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Captions
Table 1 Young’s modulus of polycrystalline aggregate of Al, Cu, Zn and Fe at room temperature,
Epoly and maximum solubility limit of Cu, Zn and Fe in Al, S.
Table 2 Lattice constants a, b, c, α, β, γ and mass density ρ of Cu, CuZn, Cu5Zn8, CuZn4 and Zn.
The present calculation results are compared with experimentally reported values.
Table 3 Elastic constants Cij of single crystal of Cu, CuZn, Cu5Zn8, CuZn4 and Zn. The present
calculation results are compared with experimentally reported and previously computed values.
Table 4 Elastic compliances Sij of single crystal of Cu, CuZn, Cu5Zn8, CuZn4 and Zn. The present
calculation results are compared with experimentally reported and previously computed values.
Table 5 Bulk modulus Bpoly, shear modulus Gpoly and Young’s modulus Epoly of polycrystalline
aggregate of Cu, CuZn, Cu5Zn8, CuZn4 and Zn. The present calculation results in Voigt, Reuss and
Hill models are compared with experimentally reported and previously computed values.
Table 6 Poisson’s ratio ν, bulk modulus to shear modulus ratio BH/GH and universal elastic
anisotropy index AU of Cu, CuZn, Cu5Zn8, CuZn4 and Zn. The present calculation results are
compared with experimentally reported and previously computed values.
Fig.1 Elemental atom distribution of (a) CuZn, (b) Cu5Zn8 and CuZn4 in (c) VCA model or (d) SQS
(3×3×3 supercell) model before geometry optimization.
Fig.2 Mole fraction dependence of (a) bulk modulus BH, (b) shear modulus GH and (c) Young’s
modulus EH of polycrystalline aggregate for Cu, CuZn, Cu5Zn8, CuZn4 and Zn. The present
calculation results in Hill model are compared with experimentally reported and previously
computed values.
Fig.3 Mass density dependence of bulk modulus BH of polycrystalline aggregate for Cu, CuZn,
Cu5Zn8, CuZn4 and Zn. The present calculation results in Hill model are compared with
experimentally reported and previously computed values.
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52
53
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58
59
60
61
62
63
64
65
Fig.4 Directional anisotropy of Young’s modulus E of single crystal for Cu, CuZn, Cu5Zn8,
CuZn4 and Zn. The magnitude of E in each direction is illustrated not only by color-coding
according to each color scale but also by the distance from the center of the three-dimensional
space.
Table
Table 1 Young’s modulus of polycrystalline aggregate of Al, Cu, Zn and Fe
at room temperature, Epoly and maximum solubility limit of Cu, Zn and Fe in
Al, S.
Epoly (GPa) [2]
S (at%) [3]
Al
70.6
Cu
129.8
2.48
Zn
104.5
66.4
Fe
211.4
0.025
Table 2 Lattice constants a, b, c, α, β, γ and mass density ρ of Cu, CuZn, Cu5Zn8, CuZn4 and Zn.
The present calculation results are compared with experimentally reported values.
Phase
Space group
a (Å)
b (Å)
c (Å)
α (deg.)
β (deg.)
γ (deg.)
ρ (g/cm3)
Cu
Fm-3m (225)
Present
3.630
8.824
Exp.
3.615
8.935
CuZn
Pm-3m (221)
Present
2.959
8.264
Exp.
2.958
8.270
Cu5Zn8
I-43m (217)
Present
8.858
8.035
Exp.
8.878
7.981
Present(VCA)
2.747
4.291
7.702
Present(SQS)
2.744
2.748
4.291
90.00
90.01
120.99
7.782
Exp.
2.738
4.294
7.744
Present
2.640
4.977
7.231
Exp.
2.665
4.947
7.138
CuZn4
P63 /mmc (194)
Zn
P63 /mmc (194)
[18]
[19]
[20]
[21]
[22]
Table 3 Elastic constants Cij of single crystal of Cu, CuZn, Cu5Zn8, CuZn4 and Zn. The present calculation
results are compared with experimentally reported and previously computed values.
Elastic constant of single crystal (GPa)
Cu
CuZn
Cu5Zn8
CuZn4
Zn
C11
C12
C13
C22
C23
C33
C44
C55
C66
Present
170.8
121.5
75.3
Cal.
176.0
118.2
81.9
[29]
Cal.
183.5
125.9
80.9
[30]
Cal.
171.2
123.1
72.4
[31]
Cal.
171.1
122.2
75.3
[32]
Exp. at 4.2 K
176.2
124.9
81.8
[33]
Exp. at RT
170.0
122.5
75.8
[34]
Exp. at RT
168.1
121.5
75.1
[35]
Present
134.8
104.7
75.4
Cal.
126.3
110.5
89.3
[36]
Cal.
130.2
112.5
83.1
[29]
Cal.
124.0
108.7
78.6
[37]
Cal.
123.4
110.9
84.3
[38]
Exp. at 4.2 K
139.6
109.2
82.3
[39]
Exp. at RT
131.1
101.5
73.8
[40]
Exp. at RT
127.1
107.1
80.3
[41]
Present
204.3
59.7
44.4
Cal.
195.6
61.5
41.7
[42]
Cal.
185.3
72.9
60.5
[43]
Present(VCA)
149.4
52.4
57.9
Present(SQS)
157.9
49.8
63.8
Present
(SQS+SBP)
145.1
57.7
Present
175.5
Cal.
182.2
58.7
170.3
76.0
63.7
170.3
69.9
43.2
51.2
58.7
36.1
170.7
29.8
40.4
61.3
44.6
[44]
Cal.
171.0
37.3
51.9
63.7
41.3
[29]
Exp. at 4.2 K
179.1
37.5
55.4
68.8
46.0
[45]
Exp. at 4.2 K
177.0
34.8
52.8
68.5
45.9
[46]
Exp. at RT
160.9
33.5
50.1
61.0
38.3
[47]
Exp. at RT
159.0
32.3
48.2
62.1
40.0
[48]
148.0
63.7
63.8
35.9
Table 4 Elastic compliances Sij of single crystal of Cu, CuZn, Cu5Zn8, CuZn4 and Zn. The present
calculation results are compared with experimentally reported and previously computed values.
Elastic compliance of single crystal (TPa-1)
Cu
CuZn
Cu5Zn8
CuZn4
Zn
S11
S12
S13
S22
S23
S33
S44
S55
S66
Present
14.35
-5.97
13.29
Cal.
12.33
-4.95
12.20
[29]
Cal.
12.34
-5.02
12.36
[30]
Cal.
14.65
-6.13
13.82
[31]
Cal.
14.44
-6.01
13.28
[32]
Exp. at 4.2 K
13.78
-5.72
12.22
[33]
Exp. at RT
14.84
-6.21
13.19
[34]
Exp. at RT
15.12
-6.34
13.31
[35]
Present
23.18
-10.14
13.26
Cal.
43.15
-20.14
11.20
[36]
Cal.
38.75
-17.97
12.04
[29]
Cal.
44.55
-20.81
12.72
[37]
Cal.
54.30
-25.70
11.86
[38]
Exp. at 4.2 K
22.86
-10.03
12.15
[39]
Exp. at RT
23.52
-10.26
13.55
[40]
Exp. at RT
34.31
-15.69
12.45
[41]
Present
5.64
-1.28
22.51
Cal.
6.02
-1.44
23.98
[42]
Cal.
6.94
-1.96
16.53
[43]
Present(VCA)
8.19
-2.12
-1.93
Present(SQS)
8.19
-2.33
-2.15
Present
(SQS+SBP)
8.94
-2.49
Present
7.64
Cal.
6.72
17.04
7.74
13.52
-2.41
7.68
14.31
0.08
-6.74
28.80
27.74
6.95
-0.15
-4.48
22.22
22.41
[44]
Cal.
7.79
0.31
-6.60
26.47
24.21
[29]
Exp. at 4.2 K
7.46
0.39
-6.32
24.72
21.76
[45]
Exp. at 4.2 K
7.35
0.32
-5.92
23.72
21.79
[46]
Exp. at RT
8.38
0.54
-7.33
28.43
26.11
[47]
Exp. at RT
8.24
0.35
-6.66
26.45
25.00
[48]
9.61
-2.80
16.10
32.59
Table 5 Bulk modulus Bpoly, shear modulus Gpoly and Young’s modulus Epoly of polycrystalline aggregate
of Cu, CuZn, Cu5Zn8, CuZn4 and Zn. The present calculation results in Voigt, Reuss and Hill models are
compared with experimentally reported and previously computed values.
Bpoly (GPa)
Cu
CuZn
Cu5Zn8
CuZn4
Zn
Gpoly (GPa)
Epoly (GPa)
BV
BR
BH
GV
GR
GH
EV
ER
EH
Present
137.9
137.9
137.9
55.0
41.3
48.1
145.7
112.6
129.4
Cal.
137.4
137.4
137.4
60.7
47.3
54.0
158.8
127.3
143.3
[29]
Cal.
145.1
145.1
145.1
60.1
46.9
53.5
158.3
127.1
142.9
[30]
Cal.
139.1
139.1
139.1
53.0
40.1
46.6
141.2
109.9
125.7
[31]
Cal.
138.5
138.5
138.5
55.0
41.1
48.0
145.6
112.2
129.2
[32]
Exp. at 4.2 K
142.0
142.0
142.0
59.3
43.6
51.5
156.3
118.7
137.8
[33]
Exp. at RT
138.3
138.3
138.3
55.0
40.4
47.7
145.6
110.4
128.3
[34]
Exp. at RT
137.0
137.0
137.0
54.4
39.8
47.1
144.1
108.7
126.7
[35]
Present
114.7
114.7
114.7
51.3
28.9
40.1
133.8
80.0
107.7
Cal.
115.8
115.8
115.8
56.7
17.4
37.1
146.3
49.8
100.5
[36]
Cal.
118.4
118.4
118.4
53.4
19.0
36.2
139.2
54.1
98.5
[29]
Cal.
113.8
113.8
113.8
50.2
16.7
33.5
131.3
47.7
91.4
[37]
Cal.
115.1
115.1
115.1
53.1
14.1
33.6
138.0
40.5
91.8
[38]
Exp. at 4.2 K
119.3
119.3
119.3
55.5
29.8
42.6
144.1
82.4
114.2
[39]
Exp. at RT
111.4
111.4
111.4
50.2
28.4
39.3
130.9
78.6
105.5
[40]
Exp. at RT
113.8
113.8
113.8
52.2
21.1
36.6
135.8
59.5
99.2
[41]
Present
107.9
107.9
107.9
55.6
52.5
54.0
142.3
135.6
138.9
Cal.
106.2
106.2
106.2
51.8
49.1
50.5
133.8
127.7
130.7
[42]
Cal.
110.4
110.4
110.4
58.8
58.7
58.7
149.8
149.6
149.7
[43]
Present(VCA)
90.8
89.8
90.3
54.0
53.5
53.8
135.2
133.9
134.6
Present
(SQS+SBP)
92.3
91.6
91.9
55.1
52.4
53.8
137.8
132.1
135.0
Present
77.9
57.8
67.8
45.2
34.0
39.6
113.7
85.2
99.4
Cal.
69.3
55.9
62.6
51.4
41.9
46.7
123.6
100.6
112.1
[44]
Cal.
76.4
61.5
69.0
47.5
36.6
42.1
118.1
91.6
104.9
[29]
Exp. at 4.2 K
80.4
66.1
73.2
51.1
39.4
45.3
126.5
98.7
112.6
[45]
Exp. at 4.2 K
78.1
64.9
71.5
51.4
40.2
45.8
126.4
100.1
113.2
[46]
Exp. at RT
72.2
59.0
65.6
44.7
34.0
39.3
111.1
85.6
98.4
[47]
Exp. at RT
70.8
59.0
64.9
45.4
35.6
40.5
112.3
89.0
100.6
[48]
Table 6 Poisson’s ratio ν, bulk modulus to shear modulus ratio BH/GH and universal
elastic anisotropy index AU of Cu, CuZn, Cu5Zn8, CuZn4 and Zn. The present calculation
results are compared with experimentally reported and previously computed values.
Cu
CuZn
Cu5Zn8
CuZn4
Zn
BH/GH
AU
Present
0.34
2.87
1.66
Cal.
0.33
2.54
1.42
[29]
Cal.
0.34
2.71
1.40
[30]
Cal.
0.35
2.99
1.61
[31]
Cal.
0.34
2.88
1.69
[32]
Exp. at 4.2 K
0.34
2.76
1.80
[33]
Exp. at RT
0.35
2.90
1.81
[34]
Exp. at RT
0.35
2.91
1.84
[35]
Present
0.34
2.86
3.87
Cal.
0.36
3.26
11.27
[36]
Cal.
0.36
3.27
9.03
[29]
Cal.
0.37
3.40
10.05
[37]
Cal.
0.37
3.43
13.87
[38]
Exp. at 4.2 K
0.34
2.80
4.32
[39]
Exp. at RT
0.34
2.83
3.82
[40]
Exp. at RT
0.35
3.11
7.39
[41]
Present
0.29
2.00
0.29
Cal.
0.29
2.10
0.28
[42]
Cal.
0.27
1.88
0.01
[43]
Present (VCA)
0.25
1.68
0.06
Present
(SQS+SBP)
0.26
1.71
0.26
Present
0.26
1.71
2.01
Cal.
0.20
1.34
1.37
[44]
Cal.
0.25
1.64
1.74
[29]
Exp. at 4.2 K
0.24
1.62
1.70
[45]
Exp. at 4.2 K
0.24
1.56
1.59
[46]
Exp. at RT
0.25
1.67
1.79
[47]
Exp. at RT
0.24
1.60
1.58
[48]
Figure
Cu
(a)
(b)
(c)
(d)
Zn
Cu0.2Zn0.8
Fig.1 Elemental atom distribution of (a) CuZn, (b) Cu5Zn8 and CuZn4 in (c)
VCA model or (d) SQS (3×3×3 supercell) model before geometry
optimization.
Present
160
Exp.(RT)
(a)
Cu
140
B /GPa
Exp.(4.2 K)
Cal.
CuZn
Cu5Zn8
120
CuZn4
100
Zn
80
60
70
(b)
G /GPa
60
50
40
Cu5Zn8
Cu
CuZn4
Zn
30
CuZn
20
160
(c)
E /GPa
140
120
Cu5Zn8
Cu
CuZn4
100
80
Zn
CuZn
60
0.0
0.2
0.4
0.6
0.8
1.0
Mole fraction of Zn
Fig.2 Mole fraction dependence of (a) bulk modulus BH, (b) shear modulus GH and (c)
Young’s modulus EH of polycrystalline aggregate for Cu, CuZn, Cu5Zn8, CuZn4 and Zn.
The present calculation results in Hill model are compared with experimentally reported
and previously computed values.
Present
Cal.
Exp.(4.2 K)
Exp.(RT)
160
B /GPa
140
Cu
120
CuZn
100
Cu5Zn8
80
CuZn4
60
Zn
40
7.0
7.5
8.0
8.5
9.0
Mass density, ρ /g/cm
Fig.3 Mass density dependence of bulk modulus BH of polycrystalline aggregate for Cu,
CuZn, Cu5Zn8, CuZn4 and Zn. The present calculation results in Hill model are
compared with experimentally reported and previously computed values.
(a) Cu, AU = 1.66
(b) CuZn, AU = 3.87
(GPa)
(GPa)
180
180
160
165
150
140
135
120
120
100
105
80
90
60
75
(e) CuZn4 (VCA), AU = 0.06
(c) Cu5Zn8, AU = 0.29
(GPa)
(GPa)
176
147
168
144
160
141
138
152
135
144
(e) CuZn4 (SQS+SBP), AU = 0.26
132
136
129
128
126
120
123
(GPa)
(f) Zn, AU = 2.01
150
(GPa)
130
145
120
110
140
100
135
90
130
125
120
115
Fig.4 Directional anisotropy of Young’s modulus E of single crystal for Cu, CuZn, Cu5Zn8,
CuZn4 and Zn. The magnitude of E in each direction is illustrated not only by color-coding
according to each color scale but also by the distance from the center of the three-dimensional
space.
80
70
60
50
40
*Author Contributions Section
Author Contributions Section
Conceptualization, H.I. and S.H.; Methodology, H.I. and S.H.; Investigation, H.I.; Writing – Original
Draft, H.I.; Writing –Review & Editing, S.H.; Funding Acquisition, S.H.; Resources, S.H.;
Supervision, S.H.
*CRediT Author Statement
Credit Author Statement
Conceptualization, H.I. and S.H.; Methodology, H.I. and S.H.; Investigation, H.I.; Writing – Original
Draft, H.I.; Writing –Review & Editing, S.H.; Funding Acquisition, S.H.; Resources, S.H.;
Supervision, S.H.
*Declaration of Interest Statement
Declaration of interests
☒ The authors declare that they have no known competing financial interests or personal relationships
that could have appeared to influence the work reported in this paper.
☐The authors declare the following financial interests/personal relationships which may be considered
as potential competing interests:
...
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