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Effect of alloying on the thermal-elastic properties of 3d high-entropy alloys
Materials Chemistry and Physics xxx (2017) 1e7
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Effect of alloying on the thermal-elastic properties of 3d high-entropy alloys Huijuan Ge, Hongquan Song, Jiang Shen, Fuyang Tian* Institute for Applied Physics, University of Science and Technology Beijing, Beijing 100083, China
h i g h l i g h t s The thermal-elastic properties are studied. The influence of alloying elements on the Curie temperature is discussed. The Young's moduli along different crystal directions are studied for CoCrFeMnNi.
a r t i c l e i n f o
a b s t r a c t
Article history: Received 3 March 2017 Received in revised form 6 October 2017 Accepted 17 October 2017 Available online xxx
Using the quasi-harmonic approximation in combination with ab initio calculations, we study the temperature dependent elasticity of single-phase face centered cubic (fcc) Co-Cr-Fe-Mn-Ni multi-principle element alloys. Results suggest that the addition of an equimolar alloying element increases the fcc stability due to the increasing valence electron concentration of alloys. The ductility of paramagnetic alloys becomes superior with the increase of temperature. For the CoCrFeMnNi high-entropy alloy, the Young's moduli along different crystal directions are consistent with experiments. The difference between ab initio predicted temperature dependent elastic moduli and experiments is discussed. © 2017 Elsevier B.V. All rights reserved.
Keywords: High-entropy alloys ab initio calculations Curie temperature Elastic moduli Thermal-elastic properties
1. Introduction Recently, a new class of materials, high-entropy alloys (HEAs) named by Yeh et al, have attracted extensive attention in scientific literature [1e3]. HEAs are composed of at least four principal alloying elements in equimolar or near-equimolar ratio [4,5], which are also known as multi-principle element alloys. The extensively available experiments suggest that the family of single-phase HEAs include fcc 3d-HEAs mainly composed of 3d transition metals [1,2], the body centered cubic (bcc) refractory-HEAs composed of refractory metals [6], and the hexagonal closed-packed (hcp) rareearth-HEAs composed of rare-earth elements [7]. Since Cantor et al [8] found the fcc equimolar CoCrFeMnNi HEA, a large number of experimental and theoretical works focused on the Co-Cr-Fe-Mn-Ni based alloys [9e15]. Laplanche [9] and Haglund [10] et al studied the temperature dependent of elastic
* Corresponding author. E-mail address: [email protected] (F. Tian).
moduli and thermal expansion coefficient (TEC) at temperature (T ¼ 200e1270 K) and the polycrystalline elastic moduli at cryogenic temperature (T ¼ 50e300 K) of CoCrFeMnNi, respectively. To better understand the CoCrFeMnNi solid solution, Wu and Bei et al [11,12] investigated mechanical properties, recovery, recrystallization, grain growth and phase stability of the Cr-Mn-Fe-Co-Ni based binary, ternary and quaternary alloys. Liu [13] took CoCrFeMnNi as an example to study the microstructure stability and grain growth of HEAs. Lucas et al [14,15] used the vibrating sample magnetometry to determine the high-temperature magnetic applications of the equimolar CoCrFeNi HEA and explored the effects of Cr content on the magnetic properties of CoFeNiCrx HEAs. Results showed that the Curie temperature of CoFeNiCrx increases with decrease of Cr content. Although single-phase HEAs have a simple crystal structure, the atomic random distribution on lattice sites in solid solutions induces the chemical disorder and magnetic disorder. For singlephase 3d HEAs composed of magnetic elements, the Co-Cr-FeMn-Ni alloys show the various magnetic behaviors and different
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H. Ge et al. / Materials Chemistry and Physics xxx (2017) 1e7
Curie temperatures [14e16], due to different magnetic states of alloying elements in their ground state (ferromagnetic (FM) for Fe, Ni, Co, while antiferromagnetic for Cr, multi-magnetic for Mn). As far as we know, the theoretically predicted intrinsic magnetic properties of the Cr-Mn-Fe-Co-Ni multi-principle element alloys are very limited. Ab initio calculations have been used successfully to predict the fcc stability [17], Young's modulus at low temperature [18] and stacking fault energy [19] for the quinary CoCrFeMnNi HEA. Experiments showed that the binary (CoNi, FeNi), ternary (CoCrNi, CoFeNi, CoMnNi, CrFeNi), quaternary (CoCrFeNi, CoCrMnNi, CoFeMnNi) and quinary (CoCrFeMnNi) alloys all adopt a singlephase fcc solid solutions [12]. From the equimolar binary alloy to equimolar quinary HEA, it is very interesting to study the effect of alloying elements on the elasticity of Co-Cr-Fe-Mn-Ni alloys. In this work we use ab initio calculations to investigate the influence of alloying elements on the equilibrium bulk properties of Co-Cr-FeMn-Ni alloys. Meanwhile the thermal-elastic properties as a function of temperature are systematically studied for the single-phase Co-Cr-Fe-Mn-Ni multi-principle element alloys based on quasiharmonic Debye-Grüneisen approximation. The rest of the paper is organized as follows. In Section 2, we present the methodology and give the most important details of the numerical calculations. In Section 3, we show the results of the equilibrium bulk properties and elastic properties of different magnetic states and discuss the effect of alloying element on the Curie temperature and the thermal-elastic properties of Co-Cr-FeMn-Ni alloys. Finally, some conclusions are drawn in Section 4. 2. Theoretical methodology The exact muffin-tin orbitals (EMTO) method is an efficient ab initio approach to solving the KohneSham equation [20] which employs large overlapping potential spheres for the exact singleelectron potential and the full charge density method further improves the accuracy of full potential total energy [21]. The substitutional disorder in both chemical and magnetic degrees of freedom [22,23], is treated by the coherent potential approximation (CPA) which use an effective medium in single site approximation. For the self-consistent calculations, we employed the Perdew-Burke-Ernzerhof (PBE) exchange correlation functional form of generalized gradient approximation (GGA) to compute the charge density and total energy [24]. The Green's function was calculated for 24 complex energy points from the bottom of the valence bands to the Fermi level. In the irreducible wedge of the fcc Brillouin zone, we used 240 inequivalent k points, allowing to maintain a 102 mRy/atom accuracy in the total energy. We employed the disorder local magnetic moment (DLM) [25] picture to describe the paramagnetic (PM) state of the present HEAs. The DLM resembles a random spin configuration with zero averaged magnetization, while allowing the presence of finite local magnetic moments on each lattice site above the magnetic transition temperature [26e28]. There are only three independent elastic constants c11, c12, and c44 in the cubic lattice. Based on the equilibrium volume and bulk modulus derived from the forth-order Birch-Murnaghan equation of state (i.e. energy as a function of volume), the elastic constants c11, c12 and c44 were obtained through by calculating the total energy as a function of volume-conserving lattice strains [29,30]. Combining the three elastic constants c11, c12, c44 with the VoigteReusseHill averaging method [31,32], we can calculate the polycrystalline elastic moduli (shear modulus G, Young's modulus E), Poisson's ratio v and Zener ratio AZ [21]. We used the quasi-harmonic Debye-Grüneisen approximation to compute the TEC based on the equation of state and the ab initio
calculated Poisson ratio, then we calculated the equilibrium volume and bulk modulus at given temperature, and further the temperature dependent elastic constants and polycrystalline elastic moduli were estimated with the method of elastic calculations [21]. The quasi-harmonic approximation is a simple and effective method to account for the lattice vibrational information, without the phonon density of state for solid solutions. In the quasi-harmonic DebyeGrüneisen model, the harmonic approximation at any given crystal structure is assumed, even if it does not correspond to the equilibrium structure. Note that we also considered the contributions of electronic entropy and magnetic entropy to Gibbs energy implemented in quasi-harmonic approximation. 3. Results and discussion 3.1. Equilibrium bulk properties Fig. 1 shows the equation of states of the paramagnetic (PM) CrMn-Fe-Co-Ni alloys. Although the Fe, Co and Ni alloying elements have similar Goldschmidt atomic radii [33], their binary solid solutions have different equilibrium Wigner-Seitz (WS) radii (see Fig. 1 and Table 1), for example the PM CoNi alloy has smaller WS radius than FeNi. Interestingly their WS radii of binary solid solutions are very close to the average WS radii derived from the experimental lattice parameters of alloying elements via Vegard's law [34]. For CoCrNi, CoFeNi and CoMnNi ternary alloys, ab initio calculations predict the similar WS radius to each other, which are slightly smaller than the average WS radii. The EMTO predicted WS radius of CrFeNi is much smaller than the experimental average one. The difference between ab initio calculations and experimental average may be due to ab initio performed at T ¼ 0 K temperature. The EMTO predicted WS radii of quaternary and quinary HEAs are very close to each other. The FM alloys have slightly larger WS than the PM ones, except for Mn-containing alloys, due to the magnetic order. 3.2. Curie temperature Within the numerous successful applications, the EMTO-CPA method has been developed an attractive tool in materials science to predict finite temperature magnetic properties. Here we use Curie temperature TC to describe the finite temperature
Fig. 1. Equation of states (energy (meV/atom) vs volume (Å3/atom)) for the at paramagnetic (PM) Cr-Mn-Fe-Co-Ni multi-principle element alloys. Left column (a) is for binary alloys, middle column (b) for ternary alloys, right column (c) for HEAs (quaternary and quinary alloys). All energies are relative to their corresponding total energy at the equilibrium volume.
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3
Table 1 Listed are the equilibrium bulk parameters, Curie temperature and elastic constants of the Co-Cr-Fe-Mn-Ni alloys. The energy difference between paramagnetic (PM) and ferromagnetic (FM) states DE ¼ EPM-EFM (mRy/atom), the Curie temperature TC (K), WignereSeitz (WS) radius u (Bohr), the elastic constants (c11 c12, c44) (GPa), the tetragonal shear modulus c΄¼(c11-c12)/2 (GPa) and the Cauchy pressure (c12-c44) (GPa). ut represents the theoretical (EMTO) Wigner-Seitz radius, ue stands for the average WS radius from the experimental radii of pure alloying elements [34]. States
DE
TC
ut
ue
PM FM
9.083
956
2.598 2.605
2.608
PM FM
5.263
554
2.623 2.642
2.635
PM FM
0.066
7
2.606 2.612
2.633
PM FM
7.959
838
2.607 2.629
2.627
PM FM
1.808
190
2.613 2.600
2.628
PM FM
1.004
106
2.622 2.629
2.650
PM FM
1.531
161
2.606 2.618
2.642
PM FM
0.24
25
2.611 2.608
2.642
PM FM
1.595
168
2.608 2.617
2.638
PM FM
0.321
34
2.607 2.613
2.647
c11 CoNi 253.4 264.7 FeNi 218.1 214.6 CoCrNi 278.8 266.7 CoFeNi 234.2 229.8 CoMnNi 211.7 216.6 CrFeNi 234.2 225.3 CoCrFeNi 272.8 237.9 CoCrMnNi 249.1 217.6 CoFeMnNi 214.5 209.8 CoCrFeMnNi 242.2 216.5
c12
c44
c'
c12-c44
158.9 167.1
180.3 172.8
47.2 48.8
21.4 5.8
146.8 164.8
158.5 135.7
35.7 24.9
11.7 29.1
188.8 182.1
183.0 177.6
44.9 42.3
5.8 4.5
144.2 174.4
177.9 147.2
45.0 27.7
33.7 27.2
116.4 102.8
184.4 202.7
47.6 56.9
67.9 99.9
167.9 168.5
167.2 158.9
33.2 28.4
0.7 9.5
180.9 161.8
185.5 172.9
46.0 38.0
4.6 11.1
161.6 141.0
185.3 159.4
43.7 38.3
23.7 18.3
116.5 116.6
190.6 179.7
49.0 46.6
74.0 63.0
140.6 113.1
192.7 184.4
50.8 51.7
52.0 71.2
magnetic properties of the Co-Cr-Fe-Mn-Ni alloys. We consider two different magnetic scenarios: the ferromagnetic (FM) state and the paramagnetic (PM) state. Within the mean field approximation, TC is directly related to the sum of the magnetic interactions. If alloyed with a small concentration of nonmagnetic elements, TC is expressed as [35]
TC ¼
2 2 PM FM Etot ¼ Etot DE; 3ð1 cÞkB 3ð1 cÞkB
(1)
PM EFM ) is the where kB represents the Boltzmann constant, ðEtot tot difference between the ground state total energies of the PM and FM alloys, c is the content of nonmagnetic elements. For the present alloys composed of magnetic elements: Cr, Mn, Fe, Co and Ni, Eq. (1) may be formulated as
TC ¼
2 PM FM Etot Etot : 3kB
(2)
For the solid-solution Cr-Mn-Fe-Co-Ni alloys, the alloying elements play an important role in the Curie temperature. Our ab initio prediction of binary alloys is Tc ¼ 956 K for CoNi and Tc ¼ 554 K for FeNi, respectively. It is reasonable, compared with the considerable TC of all three individual ferromagnetic elements (Co (1394 K), Fe (1041 K), and Ni (611 K)). Fig. 2 shows the Tc of present alloys. With the addition of equimolar alloying element of Cr and Fe, TC decreases from 956 K for CoNi to 7 K for CoCrNi (838 K for CoFeNi), from 554 K for FeNi to 106 K for CrFeNi, and from 838 K for CoFeNi to 161 K CoCrFeNi, except for the case of CoMnNi and CoFeMnNi. Also the addition of equimolar Mn obviously decreases the TC of CoNi, CoCrNi and CoFeNi. While the addition of Co slightly increases TC, from 554 K for FeNi to 838 K for CoFeNi and from 106 K for CrFeNi to 161 K CoCrFeNi. The change of Tc may derive from the
Fig. 2. Curie temperature TC for the Co-Cr-Fe-Mn-Ni multi-principle element alloys (the symbol pm presents what the paramagnetic (PM) state is more stable than the ferromagnetic (FM) one).
intrinsic magnetic property of alloying elements, for example Cr is antiferromagnetic, Co is ferromagnetic, and multi-magnetic for Mn. The present prediction of TC ¼ 161 K for CoCrFeNi is in good accordance with the experimental value of 130 K [15]. Equation (2) suggests that the large TC is from a large energy difference between PM and FM. In case of phase stability, from Fig. 2 we can find that the PM CoMnNi and CoCrMnNi are
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more stable at T ¼ 0 K. In contrary, the FM state is more stable for other alloys. 3.3. Elastic moduli Tables 1 and 2 list the three independent elastic constants (c11, c12 c44), the tetragonal shear modulus c’¼ (c11-c12)/2, Cauchy pressure (c12-c44), polycrystalline elastic moduli (B, G, E), the Zener ratio AZ, the Poisson's ratio n, and the Pugh ratio B/G as well as the elastic anisotropy ratio AVR. From the three independent elastic constants c11, c12, c44 and c’, we can find that all alloys fulfill the condition of mechanical stability (c44 > 0, c11>|c12| and c11þ2c12 > 0 [21]), i.e. they are mechanically stable. The tetragonal shear modulus c’ represents the anisotropy of elastic moduli against tetragonal distortions. The decreasing c’ suggests that the fcc stability reduces [36]. Compared with the FM state, the PM state makes the fcc phase more stable, except for CoNi, CoMnNi and CoCrFeMnNi. Compared with the tetragonal elastic modulus c’ ¼ 47.6 (56.9) GPa for PM (FM) CoMnNi, the addition of Cr slightly decreases the c’ ¼ 43.7 (38.3) GPa of CoCrMnNi. Whereas the comparison of c’ between CoNi, CoCrFeNi, FeNi and CoMnNi, CoCrFeMnNi, CoFeNi indicates that both Co and Mn increase the stability of fcc phase. It is related to the valence electron concentration (VEC). In the case of single phase HEAs, with increase of VEC, the phase transforms from bcc to fcc according to the available experiments [37]. The VEC ¼ 6, 7, 8, 9, 10 are for Cr, Mn, Fe, Co, Ni, respectively. The addition of Cr (Co) decreases (increases) the VEC of present alloys. The Cauchy pressure (c12-c44) is often used to characterize the bond properties of a material. (c12-c44) > 0 suggests that the material is dominated by the metallic bond, while (c12-c44) < 0 for the covalent bond. From Table 1, we see that for FM and PM states, CoNi, CoMnNi, CoCrFeNi, CoCrMnNi, CoFeMnNi and CoCrFeMnNi show the covalent bond, CoCrNi and CrFeNi have weak metallic bond. However, FeNi and CoFeNi show the covalent bond in the PM state, while show metallic bond in the FM state. Cr (Co or Mn) slightly reduces (strengthens) the covalent bond behavior in PM state of the present alloys. Table 2 lists the polycrystalline elastic moduli, i.e. the bulk modulus B, shear modulus G, Young's modulus E, and Poisson's ratio n, Pugh ratio B/G as well as elastic anisotropy AVR and AZ of the Cr-Fe-Mn-Co-Ni alloys. We find that the polycrystalline elastic moduli Et, Gt, and nt are in good agreement with the available experiments, except for CoMnNi. For the present alloys, the PM state has larger Young's modulus E and shear modulus G than the FM state. The brittle-ductile behavior of alloys is often indicated by Pugh ratio B/G [38]. When B/G > 1.75, the alloy is ductile, otherwise the material is brittle. The values of B/G (from 1.80 to 2.59) suggest that CoNi, FeNi, CoCrNi, FM CoFeNi, CrFeNi, CoCrFeNi and CoCrMnNi may be slightly ductile. While B/G ¼ 1.15e1.69 suggests the less ductile for PM CoFeNi, CoMnNi, CoFeMnNi, and CoCrFeMnNi. The elastic isotropy for cubic lattice is often determined by the Zener ratio AZ and the elastic anisotropy AVR. If AZ ¼ 1 or AVR ¼ 0, the material is elastically isotropy, otherwise it is elastically anisotropy [21]. From the value of AZ, AVR listed in Table 2 we see that these HEAs are elastically anisotropic. Generally speaking, the elastic modulus has empirically related to the mechanical properties. The bulk modulus B represents the ability of a material to resist the volume change under pressure. The largest bulk modulus of CoCrNi among the present alloys suggests that CoCrNi has the good capacity to resist volume change under pressure. The FM alloys have similar bulk moduli to the PM ones. It is clear from Fig. 4 and Table 2 that with the addition of equimolar Co (Cr) (from binary FeNi (CoNi) to ternary CoFeNi (CoCrNi), from ternary CrFeNi (CoMnNi) to quaternary CoCrFeNi (CoCrMnNi)),
both bulk modulus and Young's modulus become large. Whereas the change of shear modulus is small. From Table 2, we see that the theoretical Young's moduli and shear moduli at 0 K are slightly larger than experiments. Note that the above discussions are based on ab initio calculations performed at T ¼ 0 K temperature. 3.4. Thermal-elastic properties Using the thermal expansion equation uðTÞ ¼ uð0 KÞð1 þ aT TÞ, whereu(0 K) is the WS radius at 0 K, uðTÞ at temperature T, and aT is the thermal expansion coefficient (TEC) at T temperature, one may estimate the WS radius at T, and then combining with the bulk modulus, one can calculate the elastic constants and polycrystalline elastic moduli at given temperature T. Fig. 3 and Fig. 4 show the elastic constants and polycrystalline elastic moduli of these 3d multi-principle element alloys as a function of temperature. From Fig. 3, we can see that all of these alloys satisfy the criteria (c44 > 0, c11>|c12|, c11þ2c12 > 0) of mechanical stability at finite temperature, and c11 and c44 decrease obviously with the increase of temperature. Fig. 3 shows that the Cauchy pressure (c12-c44) is positively correlated with temperature. It indicates that the metal bond characteristics of these alloys are enhanced with the increasing temperature, i.e, the present alloys show the good ductility when the temperature is considered in ab initio calculations. The predicted polycrystalline elastic moduli (B, G, E, B/G and n) as a function of temperature are shown in Fig. 4. It is found that with the increase of temperature the B, G and E values decrease, the change of B is the most gentle, whereas both Pugh ratio B/G and Poisson's ratio n increase. Results suggest the toughness of the present alloys tends to be superior with the increasing temperature. From Fig. 4 we can see that CoCrFeNi (CoMnNi) has the largest (smallest) bulk modulus in Co-Cr-Fe-Mn-Ni alloy at range of temperature T ¼ 100e600 K. The quaternary and quinary HEAs have close the Young's modulus E and shear modulus G at the low temperature T ¼ 100e300, whereas with increasing temperature, the difference of elastic moduli (E and G) between quaternary and quinary HEAs becomes large. In all, the alloying element have almost no effect on the elastic moduli (E and G) of quaternary and quinary HEAs. For CoCrFeMnNi, the empirical equations of Young's modulus E and shear modulus G from experiments [9] are as follows
E ¼ 214 35
. 416 eT 1
(3)
. 448 G ¼ 85 16 e T 1
(4)
From the two above equations, we can see that both Young's modulus E and shear modulus G decrease with increasing temperature. The green lines in Fig. 5 show the elastic moduli E and G as a function of temperature calculated by using Eqns (3) and (4). As the above discussions, the elastic moduli E and G can be calculated with the temperature dependent elastic constant calculated based on the combination of the TEC with the equilibrium WS radius. Considering the differences of TEC and equilibrium WS radius between theoretical calculations and experiments, we computed the temperature dependent elastic moduli E and G by using the experimental TEC and theoretical and experimental WS radius. The experimental TEC [9] is given by
a ¼ 23:7 106 1 e299 : T
(5)
The red and blue lines shown in Fig. 5 represent the elastic modulus calculated via the experimental TEC in combination
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Table 2 Shown are the theoretical polycrystalline elastic moduli B, G, E (GPa), the Zener ration AZ, the Poisson's ratio n, the Pugh ratio B/G as well as the elastic anisotropy ratio AVR at the equilibrium bulk parameters of paramagnetic (PM) and ferromagnetic (FM) Co-Cr-Fe-Mn-Ni alloys. The experimental data of Young modulus Ee, the shear modulus Ge, and the Poisson's ratio ne are listed. State
B
Gt
Ge
Et
PM FM
190.4 199.6
105.9 104.4
84a
268.1 266.7
PM FM
170.5 181.4
88.0 70.1
62a
225.3 186.3
PM FM
218.8 210.3
104.9 100.7
87a
271.4 260.5
PM FM
174.2 192.8
103.2 76.7
60a
258.5 203.1
PM FM
148.2 140.7
107.8 122.2
77a
260.2 284.4
PM FM
190.0 187.4
88.8 81.4
e
230.4 213.3
PM FM
211.5 187.2
106.7 95.2
84a
274.1 244.3
PM FM
190.8 166.5
104.7 90.7
78a
265.6 230.2
PM FM
149.2 147.7
111.2 105.1
77a
267.2 254.9
PM FM
174.5 147.6
113.4 111.1
86c
279.7 266.5
Ee CoNi e FeNi e CoCrNi 231b
nt
ne
B/G
AVR
AZ
0.27 0.28
0.29a
1.80 1.91
0.20 0.18
3.82 3.54
0.28 0.33
0.34a
1.94 2.59
0.24 0.30
4.44 5.46
0.30 0.29
0.30a
2.09 2.09
0.22 0.23
4.07 4.20
0.25 0.32
0.35a,
1.69 2.51
0.21 0.30
3.95 5.32
0.21 0.16
0.23a
1.38 1.15
0.19 0.18
3.70 3.56
0.30 0.31
e
2.14 2.30
0.28 0.31
5.04 5.59
0.28 0.28
0.28a,
1.98 1.97
0.22 0.25
4.04 4.55
0.27 0.27
0.25a
1.82 1.84
0.23 0.22
4.24 4.16
0.20 0.21
0.22a
1.34 1.40
0.20 0.20
3.89 3.86
0.23 0.20
0.26a.c
1.54 1.33
0.20 0.18
3.79 3.57
CoFeNi
a
Ref. [11],
b
Ref. [39], c Ref. [10],
d
162 CoMnNi e CrFeNi 231b CoCrFeNi 226b CoCrMnNi e CoFeMnNi e CoCrFeMnNi 203d 215c
Ref. [9].
Fig. 3. Elastic constants as a function of temperature for PM Co-Cr-Fe-Mn-Ni alloys. The left panel is for ternary alloys and the right panel is for quaternary and quinary HEAs.
Fig. 4. Polycrystalline elastic moduli as a function of temperature for the PM Co-Cr-FeMn-Ni alloys. The left panel is for ternary alloys and the right panel is for quaternary and quinary HEAs.
with the experimental and theoretical WS radii, respectively. We can see that at T ¼ 100e500 K the similar trend of elastic moduli E and G derived from the experimental and theoretical WS radii. From Fig. 5, we can see that our ab initio results of E and G are in
good agreement with the experimental data near room temperature. Meanwhile we note that the ab initio calculated Young's modulus as a function of temperature is slightly different from the experiments. It may be due to the EMTO-CPA underestimated
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Fig. 5. Young's modulus E and shear modulus G as a function of temperature for CoCrFeMnNi: the black line represents ab initio calculated data in the present work; the red line represents estimated data gotten via the combination of the experimental TEC (Eq. (5)) with the ab initio calculated equilibrium WS radius; the blue line represents the theoretical data calculated via the combination of experimental TEC (Eq. (5)) with the experimental equilibrium WS radius 2.651 Bohr (at 300 K); the green line represents the experimental data obtained via solving Eq. (3) and Eq. (4).
equilibrium WS radius and the quasi-harmonic Debye-Grüneisen overestimated TEC. Figure 6 shows the room-temperature Young's moduli along different crystal directions. According to the elastic theorem, we can compute Young's modulus in plane (110) family as a function of crystal direction based on the three independent elastic constants (c11, c12 and c44). The estimated Young's modulus from experimental elastic constants [41] is E ¼ 201.6 GPa, which is consistent with other experimental measures [9]. The present theoretical prediction for the <110>, <331> and <100> crystal direction is in good agreement with experiments. Considering the same experimental method are employed in Refs. [40,41] and there exist some differences for Young's modulus along the same crystal direction, our theoretical calculations of Young's moduli are also consistent with experiment for <311> and <111> crystal directions. 4. Conclusions By performing the ab initio EMTO-CPA calculations in combination with the quasi-harmonic Debye-Grüneisen model, we have
Fig. 6. Theoretical and experimental Young's modulus for CoCrFeMnNi alloys plotted in plane (110) as a function of crystal direction, including the five cubic directions. Exp. a from Ref. [40], Exp. b from Ref. [41], and Exp. c from Ref. [9].
investigated the equilibrium bulk properties, the Curie temperature and thermal-elastic properties of ferromagnetic and paramagnetic Co-Cr-Fe-Mn-Ni multi-principle element alloys, from binary (CoNi, FeNi) to ternary (CoCrNi, CoFeNi, CoMnNi, CrFeNi) to quaternary (CoCrFeNi, CoCrMnNi, CoFeMnNi) and quinary (CoCrFeMnNi) fcc solid solutions. The addition of equimolar alloying element Cr, Fe or Mn decreases the Curie temperature Tc, whereas the additions of Co increases Tc of the Co-Cr-Fe-Mn-Ni multi-principle element alloys. Compared with the ferromagnetic state, the paramagnetic state increases the mechanical stability of fcc phase for the CoNi, CoMnNi and CoCrFeMnNi alloys. The calculated polycrystalline elastic moduli (Young moduli E, shear modulus G, and Poisson's ratio n) are in good agreement with the available experiments. CoCrNi has the good capacity to resist volume change under pressure among Co-Cr-Fe-Mn-Ni alloys. The elastic moduli linearly decrease with increasing temperature. The ductility of paramagnetic alloys becomes superior with the increasing temperature. For the CoCrFeMnNi high-entropy alloy, ab initio predicted Young's moduli along different crystal directions are consistent with the experiments. Based on the experimental thermal expansion coefficient, the theoretically predicted the elastic moduli (E and G) have similar trend to the experimental measures with increasing the temperature from T ¼ 100 K to T ¼ 600 K. Acknowledgment Work was supported by the National Natural Science Foundation of China with Grant No. 51771015 and Grant No. 51401014. References [1] J.W. Yeh, S.K. Chen, S.J. Lin, J.Y. Gan, T.S. Chin, T.T. Shun, C.H. Tsau, S.Y. Chang, Nanostructured high-entropy alloys with multiple principal elements: novel alloy design concepts and outcomes, Adv. Eng. Mater 6 (2004) 299e303. [2] J.W. Yeh, Recent progress in high entropy alloys, Ann. Chim. Sci. Mat. 31 (2006) 633e648. [3] J.W. Yeh, Y.L. Chen, S.J. Lin, S.K. Chen, High-entropy alloys e a new era of exploitation, Mater. Sci. Forum 560 (2007) 1e9. [4] O.N. Senkov, S.V. Senkova, D.B. Miracle, C. Woodward, Mechanical properties of low-density, refractory multi-principal element alloys of the CreNbeTieVeZr system, Mater. Sci. Eng. A 565 (2013) 51e62. Fazakas, V. Zadorozhnyy, L.K. Varga, A. Inoue, D.V. Louzguine-Luzgin, [5] E. F.Y. Tian, L. Vitos, Experimental and theoretical study of Ti 20 Zr 20 Hf 20 Nb 20 X 20 (X ¼ V or Cr) refractory high-entropy alloys, Int. J. Refract. Met. Hard Mater 47 (2014) 131e138. [6] O.N. Senkov, G.B. Wilks, D.B. Miracle, C.P. Chuang, P.K. Liaw, Refractory highentropy alloys, Intermetallics 18 (2010) 1758e1765. [7] A. Takeuchi, K. Amiya, T. Wada, K. Yubuta, W. Zhang, High-entropy alloys with a hexagonal close-packed structure designed by equi-atomic alloy strategy and binary phase diagrams, Jom 66 (2014) 1984e1992. [8] B. Cantor, I.T.H. Chang, P. Knight, A.J.B. Vincent, Microstructural development in equiatomic multicomponent alloys, Mater. Sci. Eng. A 375e377 (2004) 213e218. [9] G. Laplanche, P. Gadaud, O. Horst, F. Otto, G. Eggeler, E.P. George, Temperature dependencies of the elastic moduli and thermal expansion coefficient of an equiatomic, single-phase CoCrFeMnNi high-entropy alloy, J. Alloy. Compd. 623 (2015) 348e353. [10] A. Haglund, M. Koehler, D. Catoor, E.P. George, V. Keppens, Polycrystalline elastic moduli of a high-entropy alloy at cryogenic temperatures, Intermetallics 58 (2015) 62e64. [11] Z. Wu, H. Bei, G.M. Pharr, E.P. George, Temperature dependence of the mechanical properties of equiatomic solid solution alloys with face-centered cubic crystal structures, Acta Mater 81 (2014) 428e441. [12] Z. Wu, H. Bei, F. Otto, G.M. Pharr, E.P. George, Recovery, recrystallization, grain growth and phase stability of a family of FCC-structured multi-component equiatomic solid solution alloys, Intermetallics 46 (2014) 131e140. [13] W.H. Liu, Y. Wu, J.Y. He, T.G. Nieh, Z.P. Lu, Grain growth and the HallePetch relationship in a high-entropy FeCrNiCoMn alloy, Scr. Mater 68 (2013) 526e529. ~ oz, Y. Xiao, A.O. Sheets, S.L. Semiatin, [14] M.S. Lucas, L. Mauger, J.A. Mun J. Horwath, Z. Turgut, Magnetic and vibrational properties of high-entropy alloys, J. Appl. Phys. 109 (2011) 07E307. [15] M.S. Lucas, D. Belyea, C. Bauer, N. Bryant, E. Michel, Z. Turgut, S.O. Leontsev,
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Effect of alloying on the thermal-elastic properties of 3d high-entropy alloys Huijuan Ge, Hongquan Song, Jiang Shen, Fuyang Tian* Institute for Applied Physics, University of Science and Technology Beijing, Beijing 100083, China
h i g h l i g h t s The thermal-elastic properties are studied. The influence of alloying elements on the Curie temperature is discussed. The Young's moduli along different crystal directions are studied for CoCrFeMnNi.
a r t i c l e i n f o
a b s t r a c t
Article history: Received 3 March 2017 Received in revised form 6 October 2017 Accepted 17 October 2017 Available online xxx
Using the quasi-harmonic approximation in combination with ab initio calculations, we study the temperature dependent elasticity of single-phase face centered cubic (fcc) Co-Cr-Fe-Mn-Ni multi-principle element alloys. Results suggest that the addition of an equimolar alloying element increases the fcc stability due to the increasing valence electron concentration of alloys. The ductility of paramagnetic alloys becomes superior with the increase of temperature. For the CoCrFeMnNi high-entropy alloy, the Young's moduli along different crystal directions are consistent with experiments. The difference between ab initio predicted temperature dependent elastic moduli and experiments is discussed. © 2017 Elsevier B.V. All rights reserved.
Keywords: High-entropy alloys ab initio calculations Curie temperature Elastic moduli Thermal-elastic properties
1. Introduction Recently, a new class of materials, high-entropy alloys (HEAs) named by Yeh et al, have attracted extensive attention in scientific literature [1e3]. HEAs are composed of at least four principal alloying elements in equimolar or near-equimolar ratio [4,5], which are also known as multi-principle element alloys. The extensively available experiments suggest that the family of single-phase HEAs include fcc 3d-HEAs mainly composed of 3d transition metals [1,2], the body centered cubic (bcc) refractory-HEAs composed of refractory metals [6], and the hexagonal closed-packed (hcp) rareearth-HEAs composed of rare-earth elements [7]. Since Cantor et al [8] found the fcc equimolar CoCrFeMnNi HEA, a large number of experimental and theoretical works focused on the Co-Cr-Fe-Mn-Ni based alloys [9e15]. Laplanche [9] and Haglund [10] et al studied the temperature dependent of elastic
* Corresponding author. E-mail address: [email protected] (F. Tian).
moduli and thermal expansion coefficient (TEC) at temperature (T ¼ 200e1270 K) and the polycrystalline elastic moduli at cryogenic temperature (T ¼ 50e300 K) of CoCrFeMnNi, respectively. To better understand the CoCrFeMnNi solid solution, Wu and Bei et al [11,12] investigated mechanical properties, recovery, recrystallization, grain growth and phase stability of the Cr-Mn-Fe-Co-Ni based binary, ternary and quaternary alloys. Liu [13] took CoCrFeMnNi as an example to study the microstructure stability and grain growth of HEAs. Lucas et al [14,15] used the vibrating sample magnetometry to determine the high-temperature magnetic applications of the equimolar CoCrFeNi HEA and explored the effects of Cr content on the magnetic properties of CoFeNiCrx HEAs. Results showed that the Curie temperature of CoFeNiCrx increases with decrease of Cr content. Although single-phase HEAs have a simple crystal structure, the atomic random distribution on lattice sites in solid solutions induces the chemical disorder and magnetic disorder. For singlephase 3d HEAs composed of magnetic elements, the Co-Cr-FeMn-Ni alloys show the various magnetic behaviors and different
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H. Ge et al. / Materials Chemistry and Physics xxx (2017) 1e7
Curie temperatures [14e16], due to different magnetic states of alloying elements in their ground state (ferromagnetic (FM) for Fe, Ni, Co, while antiferromagnetic for Cr, multi-magnetic for Mn). As far as we know, the theoretically predicted intrinsic magnetic properties of the Cr-Mn-Fe-Co-Ni multi-principle element alloys are very limited. Ab initio calculations have been used successfully to predict the fcc stability [17], Young's modulus at low temperature [18] and stacking fault energy [19] for the quinary CoCrFeMnNi HEA. Experiments showed that the binary (CoNi, FeNi), ternary (CoCrNi, CoFeNi, CoMnNi, CrFeNi), quaternary (CoCrFeNi, CoCrMnNi, CoFeMnNi) and quinary (CoCrFeMnNi) alloys all adopt a singlephase fcc solid solutions [12]. From the equimolar binary alloy to equimolar quinary HEA, it is very interesting to study the effect of alloying elements on the elasticity of Co-Cr-Fe-Mn-Ni alloys. In this work we use ab initio calculations to investigate the influence of alloying elements on the equilibrium bulk properties of Co-Cr-FeMn-Ni alloys. Meanwhile the thermal-elastic properties as a function of temperature are systematically studied for the single-phase Co-Cr-Fe-Mn-Ni multi-principle element alloys based on quasiharmonic Debye-Grüneisen approximation. The rest of the paper is organized as follows. In Section 2, we present the methodology and give the most important details of the numerical calculations. In Section 3, we show the results of the equilibrium bulk properties and elastic properties of different magnetic states and discuss the effect of alloying element on the Curie temperature and the thermal-elastic properties of Co-Cr-FeMn-Ni alloys. Finally, some conclusions are drawn in Section 4. 2. Theoretical methodology The exact muffin-tin orbitals (EMTO) method is an efficient ab initio approach to solving the KohneSham equation [20] which employs large overlapping potential spheres for the exact singleelectron potential and the full charge density method further improves the accuracy of full potential total energy [21]. The substitutional disorder in both chemical and magnetic degrees of freedom [22,23], is treated by the coherent potential approximation (CPA) which use an effective medium in single site approximation. For the self-consistent calculations, we employed the Perdew-Burke-Ernzerhof (PBE) exchange correlation functional form of generalized gradient approximation (GGA) to compute the charge density and total energy [24]. The Green's function was calculated for 24 complex energy points from the bottom of the valence bands to the Fermi level. In the irreducible wedge of the fcc Brillouin zone, we used 240 inequivalent k points, allowing to maintain a 102 mRy/atom accuracy in the total energy. We employed the disorder local magnetic moment (DLM) [25] picture to describe the paramagnetic (PM) state of the present HEAs. The DLM resembles a random spin configuration with zero averaged magnetization, while allowing the presence of finite local magnetic moments on each lattice site above the magnetic transition temperature [26e28]. There are only three independent elastic constants c11, c12, and c44 in the cubic lattice. Based on the equilibrium volume and bulk modulus derived from the forth-order Birch-Murnaghan equation of state (i.e. energy as a function of volume), the elastic constants c11, c12 and c44 were obtained through by calculating the total energy as a function of volume-conserving lattice strains [29,30]. Combining the three elastic constants c11, c12, c44 with the VoigteReusseHill averaging method [31,32], we can calculate the polycrystalline elastic moduli (shear modulus G, Young's modulus E), Poisson's ratio v and Zener ratio AZ [21]. We used the quasi-harmonic Debye-Grüneisen approximation to compute the TEC based on the equation of state and the ab initio
calculated Poisson ratio, then we calculated the equilibrium volume and bulk modulus at given temperature, and further the temperature dependent elastic constants and polycrystalline elastic moduli were estimated with the method of elastic calculations [21]. The quasi-harmonic approximation is a simple and effective method to account for the lattice vibrational information, without the phonon density of state for solid solutions. In the quasi-harmonic DebyeGrüneisen model, the harmonic approximation at any given crystal structure is assumed, even if it does not correspond to the equilibrium structure. Note that we also considered the contributions of electronic entropy and magnetic entropy to Gibbs energy implemented in quasi-harmonic approximation. 3. Results and discussion 3.1. Equilibrium bulk properties Fig. 1 shows the equation of states of the paramagnetic (PM) CrMn-Fe-Co-Ni alloys. Although the Fe, Co and Ni alloying elements have similar Goldschmidt atomic radii [33], their binary solid solutions have different equilibrium Wigner-Seitz (WS) radii (see Fig. 1 and Table 1), for example the PM CoNi alloy has smaller WS radius than FeNi. Interestingly their WS radii of binary solid solutions are very close to the average WS radii derived from the experimental lattice parameters of alloying elements via Vegard's law [34]. For CoCrNi, CoFeNi and CoMnNi ternary alloys, ab initio calculations predict the similar WS radius to each other, which are slightly smaller than the average WS radii. The EMTO predicted WS radius of CrFeNi is much smaller than the experimental average one. The difference between ab initio calculations and experimental average may be due to ab initio performed at T ¼ 0 K temperature. The EMTO predicted WS radii of quaternary and quinary HEAs are very close to each other. The FM alloys have slightly larger WS than the PM ones, except for Mn-containing alloys, due to the magnetic order. 3.2. Curie temperature Within the numerous successful applications, the EMTO-CPA method has been developed an attractive tool in materials science to predict finite temperature magnetic properties. Here we use Curie temperature TC to describe the finite temperature
Fig. 1. Equation of states (energy (meV/atom) vs volume (Å3/atom)) for the at paramagnetic (PM) Cr-Mn-Fe-Co-Ni multi-principle element alloys. Left column (a) is for binary alloys, middle column (b) for ternary alloys, right column (c) for HEAs (quaternary and quinary alloys). All energies are relative to their corresponding total energy at the equilibrium volume.
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3
Table 1 Listed are the equilibrium bulk parameters, Curie temperature and elastic constants of the Co-Cr-Fe-Mn-Ni alloys. The energy difference between paramagnetic (PM) and ferromagnetic (FM) states DE ¼ EPM-EFM (mRy/atom), the Curie temperature TC (K), WignereSeitz (WS) radius u (Bohr), the elastic constants (c11 c12, c44) (GPa), the tetragonal shear modulus c΄¼(c11-c12)/2 (GPa) and the Cauchy pressure (c12-c44) (GPa). ut represents the theoretical (EMTO) Wigner-Seitz radius, ue stands for the average WS radius from the experimental radii of pure alloying elements [34]. States
DE
TC
ut
ue
PM FM
9.083
956
2.598 2.605
2.608
PM FM
5.263
554
2.623 2.642
2.635
PM FM
0.066
7
2.606 2.612
2.633
PM FM
7.959
838
2.607 2.629
2.627
PM FM
1.808
190
2.613 2.600
2.628
PM FM
1.004
106
2.622 2.629
2.650
PM FM
1.531
161
2.606 2.618
2.642
PM FM
0.24
25
2.611 2.608
2.642
PM FM
1.595
168
2.608 2.617
2.638
PM FM
0.321
34
2.607 2.613
2.647
c11 CoNi 253.4 264.7 FeNi 218.1 214.6 CoCrNi 278.8 266.7 CoFeNi 234.2 229.8 CoMnNi 211.7 216.6 CrFeNi 234.2 225.3 CoCrFeNi 272.8 237.9 CoCrMnNi 249.1 217.6 CoFeMnNi 214.5 209.8 CoCrFeMnNi 242.2 216.5
c12
c44
c'
c12-c44
158.9 167.1
180.3 172.8
47.2 48.8
21.4 5.8
146.8 164.8
158.5 135.7
35.7 24.9
11.7 29.1
188.8 182.1
183.0 177.6
44.9 42.3
5.8 4.5
144.2 174.4
177.9 147.2
45.0 27.7
33.7 27.2
116.4 102.8
184.4 202.7
47.6 56.9
67.9 99.9
167.9 168.5
167.2 158.9
33.2 28.4
0.7 9.5
180.9 161.8
185.5 172.9
46.0 38.0
4.6 11.1
161.6 141.0
185.3 159.4
43.7 38.3
23.7 18.3
116.5 116.6
190.6 179.7
49.0 46.6
74.0 63.0
140.6 113.1
192.7 184.4
50.8 51.7
52.0 71.2
magnetic properties of the Co-Cr-Fe-Mn-Ni alloys. We consider two different magnetic scenarios: the ferromagnetic (FM) state and the paramagnetic (PM) state. Within the mean field approximation, TC is directly related to the sum of the magnetic interactions. If alloyed with a small concentration of nonmagnetic elements, TC is expressed as [35]
TC ¼
2 2 PM FM Etot ¼ Etot DE; 3ð1 cÞkB 3ð1 cÞkB
(1)
PM EFM ) is the where kB represents the Boltzmann constant, ðEtot tot difference between the ground state total energies of the PM and FM alloys, c is the content of nonmagnetic elements. For the present alloys composed of magnetic elements: Cr, Mn, Fe, Co and Ni, Eq. (1) may be formulated as
TC ¼
2 PM FM Etot Etot : 3kB
(2)
For the solid-solution Cr-Mn-Fe-Co-Ni alloys, the alloying elements play an important role in the Curie temperature. Our ab initio prediction of binary alloys is Tc ¼ 956 K for CoNi and Tc ¼ 554 K for FeNi, respectively. It is reasonable, compared with the considerable TC of all three individual ferromagnetic elements (Co (1394 K), Fe (1041 K), and Ni (611 K)). Fig. 2 shows the Tc of present alloys. With the addition of equimolar alloying element of Cr and Fe, TC decreases from 956 K for CoNi to 7 K for CoCrNi (838 K for CoFeNi), from 554 K for FeNi to 106 K for CrFeNi, and from 838 K for CoFeNi to 161 K CoCrFeNi, except for the case of CoMnNi and CoFeMnNi. Also the addition of equimolar Mn obviously decreases the TC of CoNi, CoCrNi and CoFeNi. While the addition of Co slightly increases TC, from 554 K for FeNi to 838 K for CoFeNi and from 106 K for CrFeNi to 161 K CoCrFeNi. The change of Tc may derive from the
Fig. 2. Curie temperature TC for the Co-Cr-Fe-Mn-Ni multi-principle element alloys (the symbol pm presents what the paramagnetic (PM) state is more stable than the ferromagnetic (FM) one).
intrinsic magnetic property of alloying elements, for example Cr is antiferromagnetic, Co is ferromagnetic, and multi-magnetic for Mn. The present prediction of TC ¼ 161 K for CoCrFeNi is in good accordance with the experimental value of 130 K [15]. Equation (2) suggests that the large TC is from a large energy difference between PM and FM. In case of phase stability, from Fig. 2 we can find that the PM CoMnNi and CoCrMnNi are
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more stable at T ¼ 0 K. In contrary, the FM state is more stable for other alloys. 3.3. Elastic moduli Tables 1 and 2 list the three independent elastic constants (c11, c12 c44), the tetragonal shear modulus c’¼ (c11-c12)/2, Cauchy pressure (c12-c44), polycrystalline elastic moduli (B, G, E), the Zener ratio AZ, the Poisson's ratio n, and the Pugh ratio B/G as well as the elastic anisotropy ratio AVR. From the three independent elastic constants c11, c12, c44 and c’, we can find that all alloys fulfill the condition of mechanical stability (c44 > 0, c11>|c12| and c11þ2c12 > 0 [21]), i.e. they are mechanically stable. The tetragonal shear modulus c’ represents the anisotropy of elastic moduli against tetragonal distortions. The decreasing c’ suggests that the fcc stability reduces [36]. Compared with the FM state, the PM state makes the fcc phase more stable, except for CoNi, CoMnNi and CoCrFeMnNi. Compared with the tetragonal elastic modulus c’ ¼ 47.6 (56.9) GPa for PM (FM) CoMnNi, the addition of Cr slightly decreases the c’ ¼ 43.7 (38.3) GPa of CoCrMnNi. Whereas the comparison of c’ between CoNi, CoCrFeNi, FeNi and CoMnNi, CoCrFeMnNi, CoFeNi indicates that both Co and Mn increase the stability of fcc phase. It is related to the valence electron concentration (VEC). In the case of single phase HEAs, with increase of VEC, the phase transforms from bcc to fcc according to the available experiments [37]. The VEC ¼ 6, 7, 8, 9, 10 are for Cr, Mn, Fe, Co, Ni, respectively. The addition of Cr (Co) decreases (increases) the VEC of present alloys. The Cauchy pressure (c12-c44) is often used to characterize the bond properties of a material. (c12-c44) > 0 suggests that the material is dominated by the metallic bond, while (c12-c44) < 0 for the covalent bond. From Table 1, we see that for FM and PM states, CoNi, CoMnNi, CoCrFeNi, CoCrMnNi, CoFeMnNi and CoCrFeMnNi show the covalent bond, CoCrNi and CrFeNi have weak metallic bond. However, FeNi and CoFeNi show the covalent bond in the PM state, while show metallic bond in the FM state. Cr (Co or Mn) slightly reduces (strengthens) the covalent bond behavior in PM state of the present alloys. Table 2 lists the polycrystalline elastic moduli, i.e. the bulk modulus B, shear modulus G, Young's modulus E, and Poisson's ratio n, Pugh ratio B/G as well as elastic anisotropy AVR and AZ of the Cr-Fe-Mn-Co-Ni alloys. We find that the polycrystalline elastic moduli Et, Gt, and nt are in good agreement with the available experiments, except for CoMnNi. For the present alloys, the PM state has larger Young's modulus E and shear modulus G than the FM state. The brittle-ductile behavior of alloys is often indicated by Pugh ratio B/G [38]. When B/G > 1.75, the alloy is ductile, otherwise the material is brittle. The values of B/G (from 1.80 to 2.59) suggest that CoNi, FeNi, CoCrNi, FM CoFeNi, CrFeNi, CoCrFeNi and CoCrMnNi may be slightly ductile. While B/G ¼ 1.15e1.69 suggests the less ductile for PM CoFeNi, CoMnNi, CoFeMnNi, and CoCrFeMnNi. The elastic isotropy for cubic lattice is often determined by the Zener ratio AZ and the elastic anisotropy AVR. If AZ ¼ 1 or AVR ¼ 0, the material is elastically isotropy, otherwise it is elastically anisotropy [21]. From the value of AZ, AVR listed in Table 2 we see that these HEAs are elastically anisotropic. Generally speaking, the elastic modulus has empirically related to the mechanical properties. The bulk modulus B represents the ability of a material to resist the volume change under pressure. The largest bulk modulus of CoCrNi among the present alloys suggests that CoCrNi has the good capacity to resist volume change under pressure. The FM alloys have similar bulk moduli to the PM ones. It is clear from Fig. 4 and Table 2 that with the addition of equimolar Co (Cr) (from binary FeNi (CoNi) to ternary CoFeNi (CoCrNi), from ternary CrFeNi (CoMnNi) to quaternary CoCrFeNi (CoCrMnNi)),
both bulk modulus and Young's modulus become large. Whereas the change of shear modulus is small. From Table 2, we see that the theoretical Young's moduli and shear moduli at 0 K are slightly larger than experiments. Note that the above discussions are based on ab initio calculations performed at T ¼ 0 K temperature. 3.4. Thermal-elastic properties Using the thermal expansion equation uðTÞ ¼ uð0 KÞð1 þ aT TÞ, whereu(0 K) is the WS radius at 0 K, uðTÞ at temperature T, and aT is the thermal expansion coefficient (TEC) at T temperature, one may estimate the WS radius at T, and then combining with the bulk modulus, one can calculate the elastic constants and polycrystalline elastic moduli at given temperature T. Fig. 3 and Fig. 4 show the elastic constants and polycrystalline elastic moduli of these 3d multi-principle element alloys as a function of temperature. From Fig. 3, we can see that all of these alloys satisfy the criteria (c44 > 0, c11>|c12|, c11þ2c12 > 0) of mechanical stability at finite temperature, and c11 and c44 decrease obviously with the increase of temperature. Fig. 3 shows that the Cauchy pressure (c12-c44) is positively correlated with temperature. It indicates that the metal bond characteristics of these alloys are enhanced with the increasing temperature, i.e, the present alloys show the good ductility when the temperature is considered in ab initio calculations. The predicted polycrystalline elastic moduli (B, G, E, B/G and n) as a function of temperature are shown in Fig. 4. It is found that with the increase of temperature the B, G and E values decrease, the change of B is the most gentle, whereas both Pugh ratio B/G and Poisson's ratio n increase. Results suggest the toughness of the present alloys tends to be superior with the increasing temperature. From Fig. 4 we can see that CoCrFeNi (CoMnNi) has the largest (smallest) bulk modulus in Co-Cr-Fe-Mn-Ni alloy at range of temperature T ¼ 100e600 K. The quaternary and quinary HEAs have close the Young's modulus E and shear modulus G at the low temperature T ¼ 100e300, whereas with increasing temperature, the difference of elastic moduli (E and G) between quaternary and quinary HEAs becomes large. In all, the alloying element have almost no effect on the elastic moduli (E and G) of quaternary and quinary HEAs. For CoCrFeMnNi, the empirical equations of Young's modulus E and shear modulus G from experiments [9] are as follows
E ¼ 214 35
. 416 eT 1
(3)
. 448 G ¼ 85 16 e T 1
(4)
From the two above equations, we can see that both Young's modulus E and shear modulus G decrease with increasing temperature. The green lines in Fig. 5 show the elastic moduli E and G as a function of temperature calculated by using Eqns (3) and (4). As the above discussions, the elastic moduli E and G can be calculated with the temperature dependent elastic constant calculated based on the combination of the TEC with the equilibrium WS radius. Considering the differences of TEC and equilibrium WS radius between theoretical calculations and experiments, we computed the temperature dependent elastic moduli E and G by using the experimental TEC and theoretical and experimental WS radius. The experimental TEC [9] is given by
a ¼ 23:7 106 1 e299 : T
(5)
The red and blue lines shown in Fig. 5 represent the elastic modulus calculated via the experimental TEC in combination
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H. Ge et al. / Materials Chemistry and Physics xxx (2017) 1e7
5
Table 2 Shown are the theoretical polycrystalline elastic moduli B, G, E (GPa), the Zener ration AZ, the Poisson's ratio n, the Pugh ratio B/G as well as the elastic anisotropy ratio AVR at the equilibrium bulk parameters of paramagnetic (PM) and ferromagnetic (FM) Co-Cr-Fe-Mn-Ni alloys. The experimental data of Young modulus Ee, the shear modulus Ge, and the Poisson's ratio ne are listed. State
B
Gt
Ge
Et
PM FM
190.4 199.6
105.9 104.4
84a
268.1 266.7
PM FM
170.5 181.4
88.0 70.1
62a
225.3 186.3
PM FM
218.8 210.3
104.9 100.7
87a
271.4 260.5
PM FM
174.2 192.8
103.2 76.7
60a
258.5 203.1
PM FM
148.2 140.7
107.8 122.2
77a
260.2 284.4
PM FM
190.0 187.4
88.8 81.4
e
230.4 213.3
PM FM
211.5 187.2
106.7 95.2
84a
274.1 244.3
PM FM
190.8 166.5
104.7 90.7
78a
265.6 230.2
PM FM
149.2 147.7
111.2 105.1
77a
267.2 254.9
PM FM
174.5 147.6
113.4 111.1
86c
279.7 266.5
Ee CoNi e FeNi e CoCrNi 231b
nt
ne
B/G
AVR
AZ
0.27 0.28
0.29a
1.80 1.91
0.20 0.18
3.82 3.54
0.28 0.33
0.34a
1.94 2.59
0.24 0.30
4.44 5.46
0.30 0.29
0.30a
2.09 2.09
0.22 0.23
4.07 4.20
0.25 0.32
0.35a,
1.69 2.51
0.21 0.30
3.95 5.32
0.21 0.16
0.23a
1.38 1.15
0.19 0.18
3.70 3.56
0.30 0.31
e
2.14 2.30
0.28 0.31
5.04 5.59
0.28 0.28
0.28a,
1.98 1.97
0.22 0.25
4.04 4.55
0.27 0.27
0.25a
1.82 1.84
0.23 0.22
4.24 4.16
0.20 0.21
0.22a
1.34 1.40
0.20 0.20
3.89 3.86
0.23 0.20
0.26a.c
1.54 1.33
0.20 0.18
3.79 3.57
CoFeNi
a
Ref. [11],
b
Ref. [39], c Ref. [10],
d
162 CoMnNi e CrFeNi 231b CoCrFeNi 226b CoCrMnNi e CoFeMnNi e CoCrFeMnNi 203d 215c
Ref. [9].
Fig. 3. Elastic constants as a function of temperature for PM Co-Cr-Fe-Mn-Ni alloys. The left panel is for ternary alloys and the right panel is for quaternary and quinary HEAs.
Fig. 4. Polycrystalline elastic moduli as a function of temperature for the PM Co-Cr-FeMn-Ni alloys. The left panel is for ternary alloys and the right panel is for quaternary and quinary HEAs.
with the experimental and theoretical WS radii, respectively. We can see that at T ¼ 100e500 K the similar trend of elastic moduli E and G derived from the experimental and theoretical WS radii. From Fig. 5, we can see that our ab initio results of E and G are in
good agreement with the experimental data near room temperature. Meanwhile we note that the ab initio calculated Young's modulus as a function of temperature is slightly different from the experiments. It may be due to the EMTO-CPA underestimated
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H. Ge et al. / Materials Chemistry and Physics xxx (2017) 1e7
Fig. 5. Young's modulus E and shear modulus G as a function of temperature for CoCrFeMnNi: the black line represents ab initio calculated data in the present work; the red line represents estimated data gotten via the combination of the experimental TEC (Eq. (5)) with the ab initio calculated equilibrium WS radius; the blue line represents the theoretical data calculated via the combination of experimental TEC (Eq. (5)) with the experimental equilibrium WS radius 2.651 Bohr (at 300 K); the green line represents the experimental data obtained via solving Eq. (3) and Eq. (4).
equilibrium WS radius and the quasi-harmonic Debye-Grüneisen overestimated TEC. Figure 6 shows the room-temperature Young's moduli along different crystal directions. According to the elastic theorem, we can compute Young's modulus in plane (110) family as a function of crystal direction based on the three independent elastic constants (c11, c12 and c44). The estimated Young's modulus from experimental elastic constants [41] is E ¼ 201.6 GPa, which is consistent with other experimental measures [9]. The present theoretical prediction for the <110>, <331> and <100> crystal direction is in good agreement with experiments. Considering the same experimental method are employed in Refs. [40,41] and there exist some differences for Young's modulus along the same crystal direction, our theoretical calculations of Young's moduli are also consistent with experiment for <311> and <111> crystal directions. 4. Conclusions By performing the ab initio EMTO-CPA calculations in combination with the quasi-harmonic Debye-Grüneisen model, we have
Fig. 6. Theoretical and experimental Young's modulus for CoCrFeMnNi alloys plotted in plane (110) as a function of crystal direction, including the five cubic directions. Exp. a from Ref. [40], Exp. b from Ref. [41], and Exp. c from Ref. [9].
investigated the equilibrium bulk properties, the Curie temperature and thermal-elastic properties of ferromagnetic and paramagnetic Co-Cr-Fe-Mn-Ni multi-principle element alloys, from binary (CoNi, FeNi) to ternary (CoCrNi, CoFeNi, CoMnNi, CrFeNi) to quaternary (CoCrFeNi, CoCrMnNi, CoFeMnNi) and quinary (CoCrFeMnNi) fcc solid solutions. The addition of equimolar alloying element Cr, Fe or Mn decreases the Curie temperature Tc, whereas the additions of Co increases Tc of the Co-Cr-Fe-Mn-Ni multi-principle element alloys. Compared with the ferromagnetic state, the paramagnetic state increases the mechanical stability of fcc phase for the CoNi, CoMnNi and CoCrFeMnNi alloys. The calculated polycrystalline elastic moduli (Young moduli E, shear modulus G, and Poisson's ratio n) are in good agreement with the available experiments. CoCrNi has the good capacity to resist volume change under pressure among Co-Cr-Fe-Mn-Ni alloys. The elastic moduli linearly decrease with increasing temperature. The ductility of paramagnetic alloys becomes superior with the increasing temperature. For the CoCrFeMnNi high-entropy alloy, ab initio predicted Young's moduli along different crystal directions are consistent with the experiments. Based on the experimental thermal expansion coefficient, the theoretically predicted the elastic moduli (E and G) have similar trend to the experimental measures with increasing the temperature from T ¼ 100 K to T ¼ 600 K. Acknowledgment Work was supported by the National Natural Science Foundation of China with Grant No. 51771015 and Grant No. 51401014. References [1] J.W. Yeh, S.K. Chen, S.J. Lin, J.Y. Gan, T.S. Chin, T.T. Shun, C.H. Tsau, S.Y. Chang, Nanostructured high-entropy alloys with multiple principal elements: novel alloy design concepts and outcomes, Adv. Eng. Mater 6 (2004) 299e303. [2] J.W. Yeh, Recent progress in high entropy alloys, Ann. Chim. Sci. Mat. 31 (2006) 633e648. [3] J.W. Yeh, Y.L. Chen, S.J. Lin, S.K. Chen, High-entropy alloys e a new era of exploitation, Mater. Sci. Forum 560 (2007) 1e9. [4] O.N. Senkov, S.V. Senkova, D.B. Miracle, C. Woodward, Mechanical properties of low-density, refractory multi-principal element alloys of the CreNbeTieVeZr system, Mater. Sci. Eng. A 565 (2013) 51e62. Fazakas, V. Zadorozhnyy, L.K. Varga, A. Inoue, D.V. Louzguine-Luzgin, [5] E. F.Y. Tian, L. Vitos, Experimental and theoretical study of Ti 20 Zr 20 Hf 20 Nb 20 X 20 (X ¼ V or Cr) refractory high-entropy alloys, Int. J. Refract. Met. Hard Mater 47 (2014) 131e138. [6] O.N. Senkov, G.B. Wilks, D.B. Miracle, C.P. Chuang, P.K. Liaw, Refractory highentropy alloys, Intermetallics 18 (2010) 1758e1765. [7] A. Takeuchi, K. Amiya, T. Wada, K. Yubuta, W. Zhang, High-entropy alloys with a hexagonal close-packed structure designed by equi-atomic alloy strategy and binary phase diagrams, Jom 66 (2014) 1984e1992. [8] B. Cantor, I.T.H. Chang, P. Knight, A.J.B. Vincent, Microstructural development in equiatomic multicomponent alloys, Mater. Sci. Eng. A 375e377 (2004) 213e218. [9] G. Laplanche, P. Gadaud, O. Horst, F. Otto, G. Eggeler, E.P. George, Temperature dependencies of the elastic moduli and thermal expansion coefficient of an equiatomic, single-phase CoCrFeMnNi high-entropy alloy, J. Alloy. Compd. 623 (2015) 348e353. [10] A. Haglund, M. Koehler, D. Catoor, E.P. George, V. Keppens, Polycrystalline elastic moduli of a high-entropy alloy at cryogenic temperatures, Intermetallics 58 (2015) 62e64. [11] Z. Wu, H. Bei, G.M. Pharr, E.P. George, Temperature dependence of the mechanical properties of equiatomic solid solution alloys with face-centered cubic crystal structures, Acta Mater 81 (2014) 428e441. [12] Z. Wu, H. Bei, F. Otto, G.M. Pharr, E.P. George, Recovery, recrystallization, grain growth and phase stability of a family of FCC-structured multi-component equiatomic solid solution alloys, Intermetallics 46 (2014) 131e140. [13] W.H. Liu, Y. Wu, J.Y. He, T.G. Nieh, Z.P. Lu, Grain growth and the HallePetch relationship in a high-entropy FeCrNiCoMn alloy, Scr. Mater 68 (2013) 526e529. ~ oz, Y. Xiao, A.O. Sheets, S.L. Semiatin, [14] M.S. Lucas, L. Mauger, J.A. Mun J. Horwath, Z. Turgut, Magnetic and vibrational properties of high-entropy alloys, J. Appl. Phys. 109 (2011) 07E307. [15] M.S. Lucas, D. Belyea, C. Bauer, N. Bryant, E. Michel, Z. Turgut, S.O. Leontsev,
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