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
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)
69.75 26.90 18.08 72.00 21.00 36.88 23.03 57.19 0.24 1.60 36 atoms 8*8*7 400 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
67.50 22.33 18.43 70.30 16.00 35.96 20.66 52.01 0.26 1.74 36 atoms 7*7*6 294 eV (Fully relax)
65.90 24.78 18.05 73.45 17.70 36.24 21.20 53.21 0.26 1.71 36 atoms 8*8*7 400 eV (Fully relax)
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
Table S3. Calculated and experimental elastic properties (in GPa) of Mg-X alloys
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
Table S4. Calculated and experimental elastic properties (in GPa) of Mg-X alloys
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
their bond structure, (a) LPDOS curve and (b) the 0.5Δρmax charge density isosurface plotted in
prismatic plane.
12
Figure S5. Local phonon density of states (LPDOS) of atoms in each layer of EF together with
their bond structure, (a) LPDOS curve and (b) the 0.5Δρmax charge density isosurface plotted in
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
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