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Supplementary information for “Strengthen Mg Alloys by Planar

Faults and Solutes”

William Yi Wang,a, b,* Yi Wang,b Shun Li Shang,b Kristopher A. Darling,c Hongyeun Kim,b Bin

Tang,a Hong Chao Kou,a Suveen N. Mathaudhu,d Xi Dong Hui,e Jin Shan Li,a Laszlo J. Kecskes,c

and Zi-Kui Liu b, *

a State Key Laboratory of Solidification Processing, Northwestern Polytechnical University, Xi’an, Shaan Xi 710072, China

b Department of Materials Science and Engineering, The Pennsylvania State University, University Park, PA 16802, USA

c U.S. Army Research Laboratory, Weapons and Materials Research Directorate, RDRL-WMM-B, Aberdeen Proving Ground, MD 21005, USA

d Department of Mechanical Engineering, University of California - Riverside, Riverside, CA, 92521, USA

e State Key Laboratory for Advanced Metals and Materials, University of Science and Technology Beijing, Beijing 100083, China

* Corresponding authors: WYW ([email protected]) and ZKL ([email protected]).

1

mailto:[email protected]

mailto:[email protected]

Phonon calculations, Debye model and Debye temperature

From phonon density of states, the lattice vibrational contribution to Helmholtz free

energy can be calculated through 1, 2, 3, 4, 5

Equation S1Fvib(V ,T )=κBT∫ ln {2 sinh [ ℏω

2κBT ]}g (ω )dω

where κB

is the Boltzmann constant; T the temperature, g(ω )

the phonon density of states as a

function of phonon frequency ω

at volume V. Alternatively, Fvib(V ,T )

can also be described by

the Debye Temperature (ΘD

) as

Equation S

2

κBΘD (n)=hωD(n )

ωD( n)=[n+33 ∫0

ωmaxωn g(ω )dω]1/n

with n≠0, n>-3

where ωD( n)

is the Debye cutoff frequency. The nth moment Debye temperature is obtained by

Equation S3 ΘD=

ℏκBωD( n)

With different value of n, the obtained Debye temperature related to different physical meaning 6,

for instance, ΘD(2) usually links to the Debye temperature gained from the heat capacity data 4, 5.

Consequently, the vibrational contribution to Helmholtz free energy of SFs and LPSOs

can be expressed by Debye model through Debye temperature 5

Equation S4 Fvib(V ,T )=9

8κ BΘD+κBT {3 ln [1−exp(−ΘDT )−D(ΘDT )]}

2

where D(ΘD/T )

is the Debye function given by

Equation S5

D( x )= 3x3∫0

x t3

exp( t )−1dt

The vibrational entropy can be wrote as

Equation S6

Svib=3κB {43D(ΘDT )−ln [1−exp (−ΘDT )− ]}

Here, Debye temperatures of the atom in each layer of SFs and LPSOs are also determined by

phonon frequency shown in Equation S3, which are summarized in the following Table S1.

3

Table S1. Debye temperature (ΘD , K) of the atom in each layer of stacking faults (SFs) and long periodic stacking ordered structures (LPSOs) of Mg. The second moment of phonon DOS is

used to derive the ΘD in this work.

Atomic Layer

SFs LPSOsI1 I2 EF 6H 10H 14H 18R 24R

L1 327.9 326.6 337.2 333.5 311.1 338.9 332.1 292.7L2 333.8 341.1 334.6 332.9 320.3 334.7 354.3 301.7L3 328.0 344.3 330.1 337.7 323.1 341.3 357.6 309.1L4 330.9 344.9 323.4 341.3 320.3 336.9 354.3 346.7L5 330.9 341.6 320.4 337.7 311.1 341.3 334.1 379.8L6 330.9 335.1 323.4 332.9 310.1 334.7 317.4 360.9L7 329.6 335.1 330.1 318.0 338.9 332.1 317.7L8 329.6 341.6 334.6 320.5 338.9 357.5 302.7L9 344.9 334.6 318.0 334.7 359.8 292.7L10 344.3 310.1 341.3 354.3 301.7L11 341.1 336.9 334.3 308.8L12 326.6 341.3 317.5 346.7L13 339.3 334.7 332.2 380.0L14 338.9 357.4 361.0L15 359.8 317.7L16 354.3 302.7L17 334.2 292.6L18 317.4 301.7L19 308.9L20 346.6L21 380.0L22 361.1L23 317.6L24 302.6

Total 330.2 339.3 330.3 336.0 316.4 338.1 343.8 334.5HCP Mg

321.5323a, 325b, 320c

Note: SHigh: the atomic layers with high vibrational entropy at high temperaturea. Zhang, et al., derived from the second moment of phonon DOS. 4

b. Seitz F. and Trunbull D. Solid. State. Physics. New York: Academic Press;1964 (Exp.)c. Dederch ,et al., Metals: Phonon states, electron states and Fermi surfaces. Berlin: Springer-

Verlag: 1981 (Exp.)

4

Table S2. Calculated and experimental elastic properties (in GPa) of Mg-X alloys

X Conditions C11 C12 C13 C33 C44 B G E υ B/G Remark

Mg

PAW-PBE 57.55 28.225

21.33 65.35 12.1 36.34 15.45 40.59 0.31 2.35 2 atoms22*22*12 294 eV

59.15 27.93 21.3 66.30 12.4 36.34 16.23 42.38 0.31 2.24 2 atoms25*25*14 294 eV

70.00 25.60 12.90 84.55 18.80 36.14 22.49 55.88 0.24 1.61 36 atoms 7*7*6 294 eV

70.35 25.78 13.25 84.85 18.90 36.39 22.57 56.11 0.24 1.61 36 atoms 7*7*6 294 eV (Fully relax)


59.40 28.00 22.50 63.50 14.20 36.19 16.74 43.50 0.30 2.16 96 atoms5*5*4 294 eV

77.90 21.18 14.10 73.45 20.80 36.24 25.26 61.50 0.22 1.44 96 atoms6*6*5 400 eV

60.3 28.8 21.7 66.5 15.7 36.8 16.9 44.0 - 2.18 3

25*25*25 500 eV

PAW-PBEsol 57.2 26.9 20.5 62.6 14.9 37.8 16.3 42.7 - 2.31

PAW-GGA 64.57 24.14 21.63 64.78 14.83 35.84 17.87 45.98 0.29 2.00 2 atoms22*22*12 294 eV

58.1 27.6 21.6 64.7 14.2 35.8 16.1 41.9 0.31 2.23 2 atoms 7

25*25*16 294 eV

69.40 24.83 13.33 82.10 19.30 35.72 21.96 54.66 0.25 1.63 36 atoms7*7*6 294 eV



63.5 24.9 20.0 66.0 19.3 35.8 18.5 47.4 0.28 1.93 36 atoms 8

6*6*5 273 eV74.40 20.75 15.10 71.75 17.30 35.74 23.10 57.02 0.23 1.54 96 atoms

6*6*5 400 eV67.5 24.76 24.1 72.4 23.97 39.3 22.8 57.3 0.23 1.72 9

Exp. (0K) 63.5 25.9 21.7 66.5 18.4 36.9 19.4 49.5 0.28 1.90 10

Exp. (298K) 59.4 25.6 21.4 61.6 16.4 35.2 17.4 44.8 - 2.02 10

Exp. (298K) 59.5 25.9 21.8 61.6 16.4 35.6 17.3 44.6 - - 11

Note: B=Bulk modulus, G=Shear modulus, E=Young’s modulus, λ=Spring constant,

υ=Poisson’s ratio

5


X Conditions C11 C12 C13 C33 C44 B G E υ B/G RemarkAl PAW-PBE 60.1

028.68 20.75 65.8

016.70 36.1

117.59 45.4

10.29 2.05 1.04 at%

5*5*4 336 eV

72.70

20.45 18.03 71.40

14.30 36.57

21.65 54.24

0.25 1.69 1.04 at%6*6*5 400 eV

60.11

28.65 20.74 65.80

16.70 36.10

17.60 45.41

0.29 2.05 1.04 at%7*7*6 336 eV

61.10

28.58 17.18 74.30

12.90 35.73

17.21 44.49

0.29 2.08 2.77 at%7*7*6 336 eV

64.00

28.38 18.68 74.25

18.80 37.21

20.34 51.61

0.27 1.83 2.77 at%8*8*7 400 eV (Fully relax)

58.79

30.01 17.44 74.25

12.91 35.69

16.45 42.77

0.30 2.16 2.77 at%7*7*6 336 eV (ISIF=4)

PAW-GGA

65.6 25.9 19.3 69.0 13.6 36.6 18.5 47.4 0.28 1.98 2.77 at% 8

6*6*5 339 eV

Exp. (298K)

- - - - - - - 45.2 - - 2.7 at% 12

Li PAW-PBE 73.60

22.73 15.05 69.30

20.10 35.67

23.73 58.26

0.23 1.50 1.04 at%5*5*4 380 eV(Li-sv)

66.03

20.57 18.70 66.71

13.47 34.99

19.35 49.01

0.27 1.81 2.77 at%7*7*6 669 eV(Li-sv)

PAW-GGA

63.05

25.45 17.90 73.15

17.4 35.61

20.27 51.11

0.26 1.77 2.77 at%8*8*7 400 eV(Li-sv)

61.3 17.6 23.33 61.10

15.10 34.57

17.73 45.41

0.28 1.95 2.77 at%7*7*6 294 eV(Li )

58.9 24.5 23.2 54.0 15.0 34.8 16.2 42.0 0.30 2.17 2.77 at% 8

6*6*5 Exp. (298K)

59.0 25.9 21.7 61.0 16.2 35.3 17.1 44.1 0.29 2.06 3.02 at% 11

Exp. - - - - - - - 45.8 - - 12

Ti PAW-PBE 63.00

25.98 20.95 65.47

17.15 36.41

18.85 48.23

0.28 1.93 1.04 at%7*7*6 311 eV

62.65

32.58 18.80 75.80

15.80 37.90

17.93 46.46

0.30 2.11 2.77 at%7*7*6 311 eV

62.46

27.61 19.31 73.83

15.37 36.32

18.23 46.85

0.28 1.99 2.77 at%7*7*6 311 eV

6


X C11 C12 C13 C33 C44 B G E υ B/G Remark

Ca 60.64 28.92 22.94 65.98 13.64 37.47 16.31 42.72 0.31 2.30 1.04 at%

63.82 30.54 21.32

6

73.36 14.04 38.64 17.24 45.02 0.31 2.24 2.77 at%

56.4 27.6 22.3 60.4 16.0 35.2 15.4 40.4 0.31 2.28 2.77 at% Ref: 8

Cu 55.82 25.39 25.00 58.57 10.46 35.69 14.00 37.13 0.33 2.55 1.04 at%

63.30 21.15 18.26 64.63 13.40 34.06 18.50 47.00 0.27 1.84 2.77 at%

61.6 24.5 25.7 62.7 15.9 37.5 17.4 45.2 0.30 2.16 2.77 at% Ref: 8

Fe 61.41 20.96 23.71 59.69 15.64 35.57 18.02 46.23 0.28 1.97 1.04 at%

64.13 19.95 16.97 68.38 22.64 33.93 23.05 56.37 0.22 1.47 2.77 at%

K 61.44 28.53 22.22 65.09 15.23 37.21 17.12 44.52 0.30 2.17 1.04 at%

67.56 27.36 21.15 71.57 13.63 38.39 18.78 48.45 0.29 2.04 2.77 at%

54.7 23.1 21.7 56.4 13.8 33.2 15.3 39.7 0.30 2.17 2.77 at% Ref: 8

La 63.29 25.17 23.28 68.07 15.13 39.02 17.69 46.10 0.30 2.21 1.04 at%

60.81 32.52 22.15 71.06 17.29 37.65 18.40 47.46 0.29 2.04 2.77 at%

Mn 57.95 11.84 19.16 64.40 17.82 31.29 20.63 50.73 0.23 1.52 1.04 at%

55.81 23.60 20.84 62.64 19.00 33.95 18.16 46.23 0.27 1.87 2.77 at%

Na 60.19 24.57 24.90 59.94 13.39 36.59 16.50 43.02 0.30 2.22 1.04 at%

68.23 22.97 20.03 69.38 13.91 36.80 19.59 49.91 0.27 1.89 2.77 at%

Ni 56.21 21.48 26.25 52.37 11.00 35.10 14.57 38.41 0.32 2.41 1.04 at%

63.44 18.28 22.14 59.67 10.16 34.43 17.23 44.30 0.29 2.00 2.77 at%

63.5 25.9 23.8 69.6 19.2 38.2 19.6 50.3 0.28 1.94 2.77 at% Ref: 8

Si 61.58 23.88 22.95 63.77 16.69 36.24 18.29 46.98 0.28 1.98 1.04 at%

57.04 27.57 20.93 65.66 11.17 35.40 14.77 38.90 0.32 2.40 2.77 at%

65.0 27.3 19.7 70.1 14.5 37.0 18.5 47.5 0.29 2.01 2.77 at% Ref: 8

Sn 62.83 26.06 22.04 65.71 17.79 36.83 19.07 48.78 0.28 1.93 1.04 at%

63.64 26.73 20.76 72.05 12.51 37.26 17.37 45.11 0.30 2.15 2.77 at%

Sr 63.77 26.90 22.43 67.76 14.78 37.66 17.91 46.38 0.29 2.10 1.04 at%

65.64 30.11 22.35 72.56 13.87 39.31 17.69 46.15 0.30 2.22 2.77 at%

Ti 63.00 25.98 20.95 65.47 17.15 36.41 18.85 48.23 0.28 1.93 1.04 at%

7

62.46 27.61 19.31 73.83 15.37 36.32 18.23 46.85 0.28 1.99 2.77 at%

Y 62.24 26.74 22.63 67.46 14.27 37.38 17.52 45.46 0.30 2.13 1.04 at%

66.29 31.01 19.09 77.74 18.25 38.80 20.27 51.79 0.28 1.91 2.77 at%

59.5 27.3 21.6 64.5 19.0 36.1 18.3 47.1 0.28 1.97 2.77 at% Ref: 8

Zn 60.11 25.36 22.89 62.55 13.62 36.10 16.59 43.15 0.30 2.18 1.04 at%

65.23 23.78 16.08 73.57 15.92 35.14 20.25 50.97 0.26 1.73 2.77 at%

62.3 25.5 23.1 66.2 14.1 37.1 17.3 44.8 0.30 2.15 2.77 at% Ref: 8

48.0 Exp.

Zr 63.80 27.02 21.30 66.91 16.24 37.06 18.53 47.66 0.29 2.00 1.04 at%

61.30 31.63 19.92 73.11 17.19 37.66 18.07 46.74 0.29 2.08 2.77 at%

Note: PAW-GGA-PBE pseudopotential is selected.

8

Figure S1. Correlation and connection between various stacking faults (SFs) and long periodic

stacking order structures (LPSOs) revealed by the deformation electron density. (a) the square

relationship between stacking fault energy and electron redistribution (δΔρ ) 13. (b) the

correlation between the formation energy of LPSO structures and the number of fault layers 14.

The line represents the ideal linear relation between the formation energy and the number of fault

layers. (c) and (d) the bond structures of SFs and LPSOs characterized by Δρ=0 . 0021e-/Å3

isosurface, respectively.

9

Figure S2. Variation in force constants as a function of bond length between atoms up to 8 Å, (a)

growth fault; (b) deformation fault and (c) extrinsic fault. Bond length splitting of the first

nearest neighbor shown in the insert image presents the interactions between fault-fault, fault-

non-fault and non-fault-non-fault layers

10

Figure S3. Local phonon density of states (LPDOS) of atoms in each layer of I1 together with

their bond structure, (a) LPDOS curve and (b) the 0.5Δρmax charge density isosurface plotted in

prismatic plane.

11

Figure S4. Local phonon density of states (LPDOS) of atoms in each layer of I2 together with


prismatic plane.

12

Figure S5. Local phonon density of states (LPDOS) of atoms in each layer of EF together with


prismatic plane.

13

Figure S6. Prismatic plane ((100)S.C.) view of Δρ=0.0021 e-/Å3 isosurface plots Mg97Zn1Y2 with

atomic array of Y and Zn. The bond structure around the solute atoms is anomalous due to the

electron redistribution. Around the solute atoms, the Δρ of the basal plane is increased while the

Δρ along the prismatic and the pyramidal planes are decreased. The strengthened electron

density by alloying elements indicates a qualitative description of the stronger pinning effect.

Moreover, the weakened bond strength of Mg matrix in the prismatic plane by the fault layers

and solute atoms indicates a possible non-basal slip system could occur during deformation,

which could improve the ductility of Mg alloys

14

References

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3. Wrobel J, Hector Jr LG, Wolf W, Shang SL, Liu ZK, Kurzydlowski KJ. Thermodynamic and mechanical properties of lanthanum-magnesium phases from density functional theory. J Alloy Compd 512, 296-310 (2012).

4. Zhang H, Shang S-L, Wang Y, Chen L-Q, Liu Z-K. Thermodynamic properties of Laves phases in the Mg-Al-Ca system at finite temperature from first-principles. Intermetallics 22, 17-23 (2012).

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9. Ouyang YF, et al. First-principles calculations of elastic and thermo-physical properties of Al, Mg and rare earth lanthanide elements. Physica B 404, 2299-2304 (2009).

10. Slutsky LJ, Garland CW. Elastic Constants of Magnesium from 4.2 K to 300 K. Phys Rev 107, 972-976 (1957).

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12. Hardie D. ELASTIC PROPERTIES OF MAGNESIUM SOLID SOLUTIONS. Acta Metal 19, 719-& (1971).

13. Wang WY, et al. Electron localization morphology of the stacking faults in Mg: A first-principles study. Chem Phys Lett 551, 121-125 (2012).

14. Wang WY, et al. Electronic structures of long periodic stacking order structures in Mg: A first-principles study. J Alloy Compd 586, 656-662 (2014).

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