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金属Nb的Finnis-Sinclair势开发及势函数形式对材料性能的影响

高静怡 孙嘉兴 王逊 周刚 王皞 刘艳侠 徐东生

引用本文:
Citation:

金属Nb的Finnis-Sinclair势开发及势函数形式对材料性能的影响

高静怡, 孙嘉兴, 王逊, 周刚, 王皞, 刘艳侠, 徐东生

Development of Finnis-Sinclair potential of metal Nb and the influence of potential function form on the properties of material

Gao Jing-Yi, Sun Jia-Xing, Wang Xun, Zhou Gang, Wang Hao, Liu Yan-Xia, Xu Dong-Sheng
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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.
    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.
      通信作者: 刘艳侠, ldlyx@163.com
    • 基金项目: 国家重点研发计划 (批准号: 2016YFB0701304) 资助的课题.
      Corresponding author: Liu Yan-Xia, ldlyx@163.com
    • Funds: Project supported by the National Key Research & Development Program of China (Grant No. 2016YFB0701304)
    [1]

    Fellinger M R, Park H, Wilkins J W 2010 Phys. Rev. B 81 144119Google Scholar

    [2]

    Finnis M W, Sinclair J E 1984 Phil. Mag. A 50 45Google 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 152421Google Scholar

    [4]

    Johnson R A, Oh D J 1989 J. Mater. Res. 4 1195Google Scholar

    [5]

    Ackland G J 2012 J. Phys. Conf. Ser. 402 012001Google 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 109432Google Scholar

    [7]

    Pasianot R., Savino E. J. 1992 Phys. Rev. B 45 12704Google Scholar

    [8]

    Ackland G J 1992 Phil. Mag. A 66 917Google Scholar

    [9]

    Kim Y M, Lee B J, Baskes M I 2006 Phys. Rev. B 74 014101Google Scholar

    [10]

    Mendelev M I, Underwood T L, Ackland G J 2016 J. Chem. Phys. 145 154102Google Scholar

    [11]

    Farkas D, Jones C 1996 Mater. Sci. Eng. 4 23

    [12]

    Jones C, Farkas D 1996 Comput. Mater. Sci. 6 231Google 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 1873Google Scholar

    [14]

    Smirnova D E, Starikov S V 2017 Comput. Mater. Sci. 129 259Google Scholar

    [15]

    Ghosh G, Olson G B 2007 Acta Mater. 55 3281Google Scholar

    [16]

    Daw M S, Baskes M I, 1983 Phys. Rev. Lett. 50 1285Google Scholar

    [17]

    Daw M S, Baskes M I 1984 Phys. Rev. B Condens. Matter 29 6443Google Scholar

    [18]

    Yang C M, Qi L 2019 Comput. Mater. Sci. 161 351Google 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 1Google Scholar

    [21]

    Ackland G J, Thetford R 1987 Phil. Mag. A 56 15Google Scholar

    [22]

    Rebonato R, Welch D O, Hatcher R D, Bilello J C 1987 Phil. Mag. A 55 655Google Scholar

    [23]

    Zope R R, Mishin Y 2003 Phys. Rev. B 68 024102Google Scholar

    [24]

    Nguyen-Manh D, Horsfield A P, Dudarev S L 2006 Phys. Rev. B 73 020101Google Scholar

    [25]

    Derlet P M, Nguyen-Manh D, Dudarev S L 2007 Phys. Rev. B 76 054107Google Scholar

  • 图 1  bcc结构6种间隙构型

    Fig. 1.  Six interstitial configurations of bcc structure

    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}$$\dfrac{{\sqrt {{\rm{43}}} }}{{4}}{a}$
    原子数2668126
    八面体距离$\dfrac{1}{2}{a}$$\dfrac{{\sqrt 2 }}{2}{a}$$\dfrac{{\sqrt {\rm{5}} }}{2}{a}$$\dfrac{{\sqrt 6 }}{2}{a}$$\dfrac{3}{2}{a}$
    原子数248810
    四面体距离$\dfrac{{\sqrt {\rm{5}} }}{{4}}{a}$$\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448124
    下载: 导出CSV

    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aEc/eVB/(1011 Pa)Eγ100/
    (mJ·m–2)
    $E_{\rm{v}}^{\rm{f}}$/eV
    实验值3.30087.571.71020462.64
    本文计
    算值
    3.30087.571.71020502.64
    下载: 导出CSV

    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/10–8c0c1c2A/eV
    无修
    正项
    6.634470.262198–0.1389740.01844610.636219
    带修
    正项
    6.634470.262198–0.1389740.01844610.636219
    下载: 导出CSV

    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)

    C44C11C12$C' $Pc
    实验值[2]0.2812.4661.3320.5460.5255
    本文结果0.5672.3431.3930.4750.4134
    下载: 导出CSV

    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
    Cutoffc4.24.24.25.312615.31261
    d3.9153543.9153543.9153545.07095.0709
    $ \left\langle {111} \right\rangle $ crow4.8574.109.0375.2545.25515.4876.977
    $ \left\langle {111} \right\rangle $ dum4.7956.6105.2535.20310.7497.775
    $ \left\langle {110} \right\rangle $ dum4.4823.995.9305.5975.6847.1484.425
    $ \left\langle {100} \right\rangle $ dum4.8214.138.3855.9496.00513.8447.616
    Tetrahedral4.266.8935.7585.73310.6596.371
    Octahedral4.236.8506.0606.00911.0696.659
    下载: 导出CSV

    表 6  不同函数形式的势参数

    Table 6.  Potential parameters of different functional forms.

    函数形式(35), (36)式(35), (6)式(5), (6)式(5), (32), (33)式(5), (32), (34)式
    c0–20.2072–14.05430.2621980.2621980.262198
    c115.468311.0332–0.138974–0.138974–0.138974
    c2–2.81702–2.043510.01844610.01844610.0184461
    A1.287100.6369660.6362190.6362190.636219
    下载: 导出CSV

    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
    C118.198542.053662.343022.343022.34302
    C12–2.885931.538171.393491.393491.39349
    $C' $5.542350.2577450.4747670.4747670.474767
    C44–3.207761.213740.566640.566640.56664
    Pc0.1609150.1622170.4134240.4134240.413424
    Octahedral–75.925614.543211.06937.99096.65925
    Tetrahedral–80.061613.922310.65937.537376.37076
    $ \left\langle {111} \right\rangle $ crow–89.914020.932015.487111.09926.97688
    $ \left\langle {100} \right\rangle $ dum–947.48615.225013.84399.570217.61644
    $ \left\langle {110} \right\rangle $ dum–954.0525.001807.147504.563484.42502
    $ \left\langle {111} \right\rangle $ dum72.300417.923910.74908.124067.77466
    下载: 导出CSV

    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 = 0.55x = 0.70x = 0.80
    均方差1.9669 × 10–71.3307 × 10–76.6345 × 10–8
    B1.067411.067421.06742
    ${\gamma _{100}}$0.1281590.128080.12808
    $E_{\rm{v}}^{\rm{f}}$2.639982.639992.63999
    ${E_C}$7.577.577.57
    C112.335512.340812.34302
    C121.397241.39461.39349
    $C' $0.4691370.4731050.474767
    C440.5703920.5677490.56664
    Pc0.4134240.4134240.413424
    Octahedral6.934216.760736.65925
    Tetrahedral6.625076.463656.37076
    $ \left\langle {111} \right\rangle $ crow7.451717.155946.97688
    $ \left\langle {100} \right\rangle $ dum8.308977.900987.61644
    $ \left\langle {110} \right\rangle $ dum4.808784.593694.42502
    $ \left\langle {111} \right\rangle $ dum8.067177.897047.77466
    下载: 导出CSV

    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 = 0.45y = 0.50y = 0.60
    均方差1.57065 × 10–76.6345 × 10–81.08929 × 10–10
    B1.067421.067421.06742
    ${\gamma _{100}}$0.1281110.128080.127726
    $E_{\rm{v}}^{\rm{f}}$2.639992.639992.64000
    ${E_{\rm{C}}}$7.577.577.57
    C112.353412.343022.32299
    C121.388301.393491.40351
    $c'$0.4825550.4747670.45974
    C440.5335680.566640.627336
    Pc0.4273660.4134240.388087
    Octahedral6.426996.659257.07134
    Tetrahedral6.149256.370766.76314
    $ \left\langle {111} \right\rangle $ crow6.632306.976887.59196
    $ \left\langle {100} \right\rangle $ dum7.508747.616447.80542
    $ \left\langle {110} \right\rangle $ dum4.530494.425024.23158
    $111 $ dum7.288967.774669.07468
    下载: 导出CSV
  • [1]

    Fellinger M R, Park H, Wilkins J W 2010 Phys. Rev. B 81 144119Google Scholar

    [2]

    Finnis M W, Sinclair J E 1984 Phil. Mag. A 50 45Google 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 152421Google Scholar

    [4]

    Johnson R A, Oh D J 1989 J. Mater. Res. 4 1195Google Scholar

    [5]

    Ackland G J 2012 J. Phys. Conf. Ser. 402 012001Google 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 109432Google Scholar

    [7]

    Pasianot R., Savino E. J. 1992 Phys. Rev. B 45 12704Google Scholar

    [8]

    Ackland G J 1992 Phil. Mag. A 66 917Google Scholar

    [9]

    Kim Y M, Lee B J, Baskes M I 2006 Phys. Rev. B 74 014101Google Scholar

    [10]

    Mendelev M I, Underwood T L, Ackland G J 2016 J. Chem. Phys. 145 154102Google Scholar

    [11]

    Farkas D, Jones C 1996 Mater. Sci. Eng. 4 23

    [12]

    Jones C, Farkas D 1996 Comput. Mater. Sci. 6 231Google 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 1873Google Scholar

    [14]

    Smirnova D E, Starikov S V 2017 Comput. Mater. Sci. 129 259Google Scholar

    [15]

    Ghosh G, Olson G B 2007 Acta Mater. 55 3281Google Scholar

    [16]

    Daw M S, Baskes M I, 1983 Phys. Rev. Lett. 50 1285Google Scholar

    [17]

    Daw M S, Baskes M I 1984 Phys. Rev. B Condens. Matter 29 6443Google Scholar

    [18]

    Yang C M, Qi L 2019 Comput. Mater. Sci. 161 351Google 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 1Google Scholar

    [21]

    Ackland G J, Thetford R 1987 Phil. Mag. A 56 15Google Scholar

    [22]

    Rebonato R, Welch D O, Hatcher R D, Bilello J C 1987 Phil. Mag. A 55 655Google Scholar

    [23]

    Zope R R, Mishin Y 2003 Phys. Rev. B 68 024102Google Scholar

    [24]

    Nguyen-Manh D, Horsfield A P, Dudarev S L 2006 Phys. Rev. B 73 020101Google Scholar

    [25]

    Derlet P M, Nguyen-Manh D, Dudarev S L 2007 Phys. Rev. B 76 054107Google Scholar

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出版历程
  • 收稿日期:  2020-12-21
  • 修回日期:  2021-01-27
  • 上网日期:  2021-05-29
  • 刊出日期:  2021-06-05

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