.
数值 a/Å Ec/eV B/(1011 Pa) Eγ100/
(mJ·m–2)$E_{\rm{v}}^{\rm{f}}$/eV 实验值 3.3008 7.57 1.710 2046 2.64 本文计
算值3.3008 7.57 1.710 2050 2.64
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表 3 金属Nb的FS势参数及拟合均方差
Table 3. FS potential parameters of metal Nb and fitting mean square error.
均方差/10–8 c0 c1 c2 A/eV 无修
正项6.63447 0.262198 –0.138974 0.0184461 0.636219 带修
正项6.63447 0.262198 –0.138974 0.0184461 0.636219
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表 4 金属Nb的弹性常数(单位为1011 Pa)
Table 4. Elastic constants of metal Nb (in 1011 Pa)
C44 C11 C12 $C' $ Pc 实验值[2] 0.281 2.466 1.332 0.546 0.5255 本文结果 0.567 2.343 1.393 0.475 0.4134
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表 5 金属Nb的间隙形成能
Table 5. Interstitial formation energy of metal Nb.
FS[21] FS(87)[22] FS(87)未驰豫 DFT[24] DFT[25] 本文无修正项 本文有修正项 Cutoff c 4.2 4.2 4.2 5.31261 5.31261 d 3.915354 3.915354 3.915354 5.0709 5.0709 $ \left\langle {111} \right\rangle $ crow 4.857 4.10 9.037 5.254 5.255 15.487 6.977 $ \left\langle {111} \right\rangle $ dum 4.795 — 6.610 5.253 5.203 10.749 7.775 $ \left\langle {110} \right\rangle $ dum 4.482 3.99 5.930 5.597 5.684 7.148 4.425 $ \left\langle {100} \right\rangle $ dum 4.821 4.13 8.385 5.949 6.005 13.844 7.616 Tetrahedral — 4.26 6.893 5.758 5.733 10.659 6.371 Octahedral — 4.23 6.850 6.060 6.009 11.069 6.659
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表 7 不同函数形式的各物理量计算结果
Table 7. Calculation results of each physical quantity in different function forms.
函数形式 (35), (36)式 (35), (6)式 (5), (6)式 (5), (32), (33)式 (5), (32), (34)式 C11 8.19854 2.05366 2.34302 2.34302 2.34302 C12 –2.88593 1.53817 1.39349 1.39349 1.39349 $C' $ 5.54235 0.257745 0.474767 0.474767 0.474767 C44 –3.20776 1.21374 0.56664 0.56664 0.56664 Pc 0.160915 0.162217 0.413424 0.413424 0.413424 Octahedral –75.9256 14.5432 11.0693 7.9909 6.65925 Tetrahedral –80.0616 13.9223 10.6593 7.53737 6.37076 $ \left\langle {111} \right\rangle $ crow –89.9140 20.9320 15.4871 11.0992 6.97688 $ \left\langle {100} \right\rangle $ dum –947.486 15.2250 13.8439 9.57021 7.61644 $ \left\langle {110} \right\rangle $ dum –954.052 5.00180 7.14750 4.56348 4.42502 $ \left\langle {111} \right\rangle $ dum 72.3004 17.9239 10.7490 8.12406 7.77466
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表 8 不同对势截断距离下的各物理量计算结果
Table 8. Calculation results of each physical quantity under different pair potential cutoff distance.
截断距离 x = 0.55 x = 0.70 x = 0.80 均方差 1.9669 × 10–7 1.3307 × 10–7 6.6345 × 10–8 B 1.06741 1.06742 1.06742 ${\gamma _{100}}$ 0.128159 0.12808 0.12808 $E_{\rm{v}}^{\rm{f}}$ 2.63998 2.63999 2.63999 ${E_C}$ 7.57 7.57 7.57 C11 2.33551 2.34081 2.34302 C12 1.39724 1.3946 1.39349 $C' $ 0.469137 0.473105 0.474767 C44 0.570392 0.567749 0.56664 Pc 0.413424 0.413424 0.413424 Octahedral 6.93421 6.76073 6.65925 Tetrahedral 6.62507 6.46365 6.37076 $ \left\langle {111} \right\rangle $ crow 7.45171 7.15594 6.97688 $ \left\langle {100} \right\rangle $ dum 8.30897 7.90098 7.61644 $ \left\langle {110} \right\rangle $ dum 4.80878 4.59369 4.42502 $ \left\langle {111} \right\rangle $ dum 8.06717 7.89704 7.77466
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表 9 不同电子密度截断距离下的各物理量计算结果
Table 9. Calculation results of each physical quantity under different electron density cutoff distance.
截断距离 y = 0.45 y = 0.50 y = 0.60 均方差 1.57065 × 10–7 6.6345 × 10–8 1.08929 × 10–10 B 1.06742 1.06742 1.06742 ${\gamma _{100}}$ 0.128111 0.12808 0.127726 $E_{\rm{v}}^{\rm{f}}$ 2.63999 2.63999 2.64000 ${E_{\rm{C}}}$ 7.57 7.57 7.57 C11 2.35341 2.34302 2.32299 C12 1.38830 1.39349 1.40351 $c'$ 0.482555 0.474767 0.45974 C44 0.533568 0.56664 0.627336 Pc 0.427366 0.413424 0.388087 Octahedral 6.42699 6.65925 7.07134 Tetrahedral 6.14925 6.37076 6.76314 $ \left\langle {111} \right\rangle $ crow 6.63230 6.97688 7.59196 $ \left\langle {100} \right\rangle $ dum 7.50874 7.61644 7.80542 $ \left\langle {110} \right\rangle $ dum 4.53049 4.42502 4.23158 $111 $ dum 7.28896 7.77466 9.07468
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[1] Fellinger M R, Park H, Wilkins J W 2010 Phys. Rev. B 81 144119
Google Scholar
[2] Finnis M W, Sinclair J E 1984 Phil. Mag. A 50 45
Google Scholar
[3] Liao X C, Gong H F, Chen Y C, Liu G D, Liu T, Shu R, Liu Z X, Hu W Y, Gao F, Jiang C, Deng H Q 2020 J. Nucl. Mater. 541 152421
Google Scholar
[4] Johnson R A, Oh D J 1989 J. Mater. Res. 4 1195
Google Scholar
[5] Ackland G J 2012 J. Phys. Conf. Ser. 402 012001
Google Scholar
[6] Cheng C, Ma Y L, Bao Q L, Wang X, Sun J X, Zhou G, Wang H, Liu Y X, Xu D S 2020 Comput. Mater. Sci. 173 109432
Google Scholar
[7] Pasianot R., Savino E. J. 1992 Phys. Rev. B 45 12704
Google Scholar
[8] Ackland G J 1992 Phil. Mag. A 66 917
Google Scholar
[9] Kim Y M, Lee B J, Baskes M I 2006 Phys. Rev. B 74 014101
Google Scholar
[10] Mendelev M I, Underwood T L, Ackland G J 2016 J. Chem. Phys. 145 154102
Google Scholar
[11] Farkas D, Jones C 1996 Mater. Sci. Eng. 4 23
[12] Jones C, Farkas D 1996 Comput. Mater. Sci. 6 231
Google Scholar
[13] Li Y J, Wu A P, Li Q, Zhao Y, Zhu R C, Wang G Q 2019 Trans. Nonferrous Met. Soc. China 29 1873
Google Scholar
[14] Smirnova D E, Starikov S V 2017 Comput. Mater. Sci. 129 259
Google Scholar
[15] Ghosh G, Olson G B 2007 Acta Mater. 55 3281
Google Scholar
[16] Daw M S, Baskes M I, 1983 Phys. Rev. Lett. 50 1285
Google Scholar
[17] Daw M S, Baskes M I 1984 Phys. Rev. B Condens. Matter 29 6443
Google Scholar
[18] Yang C M, Qi L 2019 Comput. Mater. Sci. 161 351
Google Scholar
[19] Hu W Y, Zhang B W, Shu X L, Huang 1999 J. Alloys Compd. 289 159
[20] Li J H, Dai X D, Liang S H, Tai K P, Kong Y, Liu B X 2008 Phys. Rep. 455 1
Google Scholar
[21] Ackland G J, Thetford R 1987 Phil. Mag. A 56 15
Google Scholar
[22] Rebonato R, Welch D O, Hatcher R D, Bilello J C 1987 Phil. Mag. A 55 655
Google Scholar
[23] Zope R R, Mishin Y 2003 Phys. Rev. B 68 024102
Google Scholar
[24] Nguyen-Manh D, Horsfield A P, Dudarev S L 2006 Phys. Rev. B 73 020101
Google Scholar
[25] Derlet P M, Nguyen-Manh D, Dudarev S L 2007 Phys. Rev. B 76 054107
Google Scholar
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