基于物理模型的BaZrO3钙钛矿机器学习力场

赵亮 牛宏伟 荆宇航

赵亮,牛宏伟,荆宇航. 基于物理模型的BaZrO3钙钛矿机器学习力场[J]. 航空材料学报,2023,43(6):80-89 doi: 10.11868/j.issn.1005-5053.2023.000011
引用本文: 赵亮,牛宏伟,荆宇航. 基于物理模型的BaZrO3钙钛矿机器学习力场[J]. 航空材料学报,2023,43(6):80-89 doi: 10.11868/j.issn.1005-5053.2023.000011
ZHAO Liang,NIU Hongwei,JING Yuhang. Investigation of a physical model-based machine-learning force field for BaZrO3 perovskite[J]. Journal of Aeronautical Materials,2023,43(6):80-89 doi: 10.11868/j.issn.1005-5053.2023.000011
Citation: ZHAO Liang,NIU Hongwei,JING Yuhang. Investigation of a physical model-based machine-learning force field for BaZrO3 perovskite[J]. Journal of Aeronautical Materials,2023,43(6):80-89 doi: 10.11868/j.issn.1005-5053.2023.000011

基于物理模型的BaZrO3钙钛矿机器学习力场

doi: 10.11868/j.issn.1005-5053.2023.000011
基金项目: 国家自然科学基金项目(12172112)
详细信息
    通讯作者:

    赵亮(1976—),男,博士,高级工程师,主要从事模拟仿真及数字化方向研究,联系地址:北京市海淀区温泉镇环山村北京航空材料研究院(100095),E-mail:13811346725@139.com

  • 中图分类号: O369;TB321

Investigation of a physical model-based machine-learning force field for BaZrO3 perovskite

  • 摘要: 钙钛矿已经作为高性能航空发动机热障涂层陶瓷的备选材料之一。其在高温、高压和辐照等复杂环境中原子间的相互作用往往非常复杂。经验力场仅考虑了原子间的两体、三体或四体等相互作用,物理假设过于简单,对于复杂环境的势能面往往难以精确描述。机器学习力场能获得远比经验力场准确的势能面。本文采用机器学习方法,针对最常见的钙钛矿氧化物锆酸钡(BaZrO3),提出了基于物理模型的机器学习力场,用来描述 BaZrO3这种典型钙钛矿的静态性质、相稳定性和力学性质。使用密度泛函理论数据库训练了基于物理模型的机器学习力场,计算了静态性质、相稳定性和力学性质。对于静态性质,使用纯机器学习力场和基于物理模型的机器学习力场计算了弹性常数C11C12C44,模拟结果与DFT相比,前者的误差为0.34%、8.75%和10.71%,后者的误差为0.34%、2.5%和7.14%,远优于经验力场。对于相稳定性,发现基于物理模型的机器学习力场继承了经验力场在维持相稳定性方面的优势,优于纯机器学习力场。对于力学性能,计算了BaZrO3的四个不同晶向的杨氏模量,发现机器学习力场和基于物理模型的机器学习力场的计算结果与试验值的误差分别为9.22%和1.6%,远低于经验力场的结果。可见,将物理模型融入机器学习力场开发是提升原子模拟精准度的重要途径。

     

  • 图  1  构建基于物理模型的机器学习力场的流程

    Figure  1.  Workflow for constructing a machine learning force field based on a physical model

    图  2  训练过程中训练集体系势能和原子受力的RMSE随训练批次的变化

    Figure  2.  Variation of RMSE of potential energy and atomic forces on training set against training batches during training

    图  3  机器学习力场计算结果和密度泛函理论参考数据的比较  (a)训练集每个原子平均能量的比较;(b)测试集每个原子平均能量的比较;(c)训练集上各个原子受力的比较;(d)测试集上各个原子受力的比较;(e)训练集每个原子的平均能量和各个原子受力的分布;(f)测试集每个原子的平均能量和各个原子受力的分布

    Figure  3.  Comparisons of machine learning potential results with respect to the references of DFT data  (a) comparison of average energy per atom for training set;(b) comparison of average energy per atom for test set;(c)comparison of forces on individual atoms on a training set;(d)comparison of forces on individual atoms on test set;(e) error distributions for energies and forces for training set;(f)error distributions for energies and forces for test set

    图  4  使用基于物理模型的机器学习力场能量最小化过程中结构和RDF的变化  (a)初始时刻;(b)中间时刻;(c)最后时刻;(1)结构;(2)RDF

    Figure  4.  Changes in structure and RDF during energy minimization of empirical force fields with machine learning corrections (a)initial moment;(b)middle moment;(c)final moment;(1)structures;(2)RDF

    图  5  BaZrO3模型沿不同晶向的线弹性阶段应力-应变曲线 (a)[$ 100 $];(b)[$ 1\bar{1}0 $];(c)[$ 111 $];(d)[$ 11\bar{2} $]

    Figure  5.  Stress-strain curves of BaZrO3 in linear elastic region along different crystal directions (a)[$ 100 $];(b)[$ 1\bar{1}0 $];(c)[$ 111 $];(d)[$ 11\bar{2} $]

    表  1  用于描述BaZrO3体系的数据库总结

    Table  1.   Database summary used to describe BaZrO3 system

    Corresponding properties Structure type No. atoms No. configurations No. atomic environments
    Static properties Cubic under strain $(\mathrm{\delta },\mathrm{\delta },\mathrm{\delta },\mathrm{0,0},0)$ 40 20000 800000
    Cubic under strain $(\mathrm{\delta },\mathrm{\delta },\mathrm{\delta },\mathrm{0,0},0)$ 60 20000 1200000
    Cubic under strain $(\mathrm{\delta },\mathrm{\delta },\mathrm{\delta },\mathrm{0,0},0)$ 80 10000 800000
    Mechanical property Cubic structure stretching along$ \left[100\right] $ crystal direction 40 2000 80000
    Cubic structure stretching along$ \left[110\right] $ crystal direction 40 2000 80000
    Cubic structure stretching along$ [1\bar{1}0] $ crystal direction 60 2000 120000
    Cubic structure stretching along$ \left[111\right] $ crystal direction 60 2000 120000
    Cubic structure stretching along$ [11\bar{2}] $ crystal direction 60 2000 120000
    下载: 导出CSV

    表  2  BaZrO3体系中不同类型原子之间的Buckingham力场参数

    Table  2.   Interatomic potentials parameters for BaZrO3

    Interaction Potentials parameters
    A/eV $ \rho $/nm C/(eV∙nm−6
    Ba2+O2- 931.700 0.03949 0.000
    Zr4+O2- 985.869 0.03760 0.000
    O2-O2- 22764.300 0.01490 2.789×10−5
    下载: 导出CSV

    表  3  基于物理模型的机器学习力场的静态性质和DFT理论计算结果的比较

    Table  3.   Comparison of static properties based on empirical force field with machine learning corrections and DFT references

    Model Static properties
    a/nm C11/ GPa C12/ GPa C44/ GPa
    DFT[44] 0.4256 292 77 82
    DFT(this work) 0.422 293 80 84
    Empirical force field [36] 0.419 261 131 139
    Machine learning force field(this work) 0.423 292 73 75
    Based on empirical force field with machine learning corrections(this work) 0.423 294 82 78
    下载: 导出CSV

    表  4  基于不同模型的无序结构能量最小化结果

    Table  4.   Energy minimization results for disordered structures based on different models

    Model Applying random displacements of O-Ba bond length percentages
    5% 10% 15% 20% 25% 30%
    Empirical force field [36] Recovery Recovery Recovery Recovery Recovery Unrecovered
    Machine learning force field(this work) Recovery Recovery Recovery Recovery Unrecovered Unrecovered
    Based on empirical force field with machine
    learning corrections(this work)
    Recovery Recovery Recovery Recovery Recovery Unrecovered
    下载: 导出CSV

    表  5  基于不同模型的施加应变结构能量最小化结果(其中$ a、b、c、\alpha 、\beta \; \mathrm{和}\;\gamma $ 为施加应变后的晶格参数,${a}_{0}、{b}_{0}、 {c}_{0}、 $${\alpha }_{0}、{\beta }_{0} \; \mathrm{和} \; {\gamma }_{0} $为施加应变前的晶格参数)

    Table  5.   Energy minimization results for structures under strain based on different models( $ a{\text ,}b{\text ,}c{\text ,}\alpha {\text ,}\beta \;{ \rm and} \; \gamma $ are lattice parameters after applying strain, $ {a}_{0}{\text ,}{b}_{0}{\text ,}{c}_{0}{\text ,}{\alpha }_{0}{\text ,}{\beta }_{0}\; {\rm{and}}\;{\gamma }_{0} $ are lattice parameters before applying strain)

    Model Strain applied to the structure
    $ a/{a}_{0},b/{b}_{0},c/{c}_{0} $ $ \alpha /{\alpha }_{0},\beta /{\beta }_{0},\gamma /{\gamma }_{0} $ Energy minimization results
    Empirical force field [36] 1.50,1.50,1.50 1.00,1.00,1.00 Recovery
    Machine learning force field(this work) Recovery
    Based on empirical force field with machine learning corrections(this work) Recovery
    Empirical force field [36] 0.75,0.75,0.75 1.00,1.00,1.00 Recovery
    Machine learning force field(this work) Recovery
    Based on empirical force field with machine learning corrections(this work) Recovery
    Empirical force field [36] 1.00,1.00,1.00 0.66,0.66,0.66 Recovery
    Machine learning force field(this work) Recovery
    Based on empirical force field with machine learning corrections(this work) Recovery
    Empirical force field [36] 1.50,1.50,1.50 0.66,0.66,0.66 Recovery
    Machine learning force field(this work) Recovery
    Based on empirical force field with machine learning corrections(this work) Recovery
    Empirical force field [36] 0.75,0.75,0.75 0.66,0.66,0.66 Recovery
    Machine learning force field(this work) Unrecovered
    Based on empirical force field with machine learning corrections(this work) Recovery
    下载: 导出CSV

    表  6  BaZrO3模拟体系参数

    Table  6.   Parameters of different BaZrO3 systems

    Model Coordinate axis corresponds to crystal direction No. atoms
    x y z
    A [$100$] [$010 $] [$001 $] 750
    B [$ 1\bar{1}0 $] [$ 111 $] [$ 11\bar{2} $] 720
    下载: 导出CSV

    表  7  BaZrO3模型沿不同晶向的杨氏模量

    Table  7.   Young’s modulus of BaZrO3 along different crystal directions

    Model Crystal directions
    [$ 100 $] [$ 1\bar{1}0 $] [$ 111 $] [$ 11\bar{2} $]
    Experiment [48] 181±11
    Empirical force field [36] 128.4 269.3 261.2 301.2
    Machine learning force field(this work) 197.7 184.2 178.2 175.2
    Based on empirical force field with machine learning corrections(this work) 183.9 199.0 188.8 183.6
    下载: 导出CSV
  • [1] 温泉,李亚忠,马薏文,等. 热障涂层技术发展[J]. 航空动力,2021(5):60-64.

    WEN Q,LI Y Z,MA Y W,et al. Development of thermal barrier coating technology[J]. Aerospace Power,2021(5):60-64.
    [2] 郭芳威,张瑞吉,邢辰,等. 层级孔喷涂粉末构筑及新一代长寿命热障涂层材料的研究进展[J]. 航空材料学报,2023,43(4):1-16.

    GUO F W,ZHANG R J,XING C,et al. Review on thermal spraying powder with hierarchy pore structure and a new generation of long-life thermal barrier coating materials[J]. Journal of Aeronautical Materials,2023,43(4):1-16.
    [3] 王晶,陆杰,赵晓峰,等. 氧化钇含量对YSZ热障涂层抗CMAS腐蚀性能的影响[J]. 航空材料学报,2023,43(4):25-36

    WANG J,LU J,ZHAO X F,et al. Effect of yttria content on CMAS resistance of YSZ thermal barrier coatings[J]. Journal of Aeronautical Materials,2023,43(4):25-36.
    [4] 李浩宇,程玉贤,刘礼祥,等. 等离子喷涂工艺参数对GdPO4热障涂层组织结构和结合强度的影响[J]. 航空材料学报,2022,42(1):25-32.

    LI H Y,CHENG Y X,LIU L X,et al. Effects of air plasma spraying parameters on microstructure and bonding strength of GdPO4 thermal barrier coatings[J]. Journal of Aeronautical Materials,2022,42(1):25-32.
    [5] 辜宁霞,荆婉如,宁磊,等. 钙钛矿太阳能电池用Ag/ZrO2/C柔性纳米纤维膜电极[J]. 材料工程,2021,49(9):79-86.

    GU N X,JING W R,NING L,et al. Ag/ZrO2/C flexible nanofiber films-based counter electrode for perovskite solar cells[J]. Journal of Materials Engineering,2021,49(9):79-86.
    [6] 张勇,翟光美,郑露露,等. 籽晶层对氧化锡纳米棒生长质量及其钙钛矿太阳能电池性能的影响[J]. 材料工程,2021,49(10):55-62.

    ZHANG Y,ZHAI G M,ZHENG L L,et al. Effect of seed layers on growth of SnO2 nanorods and performance of perovskite solar cells[J]. Journal of Materials Engineering,2021,49(10):55-62.
    [7] 张少威,蒲秀好,万艳红,等. 掺杂对Sr2Fe1.5Mo0.5O6- δ 阳极材料电化学性能影响研究进展[J]. 材料工程,2021,49(9):1-13.

    ZHANG S W,PU X H,WAN Y H,et al. Research progress in effect of element doping on electrochemical properties of Sr2Fe1.5Mo0.5O6-δ based anode materials[J]. Journal of Materials Engineering,2021,49(9):1-13.
    [8] 王杰,马帅,夏丰金,等. Cu2ZnSnS4/Bi2FeCrO6半导体异质结的脉冲激光沉积法制备及其光电性能[J]. 材料工程,2021,49 (7):103-111.

    WANG J,MA S,XIA F J,et al. Cu2ZnSnS4/Bi2FeCrO6 semiconductor heterojunction grown by pulsed laser deposition and its optoelectronic properties[J]. Journal of Materials Engineering. 2021,49(7):103-111.
    [9] GOMEZ M A,CHUNDURU M,CHIGWESHE L,et al. The effect of yttrium dopant on the proton conduction pathways of BaZrO3,a cubic perovskite[J]. The Journal of Chemical Physics,2010,132(21):214709. doi: 10.1063/1.3447377
    [10] KO T W,FINKLER J A,GOEDECKER S,et al. A fourth-generation high-dimensional neural network potential with accurate electrostatics including nonlocal charge transfer[J]. Nature Communications,2021,12(1):1-11. doi: 10.1038/s41467-020-20314-w
    [11] 刘嘉玮,王建江,许宝才,等. 钙钛矿型La1- x Ba x MnO3(0≤ x≤0.5)的红外发射率和微波吸收性能[J]. 航空材料学报,2017,37(5):29-34.

    LIU J W,WANG J J,XU B C,et al. Infrared emissivities and microwave absorption properties of perovskite La1- x Ba x MnO3(0≤ x≤0.5)[J]. Journal of Aeronautical Materials,2017,37(5):29-34.
    [12] 周昌荣,刘心宇,杨桂华,等. 新型无铅压电陶瓷Bi0.5(Na0.82K0.18)0.5TiO3-LiNbO3的研究[J]. 航空材料学报,2009,29(4):94-97.

    ZHOU C R,LIU X Y,YANG G H,et al. Study of new lead-free piezoelectric ceramics of Bi0.5(Na0.82K0.18)0.5TiO3-LiNbO3[J]. Journal of Aeronautical Materials,2009,29(4):94-97.
    [13] LOFERSKI J,RAPPAPORT P. Radiation damage in Ge and Si detected by carrier lifetime changes:damage thresholds[J]. Physical Review,1958,111(2):432. doi: 10.1103/PhysRev.111.432
    [14] PLIMPTON S. Fast parallel algorithms for short-range molecular dynamics[J]. Journal of computational physics,1995,117(1):1-19. doi: 10.1006/jcph.1995.1039
    [15] BLANK T B,BROWN S D,CALHOUN A W,et al. Neural network models of potential energy surfaces[J]. The Journal of Chemical Physics,1995,103(10):4129-4137. doi: 10.1063/1.469597
    [16] HOHENBERG P,KOHN W. Inhomogeneous electron gas[J]. Physical Review,1964,136(3B):B864. doi: 10.1103/PhysRev.136.B864
    [17] KOHN W,SHAM L J. Self-consistent equations including exchange and correlation effects[J]. Physical Review,1965,140(4A):A1133. doi: 10.1103/PhysRev.140.A1133
    [18] LEACH A R . Molecular modeling:principles and applications[M]. [S. l. ]:Pearson Education,2001.
    [19] 严六明,朱素华. 分子动力学模拟的理论与实践[M]. 北京:中国社会科学出版社,2013:30-46.
    [20] 蔡文生,CHRISTOPHE C. 高性能大规模分子动力学的前沿进展——近35年生物体系的分子动力学模拟研究回顾[J]. 化学学报,2013,71(2):24-33.

    CAI W S,CHRISTOPHE C. Frontiers in high-performance,large-scale molecular dynamics—35 years of molecular-dynamics simulations of biological systems[J]. Acta Chimica Sinica,2013,71(2):159-168.
    [21] VAN DUIN A C,DASGUPTA S,LORANT F,et al. ReaxFF:a reactive forcefield for hydrocarbons[J]. The Journal of Physical Chemistry A,2001,105(41):9396-9409. doi: 10.1021/jp004368u
    [22] TERSOFF J. New empirical approach for the structure and energy of covalent systems[J]. Physical Review B,1988,37(12):6991. doi: 10.1103/PhysRevB.37.6991
    [23] STILLINGER F H,WEBER T A. Computer simulation of local order in condensed phases of silicon[J]. Physical Review B,1985,31(8):5262. doi: 10.1103/PhysRevB.31.5262
    [24] SCHRÖDINGER E. An undulatory theory of the mechanics of atoms and molecules[J]. Physical Review,1926,28(6):1049. doi: 10.1103/PhysRev.28.1049
    [25] UNKE O T,CHMIELA S,SAUCEDA H E,et al. Machine learning force fields[J]. Chemical Reviews,2021,121(16):10142-10186. doi: 10.1021/acs.chemrev.0c01111
    [26] DIEGO M,DOMINIC W,MARKUS J,et al. Building robust machine learning force fields by composite Gaussian approximation potentials[J]. Solid-State Electronics,2023,200(2):108529.
    [27] BYGGMSTAR J,NIKOULIS G,FELLMAN A,et al Multiscale machine-learning interatomic potentials for ferromagnetic and liquid iron[J]. Journal of Physics:Condensed Matter, 2022,34(30):305402.
    [28] MANGOLD C,CHEN S,BARBALINARDO G,et al. Transferability of neural network potentials for varying stoichiometry:Phonons and thermal conductivity of Mn x Ge y compounds[J]. Journal of Applied Physics,2020,127(24):244901. doi: 10.1063/5.0009550
    [29] ZUO Y,CHEN C,LI X,et al. Performance and cost assessment of machine learning interatomic potentials[J]. The Journal of Physical Chemistry A,2020,124(4):731-745. doi: 10.1021/acs.jpca.9b08723
    [30] ALBERT P B,JAMES K,NOAM B,et al. Machine learning a general-purpose interatomic potential for silicon[J]. Physical Review X,2018,8(4):041048. doi: 10.1103/PhysRevX.8.041048
    [31] NIU H W,ZHAO J Q,LI H Y,et al. A machine-learning interatomic potential to understand primary radiation damage of silicon[J]. Computational Materials Science,2023,218(3):111970.
    [32] MICHAEL J W,JAMES M R. Benchmarking structural evolution methods for training of machine learned interatomic potentials[J]. Journal of Physics:Condensed Matter,2022,34(38):385901.
    [33] KRISHNA C P,WAHYU S. Accurate Fe-He machine learning potential for studying He effects in BCC-Fe[J]. Journal of Nuclear Materials,2023,574(2):154183.
    [34] WU J,ZHOU E,QIN Z,et al. Accessing negative Poisson's ratio of graphene by machine learning interatomic potentials[J]. Nanotechnology 2022,33(27):275710.
    [35] CAR R,PARRINELLO M. Unified approach for molecular dynamics and density-functional theory[J]. Physical Review Letters,1985,55(22):2471-2474. doi: 10.1103/PhysRevLett.55.2471
    [36] STOKES S J,ISLAM M S. Defect chemistry and proton-dopant association in BaZrO3 and BaPrO3[J]. Journal of Materials Chemistry,2010,20(30):6258-6264. doi: 10.1039/c0jm00328j
    [37] EASTWOOD J W,HOCKNEY R W,LAWRENCE D. P3m3dp—the three-dimensional periodic particle-particle/particle-mesh program[J]. Computer Physics Communications,1980,19(2):215-261. doi: 10.1016/0010-4655(80)90052-1
    [38] KRESSE G,FURTHMÜLLER J. Efficiency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis set[J]. Computational Materials Science,1996,6(1):15-50. doi: 10.1016/0927-0256(96)00008-0
    [39] KRESSE G,JOUBERT D. From ultrasoft pseudopotentials to the projector augmented-wave method[J]. Physical Review B,1999,59(3):1758. doi: 10.1103/PhysRevB.59.1758
    [40] KRESSE G,FURTHMÜLLER J. Efficient iterative schemes for ab initio to total-energy calculations using a plane-wave basis set[J]. Physical Review B,1996,54(16):11169. doi: 10.1103/PhysRevB.54.11169
    [41] PERDEW J P,BURKE K,ERNZERHOF M. Generalized gradient approximation made simple[J]. Physical Review Letters,1996,77(18):3865. doi: 10.1103/PhysRevLett.77.3865
    [42] ZHANG L,HAN J,WANG H,et al. Deep potential molecular dynamics:a scalable model with the accuracy of quantum mechanics[J]. Physical Review Letters,2018,120(14):143001. doi: 10.1103/PhysRevLett.120.143001
    [43] BEHLER J. Constructing high-dimensional neural network potentials:a tutorial review[J]. International Journal of Quantum Chemistry,2015,115(16):1032-1050. doi: 10.1002/qua.24890
    [44] JAIN A,ONG S P,HAUTIER G,et al. Commentary:the materials project:a materials genome approach to accelerating materials innovation[J]. APL Materials,2013,1(1):011002. doi: 10.1063/1.4812323
    [45] HOOVER W G. Canonical dynamics:equilibrium phase-space distributions[J]. Physical Review A,1985,31(3):1695. doi: 10.1103/PhysRevA.31.1695
    [46] NOSÉ S. A unified formulation of the constant temperature molecular dynamics methods[J]. The Journal of Chemical Physics,1984,81(1):511-519. doi: 10.1063/1.447334
    [47] PARRINELLO M,RAHMAN A. Polymorphic transitions in single crystals:a new molecular dynamics method[J]. Journal of Applied Physics,1981,52(12):7182-7190. doi: 10.1063/1.328693
    [48] VASSEN R,CAO X,TIETZ F,et al. Zirconates as new materials for thermal barrier coatings[J]. Journal of the American Ceramic Society,2000,83(8):2023-2028.
  • 加载中
图(5) / 表(7)
计量
  • 文章访问数:  85
  • HTML全文浏览量:  30
  • PDF下载量:  50
  • 被引次数: 0
出版历程
  • 收稿日期:  2023-02-08
  • 录用日期:  2023-09-05
  • 修回日期:  2023-08-09
  • 刊出日期:  2023-12-08

目录

    /

    返回文章
    返回

    《航空材料学报》关于谨防假冒期刊的郑重声明

    近期,有不法分子冒充《航空材料学报》期刊及官网,谎称提供论文发表服务,发布虚假约稿信息,骗取作者发表费用,为此,本编辑部郑重声明如下:

    1、http://www.hkclxb.cn 为假冒网站,与《航空材料学报》没有任何关系。我刊没有委托任何第三方机构或个人,代表我刊约稿或提供发表服务。

    2、《航空材料学报》为中文期刊,只接收中文文章投稿,目前不刊登英文文章。

    3. 本刊官网是http://jam.biam.ac.cn/,本刊的官方投稿方式为网上投稿(登录官网首页—作者投稿)。如有不明可电话咨询,联系电话是010-62496277。

    敬请广大读者和作者认真识别,谨防上当受骗。