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
Cite this article:
DUAN Lingjie,LIU Yongchang. Relationships Between Elastic Constants and EAM/FS Potential Functions for Cubic Crystals. Acta Metall Sin, 2020, 56(1): 112-118.
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.
Table 1 Experimental data for the five typical bcc structure crystals (elastic constants with dimension 1011 Pa)[8]
Crystal
c
d
A
c0
c1
c2
ER
V
3.1628
3.8799
1.7110
18.83
-16.25
3.972
5.125×10-11
Mo
3.2317
4.3625
1.3678
262.85
-200.70
38.880
9.408×10-10
Nb
3.3927
4.0949
2.5909
26.30
-22.68
5.405
1.103×10-10
Ta
3.4863
4.2968
2.0498
25.13
-17.98
3.584
9.312×10-12
W
3.2601
4.4763
1.5395
275.77
-207.67
39.655
2.896×10-3
Table 2 Fitted values of potential parameters for the five typical bcc structure crystals
Crystal
a / nm
utot / eV
B
C44
Cp
Cu
0.3615
-3.54
1.420
0.818
0.257
γ-Fe
0.3591
-4.29
1.670
1.170
0.475
Ni
0.3518
-4.45
1.876
1.317
0.552
Table 3 Experimental data for the three typical fcc structure crystals (elastic constants with dimension 1011 Pa)[17,31]
Crystal
c
d
A
c0
c1
c2
ER
Cu
4.0205
4.0186
0.75371
2.1130
-1.4471
0.25140
0.001806
γ-Fe
3.9921
3.9972
0.71778
3.1839
-2.3086
0.41589
0.002837
Ni
3.9107
3.9166
0.76407
3.6419
-2.7079
0.50041
0.013935
Table 4 Fitted values of potential parameters for the three typical fcc structure crystals
Fig.1 utot-R curves for five typical bcc structure crystals (R—measurement of atomic spacing)
Fig.2 utot-R curves for three typical fcc structure crystals
[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
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
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
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
YU Ligen(State Key Lab. for Powder Metallurgy; Central South University of Technology; Chungsha 410083)HE Jiawen; B. C.Hendrix(State Key Lab. for Mechanical Behavior of Materials; Xi'an Jiaotong University; Xi'an 710049)Correspondent: YU Ligen; Tel: (0731)8826911-3630; Fad: (0731)8825755. THE EFFECT OF PREFERRED ORIENTATION ON X-RAY STRESS MEASUREMENT[J]. 金属学报, 1998, 34(6): 667-672.
[8]
JIANG Bingyao; LIU Xianghuai; ZOU Shichang(Ion Beam Laboratory; Shanghai Institute of Metallurgy ; Chinese Academy of Sciences; Shanghai 200050); SUN Jian(Shanghai Jiaotong University; ShanKhai 200030)(Manuscript received 94-01-18. in revised form 94-06-20). EMBEDDED-ATOM-METHOD FUNCTION FOR hcp METALS Mg, Ti AND Zr[J]. 金属学报, 1995, 31(13): 10-14.
