Modified embedded-atom method interatomic potentials for pure Zn and Mg-Zn binary system

Calphad 60 (2018) 200–207

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Modified embedded-atom method interatomic potentials for pure Zn and Mg-Zn binary system Hyo-Sun Jang, Kyeong-Min Kim, Byeong-Joo Lee

T

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Department of Materials Science and Engineering, Pohang University of Science and Technology (POSTECH), Pohang 37673, Republic of Korea

A R T I C L E I N F O

A B S T R A C T

Keywords: 2NN MEAM Interatomic potential Zn Mg-Zn

Interatomic potentials for pure Zn and Mg–Zn binary system have been developed on the basis of the second nearest-neighbor modified embedded-atom method formalism. The potentials describe fundamental material properties of pure Zn (bulk, defect, and thermal properties) reasonably and reproduce the alloy behavior (thermodynamic, structural, and elastic properties of compounds and solution phases) of Mg-Zn alloys well in good agreement with experiments, first-principles and CALPHAD. The applicability of the developed potentials to atom-scale investigations on the slip behavior of Mg-Zn alloys is also demonstrated by showing that the calculated effects of Zn on the general stacking fault energy on the basal, prismatic and pyramidal planes are consistent with first-principles calculations.

1. Introduction

To understand the effect of alloying elements on the slip behavior or GB segregation, atomic scale analysis of dislocation movement or GB concentration is needed. The experiment can observe the slip, but it is difficult to analyze the atomic motion inside the slip, and experimental observation of GB segregation remains not trivial. First-principles calculations can analyze the atomic behavior, but it is difficult to perform slip simulation or GB segregation simulation due to sample size limitation. On the other hand, atomistic simulations based on (semi-) empirical interatomic potentials can be an appropriate tool for analyzing atomic motion during slip or atomic redistribution on GBs. Atomistic simulations require interatomic potentials for the relevant systems. Representative potentials for pure Mg are the embedded-atom method (EAM) potentials developed by Sun et al. [14] and Liu et al. [15], and the second nearest-neighbor modified embedded-atom method (2NN MEAM [16–18]) potential by Kim et al. [19] and its modified version by Wu et al. [20]. It has been reported [20,21] that the two 2NN MEAM potentials predict the core structure of edge and screw dislocations on pyramidal II plane and the generalized stacking fault energy (GSFE) on basal, prismatic, pyramidal I and pyramidal II planes equally well, in good agreement with density functional theory (DFT) calculation. One of the differences between the two 2NN MEAM Mg potentials is that the potential by Kim et al. [19] has been extended to the Mg-Al [19], Mg-Ca [22], Mg-Li [23], Mg-Nd [24], Mg-Pb [24], Mg-Sn [22] and Mg-Y [22] systems, while the modified version [20] is available only for pure Mg and Mg-Ag [25]. Mg alloys generally contain two or more alloying elements [7–9]. For more realistic simulations, interatomic potentials for

Greenhouse gas emission is one of the major problems in the transportation industry [1]. To solve this problem, the use of fossil fuel should be minimized. One of the ways to reduce the fossil fuel usage is to reduce the weight of vehicles by using light weight metals. In this respect, magnesium (Mg), the lightest structural metal, may be a suitable material for light vehicle manufacture [2–4]. However, due to the lack of independent slip systems [5], the room temperature (RT) formability of Mg is low, which is one of the major obstacles to the Mg commercialization. To improve the RT formability of Mg, < c+a > slip activation is required because the < c+a > slip satisfies the Taylor criterion [6] for multiple slip and homogeneous plasticity. From this point of view, various Mg alloys have been designed [7–13] with a common purpose of improving the RT formability. Despite these efforts, the < c+a > slip activation mostly needs the addition of expensive rare earth (RE) elements such as yttrium (Y) [10–12], even though a RE-free Mg alloy with improved RT ductility has also been reported recently [13]. Thus, for Mg commercialization, it is necessary to activate the < c+a > slip without the RE elements. To this end, it is most important to first understand the effect of alloying elements on the slip behavior. In addition to the < c+a > slip activation, it is believed that the solute segregation on the grain boundaries (including twin boundaries) and the resultant grain boundary (GB) mobility also have an effect on the RT formability by affecting the texture evolution [8,9]. However, the effect of alloying elements on the GB segregation is not well understood, either.

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Corresponding author. E-mail address: [email protected] (B.-J. Lee).

https://doi.org/10.1016/j.calphad.2018.01.003 Received 15 November 2017; Received in revised form 28 December 2017; Accepted 9 January 2018 0364-5916/ © 2018 Elsevier Ltd. All rights reserved.

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However, in the case of Zn, the experimental values for hcp Zn could not be adapted directly to assign a value to the Ec, re and B parameters. This was because the c/a ratio of real Zn (1.86 [28,29]) was far away from the ideal value while only the hcp structure with an ideal c/a ratio (1.63) could be accepted as the reference structure in the (2NN) MEAM. To provide necessary information for the Ec, re and B parameters, we performed a first-principles calculation on a hypothetical Zn with an ideal c/a ratio. This calculation was based on the DFT using VASP (Vienna ab initio simulation program) [30–32] with the projector augmented wave (PAW) method [33] and the Perdew–Burke–Ernzerhof generalized-gradient approximation (GGA) [34]. The pseudo potential used in this work was Zn sv GW, a recommended standard potential in the VASP library. The cut-off energy for the plane-wave basis set was 340 eV. A 19 × 19 × 19 k-point mesh was used for conventional bcc, fcc, and hcp primitive unit cells. The first-principles calculation was used also to obtain the energy difference between the hcp and other structures (bcc and fcc), which can be valuable information for parameter optimization. The Ec and re parameters were given the values from the abovementioned first-principles calculation. However, in the case of B, it was difficult to obtain a stable value for the hypothetical Zn. Instead, we assigned an average of experimentally reported bulk modulus values to this parameter. The value of the d parameter could be determined between two, default assumed values (zero or 0.05) from a theoretical (∂B/∂P) value [35]. The Cmax parameter was given a default value of 2.8 as normally has been done in 2NN MEAM parameter optimizations. Other adjustable parameters were determined by considering various fundamental material properties of pure Zn from experimental data and first-principles calculations. The finally determined parameter set for pure Zn is listed in Table 1, with a parameter set for pure Mg [19].

multicomponent Mg alloys are needed. The most necessary multicomponent Mg potentials may be for the Mg-Al-Zn and Mg-Zn-Ca systems which constitute the representative commercial Mg alloys, AZ31 and ZX31, respectively. Because a ternary interatomic potential is developed based on those of constituent binary systems, Mg-Zn, Mg-Al, Mg-Ca binary potentials would have to be available as well as the Al-Zn and Zn-Ca binary potentials for the development of Mg-Al-Zn and MgZn-Ca ternary potentials. Among the Mg-binary potentials, the Mg-Zn binary potential needs to be developed most urgently, because the MgAl and Mg-Ca potentials are already available [19,22], as mentioned above. Even though a well-developed EAM potential [26,27] is available for the Mg-Zn binary system, this potential cannot be used in the present work because of the difference in the potential formalism from those of already developed 2NN MEAM Mg-Al and Mg-Ca potentials. Similar to the case for ternary potentials, the Mg-Zn binary potential also requires potentials for constituent elements, Mg and Zn. 2NN MEAM potential for pure Zn is not available yet, and therefore, needs to be developed for the atomistic studies on the Mg-Al-Zn and Mg-Zn-Ca alloys. As a part of a long-term project to reveal the fundamental mechanism for formability enhancement of Mg alloys, the purpose of the present work is to develop 2NN MEAM potentials for pure Zn and MgZn binary system. In this article, we will describe the procedure for determining potential parameters and demonstrate the quality of the developed potentials by comparing calculated fundamental material properties with literature data from experiments or other calculations. 2. Interatomic potentials The development of the Mg-Zn 2NN MEAM potential requires 2NN MEAM potentials for pure Mg and pure Zn. For pure Mg, the potential by Kim et al. [19] was taken without any modification. The pure Zn potential was developed in the present work and formed the basis for the development of the Mg-Zn binary potential. The details of the 2NN MEAM formalism is described in literature [16–18], is also given as a supplementary material of this article, and thus will not be repeated here.

2.2. Development of Mg-Zn potential The 2NN MEAM potential for binary systems has thirteen parameters in addition to the unary parameters for constituent elements. Four parameters constituting the universal equation of state for a selected reference structure are Ec, re, B and d. One parameter is the ratio between the atomic electron density scaling factor (ρ0) for each element. Eight parameters constituting the many-body screening are four Cmin and four Cmax. Similar to the unary potential parameterization, the optimization of binary parameters is conducted by fitting fundamental material properties from experiments and/or other calculations for relevant systems. The Mg-Zn binary phase diagram [36] involves five intermetallic compounds, Mg51Zn20, Mg21Zn25, Mg4Zn7, MgZn2, and Mg2Zn11 as well as Mg- and Zn-rich hcp solid solutions. The crystal structures of the five intermetallic compounds were too complicate to be a reference structure for the 2NN MEAM formalism. Instead, a hypothetical L12 MgZn3 compound was selected as the reference structure. Since no information was available for this compound phase, Ec, re and B parameter values were optimized so that the experimentally available enthalpy of formation, lattice constants and elastic constants of other real compound phases were reproduced equally well. The d parameter and the ρ0Mg:ρ0Zn ratio was given a default value. Other adjustable parameters, Cmin and Cmax, were determined by considering the relative stability of metastable compound phases compared to stable phases. An ideal potential would be the one that reproduces the enthalpy of formation, lattice

2.1. Development of pure Zn potential Potential development is setting interatomic potential model parameters for the relevant system. The 2NN MEAM potential for unary systems has fourteen parameters. Four parameters constituting the universal equation of state for a selected reference structure are cohesive energy (Ec), equilibrium nearest-neighbor distance (re), bulk modulus (B) and a parameter (d) related to the pressure derivative of bulk modulus (∂B/∂P). Seven parameters constituting the electron density are decay lengths (β(0), β(1), β(2), β(3)) and weight factors (t(1), t(2), t(3)). One parameter (A) is for embedding function and two parameters (Cmin, Cmax) are for many-body screening. In the (2NN) MEAM, a reference structure with a perfect crystal structure is set for formulation. Since the equilibrium crystal structure of Zn is hcp, the hcp structure could be a good candidate reference structure for Zn. The Ec, re and B parameter values that correspond to the cohesive energy, equilibrium nearest-neighbor distance and bulk modulus of the reference structure, respectively, have been determined from the corresponding experimental data for the reference structure.

Table 1 2NN MEAM potential parameter sets for pure Mg [19] and Zn. The units of the cohesive energy Ec, equilibrium nearest-neighbor distance re, and bulk modulus B are eV, Å and eV/Å3, respectively. The reference structure for Mg and Zn are hcp, respectively.

Mg Zn

Ec

re

B

A

β(0)

β(1)

β(2)

β(3)

t(1)

t(2)

t(3)

Cmin

Cmax

d

1.55 1.09

3.20 2.77

0.23 0.44

0.52 0.70

2.30 3.50

1.00 3.00

3.00 0.00

1.00 2.00

9.00 3.00

−2.00 6.00

−9.50 −10.0

0.49 1.00

2.80 2.80

0.00 0.05

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Table 2 2NN MEAM potential parameter set for the Mg-Zn binary system. The units of the cohesive energy Ec, equilibrium nearest-neighbor distance re, and bulk modulus B are eV, Å and eV/Å3, respectively.

Reference Ec re B Cmin (Mg-Zn-Mg) Cmin (Zn-Mg-Zn) Cmin (Mg-Mg-Zn) Cmin (Mg-Zn-Zn) Cmax (Mg-Zn-Mg) Cmax (Zn-Mg-Zn) Cmax (Mg-Mg-Zn) Cmax (Mg-Zn-Zn) d ρ0Mg:ρ0Zn

Mg-Zn

Procedure for determination

L12-type MgZn3 1.21 (0.25EcMg+0.75EcZn+0.005) 2.9 0.3 1 1 (CminZn) 1 0.7225 ([0.5(CminMg)1/2+0.5(CminZn)1/2]2) 2.8 (CmaxMg) 2.8 (CmaxZn) 1.44 1.44 0.0375 (0.25dMg+0.75dZn) 1:1

By fitting the By fitting the By fitting the By fitting the Assumption By fitting the Assumption Assumption Assumption By fitting the By fitting the Assumption Assumption

enthalpy of formation of C14 and hR276 structures lattice parameter of C14 and hR276 structures bulk modulus of C14 structure enthalpy of formation of D03 structure enthalpy of formation of D022 structure

enthalpy of formation of D03 structure enthalpy of formation of D03 structure

the Mg-Zn binary system is listed in Table 2.

constant and elastic property of all stable phases equally well and does not predict wrong phases as stable phases. However, the crystal structure of mC110 Mg4Zn7 was too complex to be reproduced using the present potential. The D8c Mg2Zn11 phase did not maintain its original structure during 0 K relaxations, either. Therefore, those compound phases were not considered in the parameter optimization procedure. Further, the Mg51Zn20 phase that appears on the phase diagram only in a small temperature range (615–594 K [36]) was also excluded from consideration. For these reasons, only the hR276 Mg21Zn25 and C14 MgZn2 compounds were considered as stable compound phases in the parameterization procedure. The finally determined parameter set for

3. Calculation of fundamental material properties To evaluate the reliability of the developed potentials, fundamental material properties need to be calculated and compared with experimental or first-principles calculations. The 2NN MEAM formalism considers up to second nearest-neighbor (2NN) interactions, resulting that the radial cut off distance should be larger than the 2NN distance of structures under consideration. Because the present Mg-Zn potential may be extended to ternary or quaternary systems containing calcium

Table 3 Calculated fundamental material properties of pure Zn using the present 2NN MEAM potential, in comparison with experimental data and other calculations. Values listed are the cohesive energy Ec (eV), the lattice parameter a, c (Å), elastic modulus B, C11, C12, C13, C33, C44, C66 (GPa), structural energy differences ΔE (eV/atom), the vacancy formation energy EfV 2 −6 (eV), vacancy migration energy Em /K), specific heat Cp (J/mol K), melting V (eV), the surface energy of basal and prismatic planes Esurf (erg/cm ), thermal expansion coefficient ε (10 point (K), enthalpy of melting ΔHm (kJ/mol) and the volume change on melting ΔVm/Vsolid (%). Property

Present work

Exp.

F.P. calc.

EAM[51]

ReaxFF[52]

*Ec *a *c *c/a ratio *B *C11 *C12 *C13 *C33 *C44 *C66 *(∂B/∂P) *ΔEhcp→fcc *ΔEhcp→bcc EfV Em V

−1.09 2.78 4.50 1.62 70.51 133.43 47.02 41.51 122.42 34.14 43.20 6.66 0.0092 0.079 0.44 0.65 0.80 448 465 465 41.8 26.4 613a 10.01 13.4

−1.35 [28], −1.3 [38] 2.66 [28], 2.665 [29] 4.94 [28], 4.945 [29] 1.86 [28], 1.856 [29] 80 [38], 70 [39], 66.09 [40] 179 [38], 163 [39] 38 [38], 30.6 [39] 55 [38], 48.1 [39] 69 [38], 60.3 [39] 46 [38], 39.4 [39] 65.9 [39] 6.49 [35]b 0.031 [41]c 0.030 [41]c 0.54 [42], 0.44 [43] 0.43 [44] 0.5 [44] 510 [45]d

−1.09e 2.77e, 2.65 [48] 4.52e, 5.09 [48] 1.63e, 1.92 [48] 59.7 [48], 51.8 [49] 159.5 [49] 56 [49] 51.8 [49] 57 [49] 23.2 [49]

−1.3 2.66 4.85 1.825 81 179 47 30 158 61

−1.38 2.73 4.46 1.6337 87.7 193 41 53 188 71

Esurf

ε (0–100 °C) Cp (0–100 °C) Melting point ΔHm ΔVm/Vsolid

In Out Basal (0001) Prism (1–100) Prism (11–20)

30.2 [46] 25.4 [46] 693 [47] 7.32 [46] 8.68 [46]

Properties marked with a “*” are those used for fitting during the parameter optimization. a The value is calculated using an interfacial method. b Theoretical value calculated by Rose et al. c SGTE thermodynamic data. d Average value for polycrystalline materials. e First-principles calculation on a hypothetical hcp Zn with an ideal c/a ratio performed in this work. f First-principles calculation using the LDA. g First-principles calculation using the GGA.

202

7.4 [49] 0.011e, 0.025 [48] 0.071e, 0.086 [48] 0.36 [50]f, 0.39 [50]g 0.35 [50]f, 0.27 [50]g 0.30 [50]f, 0.17 [50]g

0 0.12

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(Ca), the radial cut-off distance was selected to be 6.0 Å. This was larger than the 2NN distance of pure Ca, an element with the largest interatomic distance among frequently used alloying elements for Mg. All structural properties were obtained from 0 K molecular statics (MS) calculations with a periodic boundary condition along each direction, if not designated specifically. The calculations were performed using an in-house molecular dynamics (MD) and statics code, KISSMD, which is available online [18], and also using the large-scale atomic/molecular massively parallel simulator (LAMMPS) package [37]. 3.1. Calculated material properties of pure Zn Table 3 shows calculated fundamental material properties of pure Zn using the present 2NN MEAM potential, in comparison with experiments [28,29,35,38–47], first-principles [48–50] and other empirical potentials [51,52]. Properties marked with a “*” are those used for fitting during the parameter optimization. As already mentioned, the cohesive energy and lattice parameters by the present first-principles calculation were for a hypothetical Zn with an ideal c/a ratio. The cohesive energy and lattice parameters for pure Zn calculated using the present potential are rather close to those for the hypothetical Zn rather than real Zn. This is because the non-ideal c/a ratio of real Zn was not reproduced by the present potential. A great amount of effort was made during the parameterization to reproduce the non-ideal c/a ratio of real Zn. However, the c/a ratio of real Zn could not be reproduced without losing good agreements with experiments in many other fundamental material properties. The calculated bulk modulus, elastic constants and ∂B/∂P values are comparable with literature data. The calculated fcc/ hcp and bcc/hcp structural energy differences are close to our own firstprinciples calculations rather than thermodynamically assessed data [41]. The agreement between the present calculation and target values for the above-mentioned properties may show the fitting quality. The comparisons in Table 3 for other properties not considered during the parameter optimization may represent the transferability of the developed potential for applications to diverse studies. The 0 K properties considered for the transferability test were the vacancy formation energy, vacancy migration energy and surface energy. The calculated vacancy formation energy matches well with experimentally reported data [42,43], while the vacancy migration energy is somewhat larger than experimental data [44]. We think the difference is related to the small c/a ratio from the present potential. We also think the calculated surface energy for the low index surface planes is reasonable, considering that the experimental value is the average value for polycrystalline materials [45]. The other properties considered for the transferability test are the thermal expansion, specific heat, melting point, heat of melting and volume change of melting. The thermal expansion coefficient and specific heat were average values in the designated temperature range (0–100 °C). The melting temperature was calculated using an interfacial method and the other two, melting related property values were those obtained at the calculated melting point.

Fig. 1. Enthalpy of formation of stable and artificial, metastable Mg-Zn binary compounds at 0 K according to the present 2NN MEAM potential, in comparison with experimental data [53–56], first-principles calculations [57], and CALPHAD data [58–60]. The structure marked with an asterisk (*) was transformed from its original structure to an unknown structure.

predicted as the most stable phase at the same or similar compositions, with no other phase predicted as a stable phase. As mentioned already, the crystal structure D8C Mg2Zn11 compound was not reproduced correctly. This compound showed a structural transformation into an unknown structure during the 0 K relaxations and the resultant phase was not a stable phase. That is, its enthalpy of formation was above the common tangent line. The next property examined was the structural and elastic property. The calculated lattice parameters and elastic moduli for stable compounds using the present potential are compared with literature data [61–66] in Table 5. The calculated lattice parameters of the two stable compounds, hR276 Mg21Zn25 and C14 MgZn2, are in reasonable agreements with experiments [61–63] and first-principles [65]. The elastic moduli of C14 MgZn2 are also consistent with an experiment [64] and first-principles [65,66]. The structural property (lattice parameter) of the hcp solid solution phase, which was not included in the parameter optimization process, was the first property considered to examine the transferability of the potential. The lattice parameters of Mg-rich hcp alloys calculated at 0 K are compared with experiments at RT [67,68] in Fig. 2. The calculation was performed on eight samples (4000 atoms) of random, disordered hcp solid solutions containing 0–2.9 at% of Zn. Fig. 2 shows the calculated lattice parameters that decrease with Zn addition, in consistent with experiments. It should be mentioned here that the 0 K calculated lattice parameters were compared with RT experimental data in this work. This is because the unary model parameter that determines the lattice parameter of pure elements at 0 K was optimized by fitting to room temperature experimental data, assuming that the thermal expansion between 0 K and RT may be small. The property considered next was the thermodynamic property (heat of mixing) of the hcp and liquid solution phases. Fig. 3 shows the calculated enthalpy of mixing for hcp and liquid Mg-Zn alloys in comparison with experiments [69,70] and CALPHAD [60,71,72]. The calculations were performed using an MD simulation based on a velocity Verlet algorithm and a Parrinello–Rahman NPT ensemble to allow the change of sample dimension in each direction. Those calculations were repeated ten times using different samples to obtain an average for each composition. The MD simulation for hcp solid solution and the CALPHAD calculations for comparison were all performed at 300 K. In the case of the liquid, we raised the temperature of the hcp samples to 1300 K to assure the melting, and then cooled to 900 K, a typical temperature of experiments [69,70] being compared. Fig. 3a shows a good agreement between the present calculation and two CALPHAD calculations [60,71] out of three considered, for the enthalpy of mixing

3.2. Calculated material properties of Mg-Zn binary alloys Similar to the case for pure Zn, the material properties considered for the Mg-Zn binary system can be categorized into two groups, one for the examination of fitting quality and the other for the transferability. The first property considered for the fitting quality was the thermodynamic property. We calculated enthalpy of formation of stable and fictitious metastable (or unstable) compounds at 0 K, as presented in Fig. 1 and Table 4, in comparison with literature data [53–60]. We could ensure that the enthalpy of formation for the stable C14 MgZn2 and hR276 Mg21Zn25 compounds are correctly reproduced, in good agreement with experiments [53–56], first-principles calculation [57] and CALPHAD calculations [58–60]. Further, those phases were 203

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Table 4 Calculated enthalpy of formation ΔHf (kJ/gram-atom) of MgxZny intermetallic compounds using the present 2NN MEAM potential, in comparison with experimental data, first-principles calculations, and CALPHAD data. Compound

Structure

Present work

Mg3Zn

L12 D03 D019 D022 B1 B2 L10 hR276stable C14stable C15 C23 C36 L12 D03 D019 D022 D8Cstable,*

−2.11 −0.03 −2.89 −2.34 127.43 −4.36 −0.26 −8.58 −10.26 −10.12 −1.41 −10.22 −0.46 −0.70 −2.19 −1.13 −1.14

MgZn

Mg21Zn25 MgZn2

MgZn3

Mg2Zn11

Exp.

F.P. calc.

CALPHAD

−7.9 [53], −12.14 [54], −8.9 [55], −8.8 [56] −10.4 [53], −13.8 [54], −10.9 [55], −15.6 [56]

−13.5 [57] −13.8 [57]

−9.4792 [58], −9.6 [59], −10.4 [60] −11.7851 [58], −11.7 [59], −11.5 [60]

−8.96 [54], −6.1 [56]

−7.1 [57]

−5.6783 [58], −5.8 [59], −9.9 [60]

The structure marked with an asterisk (*) is transformed from its original structure to an unknown structure.

for hcp solid solutions, even though another CALPHAD calculation [72] yields different solution behavior. Fig. 3b shows that the present calculation for enthalpy of mixing of liquid alloys is also comparable with experimental [69,70] and CALPHAD data [60,71]. The last property considered to examine the transferability of the potential was the GSFE, which is believed to be related with the deformation behavior of metallic alloys [73]. The GSFE was calculated for the I2-type stacking fault (ABABCACA) of pure Mg and Mg–Zn alloys on basal, prismatic, pyramidal I and pyramidal II planes, as presented in Fig. 4, in comparison with first-principles [74–76]. The calculation was performed for basal, prismatic, pyramidal I and pyramidal II samples with 48 layers of (0001) plane, 40 layers of (10-10) plane, 36 layers of (10−11) plane and 40 layers of (11−22) plane, respectively. For the Mg-Zn alloys, several Mg atoms on the stacking fault (SF) plane were replaced by Zn. The Zn concentration on the fault plane was 25 at% for basal and prismatic samples and 3.125 at% for pyramidal I and pyramidal II samples, the same as in first-principles calculations [74–76] being compared. Periodic boundary conditions were set in the two inplane directions of each SF plane. The SF was formed by the displacement of the lower-half of individual samples. The displacement directions were [10] and [1−210] for basal, [1−210] for prismatic, [1−210] and [11–23] for pyramidal I and [11–23] for pyramidal II, respectively. During the displacement, atoms could only be relaxed along the normal direction of the SF plane. Both the present work and first-principles calculations [74,75] show that Zn addition (25 at% on the SF planes) only slightly reduces the GSFE of pure Mg in basal and prismatic planes (Fig. 4a-c). The effect of Zn addition is even smaller in

Fig. 2. Lattice parameters of hcp Mg-Zn alloys at 0 K according to the present 2NN MEAM potential, in comparison with experimental data [67,68].

pyramidal I and pyramidal II planes (Fig. 4d-f), because of the low Zn content (3.125 at%) [76]. It has been shown that the present Mg-Zn 2NN MEAM potential reasonably reproduces thermodynamic, structural and elastic properties of compounds and solution phases. The effect of Zn addition to the GSFE is well reproduced in good agreements with first-principles calculations. Because of not being able to reproduce the realistic c/a ratio

Table 5 Calculated lattice parameters (Å) and elastic moduli (GPa) of MgxZny intermetallic compounds using the present 2NN MEAM potential, in comparison with experimental data and firstprinciples calculations. Compound (Structure)

Property

Present work

Exp.

Mg21Zn25 (hR276)

a c a c B C11 C12 C13 C33 C44 C66

26.120 8.788 5.335 8.677 58.695 104.938 106.552 28.687 33.289 38.360 33.775

25.640 [61], 25.7758 [62] 8.714 [61], 8.7624 [62] 5.223 [61], 5.22 [63] 8.566 [61], 8.57 [63] 59.778 [64]* 107 [64] 45 [64] 27 [64] 126 [64] 28 [64]

MgZn2 (C14)

The value marked with an asterisk (*) is calculated using the Voigt method.

204

F.P. calc.

5.2040 [65] 8.5398 [65] 63.546 [65], 60.61 [66] 119.48 [65], 92 [66] 49.98 [65], 62 [66] 30.04 [65], 37 [66] 129.48 [65], 126 [66] 24.23 [65], 24 [66] 38.25 [65]

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Fig. 3. Enthalpy of mixing for (a) hcp and (b) liquid Mg-Zn alloys according to the present 2NN MEAM potential, in comparison with experimental data [69,70] and CALPHAD data [60,71,72].

and Mg-Zn-Ca, combined with other, previously developed Mg binary potentials [19,22–24]. However, most of the properties examined were 0 K properties. It should be mentioned here that many interatomic potentials that show a good performance at 0 K often cause a problem during finite temperature MD simulation with a collapse of the crystal

of pure Zn, the present potential may not be suitable for atomistic simulation on pure Zn (EAM can be considered for simulation on pure Zn). However, the potential reasonably describes the alloy behavior of Mg-Zn alloys. It is expected that the present Mg-Zn potential can be extended to higher-order multicomponent Mg systems such as Mg-Al-Zn

Fig. 4. Generalized stacking fault energy of pure Mg and Mg–Zn (a) on basal plane along [10] direction, (b) basal plane along [1−210] direction, (c) prismatic plane along [1−210] direction, (d) pyramidal I plane along [1−210] direction, (e) pyramidal I plane along [11–23] direction and (f) pyramidal II plane along [11–23] direction according to the present 2NN MEAM potential, in comparison with first-principles calculations [74–76]. The Zn concentrations at the basal and prismatic planes were 25 at% and those at the pyramidal I and pyramidal II plane were 3.125 at%.

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Fig. 5. Change in internal energy of stable compound phases after heating to (filled symbols) and rapid cooling from each temperature (open symbols).

structure or transformation into other, unexpected crystal structures. Therefore, as the final test to confirm the reliability of the potential, the thermal stability of the equilibrium compound (hR276 Mg21Zn25 and C14 MgZn2) samples was examined by heating the samples up to 900 K with 100 K intervals (equilibrating at each temperature for 60 picoseconds) and see if the original crystal structures are maintained. The heated samples (with around 2000 atoms) were rapidly cooled from each heating temperature to 0 K and further examined to see whether the initial 0 K total energy is recovered as well as the crystal structure. The change in internal energy after heating to and cooling from each equilibration temperature is plotted in Fig. 5. One can see that the hR276 Mg21Zn25 and C14 MgZn2 compounds show a monotonic increase of internal energy with temperature and recover the initial 0 K energy when rapidly cooled to 0 K from each heating temperature. Based on the finite temperature stability test, the present authors believe that the developed Mg-Zn interatomic potential can be reliably applied to atomic scale investigations of Mg-Zn binary and higher-order alloys, in the whole temperature range. 4. Conclusion The developed Mg-Zn potential reasonably reproduces the thermodynamic, structural and elastic properties of compounds and solution phases of the Mg-Zn binary system at 0 K as well as at the solid-state temperature range of the phase diagram. The potential also reproduces the GSFE on the basal, prismatic and pyramidal planes, in good agreement with first-principles calculations. With the demonstrated reliability and transferability, this potential is expected to be applicable to atom-scale investigations of slip behavior of Mg-Zn binary alloys. Moreover, combined with other previously developed Mg binary potentials, the present Mg-Zn potential can be extended to higher-order multicomponent Mg systems such as Mg-Al-Zn and Mg-Zn-Ca. These multicomponent Mg potentials will be helpful to investigate the effect of alloying elements on slip behavior to clarify the fundamental mechanism for formability enhancement of Mg alloys. Acknowledgements This research has been supported by Basic Science Research Program through the National Research Foundation of Korea (NRF), funded by the Ministry of Science, ICT & Future Planning (2016R1A2B4006680) Korea. References [1] Emission Control, Automotive World 4 10-15, 2000.

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