Journal of Physics: Condensed Matter

J. Phys.: Condens. Matter 26 (2014) 115404 (11pp) doi:10.1088/0953-8984/26/11/115404

Effect of vacancy defects on generalizedstacking fault energy of fcc metalsEbrahim Asadi1, Mohsen Asle Zaeem1, Amitava Moitra2 andMark A Tschopp3,4

1 Materials Science and Engineering Department, Missouri University of Science and Technology,Rolla, MO 65409, USA2 S N Bose National Centre for Basic Sciences, Kolkata, India3 Engility Corporation at the US Army Research Laboratory, Aberdeen Proving Ground, MD 21005,USA4 Center for Advanced Vehicular Systems, Mississippi State University, Starkville, MS 39759, USA

E-mail: AsadiE@mst.edu and Zaeem@mst.edu

Received 22 November 2013, revised 5 January 2014Accepted for publication 22 January 2014Published 3 March 2014

AbstractMolecular dynamics (MD) and density functional theory (DFT) studies were performed toinvestigate the influence of vacancy defects on generalized stacking fault (GSF) energy of fccmetals. MEAM and EAM potentials were used for MD simulations, and DFT calculationswere performed to test the accuracy of different common parameter sets for MEAM and EAMpotentials in predicting GSF with different fractions of vacancy defects. Vacancy defects wereplaced at the stacking fault plane or at nearby atomic layers. The effect of vacancy defects atthe stacking fault plane and the plane directly underneath of it was dominant compared to theeffect of vacancies at other adjacent planes. The effects of vacancy fraction, the distancebetween vacancies, and lateral relaxation of atoms on the GSF curves with vacancy defectswere investigated. A very similar variation of normalized SFEs with respect to vacancyfractions were observed for Ni and Cu. MEAM potentials qualitatively captured the effect ofvacancies on GSF.

Keywords: generalized stacking fault, vacancy, DFT, EAM, MEAM, molecular dynamics

(Some figures may appear in colour only in the online journal)

1. Introduction

Despite the fact that even brittle materials at small deforma-tion have an infinitesimally plastic behavior, most continuummodels assume that the material is either brittle or ductile,predicting two different deformation behaviors and failuremechanisms for these two types of materials. Therefore, theability to determine the ductility of the material via a reliablecriterion is important to capture when modeling materials atcontinuum scales. Such a criterion to quantify ductility may bedefined based on the energy required to nucleate dislocations,such as those proposed in [1–3], which require two materialparameters: the unstable stacking fault energy γus and intrinsic(or stable) stacking fault energy γsf . Generalized stacking faultenergies are calculated by displacing atoms above a fault

plane by a displacement vector while holding atoms belowthe fault plane fixed [4]. The required energy per unit areato displace these atoms plotted against displaced distance iscalled generalized stacking fault (GSF) [5], which is typically asinusoidal graph for most metals. As the displacement distanceincreases, the crystal configuration reaches an unstable posi-tion whereby the required energy per unit area, or stacking faultenergy (SFE), is at a maximum value (γus). This is the energybarrier that must be overcome to nucleate a dislocation. Furtherdisplacement allows the lattice structure to reach a stableconfiguration with a SFE of γsf . Since only the stable SFE(γsf) of the GSF curve can be measured experimentally, therehave been a number of efforts to calculate this curve throughmodeling and simulating methods such as molecular dynamics(MD) and density functional theory (DFT). However, many of

0953-8984/14/115404+11$33.00 1 c© 2014 IOP Publishing Ltd Printed in the UK

J. Phys.: Condens. Matter 26 (2014) 115404 E Asadi et al

the previously reported works in calculating GSF have focusedon determining GSF for perfect crystal structures or, morerecently, solutes without any point defects in the material.Since vacancy point defects in crystal structures have beenshown to significantly influence the mechanical propertiesof materials [6], the present work is prudent to simulate theeffect of vacancy defects on the GSF curves of different fccmetals using two semi-empirical interatomic potentials, theembedded atom method (EAM) and modified-embedded atommethod (MEAM), and validates these calculations with densityfunctional theory (DFT) calculations.

A number of computational studies on stacking faultenergy have been performed to better understand slip behaviorin fcc and bcc systems and, thereby, to understand how tostrengthen these materials. For instance, unstable and stablestacking fault energies were calculated for Fe, Ni, and Alusing EAM and utilized to study tension–shear coupling atcrack tips [7, 8]. Utilizing the same parameters and metals,Farkas et al [9] observed good agreement for predicting theductility of metals using both calculated unstable stackingfault energies and experimentally-measured fracture tough-ness. GSF and surface energies of Si was calculated by Juanand Kaxiras [10] using DFT simulations. They estimated thedislocation profiles, core energies, Peierls energies, and thecorresponding stresses for Si. Zimmerman et al [5] systemat-ically calculated the full GSF curves for different fcc metalsusing different EAM potentials to compare MD results withthose obtained from DFT simulations of [11]. Lu et al [12]utilized a coupled MD-EAM and DFT method to determinethe stacking fault energies of Al and extracted the core prop-erties of various dislocations using a semi-discrete variationalPeierls–Nabarro-based method. Meyer and Lewis [13] usedempirical tight-binding potentials for Ag, Cu, and Ni [14]to calculate the intrinsic stacking fault energies at 0 and300 K. They observed a substantial deviation between thecalculated intrinsic stacking fault energies and experimen-tal values while predicting the correct general trend. Ogataet al [15] utilized DFT calculations of SFE for Al and Cuand concluded that the ideal shear strength of Al is largerthan Cu at {111}〈112〉 direction because of more extendeddeformation range before softening and abnormal high intrin-sic SFE of Al. Yan et al [16] observed that GSF is sensitiveto spin state of the system for first principle calculations forbcc Fe. They performed DFT simulations and local densityapproximation (LDA), spin-polarized LDA, generalized gradi-ent approximation (GGA), and spin-polarized GGA and cameup with suggested approximations for calculating GSF ofbcc Fe for different slip systems. Utilizing MD simulationsto calculate GSF of Al, Ni, and Cu in a nanocrystallinesystem, it was found that the nature of slip in nanocrystallinemetals depend on the values of both stable and unstable SFEsand it cannot be only described in terms of absolute SFEvalues [17]. Siegel [18] calculated the GSF for Ni and itsalloys using DFT simulations to investigate the effects ofalloying on pure Ni ductility. The dependency of GSF andthe ratio of stable to unstable stacking fault energies forfcc metals were investigated using DFT simulations withoutapplied strain on the structure [19] and with isotropic and

Figure 1. Schematic view of GSF at {111}〈112〉 direction for fcccrystals with and without 1/8th atom vacancy at the first-layer:(a) perfect fcc crystal, (b) unstable configuration, (c) stableconfiguration as viewed from 〈110〉 direction, and (d) the {111}plane with vacancy location identified for a 1/8th atom vacancyfraction.

simple shear strain on the structure [20]. Shang [21] studiedthe temperature dependency of GSF for strained crystals of Niusing DFT simulations. Furthermore, Jiang et al [22] presentedsurface stacking fault criterion to predict defect nucleationfrom the surface of fcc metal nanowires based on calculatingGSF and energy to create full dislocation slip. They haveshown that this criterion predicts the initial mode of surfacenucleated plasticity at low temperature successfully. It is worthmentioning that in all of the described works, perfect crystalmodel without vacancy defects were considered, and to the bestof our knowledge, no work has been reported on the effects ofvacancy defects on GSF.

In this paper, MD and DFT studies are performed tocalculate GSF of Al, Ni, and Cu with vacancy defects. Wealso investigated Austenite-Fe (fcc) in our study, as a test caseto observe the prediction of EAM/MEAM potentials, whichwere not developed by fitting material properties towardsthe fcc-Fe, but bcc-Fe; also Austenite-Fe has an improvedstrength [23]. First, the accuracy of MEAM parameter setsand two commonly used EAM potentials for each element istested over small scale DFT calculations to determine GSF for1/8th atom vacancy fraction at the stacking fault plane (oneatom is missing from the eight atoms within the stacking faultplane, i.e., see figure 1). Then, we systematically studied theeffects of vacancy fraction, the distance of vacancy defectsfrom the stacking fault plane, the distance of vacancies fromeach other, and the lateral relaxation of atoms on (1) the GSFcurves, (2) the stable and unstable stacking fault energies, and(3) the stable to unstable stacking fault energy ratios.

2. Methodology

Stacking fault energies for fcc metals are calculated at {111}planes in the 〈112〉 sliding direction which is the primaryslip system with the lowest energy barrier for fcc structures.The schematic of the modeled GSF with vacancies is illustrated

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Table 1. Different EAM and MEAM potentials used for MDsimulations.

Element Label Reference

Al Al-EAM1 Mishin et al [28] and Becker [29]Al-EAM2 Mendelev et al [30]Al-MEAM Lee et al [31] and Jelinek [32]

Ni Ni-EAM1 Mishin et al [28]Ni-EAM2 Becker [29]Ni-MEAM Lee et al [31] and Uddin et al [33]

Cu Cu-EAM1 Mishin et al [34]Cu-EAM2 Becker [29] and Mendelev et al [30]Cu-MEAM Lee et al [31] and Jelinek [32]

Fe Fe-EAM1 Mendelev et al [35]Fe-EAM2 Ackland et al [36]Fe-MEAM Lee et al [31] and Jelinek [32]

in figure 1. The GSF path is shown in figures 1(a)–(c) and thevacancy is shown in figure 1(d). In all the simulations forvacancy defects, a single atom is eliminated from the stackingfault plane (first layer vacancy) or from the nth layer away (nthlayer vacancy) from the fault plane. All the simulations wereperformed at 0 K, and the parallel MD code LAMMPS [24]was used for molecular statics simulations, and VASP (Viennaab initio Simulation Package) [25–27] was used for DFTcalculations. Table 1 lists the different EAM and MEAMpotentials employed for MS simulations along with the labelsthat will be used for simplicity to refer to different potentialshereafter.

2.1. MD simulations

EAM method defines the total energy Ei of an atom as thesum of the embedding energy and the pair potential term [37]

E =∑

i

Fi (ρi )+12

∑j 6=i

ϕi j (Ri j )

, (1)

where Ri j is the distance between atoms i and j having originat i , ϕi j is the pair-wise potential function, ρi is the hostelectron density, and Fi is the embedding function of atom typei . Baskes [31] also developed MEAM to improve the accuracyof MD simulations in predicting certain target properties of themetals using a semi-empirical method. MEAM adds angulardependency to EAM which is important to predict the materialbehavior better near surfaces and defects.

The parameters for the MEAM potentials are based onthe original Baskes [38] paper along with Lee et al [31],Jelinek et al [39], and Lee et al [40]. The details of the MEAMformalism have been published in the literature [31, 38, 41].Here, we have used the second nearest neighbor interactionsas outlined in [42]. Over the years, some of these originalparameters have been modified, taking into consideration morerecent DFT results and responses that differ from this originalwork, such as the stacking fault energy. The Al, Fe, Cu MEAMpotentials used in the present work are taken from the work of

Jelinek et al [32], who utilized DFT calculations of formationenergies of point defects, surface energies, and generalizedstacking fault energies in fitting an Al–Si–Mg–Cu–Fe MEAMpotential. The Fe MEAM potential is a MEAM variant of theFe potential [40], which exhibits the correct low-temperaturephase stability with respect to the pressure. The fcc equilibriumenergy and volume from this Fe potential is very close tothe bcc equilibrium to allow the correct structural transitionto appear at finite temperature without magnetic contribution.The Ni MEAM potential has been most recently used in Uddinet al [33] to simulate carbon nanotubes in a nickel matrix.

For calculating GSF using MD, a simulation cell wascreated with 〈112〉, 〈111〉, and 〈110〉 directions. A free surfaceboundary condition was used in the direction normal to the{111} plane and periodic boundary conditions were used inthe lateral directions. More than 60 atomic layers were usedin the normal direction. The influence of free surfaces wasminimized by either fixing the three layers at the top andbottom free surfaces as rigid bodies or by increasing thenumber of layers in the normal direction. The simulation cellwas divided into two equal rectangular boxes in the normaldirection and an atom is eliminated from the center of thestacking fault layer (first-layer vacancy) or the nth layer belowit (nth-layer vacancy) to create vacancy defects as illustratedin figure 1. Finally, the GSF energies were determined bydisplacing the upper rectangular box at the slide direction andcalculating the energy change in the whole simulation cell. Thecalculated GSFs are the energy change in the vacancy-defectedsimulation cell under stacking fault. During the displacement,the ‘relaxed’ GSF curve was calculated by allowing the atomsat the layer to move laterally and ‘unrelaxed’ GSF wascalculated by preventing the atoms from lateral movements.

The vacancy fraction was controlled by increasing thesize of the stacking fault plane in the periodic directions;e.g. 1/8th, 1/32nd, and 1/128th vacancy areal fractions. Forinstance, figure 1(d) shows the 8 atoms in the {111} planewithout any vacancy. For the 1/8th case, one atom is removedfrom this plane, as is shown in figure 1(d). For the 1/32ndcase, the 8-atom {111} plane is replicated by a factor of 2in both in-plane dimensions and then one vacancy is created.In addition to the effect of vacancy fraction, the vacancy caninteract with itself (i.e., another vacancy) through the periodicboundary, which can be a concern for small cell dimensionatomistic simulations (as in DFT calculations and the 1/8thcase herein). However, this tends to be more of a concernfor defect nucleation/propagation problems [43–45] (i.e., adislocation interacting with itself, or the suppression of anoblique slip system) and calculations of formation energiesof defects [46–48] (where the binding energy of defectsto themselves through the periodic boundaries require anincrease to the cell dimensions so as not to bias the formationenergy calculation). In the present work, since the generalizedstacking fault energy curve is normalized with respect to theinitial energy (with or without defect), this is a minor concerneven in the case of the 1/8th vacancy fraction which is justifiedin detail in section 3.5.

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Table 2. The influence of 1/8th atom vacancy location on the stable and unstable stacking fault energies.

γus γsfLayer EAM1 EAM2 MEAM DFT EAM1 EAM2 MEAM DFT

Al No 168.6 220.5 287.2 171.5 147.0 127.4 145.4 164.2First 174.0+0.8% a 206.6−6.3% 260.6−9.3% 163.8−4.5% 150.1+2.1% 123.3−3.2% 137.1−5.7% 147.2−10.4%

Second 168.0−0.4% 222.4+0.9% 272.0−5.3% — 141.2−3.9% 121.3−4.8% 140.2−3.6% —Third 167.5−0.7% 220.0−0.2% 282.5−1.6% — 145.1−1.3% 127.40.0% 144.2−0.8% —Fourth 167.3−0.8% 219.5−0.5% 286.1−0.4% — 145.3−1.2% 126.9−0.4% 144.6−0.6% —

Ni No 368.1 381.4 465.8 585.5 125.8 195.1 124.2 109.9First 327.2−11.1% 327.6−14.1% 416.5−10.6% 550.5−6.0% 111.7−11.2% 180.6−7.4% 111.9−9.9% 101.1−8.7%

Second 368.8+0.2% 374.7−1.8% 429.7−7.8% — 115.6−8.1% 182.8−6.3% 117.1−5.7% —Third 367.4−0.2% 381.3−0.0% 465.0−0.2% — 125.3−0.4% 195.4+0.2% 123.7−0.4% —Fourth 367.5−0.2% 381.40.0% 465.3−0.1% — 126.0+0.2% 195.3+0.1% 123.7−0.4% —

Cu No 162.6 234.0 247.2 163.0 44.4 44.1 72.2 41.3First 145.3−10.6% 211.5−9.6% 225.6−8.7% 141.0−15.6% 39.5−11.0% 42.6−3.4% 65.3−9.6% 35.0−15.3%

Second 161.8−0.5% 234.7+0.3% 224.8−9.1% — 41.2−7.2% 41.7−5.4% 68.1−5.7% —Third 161.7−0.6% 233.3−0.3% 246.3−0.4% — 44.3−0.2% 44.3+0.5% 71.7−0.7% —Fourth 161.7−0.6% 233.1−0.4% 246.3−0.4% — 44.6+0.5% 44.2− 0.2% 71.9−0.4% —

Fe No — 160.3 361.0 490.0 — 0.0 −44.7 −422.0First — 122.8−23.4% 216.7−40.0% 401.0−18.1% — 0.0 −37.0−17.2% −413.0−2.1%

Second — 159.6−0.4% 387.7+7.3% — — 0.0 −39.1−12.5% —Third — 160.5+0.1% 358.6−0.7% — — 0.0 −44.4−0.7% —Fourth — 160.5+0.1% 360.5−0.1% — — 0.0 −44.4−0.7% —

a The per cent difference between the energy with a vacancy at a certain layer and the energy of the same element and potential with no vacancy.

2.2. DFT simulations

The energy calculations and geometry optimizations in ourstudy were performed using ab initio DFT as implementedin VASP. For the treatment of electron exchange and cor-relation, we used the generalized gradient approximation(GGA) with the Perdew–Burke–Ernzerhof scheme [49–53].All simulations were carried out using a supercell of 96atoms. The plane-wave cut off energy was set to 350 eV. Theconjugate-gradient [52] method was used to relax the ions,and the geometry relaxations were performed until the energydifference between two successive ionic optimizations wasless than 0.001 eV. The Brillouin zone was sampled witha density equivalent to 98 k-points using Monkhorst–Packscheme [53].

Initially, a defect-free {111} surface supercell was fullyrelaxed (both along unit cell vectors and internal atomiccoordinates). This calculation was then followed up by arigid shear along 〈112〉 to calculate the GSF. The samemethodology was adopted when a vacancy replaced one ofthe elemental atoms just underneath the plane from wherethe rigid shear took place (1/8th atom vacancy at the firstlayer). For a large unit cell the vacancy did not affect the unitcell vectors, but only the atoms adjacent to the vacancy wereperturbed. The cross-section of the supercell was chosen insuch a way that when a vacancy resides on the plane abovewhich the rigid shear takes place, the surface coverage became∼2.2× 1014 cm−2. The convergence tests with respect to thek-points (7× 14× 1 k-points was found to be sufficient) andnumber of planes confirmed that the error bar for the totalenergy was less than 1 meV/atom.

3. Results and discussions

3.1. Effect of vacancy on stacking fault energy

In the first set of examples, the influence of an 1/8th atomvacancy defect and its distance from the stacking fault planeon stable γsf and unstable γus stacking fault energies andGSF are investigated by MD simulations using the interatomicpotentials listed in table 1 and by DFT calculations. Table 2shows γsf and γus for Al, Ni, Cu, and Austenite-Fe for theperfect crystal simulation cell with no vacancy and with 1/8thatom vacancy located at first, second, third, and fourth layerfrom the stacking fault plane. For most elements, MD and DFTsimulations calculate that both γsf and γus drop significantly(∼5–15%) as a vacancy is introduced into the stacking faultplane. Additionally, MD simulations show, in general, that theSFEs monotonically approach the perfect single crystal valuesas the vacancy moves away from the first-layer, as expected.This pattern is followed in the MD simulations using MEAMpotentials and DFT calculations for Al, Cu, and Ni. However,MD simulations using EAM1 potentials for Al, Cu, and Nido a poor job in capturing this behavior for the variation ofstacking fault energies with vacancy defects. For instance, γsfand γus for 1/8th atom vacancy at the first-layer for Al arebigger than the same values where there is no vacancy at thesimulation cell (compared to DFT, which shows 4.5% in γsfand 10.4% in γus decrease). The EAM2 potentials for Al,Cu, and Ni generally predict the same behavior as MEAMpotentials for the vacancy located at the first-layer but notnecessarily a monotonic decrease in percentage differencefor vacancies located at the second, third, and fourth layers.

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Figure 2. GSF for 1/8th atom vacancy at the first layer for (a) Al, (b) Cu, (c) Ni, and (d) Austenite-Fe.

For example, γus for Al and vacancy at the second layer andfor Cu and vacancy at the third layer are bigger than the relatedγus with no vacancy. Since the cohesive energy of bcc-Fe isslightly less than Austenite-Fe and there is the possibility ofphase transformation to hcp or bcc configurations, the stackingfault energies of Fe do not generally follow the same pattern asfor Al, Cu, and Ni. We chose fcc-Fe in our study, as a test case toobserve the prediction of EAM/MEAM potentials, which werenot developed by fitting material properties towards the fcc-Fe,but bcc-Fe. As it may be seen, all potentials fail to calculate thestacking fault energies for bcc-Fe with and without vacancy.The stacking fault energies for Fe will be discussed in detailin the explanations of figure 2(d).

GSF for 1/8th atom vacancy at the first-layer are shownin figure 2 for Al, Cu, Ni, and Fe using MD simulations andDFT calculations. Our DFT calculations for GSF of Al at

figure 2(a) show a small energy barrier which is almost thesame asγsf to create the stable stacking fault configuration. MDcalculations of GSF using Al-EAM1 show the same behavioras our DFT calculations. However, GSF curves calculated byAl-EAM2 and Al-MEAM show the same qualitative behaviorof DFT calculations presented in Zimmerman et al [5] andHartford et al [11] because those DFT calculations were usedin determining their potential parameters. All MD calculationsfor Cu and Ni shown in figures 2(b) and (c), respectively,predict the same sinusoidal pattern of GSFs predicted by DFTcalculations. Cu-EAM1 which was fitted particularly to predictGSF and Ni-MEAM predict the best match to GSF of DFTcalculations. GSF calculated by DFT at figure 2(d) showsa negative big minimum value after the peak related to theenergy barrier. This implies that the hcp phase is energeticallymore favorable than the fcc phase for pure iron. An energy

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Figure 3. GSF for no vacancy and 1/8th atom vacancy located at the first, second, and third layers obtained by MEAM-MD simulations for(a) Al, (b) Cu, and (c) Ni.

calculation for Fe phases clearly shows that for pure Fe, bcc isenergetically the ground-state, followed by the hcp and then thefcc phase. Physically, a thermal activation or externally appliedstress could ease fcc iron to overcome the energy barrier; assuch fcc-Fe would prefer to be in hcp configuration.

3.2. Effect of vacancy distance from stacking fault energyplane

The influence of the distance of 1/8th atom vacancy from thestacking fault plane on GSF is investigated in figure 3 usingMEAM-MD simulations. All elements show a significant dropon their GSF when the vacancy located at the first layer andthe influence of vacancy is almost ignorable when the vacancylocated at the third layer below the stacking fault plane.

3.3. Effect of vacancy fraction on stacking fault energy

Since it was shown that the influence of atom vacancy onGSF at layers beyond second-layer is small comparing tothe vacancy influence at the first and second layers, the atomvacancy fractions 1/8th, 1/32nd, 1/128th at these layers in asufficiently big simulation box may be approximated by having12.5%, ∼3%, <1% volume fraction randomly distributedvacancy defects if the interaction between vacancy defects

is ignored. Figure 4 shows the effects of vacancy fractionon GSF using MEAM-MD simulations where the vacancylocated at the first layer. It is worth mentioning that all theelements show the same behavior regard to including thevacancy fractions. Generally, including vacancy decreases theamount of GSF at all points of the GSF curve but this decrementis negligible for vacancy fractions less than 1/128th where theGSF is identical to the no vacancy GSF.

The influence of different atom vacancy fraction at the firstand second layers on normalized stable and unusable SFEs ofAl, Cu, and Ni are shown in figure 5, by fitting quadraticfunctions to the values of SFEs for different vacancy fractions.SFEs are normalized by dividing them with the correspondingSFEs for the case of no vacancy. For all the elements andwhen vacancies are located at the first layer, as the vacancyfraction decreases, both stable and unstable SFEs increasealmost linearly toward the value of the SFEs for cases withno vacancy. This linear behavior changes to nonlinear whenthe vacancy defects are located at the second layer, except forunstable SFE of Cu which is still linear. In addition, similareffect of atom vacancies on normalized SFEs of Ni and Cu isapparent. It is worth mentioning that the energies calculated inthis study are relative energies with respect to the no stackingfault configurations but with vacancies; thus, the absolute value

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Figure 4. GSF for 1/8th, 1/32nd, and 1/128th atom vacancy located at the first layer obtained by MEAM-MD simulations for (a) Al,(b) Cu, and (c) Ni.

of the energy of the system is not decreasing with the incrementof vacancy fraction.

3.4. Effect of lateral relaxation on stacking fault energy

The effect of lateral relaxation, shown by superscript r , ofatoms versus not allowing the atoms to move at the in-planedirections, shown by superscript u, on stable and unstableSFEs using MEAM-MD simulations are investigated in thisexample. Unrelaxed SFEs are important since it was shownthat SFEs at highly strained environments such as SFEs undershear deformation are close to the unrelaxed values [15, 20].Table 3 shows variation of γ (r)sf /γ

(u)sf and unstable γ (r)us /γ

(u)us

ratios versus the variation of vacancy fractions and locations.The difference between relaxed and unrelaxed energies ofstable SFE is typically less than the unstable SFE differences. Itwas also observed that the relaxed SFEs are less than unrelaxedSFEs and atom vacancies do not change this behavior exceptfor 1/8th atom vacancy at the first layer where γ (r)sf > γ

(u)sf .

3.5. Effect of vacancy binding on stacking fault energy

The effect that vacancies in proximity to other vacancieshave on the stacking fault energy was also investigated forthe fcc potentials. As the vacancy fraction increases, the

likelihood that two vacancies will be located near to eachother also increases; the interaction energy, or binding energy,between these two vacancies may lead to a deviation fromthe stacking fault energy curve with the two vacancies apart.For the purposes of this study, figure 6 shows several vacancylocations used herein. The first vacancy is located in location(a) and the second vacancy is positioned in locations (b)–(f)to give a 1/8th atom vacancy fraction within the {111} plane.Interestingly, note that the (a)–(c) combination is equivalentto that calculated in the 1/8th vacancy case in the precedingsubsections. The (a)–(f) combination is the furthest that thetwo vacancies can be located from each other within the {111}plane for this vacancy fraction.

The vacancy formation energy (Ef) and binding ener-gies (Eb) between the various two-vacancy combinationsis calculated to quantify the degree of energetic interactionbetween the two vacancies. The vacancy formation energiesare calculated for each potential and then the binding energiesare calculated by subtracting the formation energies for twovacancies in the listed positions from that for two vacanciesseparated by several nanometers. Table 4 lists the results ofthese calculations. A positive binding energy means that thetwo vacancies are attracted, i.e., the total energy decreasesby the vacancies being in these positions. A negative bindingenergy means that the two vacancies are repelled. For the

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Figure 5. Variation of SFEs for Al, Cu, and Ni with atom vacancy fraction at the first two layers (a) unstable SFE for vacancy at thefirst-layer, (b) unstable SFE for vacancy at the second-layer, (c) stable SFE for vacancy at the first-layer, and (d) stable SFE for vacancy atthe second-layer.

Table 3. The influence of vacancy fraction and location on γ (r)sf /γ(u)sf and γ (r)us /γ

(u)us ratios using MEAM-MD simulations.

Element No vac.1/8th vac.first-layer

1/8th vac.second-layer

1/8th vac.third-layer

1/32nd -vac.first-layer

1/32nd vac.second-layer

1/32nd vac.third-layer

Al γ(r)us /γ

(u)us 0.936 0.958 0.916 0.921 0.946 0.946 0.932

γ(r)sf /γ

(u)sf 0.969 1.048 0.998 0.962 0.986 0.986 0.975

Cu γ(r)us /γ

(u)us 0.910 0.898 0.877 0.907 0.910 0.910 0.904

γ(r)sf /γ

(u)sf 0.989 1.025 0.996 0.983 0.997 0.997 0.990

Ni γ(r)us /γ

(u)us 0.897 0.886 0.868 0.896 0.897 0.897 0.891

γ(r)sf /γ

(u)sf 0.990 1.007 0.993 0.986 0.994 0.994 0.991

purposes of these calculations, the simulation cell was ex-panded to greater than 3 nm in every direction so that thevacancies were not interacting through the periodic boundary.In all cases, the (a)–(c) and (a)–(d) vacancy combinationshave similar binding energies, as would be expected sincethe distance is identical. The potentials investigated hereinhave very different binding behaviors, though. For instance,in copper and nickel, the EAM potentials predict a positiveEb (attractive) for the (a)–(b) vacancy combination while theMEAM potential predicts a negative Eb. Furthermore, the

MEAM potential predicts that there is a significantly higherbinding energy of vacancies in the (a)–(c) locations than theEAM potentials. Comparing the (a)–(c) binding energies to the(a)–(f) binding energies indicates that the level of interactionbetween a single vacancy through the periodic boundary inthe 1/8th atom vacancy case is relatively small for the case ofEAM potentials (<∼30 meV) and is significantly higher inthe MEAM potentials (∼205 meV in the case of aluminum).

An example of the effect of vacancy positions on thestacking fault energy curve is shown for two aluminum

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Table 4. The vacancy binding energies as a function of position for the various potentials.

Eb (eV) Eb (eV) Eb (eV) Eb (eV) Eb (eV)Element Potential Ef (eV) a–b a–c a–d a–e a–f

Al EAM1 0.68 −0.0111 −0.0314 −0.0111 −0.0262 −0.0006EAM2 0.66 0.0909 −0.0204 0.0767 −0.0329 0.0017MEAM 0.68 −0.0541 0.2038 −0.0542 −0.0195 −0.0016

Cu EAM1 1.27 0.1454 −0.0114 0.1452 −0.0133 −0.0008EAM2 1.11 0.1314 −0.0192 0.1319 −0.0259 −0.0015MEAM 1.10 −0.1078 0.0848 −0.1080 0.0621 −0.0004

Ni EAM1 1.60 0.1354 −0.0319 0.1356 −0.0482 −0.0013EAM2 1.76 0.3497 −0.0038 0.3507 −0.0092 0.0004MEAM 1.48 −0.2101 0.1788 −0.2101 0.0934 −0.0001

Figure 6. The {111} plane with vacancy locations identified for thetwo-vacancy study with 1/8th atom vacancy fraction.

potentials (EAM1 and MEAM) in figure 7. The legend isarranged in order of increasing distance between the twovacancies. There are a number of observations from figure 7.First, the largest deviation from the stacking fault energy curvewith two vacancies at their furthest distance is for the (a)–(b)vacancy combination, which is consistent for 8 of the 9 fccpotentials (Ni-EAM2). However, this does not necessarilyrelate to the binding energy in table 4 as both locations(a)–(b) and (a)–(d) have the same binding energy but havedifferent responses with respect to the stacking fault energycurves (with a–d having only a minimal effect in most cases).Second, notice that the interaction between vacancies heavilyaffects the unstable stacking fault energy but in comparisondoes not significantly impact the stable stacking fault energy.This observation was true in all cases. Third, although thebinding energy was relatively high for some vacancy locations(i.e., see a–c in table 4 which is comparable to a–b Eb), thisdid not necessarily manifest as a difference in the stackingfault energy behavior. Fourth (not shown), these interactionswere only significant in the first layer. Other layers show verylittle influence of vacancy interactions on the stacking faultenergy curves. Last, the results show that the stacking faultenergy curves with the (a)–(c) vacancy combination is verysimilar to that with the (a)–(f) combination. This confirmsthat the 1/8th vacancy fraction simulation cell used in the

preceding subsections (with 8 atoms in the {111} plane) didnot adversely affect the stacking fault energy curve due to thevacancy interacting with itself through the periodic boundary.

4. Conclusions

MD simulations were performed to investigate the effect ofvacancies on {111}〈112〉GSF of Al, Ni, and Cu using differentEAM and MEAM parameter sets. DFT calculations were alsoperformed to test the accuracy of EAM and MEAM potentialsfor each element. It was found that The EAM potentialof [28, 29] for Al, EAM potential of [34] for Cu, and MEAMpotential for Ni predicts the best match of GSF to our DFTcalculations. GSF of Austenite-Fe (fcc) was calculated to testthe EAM/MEAM potentials stabilized for bcc configuration.It was shown that only MEAM potentials can roughly capturethe GSF prediction of DFT for Austenite-Fe where there isthe possibility of the phase transformation to bcc or hcpconfigurations. GSFs for all elements investigated in this studytend to drop significantly with a single atom vacancy defect atthe stacking fault plane. As we moved away the vacancy fromthe stacking fault plane, the drop in energy diminishes rapidly;e.g., GSFs are identical to the perfect crystal structure for thevacancy located at the fourth layer under the stacking faultplane. We also investigated the effect of vacancy fraction onGSFs of all elements and realized that for vacancy fractionof ∼12.5% the stable and unstable stacking fault energiesdrop by ∼10% and vacancy fraction <1% has insignificanteffect on stacking fault energies. It was found that the stableand unstable normalized SFEs change almost linearly withthe vacancy fraction for vacancies at the stacking fault planeand nonlinearly for vacancies at the other planes, with theexception of unstable SFE of Cu which is still linear versusthe vacancies at the second layer. Also, similar normalizedSFEs for Ni and Cu were obtained by variation of vacancyfraction. We showed that vacancy defects also affect the ratioof relaxed to unrelaxed stacking fault energies significantly,and even in some cases, vacancy defects surprisingly makethe relaxed SFEs bigger than the unrelaxed SFEs. Finally, weshowed that the proximity of two vacancies at the stackingfault plane affect GSF curve but the general shape of GSFremains unchanged.

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J. Phys.: Condens. Matter 26 (2014) 115404 E Asadi et al

Figure 7. The effect of vacancy locations (two vacancies) in the first layer on the stacking fault energy curve for the 1/8th atom vacancyfraction in two aluminum potentials: (a) EAM1 and (b) MEAM.

Acknowledgments

The authors are grateful for computer time allocation pro-vided by the Extreme Science and Engineering DiscoveryEnvironment (XSEDE), the High Performance ComputingCollaboratory (HPC2) at Mississippi State University throughthe Center for Advanced Vehicular Systems (CAVS), andIUAC, New Delhi, India.

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