MEAM potentials for Al, Si, Mg, Cu, and Fe alloys
Bohumir Jelinek, Sebastien Groh, Mark F. Horstemeyer, Jeffery Houze, Seong-Gon Kim, Gregory J. Wagner, Amitava Moitra, Michael I. Baskes
MMEAM potentials for Al, Si, Mg, Cu, and Fe alloys
B. Jelinek, S. Groh, ∗ and M. F. Horstemeyer † Center for Advanced Vehicular Systems, 200 Research Boulevard, Starkville, MS 39759
J. Houze and S. G. Kim ‡ Department of Physics and Astronomy, Mississippi State University, Mississippi State, MS 39762
G. J. Wagner
Sandia National Laboratories, P.O. Box 969, MS 9401, Livermore, CA 94551
A. Moitra
Department of Chemical Engineering, The Pennsylvania State University, University Park, PA 16802
M. I. Baskes § Mechanical and Aerospace Engineering, University of California in San Diego, La Jolla, CA 92093 (Dated: October 22, 2018)A set of Modified Embedded Atom Method (MEAM) potentials for the interactions between Al,Si, Mg, Cu, and Fe was developed from a combination of each element’s MEAM potential in order tostudy metal alloying. Previously published MEAM parameters of single elements have been improvedfor better agreement to the generalized stacking fault energy (GSFE) curves when compared withab-initio generated GSFE curves. The MEAM parameters for element pairs were constructed basedon the structural and elastic properties of element pairs in the NaCl reference structure garneredfrom ab-initio calculations, with adjustment to reproduce the ab-initio heat of formation of the moststable binary compounds. The new MEAM potentials were validated by comparing the formationenergies of defects, equilibrium volumes, elastic moduli, and heat of formation for several binarycompounds with ab-initio simulations and experiments. Single elements in their ground state crystalstructure were subjected to heating to test the potentials at elevated temperatures. An Al potentialwas modified to avoid formation of an unphysical solid structure at high temperatures. The thermalexpansion coefficient of a compound with the composition of AA 6061 alloy was evaluated andcompared with experimental values. MEAM potential tests performed in this work, utilizing theuniversal Atomistic Simulation Environment (ASE), are distributed to facilitate reproducibility ofthe results.
PACS numbers: 61.50.Lt, 62.20.D- 61.72.J- 68.35.-p
I. INTRODUCTION
Historically, materials have been developed throughthe correlation of processing and properties. Several im-plementations of materials science principles have givenbirth to an engineering framework for materials design.Over the past two decades, more efficient computationalmethodologies have been developed and the computa-tional power have increased enormously, making the com-putational materials design an essential cost-effective toolto design materials properties. Since materials complex-ities can limit the degree of predictability, several time-and length-scale methodologies (hence spatiotemporalhierarchy) for computational materials design naturallyevolved (cf. Horstemeyer for a review). Out of severalcomputational methodologies, atomistic simulations notonly can predict the materials properties from a statisticalviewpoint, but can also quantify the mechanisms of thestructure-property relationship. One of the most criticalcomponents of atomistic simulations is the interatomic po-tential, which determines the forces on individual atoms.First-principles calculations certainly are capable of pro-viding very reliable interatomic potentials in a variety of chemical environments. However, realistic simulations ofalloy systems, which are essential to reveal many macro-scopic materials properties, often require a number ofatoms that renders these methods impractical – they ei-ther require too much computer memory or take too longto be completed in a reasonable amount of time. One al-ternative is to use (semi-)empirical interaction potentialsthat can be evaluated efficiently, so that the atomisticapproaches that use them can, in certain cases, handlesystems with more than a million atoms.The Embedded-Atom Method (EAM) is a widely usedatomic level semiempirical model for metals, covalentmaterials, and impurities . MEAM (Modified EAM) in-corporates angular dependency of electron density intoEAM. Atomistic simulations of a wide range of elementsand alloys have been performed using MEAM potentials.MEAM model was first used for silicon, germanium, andtheir alloys . It was applied to 26 single elements and tosilicon-nickel alloys and interfaces. Gall et al. have usedMEAM to model tensile debonding of an aluminum-siliconinterface. Lee and Baskes improved MEAM to accountfor the second nearest-neighbor interactions. Also, Huang et al. used MEAM and two other potentials to deter- a r X i v : . [ c ond - m a t . m t r l - s c i ] M a r mine defect energetics in beta-SiC. MEAM parametersfor nickel and molybdenum-silicon system were deter-mined by Baskes. MEAM potentials for Cu, Ag, Au,Ni, Pd, Pt, Al, and Pb based on the first and the secondnearest-neighbor MEAM were constructed by Lee et al. .Hu et al. proposed a new analytic modified EAMmany-body potential and applied it to 17 hcp metals. Thestructural properties of various polytypes of carbon weredescribed using a MEAM potential by Lee and Lee .Recent work of Lee et al. summarized available MEAMpotentials for single elements and alloys. Several of thesepotentials were then used to perform large scale atomisticsimulations to understand the intriguing nature of theductile and brittle fracture , structure-property relation-ship , dislocation dynamics , and nature of materialsfracture .Aluminum, magnesium, copper, and iron alloys arebeing used in developing materials with novel properties.Great popularity of these alloys is connected to their gen-eral functional properties, mechanical properties, massdensity, corrosion resistance, and machinability. Lightmetal alloys, such as magnesium and aluminum alloys,are now demanded for use in the automotive and avia-tion industries. They performed remarkably well for thepurpose of decreasing the operating expenses and fuelconsumption. These alloys usually contain several otherminor elements, such as silicon, nickel, and manganese,and are known to have very complex phase compositions.Assessment of such complex systems is a very challengingtask, since different constituent elements can form differ-ent phases, whose selection depends on the ratio betweenthe constituents and also on a variety of processing andtreatment factors.Contrary to DFT potentials, most of the single ele-ment semiempirical potentials do not combine easily intomulti-component alloy models. The difficulty of com-bining single element EAM potentials into alloy systemscomes from the need of their normalization . The pro-cedure to form EAM alloy parameterization from singleelement potentials was suggested , but it does notguarantee that the resulting potential will be suitable formodeling compounds . Alloy potentials usually intro-duce new parameters for each pair of elements, allowing tofit properties of their binary compounds. The number ofparameters to adjust and the number of tests to performis proportional to the square of the number of constituentelements. In the present MEAM approach, each pairinteraction is characterized by a total of 13 parameters(Table V, and the ratio of density scaling factors ρ forconstituent elements, Table I). Adoption of the defaultvalue C max = 2 . also have 9 adjustable pair parameters.While the semiempirical potentials have been developedand tested for binary alloys , binaries, similarly tosingle element potentials, may not combine easily intoternaries. Modeling of ternary systems faces a challengesince less experimental properties are available for ternary systems. Ternary potentials are usually examined only ata particular composition range—the number of possiblecompositions grows to the power of the number of con-stituent elements. It is also nontrivial to find an equilib-rium structure for complex systems of representative sizeat low temperatures. Ternary potentials are only availablefor Fe/Ni-Cr-O (MEAM), Pu-Ga-He (MEAM), Fe-Ti-C/N, Cu-Zr-Ag, Ga-In-N, Fe-Nb-C/N (MEAM),H-C-O (Reactive Force Field, ReaxFF), Ni-Al-H , Zr-Cu-Al , and Fe-Cu-Ni (EAM) systems. To extend frombinaries to ternaries, MEAM provides a ternary screen-ing parameter C XYZ . In the present work we did notexamine ternary systems. Instead, we performed thermalexpansion simulations of a compound including all speciesof the potential. The default values of C min = 2 . C max = 2 . .In the present study we develop a MEAM potential foraluminum, silicon, magnesium, copper, iron, and theircombinations. We fit the potential to the properties ofsingle elements and element pairs, but the model implic-itly allows calculations with any combination of elements.We show the applied MEAM methodology in Appendix A.The DFT calculations are described in Sec. II. In Sec. III,the single-element volume-energy curves in basic crystalstructures, and also important material properties, suchas formation energies of vacancies, self-interstitials, sur-faces, and generalized stacking fault energies from MEAMare examined and compared with DFT calculations. InSec. IV, the MEAM potential parameters for each unlikeelement pair are initialized to fit the ab-initio heat offormation, equilibrium volume, and elastic moduli of thehypothetical NaCl reference structure. Heat of formationof binary compounds in a variety of crystal structuresfrom MEAM are thereafter examined and compared withthe ab-initio and experimental results. The MEAM pa-rameters are adjusted to match the DFT formation energyof the most stable compounds. The structural and elasticproperties for several binary compounds and formationenergies of substitutional defects are compared with ab-initio and experimental results. Finally we performedthermal expansion simulations of a compound with thecomposition of an AA 6061 alloy (IV C). We concludewith a short summary. II. AB-INITIO CALCULATIONS
Ab-initio total energy calculations in this work werebased on density functional theory (DFT), using the pro-jector augmented-wave (PAW) method as implementedin the VASP code . Exchange-correlation effects weretreated by the generalized gradient approximation (GGA)as parameterized by Perdew et al. . All DFT calcula- TABLE I. Set of the MEAM potential parameters for single elements. The reference structures for Al, Si, Mg, Cu, and Feare fcc, diamond, hcp, fcc, and bcc, respectively. E c is the cohesive energy, a is the equilibrium lattice parameter, A is thescaling factor for the embedding energy, α is the exponential decay factor for the universal energy, β (0 − are the exponentialdecay factors for the atomic densities, t (0 − are the weighting factors for the atomic densities, C max and C min are screeningparameters, ρ is the density scaling factor that is relevant only for element pairs. Definition of these parameters may be foundin Ref. 4. Non-zero parameters δ r in Rose Eq. (B1–B4) were used for Al ( δ r = 0 .
1) and Fe ( δ r = 0 . δ a = 0 . E c [eV] a [˚A] A α β (0) β (1) β (2) β (3) t (0) t (1) t (2) t (3) C min C max ρ Al 3.353 4.05 1.07 4.64 2.04 3.0 6.0 1.5 1.0 4.50 -2.30 8.01 0.8 2.8 1.0Si 4.63 5.431 1.00 4.87 4.4 5.5 5.5 5.5 1.0 2.05 4.47 -1.80 2.0 2.8 2.2Mg 1.51 3.194 0.8 5.52 4.0 3.0 0.2 1.2 1.0 10.04 9.49 -4.3 0.8 2.8 0.63Cu 3.54 3.62 1.07 5.11 3.634 2.2 6.0 2.2 1.0 4.91 2.49 2.95 0.8 2.8 1.1Fe 4.28 2.851 0.555 5.027 3.5 2.0 1.0 1.0 1.0 -1.6 12.5 -1.4 0.68 1.9 1.0 tions were performed in high precision with the plane-wave cut-off energy set to 400 eV in order to achieve theconvergence of heat of formation and elastic properties.Integration over the irreducible Brillouin zone was per-formed using the Γ-centered Monkhorst-Pack scheme with the size gradually increased to 7 × × × × × ×
29 toimprove convergence of shear moduli at small strains.Elastic constants presented here were obtained withoutrelaxation of atomic positions. Since most of the examinedhigh energy structures are, at best, metastable, relaxationdoes not maintain the crystal symmetry, resulting in largeenergy changes and unphysical elastic constants.
III. MEAM PARAMETERS FOR SINGLEELEMENTS
The present MEAM parameters for single elements arelisted in Table I. The initial values of these parameterswere taken from existing MEAM potentials . The C min screening parameter for Al, Mg, and Cu was loweredfrom 2.0 to 0.8 to improve the GSFE curves (Sec. III E).The Mg potential was adjusted to reproduce the DFTvalues of hcp, bcc, and fcc energy differences, vacancyformation energy, and (10¯10) surface formation energy.The Al potential was modified to prevent formation of anunknown structure at elevated temperatures (Sec. IV C). A. Energy dependence on volume of singleelements in fcc, hcp, bcc, and simple cubic crystalstructures
The first test of the validity of MEAM potential for sin-gle elements is a comparison of the energy-volume curvesin the fcc, hcp, bcc, diamond, and simple cubic crystalstructures, shown in Fig. 1. The MEAM potentials ap-propriately capture the lowest energy structures of Al(fcc), Si (dia), Mg (hcp), Cu (fcc), and Fe (bcc). Also,the equilibrium volumes of several crystal structures fromMEAM closely match the DFT results. Better match of DFT energy differences and volume ratios can possi-bly be obtained by optimization of Si and Cu MEAMparameters. Fe MEAM potential applied in the presentwork is a MEAM-p variant of Fe potential from the recenteffort of Lee et al. , exhibiting a correct low tempera-ture phase stability with respect to the pressure. The fccequilibrium energy and volume from this Fe potential isvery close to the bcc equilibrium in order for the struc-tural transition to appear at finite temperature withoutmagnetic contribution. In general the MEAM potentialsof the present work reproduced the DFT results for theindividual elements fairly well. B. Vacancies
The formation energy of a single vacancy E vacf is definedas the energy cost to create a vacancy: E vacf = E tot [ N ] − N ε, (1)where E tot [ N ] is the total relaxed energy of a systemwith N atoms containing a vacancy and ε is the energyper atom in the bulk. Cell volume and atomic positionswere relaxed in each case. Table II shows the formationenergies of a single vacancy for the fcc Al cell, diamondSi cell, hcp Mg, fcc Cu, and bcc Fe obtained from theMEAM and DFT calculations. The MEAM systems sizeswere 5 × × × × × × × × × × × × .Overall agreement of vacancy formation energies betweenMEAM, experiment, and DFT was within a few eV, andthe present results are comparable or better than thosefrom other calculations. The reduction in volume dueto the formation of a vacancy agrees well with the DFT,except the value for Fe is somewhat low. ∆ E [ e V ] V/Vfcc Al00.10.20.30.40.50.60.7 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 ∆ E [ e V ] V/Vdia Si00.1 0.9 1 1.1 1.2 1.3 ∆ E [ e V ] V/Vhcp Mg00.10.20.30.40.50.6 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 ∆ E [ e V ] V/Vfcc Cu00.20.40.60.8 0.8 0.9 1 1.1 1.2 1.3 1.4 ∆ E [ e V ] V/Vbcc Fe fcc MEAMDFThcp MEAMDFTbcc MEAMDFTsc MEAMDFTdia MEAMDFTfcc MEAMDFTbcc MEAMDFTsc MEAMDFThcp MEAMDFTfcc MEAMDFTbcc MEAMDFTfcc MEAMDFTbcc MEAMDFTsc MEAMDFTfcc MEAMDFTbcc MEAMDFTsc MEAMDFT
FIG. 1. (Color online) Energy-volume dependence of Al, Si,Mg, Cu, and Fe in fcc, hcp, bcc, diamond, and simple cubiccrystal structures relative to the ground state. TABLE II. Calculated single vacancy properties. Single va-cancy formation energy E vacf and formation volume Ω v valuesare obtained from the relaxed structures containing single va-cancies. Here Ω is the bulk atomic volume. All energy valuesare listed in eV. The results from the MEAM calculations arecompared with the results from the DFT calculations giveninside the parentheses, other simulations, and experiments. E vacf Ω v / Ω Present Others Exp Present OthersAl 0.67 (0.5) 0.68 a b (0.55 f ) 0.67 g a b Si 3.27 (3.6) 3.56 t s (3.6 c ) 3.6 i t Mg 0.89 (0.7) 0.59 d b (0.83 k ) 0.79 e d b Cu 1.10 (1.0) 1.05 q r (1.03 h ) 1.2 j ∼ r r Fe 1.65 (2.1) 1.84 l m (1.95 n ) 1.53 o ∼ p (0.90 n ) a MEAM results by Lee et al.
Calculated using EAM parameters extracted from Liu et al.
DFT calculation by Wright
AMEAM results by Hu et al.
Experimental results by Tzanetakis et al.
DFT results by Carling et al.
Experimental value by Hehenkamp
DFT calculation by Andersson and Simak
Experimental value by Dannefaer et al. , Throwe et al. Experimental value by Hehenkamp et al.
DFT value by Krimmel and F¨ahnle
EAM value by Mendelev et al.
Finnis-Sinclair potential value by Ackland et al.
DFT value by Domain and Becquart
Experimental value by Schaefer et al.
Experimental value referenced in Ackland et al.
EAM value by Mendelev et al.
EAM value by Mishin et al.
MEAM value by Timonova et al.
MEAM value by Ryu et al. C. Self-interstitials
The formation energy of an interstitial point defect E intf is given by E intf = E tot [ N + 1] − N ε X − ε Y (2)where E tot [ N + 1] is the total energy of a system with N type-X bulk atoms plus one impurity atom of type-Yinserted at one of the interstitial sites, and ε X ( ε Y ) isthe total energy per atom of type-X (type-Y) in its moststable bulk structure. The inserted atom Y can be of thesame type as the bulk, in which case the point defect iscalled a self-interstitial defect. Self-interstitial formationenergies were calculated for Al, Si, Mg, and Cu at theoctahedral, tetrahedral, and dumbbell sites. Dumbbellorientations were [100] for fcc, [0001] for hcp, and [110] forbcc and diamond structures. Relaxations of the atomicpositions and the volume were also performed, and theDFT and MEAM results are listed in Table III. Similarto the previous calculations, the MEAM systems sizeswere 5 × × × × × × × × × × × × TABLE III. The formation energies of various Al, Si, Mg, Cu,and Fe self-interstitials. All energy values are given in eV. Theresults from the MEAM calculations are compared with theDFT results and other classical MD (CMD) simulations.Interstitial DFT MEAM CMD DFTAl Present Ref. Ref. Ref. Ref. tetrahedral 3.3 3.32 3.16 a a a Ref. Ref. Ref. split (110) 3.7 3.71 3.88 4.7 3.9 3.40Mg Present Ref. Ref. Ref. Ref. tetrahedral 2.2 1.63 1.53 a a a Cu Present Ref. Ref. Ref. Ref. tetrahedral 3.9 3.37 2.99 a octahedral 3.5 2.72 2.97 a split (100) 3.3 2.59 2.81 3.06 3.23 2.93Fe Present Ref. Ref. Ref. Ref. tetrahedral 4.2 4.31 4.16 a a a Calculated using parameters from the given reference.
In general, the DFT results are well reproduced orslightly underestimated by the MEAM potentials. Ac-cording to the present MEAM potential, the most stableform of a self-interstitial defect for fcc Al is a dumbbellalong the [100] direction, in agreement with the DFT re-sults and an experimental observation by Jesson et al. .The results for Mg are better than those published previ-ously . The present Mg potential indicates that thetetrahedral site will be most stable in agreement withthe DFT calculations. For both Cu and Fe, the newMEAM potential produces the same relative stability ofthe examined interstitial sites with the DFT calculations. D. Surfaces
A semi-infinite surface is one of the simplest forms ofdefects. To test the transferability of the new MEAMpotentials, formation energies for several 1 × × × E surf is defined as E surff = E tot [ N ] − N εA , (3)
TABLE IV. Surface formation energies for Al, Si, Mg, Cu, andFe. The units are mJ/m . The second column indicates ifthe structure was relaxed. Comparisons with other classicalMD (CMD), DFT, and experimental values for polycrystallinesurfaces and Si facets are also given.Surface Rlx DFT MEAM CMD DFT EXPAl Present Ref. Ref.
Ref. Ref. (111) No 780 820 913 1199 1143(111) Yes 780 752 629 912(110) No 990 1154 1113 1271(110) Yes 960 1135 948 1107(100) No 890 1121 1012 1347(100) Yes 890 1088 848 1002Si Present Ref. Ref. Ref. Ref. (111) No 1620 1254 1405 1820(111) Yes 1570 1196 1405 1740 1230(100) No 2140 1850 2434 2390(100) Yes 2140 1743 1489 2390(100) 2 × Ref.
Ref. Ref. (0001) No 530 780 500 792 785(0001) Yes 530 713 310 499(10¯10) No 850 878 629 782(10¯10) Yes 850 859 316 618Cu Present Ref. Ref.
Ref. Ref. (111) No 1290 1411 1185 919 1952 1825(111) Yes 1290 1411 1181 903(110) No 1550 1645 1427 1177 2237(110) Yes 1510 1614 1412 1153(100) No 1440 1654 1291 1097 2166(100) Yes 1430 1653 1288 1083Fe Present Ref. Ref.
Ref. Ref. (111) No 2760 1366 1941 2012 2660 2475(111) Yes 2700 1306 1863 1998 2580(110) No 2420 1378 1434 1651 2380(110) Yes 2420 1372 1429 1651 2370(100) No 2500 1233 1703 1790 2480(100) Yes 2480 1222 1690 1785 2470 a Value from the given reference. b Calculated using parameters from the given reference. where E tot [ N ] is the total energy of the structure withtwo surfaces, N is the number of atoms in the structure, ε is the total energy per atom in the bulk, and A is thetotal area of both surfaces. Table IV shows the surfaceformation energies of several surfaces constructed fromfcc Al, hcp Mg, fcc Cu, and bcc Fe crystals. Resultsfrom the present MEAM potentials agree, in the orderof magnitude, with the DFT calculations, except for Fevalues being underestimated.The surfaces with lowest energy without reconstructionare identified correctly by the present MEAM potentials.The 2 × .Note that surface formation energies from the presentPAW GGA calculations are lower than our previouslypublished results using ultrasoft pseudopotentials withinlocal density (LDA) approximation—it is known thatGGA leads to surface energies which are 7–16% lowerthan LDA values for jellium and 16–29% lower than theexperimental results . A procedure and new DFTfunctionals were suggested to correct the errors ofLDA and GGA approximations. Similar correction canbe applied to vacancy formation energies , but suchcorrections were not applied in the present study. E. Stacking faults
Using an assumption of a planar dislocation core, thePeierls-Nabarro model is a powerful theory to quan-tify the dislocation core properties. In that model, a dis-location is defined by a continuous distribution of shearalong the glide plane, and the restoring force acting be-tween atoms on either sides of the interface is balancedby the resultant stress of the distribution. As shownin the recent study of Carrez et al. , a solution of thePeierls-Nabarro model can be obtained numerically byidentifying the restoring force to the gradient of the gener-alized stacking fault energy (GSFE) curve . In addition,Van Swygenhoven et al. claimed that the nature of slipin nanocrystalline metals cannot be described in termsof an absolute value of the stacking fault energy, and acorrect interpretation requires the GSFE curve, whichshows the change in energy per unit area of the crystalas a function of the displacement varied on the slip plane.However, the GSFE curve is not experimentally accessible.Therefore, to model dislocation properties reliably, theGSFE curve calculated with the MEAM potential mustreproduce the DFT data.The stacking fault energy per unit area of a stackingfault E sff is defined as E sff = E tot [ N ] − N εA , (4)where E tot [ N ] is the total energy of the structure with astacking fault, N is the number of atoms in the structure, ε is the total energy per atom in the bulk, and A is thetotal area of surface.As a validation test of the MEAM potential, the GSFEcurves obtained by molecular statics (MS) were comparedwith the DFT data by Zimmerman et al. for Al and Cu,by the present authors for Fe, and by Datta et al. forMg. After lowering the C min parameter to 0.8, the GSFEcurves calculated by MS using the MEAM potential forAl, Cu, and Mg show the skewed sinusoidal shape inagreement with the DFT predictions (Fig. 2) illustratingreasonable agreement with the DFT GSFE curves. IV. MEAM PARAMETERS FOR ELEMENTPAIRS
The MEAM potential parameters for each element pairwere initialized to match the ab-initio heat of formation,equilibrium volume, bulk modulus, and elastic moduliin the hypothetical NaCl reference structure, which waschosen for its simplicity. Since the equilibrium volume,cohesive energy, and bulk modulus of the NaCl struc-ture are directly related to MEAM parameters, they canbe reproduced exactly. An improved agreement of theshear moduli from MEAM and ab-initio simulations wasachieved in some cases by adjusting the electron densityscaling factor ρ . Then, heat of formation of binary com-pounds in a variety of crystal structures from MEAMwere examined and compared with the ab-initio results.To correlate the MEAM results with the lowest forma-tion energies of the compounds from DFT calculations,the MEAM screening and (cid:52) H XYB1 parameters for elementpairs were adjusted. The final MEAM parameters aregiven in Table V. The predicted MEAM properties forthe NaCl reference structure are compared with DFTresults in Table VI, and show that in general the MEAMheat of formation, bulk modulus, and equilibrium volumereproduce the DFT results well. In contrast, the shearelastic constants are not well reproduced. In fact the signof the shear elastic constant, representing crystal stability,is frequently in disagreement with the DFT results. Thisis really not a significant problem as the NaCl structuredoes not exist in nature. A more important criteria forsuccess of these potentials is how they perform for lowerenergy crystal structures. We address this issue in thenext section.
A. Heat of formation for binary compounds
The alloy phases that the MEAM potential predicts asmost likely to form at the temperature T = 0 K are thosewith the lowest heat of formation per atom, ∆ H , whichis defined as∆ H = E tot [ N X + N Y ] − N X ε X − N Y ε Y N X + N Y , (5) E tot is the total energy of the simulation cell, N X and N Y are the numbers of type-X and type-Y atoms in the cell, ε X and ε Y are the total energies per atom for type-X andtype-Y in their ground state bulk structures, respectively.To check the validity of our new potentials, we com-puted the heat of formation per atom for many inter-metallic phases of all alloy pairs. The total energy valuesin Eq. (5) for B1, B2, B3, C1, C15, D0 , A15, L1 , andother relevant structures were evaluated at the optimalatomic volume for each structure. Heat of formation forbasic binary compounds based on the new MEAM poten-tial and DFT results were calculated and compared withexperimental values (Figures 3–5). The DFT and MEAM E s ff [ m J / m ] u / b p where b p =[112]Al 050100 0 0.2 0.4 0.6 0.8 1 u / b p where b p =[10 ¯10] Mg 050100150200250 0 0.2 0.4 0.6 0.8 1 u / b p where b p =[112]Cu 02004006008001000 0 0.2 0.4 0.6 0.8 1 u / b p where b p =[111]FeDFTMEAM DFT 6lDFT 6ldsMEAM DFTMEAM DFTMEAM FIG. 2. GSFE curves for Al, Mg, Cu, and Fe obtained with the MEAM potential and compared with the DFT data.TABLE V. The MEAM potential parameters for element pairs. (cid:52) H XYB1 is the heat of formation of the NaCl structure (reference)with the type-X and type-Y elements relative to the energies of elemental X and Y in their equilibrium reference state, r e is their equilibrium nearest neighbor distance, α is the exponential decay factor for the universal energy, C max and C min arescreening parameters ( C XYX denotes type-Y element between two type-X elements). Non-zero parameters δ r = δ a = 0 . (cid:52) H XYB1 [eV] r XYe [˚A] α XY C XYXmin C XYXmax C YXYmin C YXYmax C XXYmin C XXYmax C XYYmin C XYYmax
Al Si 0.28 2.62 4.56 0.5 2.8 2.0 2.8 2.0 2.8 2.0 2.8Al Mg 0.23 2.87 4.52 2.0 2.8 0.0 2.8 2.0 2.8 0.0 2.8Al Cu 0.19 2.53 4.65 0.0 2.8 2.0 2.8 2.0 2.8 2.0 2.8Al Fe 0.26 2.45 4.64 0.9 2.8 0.1 2.8 2.0 2.8 2.0 2.8Si Mg 0.20 2.75 4.73 1.0 2.8 1.0 2.8 2.0 2.8 2.0 2.8Si Cu 0.14 2.46 4.74 0.0 2.8 0.0 2.8 2.0 2.8 2.0 2.8Si Fe -0.07 2.39 5.17 1.0 2.8 1.0 2.8 2.0 2.8 0.0 2.8Mg Cu 0.23 2.63 4.70 2.0 2.8 0.0 2.8 2.0 2.8 2.0 2.8Mg Fe 0.60 2.61 4.96 0.65 2.8 0.0 2.8 2.0 2.8 2.0 2.8Cu Fe 0.63 2.42 5.21 2.0 2.8 0.0 2.8 2.0 2.8 2.0 2.8 results for the phases with lowest ∆ H are also shown inTables VIII–IX.The agreement between MEAM and DFT is quite sat-isfactory. In most cases, the MEAM results preserve theorder of stability predicted by the DFT results. Thedifferences in the heat of formation per atom from theMEAM and DFT results are less than 0.5 eV at most. Ingeneral the atomic volumes predicted by MEAM agree atleast qualitatively with the DFT and experimental results.The MEAM calculations of the bulk moduli also agreesemi-quantitatively with DFT and experimental results,usually within 20%. Predicted shear moduli usually fol-low the DFT and experimental results, but in some casesthere is significant disagreement. B. Substitutions
The formation energy of a substitutional point defect E subf , in the case of the substitution of a type-X atom ofthe host with a type-Y atom, is defined by E subf = E tot [( N −
1) + 1] − ( N − ε X − ε Y (6) where E tot [( N −
1) + 1] is the total energy of a systemof N − ε X and ε Y are the total energies per atom for type-X and type-Yatoms in their ground state bulk structures. Table VIIshows the results of substitutional defect calculations us-ing the MEAM potentials and the DFT results. In generalthe MEAM results qualitatively agree with the DFT re-sults. In a number of cases of small heat of formation,MEAM indicates a small heat, but of the incorrect sign.The most significant error is for Al in Si where MEAMpredicts a large endothermic heat and DFT predicts amuch smaller value, otherwise there is general agreement. C. Finite temperature tests
Real life applications of MD potentials require extensivetesting at finite temperatures. Basic finite temperaturetests of the potentials, in accord with recommendationsof Lee et al. , revealed formation of an unknown solidstructure when the temperature of fcc Al crystal was ∆ H [ e V /a t o m ] at. % Si in AlSiAlSi 00.5 20 30 40 50 60 70 80at. % Mg in AlMgAlMg00.5 20 30 40 50 60 70 80 ∆ H [ e V /a t o m ] at. % Cu in AlCuAlCu -0.19-0.18-0.17-0.16 70 75 -0.500.511.5 20 30 40 50 60 70 80at. % Fe in AlFeAlFeD0 L1 C1A15 B2C15L1 C1D0 C15B1 A15B3B1 D0 L1 C1A15 B2C15L1 C1D0 C15B1 A15B3 A12 D0 L1 C1A15 B2C15L1 C1D0 C15B1 A15B3 DFTL1 Refs. DFTA12 D0 L1 C1A15 B2C15L1 C1D0 C15B1 A15B3 MEAMEXPTD0 L1 C1A15 B2C15L1 C1D0 C15B1 A15B3C1C16C16C16 D0 D0 C16 D0 L1 C1A15 B2C15L1 C1D0 C15B1 A15D0 B3C16C16 D0 L1 D0 D0 D0 D0 L1 C1A15 B2C15L1 C1D0 C15B1C11 b A15B3B2B2C11 b D0 D0 L1 L1 D0 L1 C1A15 B2C15L1 C1D0 C15B1C11 b A15B3B2B2 D0 D0 FIG. 3. (Color online) Heat of formation of AlSi, AlMg, AlCu, AlFe binary compounds from MEAM, DFT, and experiments.References: AlSi , AlMg , AlCu , AlFe . DFT points are labeled on the left, MEAM and experimental on the right.Values for the most stable compounds are also shown in Table VIII. The inside plot is a magnified portion of a larger plot. ∆ H [ e V /a t o m ] at. % Mg in SiMgSiMg 00.51 20 30 40 50 60 70 80at. % Cu in SiCuSiCu 00.010.020.030.04 70 75-0.500.5 20 30 40 50 60 70 80 ∆ H [ e V /a t o m ] at. % Fe in SiFeSiFe -0.27-0.26-0.25-0.24-0.23-0.22-0.21 70 75 00.5 20 30 40 50 60 70 80at. % Cu in MgCuMgCuD0 L1 C1A15 B2C15L1 C1D0 C15B1 A15B3 C1 D0 L1 C1A15 B2C15L1 C1D0 C15B1 A15B3 C1 D0 L1 C1A15 B2C15L1 C1D0 C15B1 A15B3 DFT L1 Refs. DFT D0 L1 C1A15 B2C15L1 C1D0 C15B1 A15B3 MEAM L1 EXPTD0 A15DFT L1 Refs. DFT L1 MEAMC16 D0 L1 C1A15 B2C15L1 C1D0 B20 C15B1 A15B3B20C1C16 D0 D0 L1 C16 D0 L1 C1A15 B2C15L1 C1D0 B20 C15B1 A15D0 B3B20 D0 L1 A15D0 L1 D0 A15D0 D0 D0 L1 C1A15 B2C15L1 C1D0 C15B1 A15B3C15 C15C15C15C b D0 L1 C1A15 B2C15L1 C1D0 C15B1 A15B3 C15C15C b FIG. 4. (Color online) Heat of formation of SiMg, SiCu, SiFe, MgCu binary compounds from MEAM, DFT, and experiments.References: SiMg , SiCu , SiFe , and MgCu . DFT points are labeled on the left, MEAM and experimental onthe right. Values for the most stable compounds are also shown in Table IX. The inside plot is a magnified portion of a largerplot. ∆ H [ e V /a t o m ] at. % Fe in MgFeMgFe 00.511.5 20 30 40 50 60 70 80at. % Fe in CuFeCuFeL1 C1A15 B2C15L1 C1D0 C15B1 A15B3B2 D0 L1 C1A15 B2C15L1 C1D0 C15B1 A15B3 D0 L1 C1A15 B2C15L1 C1D0 C15B1 A15B3 DFTB2L1 Refs. DFT D0 L1 C1A15 B2C15L1 C1D0 C15B1 A15B3 MEAM
FIG. 5. (Color online) Heat of formation of MgFe and CuFe binary compounds from MEAM and DFT. References: MgFe ,CuFe . DFT points are labeled on the left, MEAM on the right. Values for the most stable compounds are also shown inTable IX.TABLE VI. Structural and elastic properties of element pairsin the reference NaCl (B1) crystal structure from DFT andMEAM calculations. ∆ H is the heat of formation in eV/atom, V is the volume per atom in ˚A . Elastic constants B , C ,and ( C − C ) / pair method ∆ H V B C C − C AlSi DFT 0.28 17.9 76.7 10 76MEAM 0.28 18.0 76.4 -13 8AlMg DFT 0.42 23.7 30.9 -18 36MEAM 0.23 23.6 33.9 -3 35AlCu DFT 0.19 16.1 77.5 -18 52MEAM 0.19 16.2 77.4 -19 56AlFe DFT 0.36 14.7 90.3 -25 105MEAM 0.26 14.7 92.7 -27 109SiMg DFT 0.41 20.9 50.6 -26 48MEAM 0.20 20.8 54.9 9 61SiCu DFT 0.39 14.9 99.0 -29 58MEAM 0.14 14.9 105.9 9 223SiFe DFT 0.25 12.9 100.9 -70 112MEAM -0.07 13.7 157.9 65 363MgCu DFT 0.23 18.5 48.7 -10 49MEAM 0.23 18.2 49.6 -1 61MgFe DFT 0.86 17.7 50.4 -23 83MEAM 0.60 17.8 56.5 -17 62CuFe DFT 0.78 14.1 107.4 -23 134MEAM 0.63 14.2 111.8 10 131
TABLE VII. The formation energies of substitutional pointdefects in Al, Si, Mg, Cu, and Fe. All energy values are givenin eV. DFT values are given in parentheses.
Host Substitute atomAl Si Mg Cu FeAl 0.5 (0.5) − − − − − − − − − − − − − − increased to 800 K under zero pressure conditions. Toprevent formation of this structure, β (1) and t (1) parame-ters of Al were adjusted. Heating of other elements underzero pressure conditions did not result in forming newstructures.To test a system including all components of the newpotential, an 20–100 ◦ C average thermal expansion coeffi-cient of a model system with the composition similar toAA 6061 alloy (Table X) was evaluated and compared withexperimental data. Atoms of constituents were placed inthe substitutional positions of a 20x20x20 fcc Al cell. Thesystem was heated from -200 ◦ C to 20 ◦ C (and 100 ◦ C)over the interval of 0.1 ns, and then equilibrated at 20 ◦ C1 TABLE VIII. Structural and elastic properties of element pairs in varying crystal structures from the present DFT and MEAMcalculations compared with references and measured values. ∆ H is the heat of formation in meV/atom, V is the volume peratom in ˚A , and elastic constants B , C , and ( C − C ) / compos. str. met. ∆ H V B C C − C Al Si L1 DFT 121 16.04 74.3 24.1 9.4MEAM/EAM 113 16.67 96.7 31.2 31.2Al Si C1 DFT 178 18.78 62.9 25.4 − − − Mg A12 DFT − − , − , − ,20.04 , 20.25 MEAM/EAM 49, − , − EXP − , 20.30 Al Mg L1 DFT − − , − , 17.78 − − L1 DFT − − , − , 19.89 −
46, 21 , 51 , 19.48 − −
32, 90 Cu C16 DFT − − , − , − −
75 15.80EXP − C1 DFT − − , − −
69 17.31 77.4 43.2 7.7EXP 15.63 AlCu D0 DFT − − DFT − − − − DFT − − −
136 12.77 124.7 34.5 102.8MEAM/EAM 161 13.48 111.6 200.5 92.8Al Cu L1 DFT −
40 15.22 89.9 23.4 61.5MEAM/EAM −
284 14.58 106.0 40.3 41.2AlCu B2 DFT −
139 13.45 108.6 31.4 − −
198 14.15 109.2 56.7 2.8AlFe B2 DFT − − , − , − , − , − , − , − , − , 12.07 ,11.33 , 11.22 ,11.65 , 11.89 ,11.93 , 12.19 ,183.0 ,155.0 ,156.0 ,107.0 ,38.1 MEAM/EAM − − , − , − ,13.92 , 11.45 ,138.0 ,193.0 ,110.7 ,7.9 EXP − , − , − , − ,136.0 ,127.1 , 33.7 AlFe D0 DFT − − , − , − , − , − , − , − , 12.01 ,14.65 , 11.82 ,12.09 , 11.57 ,151.0 ,170.0 MEAM/EAM 346, − , − , − ,12.86 , 10.80 ,229.0 ,126.3 ,12.6 EXP − , − L1 DFT − − , − , − , − ,14.14 , 12.35 ,168.0 − −
161 12.08 156.9 67.3 135.1MEAM/EAM 205 12.59 166.7 35.5 153.5AlFe C15 DFT − − Fe A15 DFT −
161 13.91 121.5 67.7 120.0MEAM/EAM 321 15.03 103.5 1.8 66.6L1 DFT − − , − , 13.69 −
49 14.83 108.5 59.4 20.3D0 DFT − − , − , 13.35 − − Fe C11 b DFT − − , 12.80 −
72 15.25 98.6 76.8 55.0MEAM/EAM −
76 16.12 90.4 47.5 36.4 TABLE IX. Structural and elastic properties of element pairs in varying crystal structures from the present DFT and MEAMcalculations compared with references and measured values. ∆ H is the heat of formation in meV/atom, V is the volume peratom in ˚A , and elastic constants B , C , and ( C − C ) / compos. str. met. ∆ H V B C C − C SiMg C1 DFT − − − SiMg L1 DFT −
11 19.29 50.8 29.9 37.1MEAM/EAM 24 20.70 57.5 23.5 21.8A15 DFT 69 20.09 44.1 9.3 31.8MEAM/EAM −
14 21.25 56.3 32.2 33.9SiCu L1 DFT −
22, 35
SiCu C1 DFT 60 14.26 111.9 76.3 23.1MEAM/EAM −
41 14.81 102.8 97.9 16.2SiFe B20 DFT − − , − −
132 13.11EXP − B2 DFT −
457 10.55 231.9 87.0 155.8MEAM/EAM −
222 13.09 177.7 36.2 225.3SiFe D0 DFT − − , − −
269 12.03 169.2 91.6 36.6EXP − A15 DFT −
251 11.44 173.3 72.3 125.3MEAM/EAM −
232 12.28 190.1 47.6 119.9L1 DFT − − −
149 11.80 188.6 65.1 135.8D0 DFT − − −
216 12.05Si Fe C1 DFT − − −
140 15.53 158.7 95.7 41.3C16 DFT − − C1 DFT −
12 12.95 159.6 82.6 − −
516 12.95 182.4 154.6 226.8Mg Cu C b DFT − − , − , − − MgCu C15 DFT − − , − , − −
140 14.50 104.7 21.5 − − , − MgCu B2 DFT −
117 15.65 69.3 60.3 16.9MEAM/EAM −
28 15.56 73.6 51.1 − L1 DFT −
71 13.49 96.6 62.6 21.5MEAM/EAM −
25 13.81 103.9 42.5 29.0D0 DFT −
54 13.52 96.5 75.2 0.3MEAM/EAM 107 13.43 95.7 64.1 − C15 DFT 62 13.58 91.4 72.0 53.4MEAM/EAM 471 14.79 95.8 322.9 605.7MgFe L1 DFT 181 13.11 122.6 96.9 14.0MEAM/EAM 243 13.65 116.4 64.5 45.5Mg Fe L1 DFT 204 18.29 51.6 52.1 18.9MEAM/EAM 249 19.08 54.5 30.0 23.1MgFe B2 DFT 347, 357 − L1 DFT 133 11.73 132.5 99.9 6.8MEAM/EAM 267 11.78 172.6 80.5 68.0A15 DFT 178 11.95 137.6 54.2 134.7MEAM/EAM 129 12.26 192.3 70.5 155.2Cu Fe D0 DFT 175 12.10 139.2 105.5 − DFT 224, 342 − − TABLE X. Composition limits of AA 6061 alloy and amodel system used to estimate thermal expansion coefficient.Limits Model [wt. %]Element low [wt. %] high [wt. %]Si 0.40 0.8 0.51Mg 0.8 1.2 1.00Cu 0.15 0.40 0.30Fe no 0.7 0.50Mn no 0.15 0.00Cr 0.04 0.35 0.00Zn no 0.25 0.00Ti no 0.15 0.00TABLE XI. Thermal expansion coefficient of single crystal Aland AA 6061 alloy between 20 ◦ C and 100 ◦ C in the units of µ m/m/K. .CMD Exppresent Ref. Ref.
Ref.
Al fcc 14.4 15-25 25.4 23.6AA 6061 14.6 23.6 (and 100 ◦ C) for 1 ns under zero pressure conditions.Table XI shows the values of 20–100 ◦ C average ther-mal expansion coefficients. The MEAM result for singlecrystal Al is in the lower range of other MD potentialsand experiments. Since Al is a dominant element ofthe AA 6061 alloy, the thermal expansion coefficient foralloy is similarly underestimated, possibly also due toimperfections of the structure of the real material.
V. CONCLUSIONS
In this study we developed MEAM potentials for thepair combinations of aluminum, silicon, magnesium, cop-per, and iron. The MEAM formalism allows any of thesepotentials to be combined to enable prediction of multi-component alloy properties. These potentials reproduce alarge body of elemental and binary properties from DFTcalculations at the temperature of 0 K and experimentalresults. Basic finite temperature tests of the single ele-ment potentials and their alloy combinations were alsoperformed. With focus to facilitate reproducibility ofthe presented results , and subject to further testingand improvements, these potentials are one step towardsdesigning multi-component alloys by simulations. ACKNOWLEDGMENT
The authors are grateful to the Center for AdvancedVehicular Systems at Mississippi State University for sup- porting this study. Computer time allocation has beenprovided by the High Performance Computing Collabora-tory (HPC ) at Mississippi State University. Computa-tional package lammps with ASE interface was usedto perform MD simulations. Much appreciated tests of thenew MEAM potentials, including the high temperaturesimulations of Al that revealed formation of unknown Alphase at 800 K, were performed by Chandler Becker andTanner Hamann at the Metallurgy Division of the MaterialMeasurement Laboratory, National Institute of Standardsand Technology (NIST). Comparison of ab-initio elasticconstants and related discussion with Hannes Schweigerfrom Materials Design are also appreciated. Classical MDpotentials from other authors examined in this study weredownloaded from the Interatomic Potentials RepositoryProject database .Sandia National Laboratories is a multi-program lab-oratory managed and operated by Sandia Corporation,a wholly owned subsidiary of Lockheed Martin Corpora-tion, for the U.S. Department of Energy’s National Nu-clear Security Administration under contract DE-AC04-94AL85000. Appendix A: MEAM theory
The total energy E of a system of atoms in the MEAM is approximated as the sum of the atomic energies E = (cid:88) i E i . (A1)The energy of atom i consists of the embedding energyand the pair potential terms: E i = F i (¯ ρ i ) + 12 (cid:88) j (cid:54) = i φ ij ( r ij ) . (A2) F is the embedding function, ¯ ρ i is the background electrondensity at the site of atom i , and φ ij ( r ij ) is the pairpotential between atoms i and j separated by a distance r ij . The embedding energy F i (¯ ρ i ) represents the energycost to insert atom i at a site where the backgroundelectron density is ¯ ρ i . The embedding energy is given inthe form F i (¯ ρ i ) = (cid:40) A i E i ¯ ρ i ln (¯ ρ i ) if ¯ ρ i ≥ , − A i E i ¯ ρ i if ¯ ρ i < , (A3)where the sublimation energy E i and parameter A i de-pend on the element type of atom i . The backgroundelectron density ¯ ρ i is given by¯ ρ i = ρ (0) i ρ i G (Γ i ) , (A4)where Γ i = (cid:88) k =1 t ( k ) i (cid:32) ρ ( k ) i ρ (0) i (cid:33) (A5)4and G (Γ) = (cid:40) √ ≥ − , − (cid:112) | | if Γ < − . (A6)The zeroth and higher order densities, ρ (0) i , ρ (1) i , ρ (2) i , and ρ (3) i are given in Eqs. (A9). The composition-dependentelectron density scaling ρ i is given by ρ i = ρ i Z i G (cid:0) Γ ref i (cid:1) , (A7)where ρ i is an element-dependent density scaling, Z i isthe first nearest-neighbor coordination of the referencesystem, and Γ ref i is given byΓ ref i = 1 Z i (cid:88) k =1 t ( k ) i s ( k ) i , (A8)where s ( k ) i is the shape factor that depends on the ref-erence structure for atom i . Shape factors for variousstructures are specified in the work of Baskes . Thepartial electron densities are given by ρ (0) i = (cid:88) j (cid:54) = i ρ a (0) j ( r ij ) S ij (A9a) (cid:16) ρ (1) i (cid:17) = (cid:88) α (cid:88) j (cid:54) = i ρ a (1) j r ijα r ij S ij (A9b) (cid:16) ρ (2) i (cid:17) = (cid:88) α,β (cid:88) j (cid:54) = i ρ a (2) j r ijα r ijβ r ij S ij − (cid:88) j (cid:54) = i ρ a (2) j ( r ij ) S ij (A9c) (cid:16) ρ (3) i (cid:17) = (cid:88) α,β,γ (cid:88) j (cid:54) = i ρ a (3) j r ijα r ijβ r ijγ r ij S ij − (cid:88) α (cid:88) j (cid:54) = i ρ a (3) j r ijα r ij S ij , (A9d)where r ijα is the α component of the displacement vectorfrom atom i to atom j . S ij is the screening functionbetween atoms i and j and is defined in Eqs. (A16). Theatomic electron densities are computed as ρ a ( k ) i ( r ij ) = ρ i exp (cid:20) − β ( k ) i (cid:18) r ij r i − (cid:19)(cid:21) , (A10)where r i is the nearest-neighbor distance in the single-element reference structure and β ( k ) i is element-dependentparameter. Finally, the average weighting factors aregiven by t ( k ) i = (cid:80) j (cid:54) = i t ( k )0 ,j ρ a (0) j S ij (cid:80) j (cid:54) = i (cid:16) t ( k )0 ,j (cid:17) ρ a (0) j S ij , (A11) where t ( k )0 ,j is an element-dependent parameter.The pair potential is given by φ ij ( r ij ) = ¯ φ ij ( r ij ) S ij (A12)¯ φ ij ( r ij ) = 1 Z ij (cid:20) E uij ( r ij ) − F i (cid:18) Z ij Z i ρ a (0) j ( r ij ) (cid:19) − F j (cid:18) Z ij Z j ρ a (0) j ( r ij ) (cid:19)(cid:21) (A13) E uij ( r ij ) = − E ij (cid:0) a ∗ ij ( r ij ) (cid:1) e − a ∗ ij ( r ij ) (A14) a ∗ ij = α ij (cid:32) r ij r ij − (cid:33) , (A15)where E ij , α ij and r ij are element-dependent parametersand Z ij depends upon the structure of the reference sys-tem. The background densities ˆ ρ i ( r ij ) in Eq. (A13) arethe densities for the reference structure computed withinteratomic spacing r ij .The screening function S ij is designed so that S ij = 1if atoms i and j are unscreened and within the cutoffradius r c , and S ij = 0 if they are completely screened oroutside the cutoff radius. It varies smoothly between 0and 1 for partial screening. The total screening functionis the product of a radial cutoff function and three bodyterms involving all other atoms in the system: S ij = ¯ S ij f c (cid:18) r c − r ij ∆ r (cid:19) (A16a)¯ S ij = (cid:89) k (cid:54) = i,j S ikj (A16b) S ikj = f c (cid:18) C ikj − C min ,ikj C max ,ikj − C min ,ikj (cid:19) (A16c) C ikj = 1 + 2 r ij r ik + r ij r jk − r ij r ij − (cid:16) r ik − r jk (cid:17) (A16d) f c ( x ) = x ≥ (cid:2) − (cid:0) − x ) (cid:1)(cid:3) < x < x ≤ C min and C max can be defined separately foreach i - j - k triplet, based on their element types. Theparameter ∆ r controls the distance over which the radialcutoff is smoothed from 1 to 0 near r = r c . Appendix B: Equilibrium lattice parameter and bulkmodulus
MEAM postulates the Rose universal equation ofstate E R ( a ∗ ) = − E c (cid:18) a ∗ + δ αa ∗ α + a ∗ (cid:19) e − a ∗ (B1)for the reference structure of each single element andfor each element pair. The a ∗ , scaled distance from the5equilibrium nearest neighbor position r , is a ∗ = α ( r/r − . (B2)Two δ parameters may be specified for each element/pair: δ r for negative, and δ a for positive a ∗ . Then δ = (cid:40) δ r for a ∗ < δ a for a ∗ ≥ . (B3)The MEAM potential parameter α is related to the equi-librium atomic volume Ω , the bulk modulus B , and thecohesive energy of the reference structure E c as follows α = (cid:114) B Ω E c . (B4)The DFT equilibrium energies and bulk moduli wereobtained by fitting energy-volume dependence to Mur-naghan equation of state E ( V ) = E ( V ) (B5)+ B VB (cid:48) ( B (cid:48) − (cid:34) B (cid:48) (cid:18) − V V (cid:19) + (cid:18) V V (cid:19) B (cid:48) − (cid:35) . Appendix C: Trigonal and tetragonal shear modulus
For small deformations of a cubic crystal, the changeof energy density due to straining is (cid:52) E V = 12 C (cid:0) (cid:15) + (cid:15) + (cid:15) (cid:1) + C ( (cid:15) (cid:15) + (cid:15) (cid:15) + (cid:15) (cid:15) )+ 12 C (cid:0) (cid:15) + (cid:15) + (cid:15) (cid:1) + O ( (cid:15) i ) , (C1)where (cid:15) i are strains in modified Voigt notation.The trigonal shear modulus C was determined fromrhombohedral deformation given by (cid:15) = (cid:15) = (cid:15) = 0 and (cid:15) = (cid:15) = (cid:15) = δ in C1, leading to (cid:52) E V ( δ ) = 32 C δ + O ( δ ) . (C2)The tetragonal shear modulus ( C − C ) / (cid:15) = δ, (cid:15) = δ − (cid:52) E V ( δ ) = ( C − C ) δ + O ( δ ) . (C3)The C2 and C3 were used to estimate tetragonal andtrigonal shear moduli. ∗ Presently at Institute of Mechanics and Fluid Dynam-ics, TU Bergakademie Freiberg, Lampadiusstr. 4, 09596Freiberg, Germany † Also at Department of Mechanical Engineering, Missis-sippi State University, Mississippi State, MS 39762 ‡ Also at Center for Computational Sciences, MississippiState University, Mississippi State, MS 39762 § Also at Los Alamos National Laboratory, MST-8, MSG755, Los Alamos, NM 87545 M. F. Horstemeyer, in
Pract. Aspects Comput. Chem. ,edited by J. Leszczynski and M. K. Shukla (SpringerNetherlands, 2010) pp. 87–135. M. S. Daw and M. I. Baskes, Phys. Rev. Lett. , 1285(1983); Phys. Rev. B , 6443 (1984); M. I. Baskes, Phys.Rev. Lett. , 2666 (1987); M. S. Daw, Phys. Rev. B ,7441 (1989). M. I. Baskes, J. S. Nelson, and A. F. Wright, Phys. Rev.B , 6085 (1989). M. I. Baskes, Phys. Rev. B , 2727 (1992). M. I. Baskes, J. E. Angelo, and C. L. Bisson, Modell.Simul. Mater. Sci. Eng. , 505 (1994). K. Gall, M. F. Horstemeyer, M. van Schilfgaarde, andM. I. Baskes, J. Mech. Phys. Solids. , 2183 (2000). B.-J. Lee and M. I. Baskes, Phys. Rev. B , 8564 (2000). H. Huang, N. M. Ghoniem, J. K. Wong, and M. I. Baskes,Modell. Simul. Mater. Sci. Eng. , 615 (1995). M. I. Baskes, Mater. Chem. Phys. , 152 (1997). M. I. Baskes, Mater. Sci. Eng. A , 165 (1999). B.-J. Lee, J.-H. Shim, and M. I. Baskes, Phys. Rev. B ,144112 (2003). W. Hu, B. Zhang, B. Huang, F. Gao, and D. J. Bacon, J.Phys.: Condens. Matter , 1193 (2001). W. Hu, H. Deng, X. Yuan, and M. Fukumoto, Eur. Phys.J. B , 429 (2003). B.-J. Lee and J. W. Lee, Calphad , 7 (2005). B.-J. Lee, W.-S. Ko, H.-K. Kim, and E.-H. Kim, Calphad , 510 (2010). K. Kang and W. Cai, Philos. Mag. , 2169 (2007). T. S. Gates, G. M. Odegard, S. J. V. Frankland, and T. C.Clancy, Compos. Sci. Technol. , 2416 (2005). S. J. Noronha and D. Farkas, Phys. Rev. B , 132103(2002). E. Mart´ınez, J. Marian, A. Arsensil, M. Victoria, andJ. Perlado, J. Mech. Phys. Solids. , 869 (2008). G. P. Potirniche, M. F. Horstemeyer, G. J. Wagner, andP. M. Gullett, Int. J. Plast. , 257 (2006). H. Van Swygenhoven, P. M. Derlet, and A. G. Frøseth,Nat. Mater. , 399 (2004). R. A. Johnson and D. J. Oh, J. Mater. Res. , 1195 (1989). R. A. Johnson, Phys. Rev. B , 12554 (1989). X. W. Zhou, H. N. G. Wadley, R. A. Johnson, D. J. Larson,N. Tabat, A. Cerezo, A. K. Petford-Long, G. D. W. Smith,P. H. Clifton, R. L. Martens, and T. F. Kelly, Acta Mater. , 4005 (2001). X. W. Zhou, R. A. Johnson, and H. N. G. Wadley, Phys.Rev. B , 144113 (2004). Y. Mishin, M. J. Mehl, and D. A. Papaconstantopoulos,Acta Mater. , 4029 (2005). M. I. Mendelev, M. Asta, M. J. Rahman, and J. J. Hoyt,Philos. Mag. , 3269 (2009). X. Y. Liu and J. B. Adams, Acta Mater. , 3467 (1998). F. Apostol and Y. Mishin, Phys. Rev. B , 054116 (2011). X.-Y. Liu, C.-L. Liu, and L. J. Borucki, Acta Mater. ,3227 (1999). M. I. Mendelev, D. J. Srolovitz, G. J. Ackland, and S. Han,J. Mater. Res. , 208 (2005). T. Ohira, Y. Inoue, K. Murata, and J. Murayama, Appl.Surf. Sci. , 175 (2001); T. Ohira and Y. Inoue, in
MRSProceedings , Vol. 492 (1997) pp. 401–406. S. M. Valone, M. I. Baskes, and R. L. Martin, Phys. Rev.B , 214209 (2006). H.-K. Kim, W.-S. Jung, and B.-J. Lee, Acta Mater. ,3140 (2009). K.-H. Kang, I. Sa, J.-C. Lee, E. Fleury, and B.-J. Lee,Scr. Mater. , 801 (2009). E. C. Do, Y.-H. Shin, and B.-J. Lee, J. Phys.: Condens.Matter , 325801 (2009). H.-K. Kim, W.-S. Jung, and B.-J. Lee, J. Mater. Res. ,1288 (2010). K. Chenoweth, A. C. T. van Duin, and W. A. Goddard,J. Phys. Chem. A , 1040 (2008). J. E. Angelo, N. R. Moody, and M. I. Baskes, Modell.Simul. Mater. Sci. Eng. , 289 (1995); M. I. Baskes, X. Sha,J. E. Angelo, and N. R. Moody, Modell. Simul. Mater.Sci. Eng. , 651 (1997). Y. Q. Cheng, E. Ma, and H. W. Sheng, Phys. Rev. Lett. , 245501 (2009). G. Bonny, R. C. Pasianot, N. Castin, and L. Malerba,Philos. Mag. , 3531 (2009). B. Jelinek, “ASE Atomistic Potential Tests,” http://code.google.com/p/ase-atomistic-potential-tests (2011). P. E. Bl¨ochl, Phys. Rev. B , 17953 (1994); G. Kresseand D. Joubert, Phys. Rev. B , 1758 (1999). G. Kresse and J. Hafner, Phys. Rev. B , 558 (1993);G. Kresse and J. Furthm¨uller, Phys. Rev. B , 11169(1996). J. P. Perdew, J. A. Chevary, S. H. Vosko, K. A. Jackson,M. R. Pederson, D. J. Singh, and C. Fiolhais, Phys. Rev.B , 6671 (1992). H. J. Monkhorst and J. D. Pack, Phys. Rev. B , 5188(1976). B. Jelinek, J. Houze, S. Kim, M. F. Horstemeyer, M. I.Baskes, and S.-G. Kim, Phys. Rev. B , 054106 (2007). T. Lee, M. I. Baskes, S. M. Valone, and J. D. Doll,“Atomistic modeling of thermodynamic equilibrium andpolymorphism of iron,” (2011), Los Alamos Preprint:LA-UR 11-03286. X.-Y. Liu, P. Ohotnicky, J. Adams, C. Rohrer, and R. Hy-land, Surf. Sci. , 357 (1997). A. F. Wright, Phys. Rev. B , 165116 (2006). P. Tzanetakis, J. Hillairet, and G. Revel, Phys. StatusSolidi B , 433 (1976). K. M. Carling, G. Wahnstr¨om, T. R. Mattsson, N. Sand-berg, and G. Grimvall, Phys. Rev. B , 054101 (2003). T. Hehenkamp, J. Phys. Chem. Solids , 907 (1994). D. A. Andersson and S. I. Simak, Phys. Rev. B , 115108(2004). S. Dannefaer, P. Mascher, and D. Kerr, Phys. Rev. Lett. , 2195 (1986); J. Throwe, T. C. Leung, B. Nielsen,H. Huomo, and K. G. Lynn, Phys. Rev. B , 12037(1989). T. Hehenkamp, W. Berger, J.-E. Kluin, C. L¨udecke, andJ. Wolff, Phys. Rev. B , 1998 (1992). H. Krimmel and M. F¨ahnle, Phys. Rev. B , 5489 (2000). M. I. Mendelev, D. J. Srolovitz, G. J. Ackland, D. Y. Sun,and M. Asta, Philos. Mag. , 3977 (2003). G. J. Ackland, D. J. Bacon, A. F. Calder, and T. Harry,Philos. Mag. A , 713 (1997). C. Domain and C. S. Becquart, Phys. Rev. B , 024103(2001). H.-E. Schaefer, K. Maire, M. Weller, D. Herlach, A. Seeger,and J. Diehl, Scr. Metall. , 803 (1977). M. I. Mendelev, M. J. Kramer, C. A. Becker, and M. Asta,Philos. Mag. , 1723 (2008). Y. Mishin, M. J. Mehl, D. A. Papaconstantopoulos, A. F.Voter, and J. D. Kress, Phys. Rev. B , 224106 (2001). M. Timonova, B.-J. Lee, and B. J. Thijsse, Nucl. Instrum.Methods Phys. Res., Sect. B , 195 (2007). S. Ryu, C. R. Weinberger, M. I. Baskes, and W. Cai,Modell. Simul. Mater. Sci. Eng. , 075008 (2009). S. Ramos de Debiaggi, M. de Koning, and A. M. Monti,Phys. Rev. B , 104103 (2006). G. P. Purja Pun and Y. Mishin, Acta Mater. , 5531(2009). N. Sandberg, B. Magyari-K¨ope, and T. R. Mattsson, Phys.Rev. Lett. , 065901 (2002). B. A. Gillespie, X. W. Zhou, D. A. Murdick, H. N. G.Wadley, R. Drautz, and D. G. Pettifor, Phys. Rev. B ,155207 (2007). J. Tersoff, Phys. Rev. B , 9902 (1988). P. Erhart and K. Albe, Phys. Rev. B , 035211 (2005). O. K. Al-Mushadani and R. J. Needs, Phys. Rev. B ,235205 (2003). S.-G. Kim, M. F. Horstemeyer, M. I. Baskes, M. Rais-Rohani, S. Kim, B. Jelinek, J. Houze, A. Moitra, andL. Liyanage, J. Eng. Mater. Technol. , 041210 (2009). M. I. Mendelev, S. Han, D. J. Srolovitz, G. J. Ackland,D. Y. Sun, and M. Asta, Philos. Mag. , 3977 (2003). S. L. Dudarev and P. M. Derlet, J. Phys.: Condens. Matter , 7097 (2005). C.-C. Fu, F. Willaime, and P. Ordej´on, Phys. Rev. Lett. , 175503 (2004). B. J. Jesson, M. Foley, and P. A. Madden, Phys. Rev. B , 4941 (1997). L. Vitos, A. V. Ruban, H. L. Skriver, and J. Koll´ar, Surf.Sci. , 186 (1998). F. De Boer, R. Boom, W. Mattens, A. Miedema, andA. Niessen,
Cohesion in Metals: Transition Metal Alloys ,edited by F. R. de Boer and D. G. Pettifor, Vol. 1 (North-Holland, Amsterdam, 1988). J. H. Wilson, J. D. Todd, and A. P. Sutton, J. Phys.:Condens. Matter , 10259 (1990). M. Timonova and B. J. Thijsse, Modell. Simul. Mater. Sci.Eng. , 015003 (2011). A. A. Stekolnikov, J. Furthm¨uller, and F. Bechstedt, Phys.Rev. B , 115318 (2002). D. J. Eaglesham, A. E. White, L. C. Feldman, N. Moriya,and D. C. Jacobson, Phys. Rev. Lett. , 1643 (1993). X. Y. Liu, C. L. Liu, and L. J. Borucki, Acta Mater. ,3227 (1999). P. B(cid:32)lo´nski and A. Kiejna, Surf. Sci. , 123 (2007). H. Balamane, T. Halicioglu, and W. A. Tiller, Phys. Rev.B , 2250 (1992). A. E. Mattsson and W. Kohn, J. Chem. Phys. , 3441(2001). R. Armiento and A. E. Mattsson, Phys. Rev. B , 085108(2005). J. P. Perdew, A. Ruzsinszky, G. I. Csonka, O. A. Vydrov,G. E. Scuseria, L. A. Constantin, X. Zhou, and K. Burke,Phys. Rev. Lett. , 136406 (2008). A. E. Mattsson, R. Armiento, P. A. Schultz, and T. R.Mattsson, Phys. Rev. B , 195123 (2006). R. Peierls, Proc. Phys. Soc. London , 34 (1940). F. R. N. Nabarro, Proc. Phys. Soc. London , 256 (1947). P. Carrez, D. Ferr´e, and P. Cordier, Modell. Simul. Mater.Sci. Eng. , 035010 (2009). V. V´ıtek, Phys. Status Solidi , 687 (1966). J. A. Zimmerman, H. Gao, and F. F. Abraham, Modell.Simul. Mater. Sci. Eng. , 103 (2000). A. Datta, U. V. Waghmare, and U. Ramamurty, ActaMater. , 2531 (2008). M. Mihalkovic and M. Widom, “Alloy database,” http://euler.phys.cmu.edu/alloy/ (2009). C. Ravi, C. Wolverton, and V. Ozoli¸nˇs, Europhys. Lett. , 719 (2006). C. Wolverton and V. Ozoli¸nˇs, Phys. Rev. Lett. , 5518(2001). J. L. Murray, Int. Met. Rev. , 211 (1985). C. Ravi and C. Wolverton, Metall. Mater. Trans. A ,2013 (2005). R. Besson and J. Morillo, Phys. Rev. B , 193 (1997). L. Shaojun, D. Suqing, and M. Benkun, Phys. Rev. B ,9705 (1998). W.-q. Zhang, Q. Xie, X.-j. Ge, and N.-x. Chen, J. Appl.Phys. , 578 (1997). S. Meschel and O. Kleppa, Metall. Mater. Trans. A ,2162 (1991). S. J. Solares,
Multi-scale simulations of single-walled car-bon nanotube atomic force microscopy and density func-tional theory characterization of functionalized and non-functionalized silicon surfaces , Ph.D. thesis, CaliforniaInstitute of Technology (2006).
E. G. Moroni, W. Wolf, J. Hafner, and R. Podloucky,Phys. Rev. B , 12860 (1999). S. Zhou, Y. Wang, F. Shi, F. Sommer, L.-Q. Chen, Z.-K.Liu, and R. Napolitano, J. Phase Equilib. Diffus. , 158(2007). S. Narasimhan and J. W. Davenport, Phys. Rev. B ,659 (1995). Y. Zhong, M. Yang, and Z.-K. Liu, Calphad , 303(2005). N. Wang, W.-Y. Yu, B.-Y. Tang, L.-M. Peng, and W.-J.Ding, J. Phys. D: Appl. Phys. , 195408 (2008). Y.-M. Kim, N. J. Kim, and B.-J. Lee, Calphad , 650(2009). X.-Y. Liu, P. P. Ohotnicky, J. B. Adams, C. L. Rohrer,and R. W. Hyland, Surf. Sci. , 357 (1997).
J. Murray, J. Phase Equilib. , 60 (1982). P. Villars and L. D. Calvert,
Pearson’s handbook of crystal-lographic data for intermetallic phases (American Societyfor Metals, Materials Park, OH, 1985).
D. Singh, C. Suryanarayana, L. Mertus, and R.-H. Chen,Intermetallics , 373 (2003). S. G. Fries and T. Jantzen, Thermochim. Acta , 23(1998).
X.-Y. Liu,
The development of empirical potentials fromfirst-principles and application to Al alloys , Ph.D. thesis,University of Illinois at Urbana-Champaign (1997).
P. Maugis, J. Lacaze, R. Besson, and J. Morillo, Metall.Mater. Trans. A , 3397 (2006). P. G. Gonzales-Orme˜no, H. M. Petrilli, and C. G. Sch¨on,Calphad , 573 (2002). R. E. Watson and M. Weinert, Phys. Rev. B , 5981(1998). N. I. Kulikov, A. V. Postnikov, G. Borstel, and J. Braun,Phys. Rev. B , 6824 (1999). M. Fri´ak and J. Neugebauer, Intermetallics , 1316(2010). F. Lechermann, M. F¨ahnle, and J. M. Sanchez, Inter-metallics , 1096 (2005). C. L. Fu and M. H. Yoo, Acta Metall. Mater. , 703(1992). E. Lee and B.-J. Lee, J. Phys.: Condens. Matter , 175702(2010). X. SHU, W. HU, H. XIAO, H. DENG, and B. ZHANG,J. Mater. Sci. Technol. (Shenyang, China) , 601 (2001). C. Vailh´e and D. Farkas, Acta Mater. , 4463 (1997). O. Kubaschewski and W. A. Dench, Acta Metall. , 339(1955). P. D. Desai, J. Phys. Chem. Ref. Data , 109 (1987). R. Hultgren, P. D. Desai, D. T. Hawkins, M. Gleiser,and K. K. Kelley,
Selected values of the thermodynamicproperties of binary alloys (American Society for Metals,Metals Park, OH, 1973).
W. Gale, C. Smithells, and T. Totemeier,
Smithells metalsreference book (Butterworth-Heinemann, 2004).
W. B. Pearson,
A Handbook of lattice Spacings and Struc-tures of Metals and Alloys (Pergamon Press, New York,1958).
M. H. Yoo, T. Takasugi, S. Hanada, and O. Izumi, Mater.Trans., JIM , 435 (1990). G. Simmons and H. Wang,
Single crystal elastic constantsand calculated aggregate properties (MIT Press, 1971).
M. Krajˇc´ı and J. Hafner, J. Phys.: Condens. Matter ,5755 (2002). N. P. Bailey, J. Schiøtz, and K. W. Jacobsen, Phys. Rev.B , 144205 (2004). R. C. King and O. J. Kleppa, Acta Metall. , 87 (1964). B. Predel and H. Ruge, Mater. Sci. Eng. , 333 (1972). J. R. Davis, ed.,
Metals Handbook Desk Edition , 2nd ed.(ASM International, 1998).
J. J. Chu and C. A. Steeves, J. Non-Cryst. Solids ,3765 (2011).
A. J. C. Wilson, Proc. Phys. Soc. , 235 (1941). S. J. Plimpton, J. Comput. Phys. , 1 (1995).
S. R. Bahn and K. W. Jacobsen, Comput. Sci. Eng. , 56(2002). C. A. Becker, in
Models, Databases, and Simulation ToolsNeeded for the Realization of Integrated ComputationalMaterials Engineering , edited by S. M. Arnold and T. T.Wong (ASM International, 2011) Atomistic simulationsfor engineering: Potentials and challenges, . J. H. Rose, J. R. Smith, F. Guinea, and J. Ferrante, Phys.Rev. B , 2963 (1984). F. D. Murnaghan, Proc. Natl. Acad. Sci. U. S. A.30