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Acta Materialia 67 (2014) 168–180

Generalized stacking fault energy, ideal strength and twinnabilityof dilute Mg-based alloys: A first-principles study of shear deformation

S.L. Shang a,⇑, W.Y. Wang a, B.C. Zhou a, Y. Wang a, K.A. Darling b, L.J. Kecskes b,S.N. Mathaudhu c, Z.K. Liu a

a Department of Materials Science and Engineering, The Pennsylvania State University, University Park, PA 16802, USAb Materials and Manufacturing Sciences Division, US Army Research Laboratory, Aberdeen Proving Ground, MD 21005, USA

c Materials Science Division, US Army Research Office, Research Triangle Park, NC 27709, USA

Received 20 November 2013; received in revised form 9 December 2013; accepted 11 December 2013Available online 25 January 2014

Abstract

In an effort to establish a scientific foundation for the computational development of advanced Mg-based alloys, a systematic study ofthe generalized stacking fault (GSF) energy curves has been undertaken. Additionally, the associated stable and unstable stacking andtwinning fault energies, ideal shear strengths, and comparative twinnability have been investigated in terms of first-principles calculationsfor dilute Mg-based alloys of type Mg95X. These GSF properties are predicted using the simple and especially the pure alias shear defor-mations on the basal (00 01) plane and along the ½1 0 �1 0� direction of the hexagonal close-packed (hcp) lattice. Fourteen alloyingelements (X) are considered herein, namely Al, Ca, Cu, La, Li, Mn, Sc, Si, Sn, Sr, Ti, Y, Zn and Zr. The following conclusions areobtained: (i) the fault energies and the ideal shear strengths of Mg95X alloys decrease approximately linearly with an increasing equilib-rium volume of X (or Mg95X), with the exceptions being for alloying elements Al, Cu, Si and Zn; (ii) alloying elements Sr and La greatlyincrease the twin propensity of hcp Mg, while Mn, Ti and Zr exhibit opposite trends; and (iii) the observed variation in GSF propertiesfor hcp Mg caused by alloying elements X can be directly traced to the distribution of the differential charge density (Dq)—a sphericaldistribution of Dq facilitates the redistribution of charge and shear deformation, resulting in lower shear-related properties, such as stack-ing fault energy and ideal shear strength. Computed GSF properties of Mg95X are shown to agree with available experimental and othertheoretical results in the literature.� 2013 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

Keywords: Stacking and twin fault energies; Twin propensity; Ideal shear strength; First-principles calculations; Pure alias shear deformation

1. Introduction

As the lightest metallic structural materials, Mg alloys,which have densities that are approximately two-thirds thatof pure aluminum and one-quarter of steel, hold greatpotential for considerably reducing the weight of vehicles,improving their fuel efficiency, and making them moreenvironmentally friendly [1]. However, Mg-based alloyshave poor formability at room temperature due to theirlimited number of independent slip systems: for instance

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http://dx.doi.org/10.1016/j.actamat.2013.12.019

⇑ Corresponding author. Tel.: +1 814 8639957; fax: +1 814 8652917.E-mail address: sus26@psu.edu (S.L. Shang).

only two active, independent slip systems on the basalplane exist for Mg and its alloys at room temperature [2].For the computationally and data-driven development ofadvanced Mg-based alloys, a complete set of fundamentalproperties is essential, including assessment of ideal shearstrengths and the associated stable and unstable stackingand twin fault energies. Here, the ideal shear strength is afundamental indicator of material strength [3,4], beingrelated to both the minimum stress needed to plasticallydeform a perfect single crystal, and the stress required forthe formation of stacking faults [4–6]. In addition, the idealshear strength is also a key parameter to predict thePeierls–Narbarro stress rP [7], or the minimum shear stress

rights reserved.

Fig. 1. Alias shear deformation of hcp Mg95X along f0 0 0 1gh1 0 �1 0i.The “I2-shear” indicates the shear deformation from the perfect (P) hcp tothe I2 stacking fault, and the “T2-shear” indicates the shear deformationfrom the I02 stacking fault (which is slightly different from I2 due todifferent cells being used in the calculations, see the red solid boxes) to theT2 twin fault. The letters A, B and C represent different (0001) planes ofhcp lattice, and the right panel shows the projected (0001) planes alongthe [0001] direction. The red solid box indicates the unit cell used in thepresent calculations, and the black dashed box indicates the adjacent unitcell. The movements of different (0001) planes along the ½1 0 �1 0� directionduring the alias shear deformation are shown by the arrows (!), while theatoms in the red solid boxes remain in their original positions during thealias shear deformation. (For interpretation of the references to color inthis figure legend, the reader is referred to the web version of this article.)

S.L. Shang et al. / Acta Materialia 67 (2014) 168–180 169

required to move an individual dislocation. The value of rP

has been used to approximate the critical resolved shearstress (CRSS), specifically at 0 K [8]. Furthermore, thesestresses (rP and CRSS) are required inputs for continuummodels such as the finite-element and phase-field simula-tions of plastic deformation and recrystallization [9,10].The ideal shear strength has become even more importantwith the progress of nanotechnology, which allows the nearideal strength to be studied quantitatively [4,11]. Experi-mentally, well-controlled nanoindentation tests have beenused to probe what the ideal strengths of metals can be[12,13].

Regarding hexagonal close-packed (hcp) Mg, its idealshear strength has been very recently estimated usingmicropillar compression and indentation experiments[12,14–16], and found to be �0.6–1.2 GPa (see details inSection 3). However, reliable experimentally determinedideal shear strengths of Mg-based alloys have not beenreported [17,18]. Ideal shear strengths for some cubic met-als such as Al, Cu and Ni [3,4,6] have been predicted theo-retically using, for example, pure alias shear deformation interms of first-principles calculations. However, first-princi-ples ideal shear strengths have not been undertaken for Mgand its alloys. Here, an alias shear deformation involvesonly one deformed atomic plane, while the atoms in otherlayers remain in their original positions—see details inRefs. [3,4,6]. In addition, pure shear indicates that relax-ations of cell shape, cell volume and atomic positions areallowed with the shearing angle fixed [3,4,6]. Note thatthe alias shear deformation generates both ideal shearstrengths and fault energies during shear deformation toform stacking and twin faults [3,4,6].

The generalized stacking fault (GSF) energy is a mea-sure of the energy penalty between two adjacent planesduring shear deformation in a specific slip direction on agiven slip plane, representing the nature of slip and involv-ing the stable and unstable stacking and twin fault energies[3,4,19]. The GSF curve and the associated energy proper-ties can be used to model a vast number of phenomenalinked to dislocations, plastic deformation, crystal growth,phase transition [4,20] and twin–twin interactions [21]. Forinstance, the kinetic process of partial dislocation move-ments such as cross-slip and climb is controlled by stackingfault energy (c), and c / 1/R, where R is the width of sep-aration between partial dislocations [22,23]. Hence, a lowerstacking fault energy corresponds to a larger distance ofseparation between dissociated partials, resulting inreduced cross-slip and climb, thereby decreasing thesteady-state creep rate [24–26]. A lower stacking faultenergy also results in a higher strain-hardening coefficient,lower twinning stress and a higher twin propensity [4,20].In particular, the GSF energy can be used to predictsolid-solution strengthening and the thermal cross-slipstress as demonstrated for Mg alloys by Yasi et al. [27–29]. Furthermore, these solute–dislocation interactions asa function of position on the slip plane can serve as inputto predict, for example, finite-temperature flow stresses

[30] and finite-element simulations of forming [31]. Basedon the Peierls concept, Rice [32] indicated that the disloca-tion nucleation from a crack tip is directly proportional toffiffiffiffiffiffiffi

cUI

p, with cUI being the unstable stacking fault energy. In

the case of face-centered cubic (fcc) metals, Tadmor andBernstein [33,34] proposed a measure of twinnability (Kfcc)as a function of GSF energies including the intrinsic stack-ing fault energy (cI), the unstable stacking fault energy(cUI), and the unstable twin fault energy (cUT):

Kfcc ¼ 1:136� 0:151cI

cUI

� � ffiffiffiffiffiffifficUI

cUT

r: ð1Þ

The ideal stacking sequence of hcp metals is � � �ABA-

BAB� � � with A and B (and C) being the closed-packedplanes ({000 1} for hcp and {111} for fcc), while theideal stacking sequence for fcc is � � �ABCABC� � � (seeFig. 1). The commonly observed planar faults in a hcplattice include two intrinsic stacking faults of I1 (thegrowth fault of � � �ABABCBC� � �) and I2 (the deformationfault of � � �ABABCACA� � �), one extrinsic stackingfault E (� � �ABABCABAB� � �), and one twin faultT2 (� � �ABABCBABA� � �) [35]. In fact, all these hcp planar

170 S.L. Shang et al. / Acta Materialia 67 (2014) 168–180

faults can be generated by (alias) shear deformation onthe {0001} planes along the h1 0 �1 0i directions [36,37].It is worth mentioning that although the primary slipsystem of hcp metals is f0 0 0 1gh1 1 �2 0i, energeticsfavors the dissociation of perfect dislocations into partialdislocations [38–40]:

1

3h2 �1 �1 0i ! 1

3h1 �1 0 0i þ 1

3h1 0 �1 0i: ð2Þ

Therefore, deformation by partial dislocations of1=3h1 0 �1 0i type is preferred. As an example of variousslips on f0 0 0 1gh1 0 �1 0i, Fig. 1 illustrates the I2 stackingfault generated from the perfect hcp structure via the aliasshear deformation, as well as the twin fault T2 from I2.Note that the growth stacking fault I1 can be formed byremoving an A plane above the B plane, and then shearingthe remaining planes above the B plane by the displace-ment 1=3h1 0 �1 0i [35]. Alternatively, the fault I1 can besheared from T2 and the fault E from I1 based on the sim-ilar way that T2 sheared from I2 (see Fig. 1). For hcp met-als, the I2 stacking fault plays a crucial role because it isclosely related to dislocation configuration, dislocation dis-sociation, and thus intrinsic ductility [38].

Unlike ideal shear strength, much experimental workand in particular first-principles calculations of stackingfault energies (and GSF energy curves from calculations)have been performed for hcp Mg. Note that the stackingfault energy is difficult to measure precisely for manyreasons, including the small differences in the separationdistances between partial dislocations as observed by elec-tron microscope imaging [41], hence the measured stackingfault energies of hcp Mg are quite scattered: <50 [42], >90[43], and 74, 102, 125 and 280 mJ m�2 [44]. Alternatively,first-principles calculations reported a lower energy of theI2 stacking fault for hcp Mg: �21–44 mJ m�2 (see detailsin Section 3). Regarding Mg-based alloys, a few first-principles calculations have analyzed the stacking faultenergies and the GSF energy curves affected by alloyingelements. For instance, Han et al. [36] reported the first-principles GSF energy curves of Mg–Al and Mg–Li alloysusing the slab deformation method integrated with theclimbing image nudged elastic band (CINEB) method [45].Muzyk et al. [46] reported the effects of 13 alloyingelements (Ag, Al, Cu, Fe, Li, Mn, Ni, Pb, Sn, Ti, Y, Znand Zr) on the GSF energy properties of hcp Mg usingthe slab deformation method. In addition to these I2 faultrelated GSF energies for Mg-based alloys [36,45,46], theI1 fault related GSF energies of Mg-based alloys have beencalculated by Zhang et al. [47] using the energy differencemethod, the GSF energies in the f0 0 0 1gh1 1 �2 0i,f1 0 �1 0gh1 1 �2 0i, f1 0 �1 1gh1 1 �2 0i and f1 1 �2 2gh1 1 �2 3isystems of Mg-based alloys have been calculated by Wanget al. [48] using the slab deformation method. Except forthe first-principles slab deformation method (see Section 2for more details), the pure alias shear deformation method,which can predict both energy and stress, has not been usedto study the GSF properties of Mg and its alloys.

In light of the few studies on calculating GSF energies,specifically while employing ideal shear stresses analysisfor Mg-based alloys, the present work aims to investigatehow the GSF properties of hcp Mg are affected by 14 alloy-ing elements X (Al, Ca, Cu, La, Li, Mn, Sc, Si, Sn, Sr, Ti,Y, Zn and Zr) in terms of the simple and the pure aliasshear deformations via first-principles calculations, whilerare earth alloying elements (except for La) will be studiedin the future. Computational approaches are discussed inSection 2 including (i) evaluation of GSF properties(namely, the stable and unstable stacking and twin faultenergies, ideal shear strength, and the combined propertyof twinnability) in terms of the alias shear deformationon the {000 1} slip plane and along the h1 0 �1 0i direction,as well as (ii) details of first-principles calculations. In Sec-tion 3, the computed GSF properites for hcp Mg and thedilute Mg95X alloys are presented along with discussionsabout the distribution of differential charge density. Theprincipal results are summarized in Section 4.

2. Computational approach

2.1. Shear deformation and GSF properties

Both the affine and the alias shear deformations can beused to deform the lattice. Here, the affine shear indicatesthat all atomic planes are involved in the shear deforma-tion, while the alias shear indicates only one atomic planeis deformed and others remain in their original positionsduring shear (see details in Refs. [3,4,6]). The alias shearinstead of the affine shear is employed in the present workto study the GSF properties since the alias shear, especiallythe pure alias shear, is a more realistic description of theshearing processes relative to the affine shear deformationdue to the interaction between the atoms and, in turn,the displacement propagates through the as-constructedcell [4,6]. In both the alias and the affine shears, three relax-ation schemes after shear can be used: (i) simple shear with-out relaxations of cell shape, cell volume and atomicpositions; (ii) simple shear with relaxations of atomic posi-tions only; and (iii) pure shear with relaxations of cellshape, cell volume and atomic positions except for theshearing angle being fixed [3,4,6]. The GSF energies(including the stable and the unstable stacking and twinfault energies) can be predicted by the above three schemesduring the alias shear, while the stresses can only be pre-dicted from schemes (ii) and (iii). It should be remarkedthat the stresses from the alias shear without relaxations(scheme (i)) depend on the size and the shape of supercell[4]. In the present work, the simple alias shear withoutrelaxations (scheme (i)) and the pure alias shear (scheme(iii)) are employed to study the GSF properties of hcpMg and its alloys Mg95X, while the results from scheme(i) are only for reference.

To facilitate the alias shear deformation onf0 0 0 1gh1 0 �1 0i for an hcp lattice, a four-atom ortho-rhombic cell is employed to represent the hcp structure.

S.L. Shang et al. / Acta Materialia 67 (2014) 168–180 171

The lattice vectors e1, e2 and e3 of this orthorhombic cellare parallel to the ½1 1 �2 0�, ½1 0 �1 0� and [0001] directionsof the conventional hcp cell, respectively, and their lengthsare a0, a0

ffiffiffi3p

and c0 (a0 and c0 are the hcp lattice parame-ters), respectively. In order to study the dilute Mg-basedalloys, a (3 � 2 � 4) supercell with respect to thisfour-atom orthorhombic cell is employed, i.e. a 96-atomsupercell with 8 layers and 12 atoms in each layer (note thatthese 8 layers are illustrated by the red solid box in Fig. 1).One alloying atom of X (Al, Ca, Cu, La, Li, Mn, (Mg), Sc,Si, Sn, Sr, Ti, Y, Zn or Zr) is laid on the fault plane, i.e. themole fraction of X is 1/12 on the fault plane and the globalmole fraction of X is 1/96. Note that the composition of analloying element on the fault plane is more meaningful thanits global composition, in particular for dilute alloys.

After alias shear, the deformed lattice vectors matrix canbe expressed by:

R ¼ RD; ð3Þwhere the undeformed lattice vectors matrix R for the pres-ent 96-atom (3 � 2 � 4) supercell and the deformation ma-trix D relative to the f0 0 0 1gh1 0 �1 0i shear are as follows:

R¼a0 0 0

0 a0

ffiffiffi3p

0

0 0 c0

264

375�

3 0 0

0 2 0

0 0 4

264

375¼

3a0 0 0

0 2a0

ffiffiffi3p

0

0 0 4c0

264

375;

ð4Þ

D ¼1 0 0

0 1 0

0 e 1

264

375: ð5Þ

Here, a0 and c0 are the hcp lattice parameters as mentionedabove, and e is the shear magnitude corresponding to theengineering shear strain, i.e. the ratio of displacement withrespect to the height of the unit cell. In the present case,jej=ð2a0

ffiffiffi3pÞ ¼ 1=6 corresponds to the movements of the

closed-spaced atomic plane A to C (or B! A, etc.) in othersupercells instead of the original supercell due to the aliasshear used. Fig. 1 illustrates the details associated withthe generation of the I2 fault and the T2 fault via thef0 0 0 1gh1 0 �1 0i alias shears (the I2-shear and theT2-shear). Here, the selected supercell of the I02 structureis slightly different from that of the I2 structure (see thered solid boxes in Fig. 1), but the energy difference betweenthese two supercells is negligible during the analyses of theGSF energies, since the more planes used, the smaller theenergy difference should be. In the present practice offirst-principles calculations, atomic positions for the aliasshear deformation are represented by Cartesian coordi-nates. Relaxations of atomic positions, cell shape and cellvolume in the pure shear deformations were performedby an external optimizer GADGET developed by Buckoet al. [49], while still employing the VASP code as the com-putational engine. The only constraint during pure sheardeformation is the fixed shearing angle, i.e. the anglebetween the ½1 0 �1 0� and [00 01] directions for

f0 0 0 1gh1 0 �1 0i shear (see Eq. (5)). When the shear defor-mation is approaching the ideal shear strengths and theunstable fault energies, a minimum shear step about1/400 of the Burgers vector (e.g. from site A to site B asshown in Fig. 1) is adopted.

In addition to the alias shear deformation, another com-monly used scheme to predict GSF energies is the slabshear deformation method [45,46]. As a test, the slab defor-mation scheme is also employed herein to examine the GSFenergies of hcp Mg by using a 112-atom slab including 28closed-packed (0001) planes and 4 atoms per plane, andthe gap between the slabs is >10 A (more details aboutthe shear deformations and relaxations of atoms are givenin Refs. [45,46]). It is worth mentioning that the strainenergy due to lattice mismatch plays an important role inestimating the interfacial energies [50], especially for thecases with small interfacial energies (such as the presentstable fault energies and the interfacial energy of c-Ni/c0-Ni3Al [50]). For the schemes of the alias shear deformationmethod and the slab shear deformation method, the strainenergy is absent due to the same fault plane being used forthe fault structure and the reference structure; while for theschemes in which energy differences are compared betweentwo structures (i.e. the fault and the reference structures)and empirical methods (e.g. the ANNNI model [51]), thestrain energy is usually included in the predicted interfacialenergy due to different fault planes being used for the faultand the reference structures. Therefore, more accurateinterfacial energy can be predicted by the shear deforma-tion method rather than the method involving direct com-parison of energy differences.

2.2. First-principles calculations

All density functional theory (DFT)-based first-princi-ples calculations in the present work are performed bythe VASP code [52]. The ion–electron interaction isdescribed by the projector augmented wave (PAW) method[53] and the exchange-correction (X-C) functional isdescribed by the generalized gradient approximation ofPerdew–Burke–Ernzerhof (GGA-PBE) [54]. Here, theGGA-PBE is selected since it can predict more accuratestructural properties of hcp Mg, e.g. the lattice parameters,with respect to other X-C functionals such as the PBEsol[55]—see details in Ref. [56]. Note that the improvedGGA of PBEsol has a reduced gradient dependence, andpredicts accurately the surface-related properties for metalssuch as the diffusion coefficients for Mg-based alloys [57].The recommended core configurations by VASP areadopted for each element in the present work, i.e. emptyfor Li; [Ne] for Al, Mg, Sc, and Si; [Ne]3s2 for Ca andTi; [Ar] for Cu, Mn, and Zn; [Ar]3d10 for Sr, Y, and Zr;[Kr] for Sn; and [Kr]4d10 for La. In the VASP calculations,a 5 � 5 � 2 (or 6 � 7 � 1 for the slab case) k-point meshand a plane wave energy cutoff energy that is 1.3 timesthe largest cutoff energy associated with the elements ofinterest recommended by VASP (i.e. the high precision in

172 S.L. Shang et al. / Acta Materialia 67 (2014) 168–180

VASP) are employed based on our tests. The energyconvergence criterion of the electronic self-consistency ischosen as 10�5 eV per supercell for all calculations. Thereciprocal-space energy integration is performed by theMethfessel–Paxton (MP) [58] technique for structuralrelaxations with a 0.2 eV smearing width. The MP methodalso results in a very accurate description of the totalenergy, and it is found that the difference between the freeenergy and the total energy (when the smearing parameteris close to zero) is negligible in the present work(<0.15 meV per atom). Regarding the pure shears, therelaxed stresses, except for the fixed shear angle, are<0.015 GPa, and the forces acting on atoms are<0.01 eV A�1. Due to the ferromagnetic nature of Mn,the calculations for Mg95Mn are performed within thespin-polarized approximation.

3. Results and discussion

In this section, we show first the results of hcp pure Mgincluding (i) the GSF energy curves predicted by the simplealias shear and the slab shear; and (ii) the GSF fault ener-gies and the ideal shear strengths together with availableexperiments and other calculations (Section 3.1). Second,the GSF properties of Mg95X are presented as a functionof the equilibrium volume of X in the hcp structure, includ-ing the stable and the unstable fault energies of I2 and T2,the ideal shear strengths to generate I2 and T2, and thecombined property of twinnability for the dilute Mg95X(Section 3.2). Finally in Section 3.3, the variation of GSFproperties of Mg95X is discussed in terms of the differentialcharge densities.

3.1. GSF properties of hcp Mg

As a test, Fig. 2 illustrates a comparison of the GSFenergy curves of hcp Mg to generate the I2 and the T2 faultspredicted by (i) simple alias shear without relaxations and(ii) slab shear deformation. No significant differences are

140

120

100

80

60

40

20

0

Faul

t ene

rgy

(mJ/

m2 )

2.01.51.00.50.0Shear deformation

Stacking fault (slab) Stacking fault (alias) Twinning fault (slab) Twinning fault (alias)

hcp Mg

γI2

γUI2

γUT2

γT2

Fig. 2. GSF energy curves of hcp Mg to generate the I2 stacking fault andthe T2 twin fault predicted by the simple alias shear and the slab shear. Theenergy difference between the I2 and the I02 structure (see Fig. 1) isnegligible and can be ignored. The stable and the unstable I2 and T2 faultenergies are also indicated.

observed between these two GSF energy curves. The keyfault energies shown in Fig. 2 are listed in Table 1, includ-ing the stable and the unstable I2 stacking fault energies (cI2

and cUI2, respectively), and the stable and the unstable T2

twin fault energies (cT2and cUT2

, respectively). The maxi-mum difference between the simple alias shear and the slabshear is for cI2

(26.1 vs. 27.7 mJ m�2, see Table 1). Regard-ing the fault energies, the alias shear predicts values similarin magnitude to those calculated by the commonly usedslab shear method. However, for the same compositionof alloying element (X) on the fault plane, only half theatoms are required in the alias shear deformation to gener-ate the same degree of deformation as in the slab shearmethod. Furthermore, the stresses can be predicted directlyfrom the alias shear instead of the slab shear. These advan-tages determine our choice of using the alias shear ratherthan the slab shear to study the GSF properties.

As an example, Fig. 3 shows the I2 stacking fault energycurves and the corresponding stress curves of hcp Mg interms of different alias shear deformations. The I2 faultenergy curve from simple shear (without relaxations, shearscheme (i)) possesses large difference compared to the othertwo: simple shear with relaxations of atomic positions(shear scheme (ii)) and pure alias shear (shear scheme(iii)). However, the differences for the energy and the stresscurves are not significant between schemes (ii) and (iii),indicating that the results from simple shear with relax-ations are comparable with those from pure shear. Notethat the shear steps are not dense enough after generationsof unstable fault energy and ideal shear strength. Concern-ing the stable I2 stacking fault energy, the predicted valuesfrom these three shear schemes show similar results due tothe metastable I2 fault. As mentioned above, the GSFproperties from shear scheme (i) (simple shear) and shearscheme (iii) (pure shear) are reported in the present workin order to compare the results without relaxations andfully relaxed. The results from shear scheme (ii) shouldbe in the middle but close to the pure shear ones. The pre-dicted stable and unstable I2 stacking fault energies inFig. 3 are given in Table 1, while the predicted ideal shearstrength in Fig. 3 is listed in Table 2.

Table 1 compares the presently predicted stable andunstable I2 and T2 fault energies with other first-principlespredictions [35–39,46,59–65] using difference X-C function-als of the local density approximation (LDA) [66], theGGA-PW91 [67] and the GGA-PBE [54], as well asavailable experiments measured at room temperature [42–44,68]. The present stable I2 stacking fault energies (�26–30 mJ m�2) agree well with other first-principles results(�21–44 mJ m�2), but all of these first-principles resultsare lower than the scattered experimental values measuredat room temperature (<50 [42], >90 [43], 74, 102, 125 and280 mJ m�2 [44]). It is worth mentioning that most of themeasured values of stacking fault energy in the literatureare not reliable since the stacking fault energy is difficultto measure precisely for many reasons, e.g. (i) the applica-tion of less adequate theoretical relationships between

Table 1First-principles (F-P) predicted stable and unstable stacking fault and twin fault energies (symbol c in mJ m�2) of hcp Mg at 0 K, together with other F-Ppredictions and experimental results measured at room temperature from the literature. The subscript “U” represents the unstable properties, symbol “I2”

the I2 deformation stacking fault, and symbol “T2” the T2 twin fault. Note that some of the present predictions are also shown in Table 3.

cI2cUI2

cT2cUT2

Note

30.0 84.8 43.1 104.0 This work, pure shear, GGA-PBE26.1 94.9 37.1 111.2 This work, simple shear, GGA-PBE27.7 93.5 38.7 112.9 This work, slab shear, GGA-PBE33 92 42 110 F-P, slab shear and CINEB [36]36 92 39 111 F-P, slab shear, GGA-PBE [46]34 97 38 181 F-P, shear deformation, GGA-PW91 [37]21 88 F-P, slab shear, GGA-PW91 [59]29.2 58.2 F-P, energy difference, LDA [39]34 >92 F-P, shear deformation, GGA-PW91 [60]33.8 87.6 F-P, slab shear, GGA-PW91 [38]16 116 F-P, shear deformation, GGA-PW91 [61]32 93 F-P, slab shear, GGA-[62]44 51 F-P, energy difference, LDA [35]33.8 40.6 F-P, energy difference, GGA-PW91 [63]21.8, 27.7, 38.3 F-P, energy difference, GGA-PW91 [64]36 F-P, energy difference, GGA-PBE [65]78 ± 15 Expt. [68]<50 Expt. [42]>90 Expt. [43]74, 102, 125, 280 Expt. review [44]

0.8

0.4

0.0

-0.4

-0.8

Stre

ss (G

Pa)

1.00.80.60.40.20.0Shear deformation

Simple with relaxation Pure

100

80

60

40

20

0

Faul

t ene

rgy

(mJ/

m2 )

Simple Simple with relaxation Pure

hcp Mg

γI2

γUI2

τI2

Fig. 3. GSF energy and stress curves of hcp Mg to generate the I2 stackingfault based on the simple alias shear deformations with and withoutrelaxations of atomic positions, and the pure alias shear deformations.The shear steps are not dense enough after generation of the unstablestacking fault and the ideal shear strength.

S.L. Shang et al. / Acta Materialia 67 (2014) 168–180 173

experimental measurements and stacking fault energy; and(ii) the small differences in the separation distances betweenpartial dislocations in electron microscope images [4,69]. Inaddition, measurements are usually performed at roomtemperature, while first-principles calculations are carriedout at 0 K. The present results of other GSF energiesincluding the unstable I2 stacking fault energies (�85–

95 mJ m�2), the stable T2 twin fault energies (�37–43 mJ m�2) and the unstable T2 twin fault energies(�104–113 mJ m�2), are all in good agreement with otherfirst-principles results (see Table 1). It is worth mentioningthat the more accurate results of the present work are theones from the pure alias shear deformation.

Table 2 compares the present ideal shear strengths (s)predicted at 0 K to generate the I2 and the T2 faults ofhcp Mg with other results from the molecular dynamics(MD) simulations [70] and the very recent microcompres-sion and nanoindentation experiments performed at roomtemperature [12,14–16]. The present results (0.77 GPa forsI2

and 0.67 GPa for sT2) indicate that it is easy to generate

the T2 twin after the generation of the I2 stacking fault. Thepresent predictions agree with the scattered results fromMD simulations (�0.67–1.3 GPa) and experimental values(�0.6–1.2 GPa), providing the theoretical shear strengthsof hcp Mg for further experimental validations. Moredetails regarding the experimental shear strengths are givenin Table 2.

3.2. Effects of alloying elements on the GSF properties of hcp

Mg

Table 3 summarizes the predicted GSF properties ofMg95X during the generations of I2 and T2 faults via thesimple and the pure alias shears, including the stable andthe unstable fault energies and the ideal shear strengths.For a better understanding, these GSF properties are plot-ted as a function of the equilibrium volume of alloyingelement X with hcp structure obtained previously fromfirst-principles calculations [71]. Fig. 4 shows that a roughlinear relationship exists between the equilibrium volumes

Table 2First-principles predicted ideal shear strengths (symbol s in GPa) of hcpMg at 0 K, together with the estimated results from molecular dynamics(MD) simulations and measurements (at room temperature) from theliterature. Note that the present predictions are also shown in Table 3.

sI2sT2

Note

0.767 0.674 This work, pure shear, GGA-PBE�0.67–

1.3MD: nucleation stress of basal slip [70]

0.655 Nanocompression expt. of basal plane sliding [14]�0.8 Nanocompression and tension expt. of deformation

twin initiated [15]�0.6–

1.2Expt. pop-in shear stress for (0001) indentation [12]

>0.6 Expt. uniaxial compression [16]

23.1

23.0

22.9

22.8

22.7

Volu

me

of M

g 95X

(Å3 /a

tom

)

605040302010

Volume of hcp metal (Å3/atom)

Mn

CuSi

ZnAl

Ti

Li

Mg

Zr

ScSn

Y

La

Ca

Sr

Fig. 4. First-principles predicted equilibrium volumes of alloying elementsX with hcp structure [71] compared with the present predictions ofequilibrium volumes for Mg95X.

100

90

80

70

60

γ UI 2 (m

J/m

2 )

5040302010

Volume of hcp metal (Å3/atom)

Simple shear Pure shear

MnCu

SiZn

AlTi

LiMg

ZrSc

Sn Y La Ca Sr

Fig. 5. Comparison of the unstable I2 stacking fault energies cUI2of

Mg95X predicted by the simple shear deformations (without relaxations ofatomic positions) and the pure shear deformations. Alloying elements Xcorresponding to each Mg95X are marked; see Table 3 for the values.

174 S.L. Shang et al. / Acta Materialia 67 (2014) 168–180

of alloying elements X with hcp structure [71] and the equi-librium volumes of Mg95X from the present work afterpure shears. As an example, Fig. 5 shows a comparisonof the unstable I2 stacking fault energies cUI2

of Mg95X pre-dicted from the simple shear (without relaxations, shearscheme (i)) and the pure shear deformations as a functionof the equilibrium volumes of alloying elements X withhcp structure. The cUI2

energies from simple shear for allMg95X show a constant value around 95 mJ m�2, whilethe cUI2

energies from pure shear decrease roughly linearlywith increasing equilibrium volumes of alloying elements Xwith hcp structure, implying that a more reasonable cUI2

can be predicted from the pure alias shear rather than thesimple alias shear.

Fig. 6 illustrates the stable and the unstable I2 and T2

energies (c) of Mg95X from the pure alias shears, andFig. 7 shows the I2 and the T2 ideal shear strengths (s) ofMg95X from the pure alias shears. With increasing equilib-rium volumes of alloying elements X with hcp structure, all

Table 3First-principles fault properties of Mg95X predicted by the X-C functional of GGA-PBE, including the stable and the unstable stacking and twin faultenergies (symbol c in mJ m�2), the ideal shear strengths (symbol s in GPa), and the twinnabilities (symbol Kpur

fcc defined by Eq. (1) and Kpurratio defined by Eq.

(7)). The superscript “sim” indicates simple shear deformation, and “pur” pure shear deformation. The unstable energy difference (DcpurU ¼ cpur

UT2� cpur

UI2) can

also be used to define the inverse of twinnability, see also Eq. (8). Vhcp (A3 atom�1) is the volume of alloying element X in hcp structure according to first-principles calculations [71]. Note that the global composition for each alloying element X is 1/96, and it is 1/12 on the fault plane.

Mg95X Vhcp csimI2

cpurI2

csimUI2

cpurUI2

spurI2

csimT2

cpurT2

csimUT2

cpurUT2

spurT2

Kpurfcc Kpur

ratio DcpurU

Al 16.75 24.8 29.4 92.6 84.0 0.787 36.0 41.2 107.6 103.0 0.684 0.9779 0.7414 19.1Ca 41.93 24.5 23.8 96.3 72.2 0.739 33.3 34.2 110.7 89.0 0.624 0.9787 0.7431 16.7Cu 12.04 31.7 35.3 96.3 81.4 0.769 41.9 47.6 113.7 101.7 0.641 0.9580 0.6948 20.3La 37.26 19.9 15.1 94.7 64.0 0.740 25.5 23.0 106.1 77.3 0.601 1.0013 0.7862 13.3Li 20.33 37.5 35.6 99.0 88.8 0.787 45.6 47.7 120.0 108.9 0.653 0.9710 0.7254 20.1Mg 22.89 26.1 30.0 94.9 84.8 0.767 37.1 43.1 111.2 104.0 0.674 0.9774 0.7402 19.2Mn 10.75 29.6 46.1 97.3 98.0 0.899 42.5 48.7 115.2 120.1 0.690 0.9617 0.7006 22.2Sc 24.47 24.9 26.5 97.2 82.4 0.809 36.9 41.4 112.5 102.5 0.659 0.9754 0.7363 20.0Si 14.60 23.8 26.3 89.9 76.7 0.745 33.5 36.7 104.0 94.3 0.664 0.9774 0.7404 17.7Sn 27.79 23.0 25.5 92.1 80.8 0.763 33.3 37.8 105.9 100.0 0.688 0.9786 0.7429 19.1Sr 54.72 20.2 17.7 93.2 62.7 0.721 27.4 25.4 105.6 76.6 0.598 0.9899 0.7654 13.8Ti 17.29 30.4 33.2 98.4 91.7 0.843 41.8 49.3 115.0 113.3 0.656 0.9730 0.7308 21.6Y 32.67 22.9 21.7 97.8 72.9 0.778 33.5 34.5 111.6 90.9 0.630 0.9770 0.7397 18.0Zn 15.40 25.5 31.1 91.3 81.9 0.772 37.2 43.3 106.9 100.1 0.665 0.9754 0.7357 18.3Zr 23.44 30.2 31.3 101.0 87.2 0.843 40.8 46.7 116.7 108.9 0.638 0.9679 0.7201 21.7

120

110

100

90

80

70

60

Uns

tabl

e fa

ult e

nerg

ies

(mJ/

m2 )

5040302010

Volume of hcp metal (Å3/atom)

γUI2 (SF)γUT2

(twin)

Mn Si Al Li ZrSn Y La Ca SrCu Zn Ti Mg Sc

50

40

30

20

10

Stab

le fa

ult e

nerg

ies

(mJ/

m2 )

γI2 (SF)γT2

(twin)

Mn Si Al Li ZrSn Y La Ca SrCu Zn Ti Mg Sc

(a)

(b)

Fig. 6. Predicted stable and unstable I2 stacking fault (SF) and T2 twinfault energies of Mg95X based on the pure alias shear deformations.Alloying elements X corresponding to each Mg95X are marked; seeTable 3 for the values.

0.90

0.85

0.80

0.75

0.70

τ I 2 (G

Pa)

5040302010

Volume of hcp metal (Å3/atom)

0.75

0.70

0.65

0.60

0.55

τT2 (G

Pa)

τI2 (SF)τT2

(twin)

Mn Si Al Li ZrSn Y La Ca SrCu Zn Ti Mg Sc

Fig. 7. Predicted ideal shear stable strengths to generate the I2 stackingfault (SF) and the T2 twin fault of Mg95X based on the pure alias sheardeformations. Alloying elements X corresponding to each Mg95X aremarked; see Table 3 for the values.

S.L. Shang et al. / Acta Materialia 67 (2014) 168–180 175

these energies (cI2, cUI2

, cT2, cUT2

) and strengths (sI2and sT2

)decrease approximately linearly except for Al, Cu, Si andZn. Fig. 6 shows that the twin energies (cT2

and cUT2) are

higher than the corresponding stacking fault energies (cI2

and cUI2) since the formation of the twin structure arises

from stacking faults (see Fig. 2 and also Fig. 1). Concern-ing the ideal shear strengths, the formation of stackingfaults (sI2

) needs more stress (�0.1–0.2 GPa) than the for-mation of twins (sT2

) (see Fig. 7 and Table 3). Figs. 6and 7 plus Table 3 show that the highest values of the

GSF properties from the pure alias shear are for alloyingelement Mn due to it possessing the smallest equilibriumvolume, followed by alloying elements Ti, Li, Cu and Zr,etc., while the lowest values of the GSF properties forMg95X are for alloying elements La and Sr due to the largeequilibrium volumes of these elements.

Regarding the stable I2 stacking fault energies (cI2) for

the binary Mg–X alloys, Table 4 lists the present predic-tions from the alias shears and other first-principles predic-tions [17,36,38,46,72,73]. The available globalcompositions of alloying elements X (xglobal%) and thecompositions of X on the stacking fault plane (xSF%) arealso listed in Table 4. It was found that the stacking faultenergy is roughly linear in relationship with respect to theamount of alloying element present, and in particular tothe composition on the fault plane [20,51,72]. In order tocompare these I2 stacking fault energies in Table 4, a linearfitting of cI2

with respect to compositions (xglobal% andxSF%) is performed:

c ¼ c0 þ kx%; ð6Þwhere c0 is the stacking fault energy of hcp Mg, k is the fit-ting parameter, and x% is the atomic per cent of alloyingelement X. The fitted kglobal (for global composition) andkSF (for composition on the stacking fault plane) arereported in Table 4. Note that for high levels of alloyingelements X on the fault plane (xSF% = 25 and 50, seeTable 4), the reported stacking fault energies are not repre-sentative of dilute Mg-based alloys due to interactionsexisting between alloying elements on the fault plane. Tofacilitate comparison, Fig. 8 plots the present kSF fromthe pure alias shear (the accurate results) with respect tothe present kSF from the simple alias shear (the less accu-rate results) and other first-principles predictions by Zhanget al. [38], Han et al. [36], Muzyk et al. [46] and Pei et al.[73]. These predictions are approximately self-consistentwith the large deviations observed for alloying elementsMn, Li, Y, Sr and La due to the different models and dif-ferent X-C functionals used to predict the stacking faultenergy. The larger differences of kSF for alloying elementsMn, Li, Y, Sr and La from the present simple shear andpure shear contribute to the relaxations of atomic posi-tions, cell shape and cell volume.

Based on the GSF properties, it is possible to analyzethe twining propensity using the model derived by Tadmorand Bernstein [33,34] for fcc metals (see Eq. (1)—here weassume the twin behavior of fcc lattice is similar to thatof hcp lattice). Similarly to Eq. (1), the twinnability canbe analyzed using the following relative energies [36]:

Kratio ¼cUI2� cI2

cUT2� cI2

: ð7Þ

The larger the value of Kratio (and also for Kfcc, see Eq. (1)),the higher the propensity to twin will be. In addition, it isproposed that the (inverse of) twinnability can be under-stood via the difference between the unstable twin faultand the unstable stacking fault energies [32]:

Table 4Predicted stable I2 stacking fault energies cI2

(mJ m�2) for Mg–X compared with other first-principles results in the literature; the corresponding cI2of hcp

Mg are listed in parentheses. xglobal% and xSF% are the global atomic per cent and the atomic per cent in the stacking fault (SF) plane, respectively, foralloying element X. Dkglobal and DkSF are the changes of stacking fault energy (mJ m�2) with respect to xglobal% and xSF%, respectively. The present stablestacking energies are also listed in Table 3.

Mg–X xglobal% xSF% cI2Dkglobal DkSF Note

Mg–Al 1.04 8.33 29.4 (30.0) �0.58 �0.07 This work, pure shear1.04 8.33 24.8 (26.1) �1.25 �0.16 This work, simple shear2.71 27.8 (38.6a) �4.02 [72]5.44 16.4 (38.6a) [72]8.18 5.8 (38.6a) [72]1.67 50 23 (33) �6.00 �0.20 [36]

25 (41) [17]2.08 25 21 (36) �7.20 �0.60 [46]

Mg–Ca 1.04 8.33 23.8 (30.0) �5.95 �0.74 This work, pure shear1.04 8.33 24.5 (26.1) �1.54 �0.19 This work, simple shear

16 (41) [17]

Mg–Cu 1.04 8.33 35.3 (30.0) 5.09 0.64 This work, pure shear1.04 8.33 31.7 (26.1) 5.38 0.67 This work, simple shear2.08 25 53 (36) 8.16 0.68 [46]

Mg–La 1.04 8.33 15.1 (30.0) �14.30 �1.79 This work, pure shear1.04 8.33 19.9 (26.1) �5.95 �0.74 This work, simple shear

Mg–Li 1.04 8.33 35.6 (30.0) 5.38 0.67 This work, pure shear1.04 8.33 37.5 (26.1) 10.94 1.37 This work, simple shear1.67 50 46 (33) 7.80 0.26 [36]

43 (41) [17]2.08 25 47 (36) 5.28 0.44 [46]

Mg–Mn 1.04 8.33 46.1 (30.0) 15.46 1.93 This work, pure shear1.04 8.33 29.6 (26.1) 3.36 0.42 This work, simple shear2.08 25 38 (36) 0.96 0.08 [46]

Mg–Sc 1.04 8.33 26.5 (30.0) �3.36 �0.42 This work, pure shear1.04 8.33 24.9 (26.1) �1.15 �0.14 This work, simple shear

Mg–Si 1.04 8.33 26.3 (30.0) �3.55 �0.44 This work, pure shear1.04 8.33 23.8 (26.1) �2.21 �0.28 This work, simple shear

Mg–Sn 1.04 8.33 25.5 (30.0) �4.32 �0.54 This work, pure shear1.04 8.33 23 (26.1) �2.98 �0.37 This work, simple shear2.08 25 2 (36) �16.32 �1.36 [46]

Mg–Sr 1.04 8.33 17.7 (30.0) �11.81 �1.48 This work, pure shear1.04 8.33 20.2 (26.1) �5.66 �0.71 This work, simple shear

Mg–Ti 1.04 8.33 33.2 (30.0) 3.07 0.38 This work, pure shear1.04 8.33 30.4 (26.1) 4.13 0.52 This work, simple shear2.08 25 36 (36) 0 0 [46]

Mg–Y 1.04 8.33 21.7 (30.0) �7.97 �1.00 This work, pure shear1.04 8.33 22.9 (26.1) �3.07 �0.38 This work, simple shear0.93 11.11 15.7 (33.8) �19.50 �1.63 [38]

15 (41) [17]2.08 25 25 (36) �5.28 �0.44 [46]2.08 25 30 (37) �3.36 �0.28 [73]

Mg–Zn 1.04 8.33 31.1 (30.0) 1.06 0.13 This work, pure shear1.04 8.33 25.5 (26.1) �0.58 �0.07 This work, simple shear0.93 11.11 35.1 (33.8) 1.40 0.18 [38]

35 (41) [17]2.08 25 37 (36) 0.48 0.04 [46]

Mg–Zr 1.04 8.33 31.3 (30.0) 1.25 0.16 This work, pure shear1.04 8.33 30.2 (26.1) 3.94 0.49 This work, simple shear2.08 25 26 (36) �4.8 �0.4 [46]

a Results from linear fitting.

176 S.L. Shang et al. / Acta Materialia 67 (2014) 168–180

-2

-1

0

1

2Δ k

SF

(m

J/m

2 per

ato

mic

per

cent

)

-2 -1 0 1 2

ΔkSF via pure shear (mJ/m 2 per atomic percent)

Simple shear Zhang et al. Han et al. Muzyk et al. Pei et al.

La SrY

Ca

Sn Si

Sc

Al

Mg

Zn

Zr TiCu

Li

Mn

of o

ther

s

Fig. 8. Comparison of the changes of the stable I2 stacking fault energiesof Mg–X alloys with respect to the composition of alloying element X onthe staking fault plane, i.e. the present results of DkSF (see Eq. (6)) fromthe pure shear and the simple shear (without relaxations of atomicpositions), and other first-principles predictions by Zhang et al. [38], Hanet al. [36], Muzyk et al. [46], and Pei et al. [73]. Alloying elements Xcorresponding to each Mg–X alloy are marked; see Table 4 for the values.

0.78

0.76

0.74

0.72

0.70

Λra

tio

1.000.990.980.970.96Λfcc

22

20

18

16

14

ΔγU (mJ/m

2)

ΛratioΔγU

Mn

SiAl

LiZr

SnY

La

Ca

Sr

Cu

ZnTi

Sc

Mg

Mg

Mg

Cu

MnZr

Li

Ti

ScAlSn

LaSr

Ca

Zn

Fig. 9. Comparison of twinnabilities of Mg95X predicted by differentdefinitions, i.e. Kfcc by Eq. (1), Kratio by Eq. (7), and DcU by Eq. (8). Notethat the input values for these equations are from the pure alias sheardeformations as shown in Table 3, and the DcU defines the inverse oftwinnability.

1.60

1.55

1.50

1.45

1.40

1.35

1.30Sum

of i

deal

she

ar s

treng

ths

Στ(G

Pa)

2220181614

ΔγU (mJ/m2)

La

Sr

Ca

Si

Y

Zn

Al

Sn

Mg

Mg

Sc

Li

Cu

Ti

Zr

Mn

Fig. 10. Comparison of twinnabilities of Mg95X predicted by differentdefinitions, i.e. DcU by Eq. (8) and

Ps by Eq. (9). Note that the input

values for these equations are from the pure alias shear deformations asshown in Table 3, and both

Ps and DcU define the inverse of twinnability.

S.L. Shang et al. / Acta Materialia 67 (2014) 168–180 177

DcU ¼ cUT2� cUI2

: ð8Þ

In addition to using the combined GSF energies, thesummation of the ideal shear strengths of the I2 and theT2 faults can also be used to probe the (inverse of)twinnability:X

s ¼ sI2þ sT2

: ð9Þ

Note that unlike Kratio and Kfcc, the smaller the values ofDcU and

Ps, the higher the propensity to twin will be.

The combined GSF energies for Mg95X can be used toassess the propensity to twin and are listed in Table 3,including the Kpur

fcc via Eq. (1), the Kpurratio via Eq. (7), and

the DcpurU via Eq. (8), where the superscript “pur” indicates

the GSF energies computed from the pure alias shear areused. Fig. 9 illustrates the twinnability of Mg95X predictedfrom Kpur

fcc in comparison with the ones from Kpurratio and

DcpurU . A linear relationship is observed between Kpur

fcc andKpur

ratio, but a roughly inversely linear relationship is shownbetween Kpur

fcc and DcU. Similarly, Fig. 10 plots the compar-ison between DcU and

Ps, showing a quite good linear

relationship. By examining Figs. 9 and 10 (see also Table 3),it is found that alloying elements Sr and La increase greatlythe twin propensity of hcp Mg, while Mn, Ti and Zr exhibitopposite trends.

It is known that the coarse-grained hcp metals usuallyneed twinning to accommodate plastic deformation inaddition to dislocation slip due to their lack of sufficientslip systems [74]. Unlike fcc metals, twins are rarelyobserved in nanocrystalline (nc) hcp metals and alloys.Recently, twins were also observed for nc Mg and its alloysprocessed by ball milling, cryomilling or under high strainrates, such as pure Mg [75], Mg–30 wt.% Al [76], Mg–10%Ti [74] and AZ80 (Mg–8.0 wt.% Al–0.5 wt.% Zn) alloys[77]. In addition, it is worth mentioning that some alloyingelements have lower solubility or are largely immiscible in

Mg such as Ca, Si, Sn, Sr, Ti and Zn [78]. Similarly tothe conclusions drew by Yasi et al. [27], the present results(see Table 3) also indicate that the larger misfit of GSFproperties caused by alloying element X corresponds tothe lower solute solubility of X in Mg-based solid solutions[78]. However, the solute solubilities of these alloying ele-ments in metastable Mg-based alloys may be high. Forexample, the Mg–Ti phase diagram [78] indicates that Mgand Ti are almost immiscible, since their enthalpy of mix-ing is positive. However, it was found that alloying elementTi has a high solubility in metastable Mg when processedby ball milling, physical vapor deposition, electron beamdeposition or sputtering of thin films [74,79]. Therefore,the present GSF results are useful as inputs for continuummodels to study Mg alloys and can be used to guide thedevelopment of stable and metastable Mg alloys even whenthe alloying elements have lower solubility or are almost

Fig. 11. Isosurfaces of the differential charge density contours (Dq) of Mg95X, where the reference (or non-interacting) charge density is calculated fromone electronic step—see Eq. (10). These differential charge densities are parallel to the {0001} planes and close to the alloying elements X in Mg95X, andthe same isosurface levels of 0.002 eV�3 are used to draw these plots.

178 S.L. Shang et al. / Acta Materialia 67 (2014) 168–180

immiscible in Mg according to the equilibrium phasediagrams [78].

3.3. Differential charge density

In general, variations of the GSF energies and the idealshear strengths of Mg95X can be analyzed qualitatively interms of the charge density distributions [3,41,80–83].Based on the principles of the DFT theory as well as exper-iments, it is believed that (i) the denser the charge density,the stronger the bonding between atoms; (ii) anisotropy ofcharge density leads to elastic (or bonding) anisotropy asdemonstrated for fcc Al [82]; and (iii) non-spherical distri-bution of charge density retards the redistribution ofcharge density after deformation, and thus hinders theshear deformation, resulting in larger stacking fault energyand ideal shear strength. The differential charge density forMg95X can be estimated by [84]:

DqðMg95XÞ ¼ qinterðMg95XÞ � qnon-interðMg95XÞ ð10Þand contour values are in Dq/eV3. qinterðMg95XÞ is thecharge density after electronic relaxations, and the refer-ence (or non-interacting) charge density qnon-interðMg95XÞis calculated from one electronic step. Fig. 11 illustratesthe DqðMg95XÞ. These differential charge densities areparallel to the {0001} planes and close to the alloying ele-ments X for Mg95X (without ionic relaxations for simplic-ity). Based on the present differential charge densitycontours (plotted using the same levels), it is shown thatalloying elements Ti, Zr, and Sc as well as Al and Sn resultin higher GSF values due to the charge gains (yellowcolor1) around them. For alloying elements Al and Cu,

1 For interpretation of color in Fig. 11, the reader is referred to the webversion of this article.

the shear strengths and the unstable fault energies of Mg95-

Cu are lower than those of Mg95Al due to the spherical dis-tribution of Dq around Cu, although the equilibriumvolume of Cu is smaller than that of Al. Although thealloying elements La, Ca and Sr possess larger atomic vol-umes, there are no connections of the {0001} planes viaDq, resulting in lower GSF values. Regarding alloying ele-ment Mn, this possesses the smallest equilibrium volume(see Table 3) and quite high shear modulus [71], resultingin the highest GSF values of Mg95Mn as shown in Table 3.Here, it is the smallest equilibrium volume exhibited by Mnthat is mainly responsible for the highest GSF values.

In addition to the qualitative explanation of bondinginteractions in terms of the differential charge density, itshould be remarked that the quantitative bond strengths/interactions between atoms can be analyzed using the forceconstants in terms of first-principles phonon calculations[81,85,86]. Force constants could quantify the extent ofinteraction or bonding between atoms. A large positiveforce constant suggests bonding, while a negative forceconstant suggests that the two atoms in question wouldprefer to move apart. A zero or near-zero force constantindicates that the interactions between two atoms are neg-ligible [81,85,86].

4. Conclusions

As a “materials genome” project for computational anddata-driven development of advanced Mg-based alloys, asystematic investigation of the GSF curves has been per-formed in terms of first-principles calculations for diluteMg-based alloys, i.e. Mg95X. Fourteen alloying elements(X) are considered, i.e. Al, Ca, Cu, La, Li, Mn, Sc, Si,Sn, Sr, Ti, Y, Zn and Zr. The GSF curves and the associ-ated stable and unstable stacking and twin fault energies,

S.L. Shang et al. / Acta Materialia 67 (2014) 168–180 179

ideal shear strengths, and twinnabilities are predicted interms of the simple and in particular the pure alias sheardeformations on the {0001} plane and along theh1 0 �1 0i direction of the hcp lattice. It is found that (i)the fault energies (including the stable and the unstable I2

stacking fault and the T2 twin fault) and the ideal shearstrengths (during the generations of I2 and T2 faults) ofMg95X decrease with increasing equilibrium volume ofalloying element X or Mg95X—however, large deviationsfrom the linear relationships are found for alloying ele-ments Al, Cu, Si and Zn; (ii) alloying elements Sr and Laincrease greatly the twinning propensity of hcp Mg, whileMn, Ti, and Zr show opposite trends; and (iii) the changeof the GSF values of hcp Mg caused by alloying elementsX is traceable from the distribution of differential chargedensity (Dq), since a spherical distribution of Dq facilitatesredistribution of charges after shear deformation, resultingin lower values of shear-related properties. Computed GSFproperties for the dilute Mg-based alloys are in goodaccord with available experimental and theoretical resultsin the literature. It is worth emphasizing that critical exper-iments such as measurements of GSF properties will beneeded as part of the “materials genome” project, and thisis especially important for alloy design to acquire certainproperties such as superior room temperature formabilityfor as-of-yet unavailable wrought Mg alloys.

Acknowledgements

This work was financially supported by the ArmyResearch Lab under project no. W911NF-08-2-0064, theNational Science Foundation (NSF) with grant no.DMR-1006557, and the Center for Computational Materi-als Design (CCMD), a joint NSF Industry/UniversityCooperative Research Center at Penn State (IIP-1034965)and Georgia Tech (IIP-1034968). First-principles calcula-tions were carried out partially on the LION clusters sup-ported by the Materials Simulation Center and theResearch Computing and Cyber infrastructure unit at thePennsylvania State University, and partially on the re-sources of NERSC supported by the Office of Science ofthe US Department of Energy under contract no. DE-AC02-05CH11231. The authors also would like to thankthe reviewer for constructive comments and suggestions.

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