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金属学报  2020, Vol. 56 Issue (1): 112-118    DOI: 10.11900/0412.1961.2019.00257
  研究论文 本期目录 | 过刊浏览 |
立方晶体弹性常数和EAM/FS势函数的关系
段灵杰,刘永长()
天津大学材料科学与工程学院水利安全与仿真国家重点实验室  天津 300354
Relationships Between Elastic Constants and EAM/FS Potential Functions for Cubic Crystals
DUAN Lingjie,LIU Yongchang()
State Key Lab of Hydraulic Engineering Simulation and Safety, School of Materials Science and Engineering, Tianjin University, Tianjin 300354, China
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摘要: 

为了确定bcc和fcc晶体的EAM/FS势函数参数,研究了EAM/FS势函数与弹性常数的关系。对bcc和fcc结构晶体,分别导出压强(P)和体积弹性模量(B)、弹性常数(C44)和剪切弹性模量(Cp=(C11-C12)/2)的利用嵌入函数、对势函数和电子密度分布函数的表达式。发现C44Cp的大小不仅取决于所考虑的原子和周围原子的距离,还受近邻原子排列状况的影响。将关于结合能(utot)和PBC44Cp的5个拟合方程转化为求最小值的优化模型,给出了5种典型bcc结构晶体(V、Mo、Nb、Ta、W)和3种典型fcc结构晶体(Cu、γ-Fe、Ni)的结合能中待定参数的值。采用上述参数计算得到的最小结合能与实验结果一致,此时所对应的原子间距与晶格常数相同,表明了方法的有效性。

关键词 立方晶体弹性常数EAM势FS势    
Abstract

Potential functions are extensively applied in molecular dynamics (MD) simulation of metals. Selection of them is a very important step in MD simulations due to its effects of the precision and reliability of the simulations. They are one of the most important reference data during the process of calculation. In order to cover the shortage of pairwise potentials for modelling transition metals, EAM/FS many-body potentials have been introduced since 80's of last century. For the sake of determining parameters in the EAM/FS potential functions of bcc and fcc crystals through macro mechanical properties, relations between the EAM/FS potential functions and elastic constants were investigated in this work. Expressions of the pressure (P) and the bulk modulus (B), elastic constant (C44) and shear elastic modulus (Cp=(C11-C12)/2) in terms of the embedding function, pair potential function and the electron density distribution function were deduced for bcc and fcc structures, respectively. It was found that the magnitude of the C44 and Cp depends on the distances between the considered atom and surrounding atoms, but also the configuration of surrounding atoms. Finally, by converting five fitting equations about the cohesive energy (utot) and P, B, C44, Cp into an optimization model of finding minimum value, the values of the six undetermined parameters in the cohesive energy were given for five typical bcc crystals (V, Mo, Nb, Ta and W) and three typical fcc crystals (Cu, γ-Fe, Ni), respectively. For each crystal, calculation errors show accuracy of parameter values. The obtained calculation results, for the minimum cohesive energy and the corresponding atomic distance, fit well with the reported experimental data, by adopting the above values of the parameters, which indicates the effectiveness for our method.

Key wordscubic crystal    elastic constant    EAM potential    FS potential
收稿日期: 2019-08-01     
ZTFLH:  TG111.3  
基金资助:国家自然科学基金项目(U1660201);国家磁约束核聚变能源研究专项课题项目(2015GB119001)
通讯作者: 刘永长     E-mail: ycliu@tju.edu.cn
Corresponding author: Yongchang LIU     E-mail: ycliu@tju.edu.cn
作者简介: 段灵杰,男,1992年生,博士生

引用本文:

段灵杰,刘永长. 立方晶体弹性常数和EAM/FS势函数的关系[J]. 金属学报, 2020, 56(1): 112-118.
Lingjie DUAN, Yongchang LIU. Relationships Between Elastic Constants and EAM/FS Potential Functions for Cubic Crystals. Acta Metall Sin, 2020, 56(1): 112-118.

链接本文:

https://www.ams.org.cn/CN/10.11900/0412.1961.2019.00257      或      https://www.ams.org.cn/CN/Y2020/V56/I1/112

Crystala / nmutot / eVBC44Cp
V0.30399-5.311.5510.4260.546
Mo0.31472-6.822.6261.0891.516
Nb0.33008-7.571.7100.2810.567
Ta0.33058-8.101.9610.8240.524
W0.31652-8.903.1041.6061.590
表1  5种典型bcc结构晶体的实验数据(弹性常数的量纲为1011 Pa)[8]
CrystalcdAc0c1c2ER
V3.16283.87991.711018.83-16.253.9725.125×10-11
Mo3.23174.36251.3678262.85-200.7038.8809.408×10-10
Nb3.39274.09492.590926.30-22.685.4051.103×10-10
Ta3.48634.29682.049825.13-17.983.5849.312×10-12
W3.26014.47631.5395275.77-207.6739.6552.896×10-3
表2  5种典型bcc结构晶体的势参数的拟合值
Crystala / nmutot / eVBC44Cp
Cu0.3615-3.541.4200.8180.257
γ-Fe0.3591-4.291.6701.1700.475
Ni0.3518-4.451.8761.3170.552
表3  3种典型fcc结构晶体的实验数据(弹性常数的量纲为1011 Pa)[17,31]
CrystalcdAc0c1c2ER
Cu4.02054.01860.753712.1130-1.44710.251400.001806
γ-Fe3.99213.99720.717783.1839-2.30860.415890.002837
Ni3.91073.91660.764073.6419-2.70790.500410.013935
表4  3种典型fcc结构晶体的势参数的拟合值
图1  5种典型bcc结构晶体的结合能随原子间距(utot-R)的变化曲线
图2  3种典型fcc结构晶体的结合能随原子间距(utot-R)的变化曲线
[1] Leach A R. Molecular Modelling: Principles and Applications [M]. 2nd Ed., New York: Prentice Hall, 2001: 145
[2] Chang L, Zhou C Y, Liu H X, et al. Orientation and strain rate dependent tensile behavior of single crystal titanium nanowires by molecular dynamics simulations [J]. J. Mater. Sci. Technol., 2018, 34: 864
[3] Zhang H F, Yan H L, Jia N, et al. Exploring plastic deformation mechanism of multilayered Cu/Ti composites by using molecular dynamics modeling [J]. Acta Metall. Sin., 2018, 54: 1333
[3] (张海峰, 闫海乐, 贾 楠等. Cu/Ti纳米层状复合体塑性变形机制的分子动力学模拟研究 [J]. 金属学报, 2018, 54: 1333)
[4] Yuan S L, Zhang H, Zhang D J. Molecular Simulation: Theory and Experiment [M]. Bejing: Chemical Industry Press, 2016: 28
[4] (苑世领, 张 恒, 张冬菊. 分子模拟: 理论与实验 [M]. 北京: 化学工业出版社, 2016: 28)
[5] Wang J, Yu L M, Huang Y, et al. Effect of crystal orientation and He density on crack propagation behavior of bcc-Fe [J]. Acta Metall. Sin., 2018, 54: 47
[5] (王 瑾, 余黎明, 黄 远等. 晶体取向和He浓度对bcc-Fe裂纹扩展行为的影响 [J]. 金属学报, 2018, 54: 47)
[6] Zhang X, Li H W, Zhan M. Mechanism for the macro and micro behaviors of the Ni-based superalloy during electrically-assisted tension: Local Joule heating effect [J]. J. Alloys Compd., 2018, 742: 480
[7] Daw M S, Baskes M I. Embedded-atom method: Derivation and application to impurities, surfaces, and other defects in metals [J]. Phys. Rev., 1984, 29B: 6443
[8] Finnis M W, Sinclair J E. A simple empirical N-body potential for transition metals [J]. Philos. Mag., 1984, 50A: 45
[9] Zhang B W, Hu W Y, Shu X L. Thery of Embedded Atom Method and Its Application to Materials Science: Atomic Scale Materials Design Theory [M]. Changsha: Hunan University Press, 2003: 71
[9] (张邦维, 胡望宇, 舒小林. 嵌入原子方法理论及其在材料科学中的应用: 原子尺度材料设计理论 [M]. 长沙: 湖南大学出版社, 2003: 71)
[10] Zhang B W, Ouyang Y F, Liao S Z, et al. An analytic MEAM model for all BCC transition metals [J]. Physica, 1999, 262B: 218
[11] Johnson R A. Relationship between defect energies and embedded-atom-method parameters [J]. Phys. Rev., 1988, 37B: 6121
[12] Johnson R A. Alloy models with the embedded-atom method [J]. Phys. Rev., 1989, 39B: 12554
[13] Wang H P, Zheng C H, Zou P F, et al. Density determination and simulation of Inconel 718 alloy at normal and metastable liquid states [J]. J. Mater. Sci. Technol., 2018, 34: 436
[14] Wang H P, Zhao J F, Liu W, et al. An anomalous thermal expansion phenomenon induced by phase transition of Fe-Co-Ni alloys [J]. J. Appl. Phys., 2018, 124: 215107
[15] Zou P F, Wang H P, Yang S J, et al. Density measurement and atomic structure simulation of metastable liquid Ti-Ni alloys [J]. Metall. Mater. Trans., 2018, 49A: 5488
[16] Ouyang Y F, Zhang B W, Liao S Z, et al. A simple analytical EAM model for bcc metals including Cr and its application [J]. Z. Phys., 1996, 101B: 161
[17] Zhang Y, Ashcraft R, Mendelev M I, et al. Experimental and molecular dynamics simulation study of structure of liquid and amorphous Ni62Nb38 alloy [J]. J. Chem. Phys., 2016, 145: 204505
[18] Lü P, Zhou K, Cai X, et al. Thermophysical properties of undercooled liquid Ni-Zr alloys: Melting temperature, density, excess volume and thermal expansion [J]. Comput. Mater. Sci., 2017, 135: 22
[19] Dai X D, Li J H, Liu B X. Atomistic modeling of crystal-to-amorphous transition and associated kinetics in the Ni-Nb system by molecular dynamics simulations [J]. J. Phys. Chem., 2005, 109B: 4717
[20] Baskes M I. Modified embedded-atom potentials for cubic materials and impurities [J]. Phys. Rev., 1992, 46B: 2727
[21] Fan Q N, Wang C Y, Yu T, et al. A ternary Ni-Al-W EAM potential for Ni-based single crystal superalloys [J]. Physica, 2015, 456B: 283
[22] Lei Y W, Sun X R, Zhou R L, et al. Embedded atom method potentials for Ce-Ni binary alloy [J]. Comput. Mater. Sci., 2018, 150: 1
[23] Yang L, Zhang F, Wang C Z, et al. Implementation of metal-friendly EAM/FS-type semi-empirical potentials in HOOMD-blue: A GPU-accelerated molecular dynamics software [J]. J. Comput. Phys., 2018, 359: 352
[24] Srinivasan P, Nicola L, Simone A. Modeling pseudo-elasticity in NiTi: Why the MEAM potential outperforms the EAM-FS potential [J]. Comput. Mater. Sci., 2017, 134: 145
[25] Kelly A, Knowles K M. Crystallography and Crystal Defects [M]. 2nd Ed., Chichester: Wiley, 2012: 181
[26] Jamal M, Asadabadi S J, Ahmad I, et al. Elastic constants of cubic crystals [J]. Comput. Mater. Sci., 2014, 95: 592
[27] Wen M, Barnoush A, Yokogawa K. Calculation of all cubic single-crystal elastic constants from single atomistic simulation: Hydrogen effect and elastic constants of nickel [J]. Comput. Phys. Commun., 2011, 182: 1621
[28] Zope R R, Mishin Y. Interatomic potentials for atomistic simulations of the Ti-Al system [J]. Phys. Rev., 2003, 68B: 024102
[29] Pun G P P, Mishin Y. Development of an interatomic potential for the Ni-Al system [J]. Philos. Mag., 2009, 89: 3245
[30] Lai Z H. Crystal Defects and Mechanical Properties of Metals [M]. Bejing: Metallurgical Industry Press, 1988: 22
[30] (赖祖涵. 金属的晶体缺陷与力学性质 [M]. 北京: 冶金工业出版社, 1988: 22)
[31] Etesami S A, Asadi E. Molecular dynamics for near melting temperatures simulations of metals using modified embedded-atom method [J]. J. Phys. Chem. Solids, 2018, 112: 61
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