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39
INTERATOMIC INTERACTION IN FCC METALS
V. E. Zalizniak
Siberian Federal University, 79 Svobodny Prospect, Krasnoyarsk 660041, Russia
vzalizniak@sfu-kras.ru
PACS 34.20.Cf, 61.50.Ah
The parameters of interatomic potential for 10 fcc metals are presented in this paper. The potential is based on the embedded atom method [6]. Parameters are determined empirically by fitting to the equilibrium lattice constant, cohesion energy, vacancy formation energy, bulk modulus and three elastic constants. The proposed potentials are suitable for atomistic computer simulations of practical applications in areas of material science and engineering.
Keywords: interatomic potential, embedded atom method. Notation
a — equilibrium lattice constant,
Ec, Evf — cohesion energy per atom and unrelaxed vacancy formation energy,
B — bulk modulus,
c11, c12, c44 — crystal elastic constants,
cn , ci2, c4i, B(a) — calculated values of elastic constants.
1. Introduction
Computer simulations have become an increasingly powerful tool for studying material properties. While first principle quantum methods generally give the most accurate results, they can rarely be applied to complex systems, which require a large number of atoms or longer calculations. However, empirical potentials have proven to be efficient for investigating the structure and properties of materials in many fields, though these results are less accurate than first principle quantum calculations. The embedded-atom method (EAM) is widely used to represent the interaction between metal atoms. A general description of the method was done by Daw and Baskes [1, 2]. In the framework of EAM, the total energy of a system can be written as
N
E
tot
J^En , En = F (pn) +
n=1
N
N
y^ y (rnm) m =1 m = n
Pn
^ ^ p (rnm) ■
m =1
m = n
where Etot — total energy of the system of N atoms, En — the internal energy associated with atom n, pn — the electron density at atom n due to all other atoms, p(rnm) — the contribution to the electron density at atom n due to atom m at the distance rnm from atom n, F(pn) — the embedding energy of the atom into the electron density pn, y(rnm) — the two body central potential between atoms n and m separated by rnm. Interpretation and functional form of y(r), p(r), and F(p) depend on a particular method.
The popularity of the EAM model results from its quantum mechanical justification, as well as its mathematical simplicity, which makes this model conducive to large-scale computer modeling.
In recent years, a number of EAM potential models for fcc metals have been proposed. For example, Sheng et al. [3] have developed EAM potentials for fourteen fcc metals. The potentials were developed by fitting the potential-energy surface of each element derived from high-precision first-principles calculations. The three determining functions were expressed with quintic spline functions for each element. Typically, 15 equidistant spline knots were used for both the density and the pair functions, and 6 spline knots were used for the embedding function. This results in a great quantity of fitting parameters. Hijazi and Park [4] have proposed potential for seven fcc metals: Ag, Al, Au, Cu, Ni, Pd and Pt. They have used the following potential functions:
p(r) = po exp (-a (r - re)), V (r) = 1 + - l)) exp (-P^ - 1
F (P) = ' (Pe)^ - ^ • K ff) (j
where re is the equilibrium nearest distance. This potential has six adjustable parameters, a, 3, 7, a and pe. Dai et al. [5] have proposed an extended Finnis-Sinclair potential for six fcc metals: Ag, Au, Cu, Ni, Pd and Pt. The following potential functions have been employed
(r - r\) + a2 (r - ri) , r ^ ri
p(r) , n
w i 0 , r > ri
( ) = f (r - r'2)2 (Co + Cir + C2r2 + C3r3 + C4r4) , r ^ r2 V (r) = \ 0 , r>r2 '
F (p) = Fo^J'
where ri and r2 are cut-off parameters assumed to lie between the second and third neighbor atoms. By this means, one needs to fit nine parameters.
The above mentioned potential models do not provide an equally accurate description of basic properties for all fcc metals to which those potentials have been applied to. The purpose of this paper is to present potential parameters for a consistent and practicable EAM model [6], which can be applied to widely used fcc metals.
2. Embedded atom potential
Zalizniak and Zolotov [6] have assumed that the atomic electron density has the following functional form:
p(r) = po(1 + 3r)2 exp(-ar), (1)
where a and 3 are parameters of the atomic electron density distribution. Pair potential follows from the electrostatic interaction of two atoms that have positively charged nuclei and the electron densities defined by expression (1). The resulting expression is rather cumbersome and it can be written in a concise form as follows [6]:
6
V (r) = £ • exp (-ar) ^^ an (ar)n, (2)
n= — i
where parameters an depend on a and 3. The embedding function F(p) is taken in the polynomial form
4
cn
n=0
where pe — equilibrium electron density.
n
F (P) = Y1 °n( fe - 1) » (3)
3. Results of potential fitting
In order to define the potential of interaction between the same metal atoms one need to fit only two parameters: a and /3. The experimental data used in the fitting procedure consist of the equilibrium lattice constant, the cohesive energy, the vacancy formation energy, the bulk modulus and three elastic constants, given in Table 1.
Table 1. Pure metal properties used in fitting
a, A Ec, eV EVf, eV B, eV/A3 Cll, eV/A3 Cl2, eV/A3 C44, eV/A3
Al 4.05 [7] 3.34 [7] 0.64 [8] 0.474 [12] 0.666 [12] 0.377 [12] 0.177 [12]
Ca 5.58 [7] 1.84 [7] 0.70 [9] 0.133 [13] 0.173 [13] 0.114 [13] 0.102 [13]
Ni 3.52 [7] 4.44 [7] 1.79 [10] 1.161 [12] 1.548 [12] 0.967 [12] 0.775 [12]
Cu 3.61 [7] 3.49 [7] 1.28 [10] 0.863 [14] 1.042 [14] 0.754 [14] 0.466 [14]
Pd 3.89 [7] 3.89 [7] 1.85 [11] 1.205 [12] 1.417 [12] 1.099 [12] 0.447 [12]
Ag 4.09 [7] 2.95 [7] 1.10 [10] 0.632 [12] 0.763 [12] 0.566 [12] 0.283 [12]
Ir 3.84 [7] 6.94 [7] 1.97 [8] 2.216 [15] 3.683 [15] 1.554 [15] 1.635 [15]
Pt 3.92 [7] 5.84 [7] 1.35 [10] 1.765 [12] 2.164 [12] 1.565 [12] 0.478 [12]
Au 4.08 [7] 3.81 [7] 0.90 [10] 1.083 [12] 1.204 [12] 1.022 [12] 0.259 [12]
Pb 4.95 [7] 2.03 [7] 0.58 [10] 0.279 [12] 0.310 [12] 0.264 [12] 0.094 [12]
Table 2. Parameters of the atomic electron density distribution (1)
a, 1/A ß, 1/A Po, e/A3
Al 1.8008 -2.5380 0.1844
Ca 1.0958 -6.0630 0.0031
Ni 1.5900 14.6900 0.0041
Cu 1.6300 -28.0390 0.0014
Pd 1.5239 -18.3850 0.0039
Ag 1.5568 -5.0950 0.0642
Ir 2.6116 -1.4845 37.2065
Pt 2.2320 -1.7440 9.4854
Au 2.0585 -1.7950 5.6041
Pb 1.2000 -13.2940 0.0040
As the fitting procedure [6] suggests, the equilibrium lattice constant, cohesive energy, vacancy formation energy and bulk modulus are reproduced exactly. The fitting procedure is performed using a cutoff distance of 2a, so that long-range interactions are included.
The results of fitting for ten fcc metals are presented below. Table 2 lists the parameters of the atomic electron density distribution. Parameters of pair potential are
Table 3. Parameters of pair potential (2)
Al Ca Ni Cu Pd
e, eV 4.4736 4.2963 9.1686 7.2840 10.3465
a_i 1 1 1 1 1
ao 0.7236 0.6932 0.6747 0.6854 0.6867
a1 0.8981 0.5757 0.4606 0.5229 0.5331
a2 5.6170E-02 2.2931E-02 1.0457E-02 1.7243E-02 1.8342E-02
a3 -1.4951E-02 -1.3498E-02 -1.2742E-02 -1.3173E-02 -1.3238E-02
a4 -1.9584E-03 -1.6589E-03 -1.5067E-03 -1.5930E-03 -1.6062E-03
a5 -1.9226E-04 -1.4436E-04 -1.2237E-04 -1.3462E-04 -1.3654E-04
a6 -1.3088E-05 -7.4251E-06 -5.5519E-06 -6.5523E-06 -6.7185E-06
Table 4. Parameters of pair potential (2)
Ag Ir Pt Au Pb
e, eV 7.2321 28.0893 10.4249 6.4193 2.6628
a_1 1 1 1 1 1
ao 0.7010 0.6979 0.7351 0.7360 0.6874
a1 0.6366 2.4252 1.5272 1.3442 0.5361
a2 2.9404E-02 0.19029 0.1149 9.8534E-02 1.8674E-02
a3 -1.3835E-02 -1.8071E-02 -1.6559E-02 -1.6179E-02 -1.3258E-02
a4 -1.7277E-03 -2.4684E-03 -2.2677E-03 -2.2009E-03 -1.6101E-03
a5 -1.5489E-04 -2.2692E-04 -2.3969E-04 -2.3129E-04 -1.3711E-04
ao -8.4546E-06 -4.3189E-05 -2.5063E-05 -2.1497E-05 -6.7690E-06
Table 5. Parameters of embedding function (3)
Al Ca Ni Cu Pd
Pe, e/A3 0.6726 0.4105 2.3029 2.2059 2.7952
Co, eV -3.3376 -1.8389 -4.4357 -3.4868 -3.8874
ci, eV -0.5328 -0.6987 -1.7850 -1.2768 -1.7893
C2, eV 1.0845 0.0032 0.0098 0.0006 0.8255
C3, eV -0.6668 0.1761 0.5071 0.2571 1.8814
c4, eV 1.0536 1.3131 3.1479 2.4665 3.1539
Table 6. Parameters of embedding function (3)
Ag Ir Pt Au Pb
Pe, e/A3 2.4369 3.8042 4.0881 3.7648 2.4181
co, eV -2.9485 -6.9379 -5.8390 -3.8094 -2.029765
ci, eV -1.0896 -1.9624 -0.9382 -0.6167 -0.4593
C2, eV 0.1118 0.0404 3.4135 2.4495 1.7075
C3, eV 0.4725 -1.8474 -0.0709 0.3350 0.6488
c4, eV 2.2196 3.0877 1.4164 1.0778 0.5118
Table 7. Calculated and experimental properties of pure metals, the proposed potential (1)—(3). The first lines present the experimental values of the three elastic constants (they are used in fitting procedure) and the commonly accepted values of vacancy formation energies. The second lines present the values predicted by the potential
Cll, eV/A3 C12, eV/A3 c44, eV/A3 B, eV/A3 EVf, eV d, %
Al 0.666 0.377 0.177 0.474 0.62-0.66 [8] 0.14
0.666 0.377 0.176 0.474 0.64
Ca 0.173 0.114 0.102 0.133 0.7 [9] 10
0.177 0.111 0.066 0.133 0.70
Ni 1.548 0.967 0.775 1.161 1.6, 1.79 [16, 10] 7.1
1.583 0.950 0.586 1.161 1.79
Cu 1.042 0.754 0.466 0.863 1.28, 1.3 [10, 17] 5.3
1.130 0.729 0.422 0.863 1.28
Pd 1.417 1.099 0.447 1.205 1.7, 1.85 [10, 11] 1.9
1.422 1.096 0.478 1.205 1.85
Ag 0.763 0.566 0.283 0.632 1.1 [10, 17] 0.25
0.762 0.567 0.281 0.632 1.1
Ir 3.683 1.554 1.635 2.216 1.79, 2.27*[10, 18] 5.5
3.823 1.412 1.484 2.216 1.97
Pt 2.164 1.565 0.478 1.765 1.35, 1.5 [10, 17] 0.05
2.165 1.565 0.479 1.765 1.35
Au 1.204 1.022 0.259 1.083 0.89, 0.93 [11, 10] 0.6
1.217 1.016 0.257 1.083 0.9
Pb 0.310 0.264 0.094 0.279 0.58 [10] 9
0.304 0.266 0.060 0.279 0.58
*_ result of ab initio calcu ations
listed in Tables 3 and 4, while coefficients of the embedding function F(p) are given in Tables 5 and 6.
The calculated properties of pure metals from the proposed potential were compared with the experimental values, to which they were fitted in Table 7. The first lines contain the experimental values, while the second lines contain the values predicted by the potential. The last column presents the average discrepancy
d
1
4
(a) c11
- C11
C11
+
(a) c12
- C12
C12
+
c(a) c44
- C44
c44
+
\B(a) - B\
B
computed for every metal. For softer materials, such as Ca, Pb and Ni, the average discrepancies between the calculated and experimental results were found to be relatively large, but for other metals, the match between experiment and the proposed EAM model was good.
For comparative purposes, pure metal properties derived for ten fcc metals using the optimized EAM potential [3] and analytic EAM potential [4] are shown in Tables 8 and 9. The equilibrium lattice constant and the cohesive energy were reproduced exactly by all potentials. Generally, the proposed potential and the optimized EAM potential [3]
Table 8. Calculated and experimental properties of pure metals, optimized EAM potential [3]. The first lines present the experimental values of the three elastic constants (they are used in fitting procedure) and the commonly accepted values of vacancy formation energies. The second lines present the values predicted by the potential
Cll, Cl2, C44, B, Evf, eV d, %
GPa GPa GPa GPa
Al 114 61.9 31.6 76 0.62-0.66 [8] 1
113 61.6 32 77 0.67
Ca 28 18.2 16.3 14.1-19.3 0.7 [9] 7.8
28 18 17 21 0.95
Ni 261 151 132 180 1.6, 1.79 [16, 10] 2.5
263 154 127 186 1.12
Cu 176 125 82 140 1.28, 1.3 [10, 17] 1.4
175 124 79 141 0.99
Pd 234 176 71.2 180 1.7, 1.85 [10, 11] 5.7
235 180 82 188 1.44
Ag 132 97 51 100 1.1 [10, 17] 0.7
131 97 51 98 1.17
Ir 582 241 262 320 1.79, 2.27*[10, 18] 4.3
578 241 243 350 1.67
Pt 347 251 77 228-275 1.35, 1.5 [10, 17] 3
347 253 78 282 1.50
Au 193 163 42 180.3 0.89, 0.93 [11, 10] 2.9
197 165 45 178 0.98
Pb 49.4 42.1 14.9 46 0.58 [10] 1.5
50.1 42 15.2 45 0.45
* _ result of ab initio calculations
provide similar descriptions of the elastic properties for ten fcc metals (see Tables 7 and 8). The values of vacancy formation energy estimated by the EAM potential [3] were not in satisfactory agreement with the data measured for most metals. Analytic EAM potential [4] provided a better description of elastic properties for Cu and Ni in comparison with the proposed potential, but for the other metals, the proposed potential gave a better fit to the experimental data (see Tables 7 and 9).
4. Conclusion
This paper presents parameters of a new EAM potential model to describe pure fcc metals. The potential model has a simple function form with two adjustable parameters and is easy to use in computer simulations. The potential parameters were determined by fitting the pure metal bulk properties: equilibrium lattice constant, cohesive energy, bulk modulus, three elastic constants and vacancy formation energy. The fitting procedure was applied to ten fcc metals: Al, Ca, Ni, Cu, Pd, Ag, Ir, Pt, Au, and Pb. The equilibrium lattice constant, cohesive energy, bulk modulus and vacancy formation energy were reproduced exactly. The agreement between the calculated elastic constants and the
Table 9. Calculated and experimental properties of pure metals, analytic EAM potential [4]. The first lines present the experimental values of the three elastic constants (they are used in fitting procedure) and the commonly accepted values of vacancy formation energies. The second lines present the values predicted by the potential
Cll, C12, c44, B, Evf, eV d, %
GPa GPa GPa GPa
Al 114 61.9 31.6 79 0.62-0.66 [8] 17.1
98 69.9 44.7 79 0.866
Ni 246.5 147.3 124.7 180.4 1.6, 1.79 [16, 10] 3.3
232.4 154.8 127.6 180.2 1.7
Cu 170 122.5 75.8 138 1.28, 1.3 [10, 17] 1.1
167 124.3 77.3 138 1.3
Pd 234.1 176 71.2 195 1.7, 1.85 [10, 11] 3.8
225.5 180 77.7 195 1.54
Ag 124 93.4 46.1 104 1.1 [10, 17] 1.4
122 94.2 47.5 103 1.1
Pt 347 251 76.5 283 1.35, 1.5 [10, 17] 8.9
324 262 95.4 283 1.6
Au 186 157 42 167 0.89, 0.93 [11, 10] 0.9
184 157 43 167 0.9
experimental data was good. The proposed EAM potentials are believed to find applications in diverse areas of materials science and engineering.
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