Energetic Analysis of Deformation Twins and Twinning Dislocations in Magnesium

Daisuke Matsunaka1,2,+, Akira Kanoh1 and Yoji Shibutani1,2

1Department of Mechanical Engineering, Osaka University, Suita 565-0871, Japan2Center for Atomic and Molecular Technologies, Osaka University, Suita 565-0871, Japan

Twin boundaries and twinning dislocations for ð10�12Þ and ð10�11Þ twins in Mg are investigated, using the generalized-embedded-atom-method interatomic potential. The twin boundary energy for the ð10�12Þ twin is found to be larger than that for the ð10�11Þ twin. On the otherhand, both the dislocation energy and the Peierls barrier of the twinning dislocation are low for the ð10�12Þ twin. This implies that the ð10�12Þ twinis capable of having a winding morphology and can grow easily. [doi:10.2320/matertrans.M2013095]

(Received March 11, 2013; Accepted June 3, 2013; Published July 25, 2013)

Keywords: magnesium, twin boundary, twinning dislocation, generalized-embedded-atom-method interatomic potential

1. Introduction

Deformation twinning is an elementary deformationmechanism that occurs during plastic deformation ofhexagonal close-packed (hcp) crystals.1,2) A detailed under-standing of the operative deformation twinning under aspecified deformation condition still remains a challenge.

There are two typical twinning modes for Mg: ð10�12Þtensile twinning and ð10�11Þ compressive twinning, for whichthe twinning shear is the result of tensile or compressivedeformation along the hcp c axis,1) respectively. The ð10�12Þtwin is generated during an early stage of plastic deformationand contributes to the total strain. On the other hand, theð10�11Þ twin appears at a late or final stage and serves to relaxthe stress concentration.3) While the critical resolved shearstress (CRSS) for ð10�12Þ twinning is only a few MPa,4) theCRSS for ð10�11Þ twinning is about 100MPa.5) The muchlower CRSS for the ð10�12Þ tensile twinning and the earlygeneration of the ð10�12Þ twin are responsible for themechanical anisotropy exhibited by Mg.

The fundamental processes involved in deformationtwinning, including twin nucleation, propagation and thick-ening of a twin lamella, are associated with the motion oftwinning dislocations at twin boundaries.6­11) A twinningdislocation has characteristics of both a dislocation with theBurgers vector b and a step of height h. Based on atomicsimulations with a two-body interatomic potential, Serraet al. showed that twinning dislocations for the ð10�12Þ twincould more easily glide than those for the ð10�11Þ twin in hcpmetals, and concluded that there was a correlation betweenthe mobility of the twinning dislocations and their corewidth.8,9) Recently, Wang et al. carried out an atomisticinvestigation of twinning dislocations for ð10�11Þ and ð10�13Þcompressive twins in Mg using an embedded-atom methodpotential, and estimated the Peierls barrier for the motion ofthe twinning dislocations.11)

In the present study, we investigate twin boundaries andtwinning dislocations for the ð10�12Þ and ð10�11Þ twins inMg using a generalized embedded-atom method (GEAM)potential. For the ð10�12Þ and ð10�11Þ twins, the twin boundaryenergy was calculated. The dislocation energy and the Peierlsbarrier of twinning dislocations were also evaluated.

2. Twin Boundary Energy

The mirror twin structures were modeled using analternately layered geometry of parent and twin phases in athree-dimensional periodic supercell. To describe interatomicinteractions in Mg, the GEAM potential developed byZhou et al.,12) was adopted. The geometry optimization wasachieved until the forces on all atoms were smaller than1 µeV/¡. The optimized atomic structures of the ð10�12Þand ð10�11Þ twin boundaries are shown in Fig. 1. As foundin several previous studies,8,9) the internal stress, whichalternated in sign from atom to atom, was generated at theboundaries. The atoms in the boundary plane were subject toexpansive or compressive hydrostatic pressure, depending onthe atomic volume. For the ð10�12Þ twin boundary, changesin the excess potential energy extended to the second atomiclayer, whereas for the ð10�11Þ twin boundary, they wererestricted to the boundary plane itself.

The twin boundary energy £T for the ð10�12Þ and ð10�11Þtwins was calculated as follows;

£T ¼ ðETB � N®Þ=2A; ð1Þwhere ETB is the total energy of the coherent twin model withtwo twin boundaries in the supercell, N is the number ofatoms in the supercell, ® is the hcp bulk energy per Mg atom,and A is the area of the twin boundary. Since the atomicstructure in the twin model was optimized, £T was evaluatedat absolute zero temperature. Figure 2 shows £T as a functionof the twin thickness. Almost no dependence of £T on thetwin thickness is seen, implying that the elastic field due tothe twin boundary was just weak. For the ð10�12Þ twin, £Twas larger than that for the ð10�11Þ twin; at a twin thicknessof 20 atomic K1 planes, 106mJ/m2 for the ð10�12Þ twin and73mJ/m2 for the ð10�11Þ twin.

Density functional theory (DFT) calculations were alsocarried out to evaluate £T. The DFT calculations wereperformed using the projector augmented wave method13)

as implemented in the ab initio total-energy and molecular-dynamics program VASP.14,15) The exchange-correlationpotential was calculated within the generalized gradientapproximation using the Perdew­Wang parametrization.16)

The electron wavefunctions were expanded by plane wavesup to an energy cutoff of 212 eV. For the ð10�12Þ and ð10�11Þtwin boundaries, the supercell sizes were ð1=3Þ½�12�10� �+Corresponding author, E-mail: matsunaka@mech.eng.osaka-u.ac.jp

Materials Transactions, Vol. 54, No. 8 (2013) pp. 1524 to 1527©2013 The Japan Institute of Metals and Materials EXPRESS REGULAR ARTICLE

½10�1�1� � 2wt and ð1=3Þ½�12�10� � ½10�1�2� � 2wt where wt isthe twin thickness, and the Brillouin zone was sampledusing 25 © 13 © 5 and 25 © 7 © 5 k-point grids with theMonkhorst­Pack scheme, respectively. In the DFT calcu-lations, the atomic geometry was optimized until the forceson all atoms were smaller than 0.02 eV/¡. The DFT resultsof £T are also shown in Fig. 2. The GEAM and DFT resultsof £T were qualitatively consistent, but the £T valuesevaluated with the GEAM potential were some under-estimated in comparison with the DFT ones. This is becausethe GEAM potential was fitted to a perfect crystal, andalthough it describes the elastic properties, it does notnecessarily give the exact interactions at inhomogeneousregions. The DFT results of £T, obtained for a twinthickness of 20 atomic K1 planes, were 125mJ/m2 for theð10�12Þ twin and 84mJ/m2 for the ð10�11Þ twin, respectively.These £T values are in agreement with previous DFTstudies.17,18)

3. Twinning Dislocation

Twinning dislocations were next incorporated into thealternately layered twin model, so that each twinningboundary in the supercell contained one twinning dislocation.

For the ð10�12Þ twin, twinning dislocations considered hadthe step height of two K1 layers and the Burgers vector of

bð10�12Þ

2 ¼ 3� r2

3þ r2½�1011�: ð2Þ

For the ð10�11Þ twin, the twinning dislocation with the stepheights of four and two K1 layers and the Burgers vectors of

bð10�11Þ

4 ¼ 4r2 � 9

4r2 þ 3½10�12� ð3Þ

and

bð10�11Þ

2 ¼ ð1=2Þbð10�11Þ4 � ð1=6Þ½1�210� ð4Þrespectively. The dislocation lines of the bð10

�12Þ2 and bð10

�11Þ4

twinning dislocations were parallel to ½1�210� and normal totheir Burgers vectors. In this defective twin model, the twinboundary was shifted by the step, but the periodic conditionwas preserved. In the same manner as for the coherent twinmodel, geometry optimization was carried out. Figure 3shows the obtained structures of the twinning dislocations fora twin thickness of 20 atomic K1 planes. The motion of thetwinning dislocation at the twin boundary was evaluated bycalculating the minimum energy path for the gliding using thenudged elastic band (NEB) method.19) With respect to the

0 5 10 15 20 2550

60

70

80

90

100

110

120

130

Twin thickness / units of K1 plane

Twin

bou

dary

ene

rgy,

γT

/ m

Jm−

2

DFT (10−12)

GEAM (10−12)

DFT (10−11)

GEAM (10−11)

Fig. 2 Twin boundary energies for ð10�12Þ and ð10�11Þ twins as a function ofthe twin thickness.

(a) (b)

Fig. 1 Atomic structures of twin boundaries of (a) ð10�12Þ and (b) ð10�11Þtwins. Atoms are colored by their excess potential energies.

(a) (b) (c)

Fig. 3 Atomic structures of (a) bð10�12Þ

2 , (b) bð10�11Þ

4 and (c) bð10�11Þ

2 twinning dislocations. Atoms are colored by their excess potentialenergies. The twin thickness for the defective twin models with twinning dislocations is 20 atomic K1 planes.

Energetic Analysis of Deformation Twins and Twinning Dislocations in Magnesium 1525

initial and final states, one of the twinning dislocations inthe supercell moves in the ©1 direction. Figure 4 shows thedislocation energy of the twinning dislocation along theminimum energy paths, calculated as

W ¼ ðETD � £TA�N®Þ=2L; ð5Þwhere ETD is the total energy of the defective twin model inthe supercell and L is the length of the twinning dislocation.Although a slight difference in the dislocation energybetween the initial and final states was due to the elasticinteraction of the twinning dislocations, the lower state wastaken to be the equilibrium state. The difference in thedislocation energy between the equilibrium and transitionstates along the minimum energy path is evaluated as thePeierls barrier, which is an energy barrier against movementof a straight dislocation to a neighboring site in the ©1direction. The obtained results for the dislocation energyand the Peierls barrier are summarized in Table 1. For theð10�12Þ twin, the dislocation energy at the equilibrium initialor finial state was considerably small, compared with that forthe ð10�11Þ twin. Moreover, the Peierls barrier of the twinningdislocation for the ð10�12Þ twin was extremely low, inagreement with the results of atomic simulations reported

by Serra et al.8,9) Therefore, the ð10�12Þ twin boundaries cancontain many twinning dislocations, which can easily glide.This implies that the ð10�12Þ twin is capable of having awinding morphology, consistent with a thick lenticularð10�12Þ twin observed experimentally. Furthermore, thefundamental processes involved in deformation twinningare attributed to the motion of twinning dislocations,although the detailed nucleation mechanisms are controver-sial. Thus, the low dislocation energy and Peierls barrier ofthe bð10

�12Þ2 twinning dislocation are responsible for the low

CRSS for ð10�12Þ twinning.As seen in Fig. 4(b), for the bð10

�11Þ4 twinning dislocation, a

stable state also existed at the center of the minimum energypath. At this intermediate state, one of the two vertices of thetwinning dislocation step had moved and its riser plane wasinclined. This broadened structure of the bð10

�11Þ4 twinning

dislocation can be recognized as the consequence of partialglide, similar to the case for an extended dislocation.bð10

�11Þ4 is identified as the sum of two bð10

�11Þ2 Burgers vectors

with the opposite screw components of ð1=6Þ½1�210� andð1=6Þ½�12�10�.11) The bð10

�11Þ2 twinning dislocation can glide

more easily than the bð10�11Þ

4 twinning dislocation, because ofless atomic shuffles. Meanwhile, the magnitude of bð10

�11Þ4 is

smaller than bð10�11Þ

2 in Mg. According to the Frank’s rule,complete dissociation of one bð10

�11Þ4 twinning dislocation

into two bð10�11Þ

2 twinning dislocations is unfavorable. Thus,the motion of the bð10

�11Þ4 twinning dislocation was comprised

of glide of each of its two constituent bð10�11Þ

2 twinningdislocations rather than as a perfect dislocation.

4. Summary

In summary, we have investigated twin boundaries andtwinning dislocations for ð10�12Þ and ð10�11Þ twins in Mg,using the GEAM interatomic potential. The twin boundaryenergy is found to be larger for the ð10�12Þ twin than for theð10�11Þ twin based on both the GEAM and DFT results. Thecoherent twin boundary of the ð10�12Þ twin is less stablethan that of the ð10�11Þ twin. On the other hand, both thedislocation energy and the Peierls barrier of the bð10

�12Þ2

twinning dislocation for the ð10�12Þ twin are low. This impliesthat the ð10�12Þ twin is capable of having a windingmorphology. Furthermore, given that twinning mechanismsinvolve glide of twinning dislocations, the low dislocationenergy and Peierls barrier of the bð10

�12Þ2 twinning dislocation

103.948

3.949

3.95

3.951

3.952

3.953

Dis

loca

tion

ener

gy /

meV

Å-1

Reaction coordinate

(a)

10100

105

110

115

120

Dis

loca

tion

ener

gy /

meV

Å-1

Reaction coordinate

(b)

1070.2

70.3

70.4

70.5

70.6

70.7

70.8

70.9

Dis

loca

tion

ener

gy /

meV

Å-1

Reaction coordinate

(c)

Fig. 4 Variation in the dislocation energy of the twinning dislocation alongthe minimum energy path for (a) bð10

�12Þ2 , (b) bð10

�11Þ4 and (c) bð10

�11Þ2 twinning

dislocations.

Table 1 Dislocation energies and Peierls barriers of the twinningdislocations for the ð10�12Þ and ð10�11Þ twins in Mg.

TwinBurgersvector

Twinthickness

[units of K1

plane]

dislocationenergy

[meV/¡]

Peierlsbarrier

[meV/¡]

ð10�12Þ bð10�12Þ

2 16 3.82 4.03 © 10¹3

20 3.95 3.74 © 10¹3

ð10�11Þ bð10�11Þ

4 16 92.0 1.16 © 10

20 105 1.10 © 10

bð10�11Þ

2 16 62.4 5.86 © 10¹1

20 70.2 6.16 © 10¹1

D. Matsunaka, A. Kanoh and Y. Shibutani1526

are responsible for the low CRSS for ð10�12Þ twinning. Forthe ð10�11Þ twin, the bð10

�11Þ2 twinning dislocation is more

mobile than the bð10�11Þ

4 twinning dislocation. The motion ofthe bð10

�11Þ4 twinning dislocation is comprised of glide of each

of the two constituent bð10�11Þ

2 twinning dislocations rather thanas a perfect dislocation.

Acknowledgments

The authors are grateful to Mr. T. Teramoto for hiscomputational help. This work was supported by the ToyotaInitiative on Magnesium alloys.

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Energetic Analysis of Deformation Twins and Twinning Dislocations in Magnesium 1527