Key Laboratory for Anisotropy and Texture of Materials (Ministry of Education), School of Material Science and Engineering, Northeastern University, Shenyang 110819, China
2)
Key Laboratory of Dielectric and Electrolyte Functional Material Hebei Province, School of Resources and Materials, Northeastern University at Qinhuangdao, Qinhuangdao 066004, China
3)
State Key Laboratory of Metastable Materials Science and Technology, Yanshan University, Qinhuangdao 066004, China
4)
Beijing Advanced Innovation Center for Materials Genome Engineering, State Key Laboratory for Advanced Metals and Materials, University of Science and Technology Beijing, Beijing 100083, China
The martensitic transformation, mechanical, and magnetic properties of the Ni2Mn1.5−xCuxTi0.5 (x = 0.125, 0.25, 0.375, 0.5) and Ni2−yCoyMn1.5−xCuxTi0.5 [(x = 0.125, y = 0.125, 0.25, 0.375, 0.5) and (x = 0.125, 0.25, 0.375, y = 0.625)] alloys were systematically studied by the first-principles calculations. For the formation energy, the martensite is smaller than the austenite, the Ni–(Co)–Mn–Cu–Ti alloys studied in this work can undergo martensitic transformation. The austenite and non-modulated (NM) martensite always present antiferromagnetic state in the Ni2Mn1.5−xCuxTi0.5 and Ni2−yCoyMn1.5−xCuxTi0.5 (y < 0.625) alloys. When y = 0.625 in the Ni2−yCoyMn1.5−xCuxTi0.5 series, the austenite presents ferromagnetic state while the NM martensite shows antiferromagnetic state. Cu doping can decrease the thermal hysteresis and anisotropy of the Ni–(Co)–Mn–Ti alloy. Increasing Mn and decreasing Ti content can improve the shear resistance and normal stress resistance, but reduce the toughness in the Ni–Mn–Cu–Ti alloy. And the ductility of the Co–Cu co-doping alloy is inferior to that of the Ni–Mn–Cu–Ti and Ni–Co–Mn–Ti alloys. The electronic density of states was studied to reveal the essence of the mechanical and magnetic properties.
Shape memory alloys, such as Ni–Ti-based [1] and Ni–Mn-based Heusler alloys [2–4], are environmentally friendly and highly efficient, and are widely used in solid-state refrigeration [5], medical treatment [6], and aerospace [7]. Ni–Mn–Z-based (Z = Ga, In, Sn, etc.) Heusler alloys possess magnetic-field induced shape memory effect [8–11], larger magnetocaloric [12–14] and elastocaloric effect [14–16] during martensitic transformation (MT). However, intrinsic brittleness resulting from the p–d covalent hybridization between main group elements (Ni and Mn) and transitional element (Z) greatly limits its practical application. Wei et al. [17] found that the compressive strength of the novel Ni50Mn32Ti18 all-d-metal Heusler alloy overtopped 900 MPa and the change of adiabatic temperature was 10.7 K under the 3.9% strain level. Yan et al. [18] found the 1.1 GPa maximum compressive strength and the 13% directionally solidified strain in the Ni50Mn31.75Ti18.25 alloy. Due to the d–d hybridization among transition-metal elements in the Ni–Mn–Ti alloys, the mechanical properties of ternary Ni–Mn–Ti alloy were significantly improved compared with the conventional Ni–Mn-based alloys, but the magnetocaloric effect was absent because the magnetization difference (ΔM) between the austenite and martensite is almost zero.
Composition alloying is an effective method to enhance the MT and magnetic properties for the Ni–Mn–Ti alloy. Some researchers try to dope Co element in Ni–Mn–Ti alloy to enhance the magnetism. Wei et al. [19] found that under the 0.01 T magnetic field, the ΔM between martensite and austenite of the Ni35Co15Mn35Ti15 alloy can reach 15 emu·g−1. Liu et al. [20] found that annealed Ni36.5Co13.5Mn35Ti15 alloy has a large magnetic entropy change of about 25 J·kg−1·K−1 under a 5 T magnetic field. Guan et al. [21] found that the magnetization change and magnetic entropy change in the Ni36Co14Mn35Ti15 alloy under the magnetic field of 5 T reached 106 emu·g−1 and 19.3 J·kg−1·K−1, respectively. Researchers [22] proved that the magnetic entropy change of the Ni37Co13Mn34Ti16 alloy reached 38 J·kg−1·K−1 at 2 T magnetic field. The above results proved that Co doping can significantly improve ferromagnetism [23].
In this paper, the MT, mechanical and magnetic properties in Ni–(Co)–Mn–Cu–Ti alloys were studied according to the first-principles calculations, aiming to provide theoretical support for composition design.
2.
Calculation methods
The Vienna Ab-initio Simulation Package (VASP) [24–25] was used for the first-principles calculations. The projector-augmented wave (PAW) [26–27] method described the interaction between ions and electrons and the Perdew–Burke–Ernzerhof implementation of generalized gradient approximation (GGA) [28] processed the exchange-correlation potential. The valence electronic states were Ni-3d84s2, Mn-3d64s1, Ti-3d24s2, Co-3d84s1, and Cu-3d104s1. The kinetic energy cutoff was 384 eV. The k-point mesh was generated using the Monkhorst–Pack grid [29] in the Brillouin zone and 10 × 10 × 5 and 7 × 7 × 11 k-points for the 32-atom austenite and NM martensite supercell were used, respectively. The convergence criteria for the total energies were smaller than 1 meV and the total and atomic forces were smaller than 0.2 eV/nm, respectively. The elastic constants were calculated by the stress–strain method. The formation energy (Eform) was calculated by Eq. (1).
where Etotal is the compound total energy and N is the total atomic number in the unitcell, and the Ni (i = Ni, Mn, Ti, Co, Cu) is the atomic number in the unitcell and Ei is the ground state energy per atom in its reference bulk state, respectively.
For easy expression, the abbreviations of AFA, FA, AFM, and FM were used to describe the antiferromagnetic austenite, ferromagnetic austenite, antiferromagnetic martensite, and ferromagnetic martensite, respectively. The abbreviations of Ni2Mn1.5−xCuxTi0.5 (x = 0, 0.125, 0.25, 0.375, 0.5) and Ni2−yCoyMn1.5−xCuxTi0.5 [(x = 0.125, y = 0.125, 0.25, 0.375, 0.5) and (x = 0.125, 0.25, 0.375, y = 0.625)] alloys are shown in Table 1.
Table
1.
Abbreviations of Ni2Mn1.5−xCuxTi0.5 (x = 0, 0.125, 0.25, 0.375, 0.5) and Ni2−yCoyMn1.5−xCuxTi0.5 [(x = 0.125, y = 0.125, 0.25, 0.375, 0.5) and (x = 0.125, 0.25, 0.375, y = 0.625)] alloys
3.1
Phase stability and martensitic transformation
When doping a Cu atom in the supercell, the preferential site occupation of Cu should be considered. Cu substituting Ni, Mn, Ti sites are referred to as CuNi, CuMn, and CuTi, respectively. The structural model and the corresponding Eform of the austenite for CuNi, CuMn, and CuTi cases are shown in Fig. 1(a)–(c). The sequence of the formation energy is CuMn < CuNi < CuTi, therefore, the doped Cu prefers to occupy the Mn sublattice.
Fig.
1.
Structural model and corresponding formation energy of Cu substituting (a) Ni, (b) Mn, and (c) Ti site in Ni2Mn1.5Ti0.5 alloy; structural model and corresponding formation energy of (d) aggregated and (e) dispersive distribution of doped Cu atoms in Ni2Mn1.25Cu0.25Ti0.5 alloy.
When more Cu atoms are doped in a supercell, the effect of Cu–Cu distance should be considered. The doping Cu atoms are dispersive and aggregated distribution in the supercell that are represented by Cu2(D) and Cu2(A), respectively. The structural model and the corresponding Eform of the austenite for Cu2(A) and Cu2(D) cases of the doped Cu in the Ni2Mn1.25Cu0.25Ti0.5 alloy are shown in Fig. 1(d) and (e). The Eform of the Cu2(A) is much lower than that of the Cu2(D), which means doping Cu atoms prefer to be aggregated to reduce the total energy in the Ni–Mn–Cu–Ti alloy.
The Eform for the martensite and austenite of the Ni2Mn1.5−xCuxTi0.5 (x = 0.125, 0.25, 0.375, 0.5) and Ni2–yCoyMn1.5–xCuxTi0.5 [(x = 0.125, y = 0.125, 0.25, 0.375, 0.5) and (x = 0.125, 0.25, 0.375, y = 0.625)] alloys is shown in Fig. 2. For the Eform, the antiferromagnetic state is always smaller than ferromagnetic state. It means that the alloys are always present antiferromagnetic state no matter austenite or martensite. While for Eform of the Ni2−yCoyMn1.5−xCuxTi0.5 (x = 0.125, 0.25, 0.375, y = 0.625) alloys, the antiferromagnetic martensite is always smaller than the ferromagnetic case, however, the Eform of the ferromagnetic austenite is always lower than that of antiferromagnetic one, which indicates that the austenite prefers to be ferromagnetic and the martensite tends to be antiferromagnetic.
Fig.
2.
Formation energy of austenite and martensite for Ni2Mn1.5−xCuxTi0.5 (x = 0.125, 0.25, 0.375, 0.5) and Ni2−yCoyMn1.5−xCuxTi0.5 [(x = 0.125, y = 0.125, 0.25, 0.375, 0.5) and (x = 0.125, 0.25, 0.375, y = 0.625)] alloys.
Besides, for the Eform, the martensite is smaller than the austenite, so all the investigated Ni–(Co)–Mn–Cu–Ti alloys can undergo MT. The change of magnetism occurs during MT in the Ni2−yCoyMn1.5−xCuxTi0.5 (x = 0.125, 0.25, 0.375, y = 0.625) alloy system. It should be noted that the Eform increases with the increase in Cu content in the Ni2−yCoyMn1.5−xCuxTi0.5 (x = 0.125, 0.25, 0.375, y = 0.625) system, which means the stability of the martensite and austenite gradually reduces, but the austenite Eform of the Ni–Mn–Cu–Ti alloy decreases with doping Cu content, which reveals the increasing stability of the austenite.
The equilibrium lattice constants, volume, and volume change for the martensite and austenite for the Ni2Mn1.5−xCuxTi0.5 and Ni2−yCoyMn1.5−xCuxTi0.5 systems are shown in Table 2. According to Table 2, the equilibrium lattice constants of the austenite tend to decrease with the increase in Cu and Co content. The volume of the AFA and AFM phases of the Ni2−yCoyMn1.375Cu0.125Ti0.5 (y = 0.125, 0.25, 0.375, 0.5) alloys generally reduces with the increase in Co content.
Table
2.
Equilibrium lattice constants, volume, and volume change of austenite and martensite for Ni2Mn1.5−xCuxTi0.5 (x = 0, 0.125, 0.25, 0.375, 0.5) and Ni2−yCoyMn1.5−xCuxTi0.5 [(x = 0.125, y = 0.125, 0.25, 0.375, 0.5) and (x = 0.125, 0.25, 0.375, y = 0.625)] systems
Component
AFA
Non-modulated AFM
a / nm
Cell volume / nm3
a / nm
b / nm
c / nm
Cell volume / nm3
∆V / nm3
Cu0
0.58483
0.200027
0.89960
0.41761
0.51238
0.192492
0.007535
Cu1
0.58397
0.199146
0.86000
0.43085
0.52817
0.195703
0.003443
Cu2
0.58353
0.198696
0.84016
0.44325
0.51792
0.192874
0.005822
Cu3
0.58258
0.197727
0.87623
0.42103
0.52404
0.193328
0.004399
Cu4
0.58013
0.195243
0.86075
0.42927
0.52416
0.193674
0.001569
Cu1Co1
0.58369
0.198860
0.81653
0.40860
0.59234
0.197625
0.001235
Cu1Co2
0.58127
0.196396
0.74669
0.37374
0.69185
0.193073
0.003323
Cu1Co3
0.58123
0.196356
0.74698
0.37448
0.68892
0.192711
0.003645
Cu1Co4
0.58029
0.195405
0.74651
0.37483
0.68688
0.192199
0.003206
Component
FA
Non-modulated AFM
a / nm
Cell volume / nm3
a / nm
b / nm
c / nm
Cell volume / nm3
∆V / nm3
Cu1Co5
0.58309
0.198247
0.74685
0.37615
0.68292
0.191851
0.006396
Cu2Co5
0.58229
0.197432
0.75242
0.37926
0.67376
0.192266
0.005166
Cu3Co5
0.58108
0.196204
0.75695
0.38176
0.66582
0.192404
0.003800
Notes: a, b, and c are the side lengths of the cell along the x, y, and z axes, respectively.
The geometrical compatibility between the austenite and martensite has a great influence on the thermal hysteresis (∆THys) [30–31], and the ∆THys can be qualitatively described by volume change (∆V) [32]. Generally, the ∆THys increases with the increasing ∆V. According to Table 2, the ∆V of the Ni2Mn1.5Ti0.5 alloy is the largest. When doped with five Co atoms in the 32-atom Ni2−yCoyMn1.5−xCuxTi0.5 alloy, the magnetism of the austenite reverses, and the ∆V increases. The volume for FA decreases with doping Cu, but AFM increases, so the ∆V decreases. In addition, researchers [33–36] proved that Cu doping can reduce ∆THys. Therefore, the Cu doping and Cu–Co co-doping can effectively reduce the ∆V and thus the ∆THys.
In addition, the volume of austenite is always larger than that of the martensite phase, as shown in Table 2, which implies that the MT in the Ni–(Co)–Mn–Cu–Ti alloy is accompanied by volume contraction.
3.2
Mechanical and magnetic properties
The Young’s modulus Y=9BG3B+G, shear modulus G=3C11−3C12+4C4410, bulk modulus B=C11+2C123, and Poisson’s ratio ν=3B−2G2G+6B of the Ni2Mn0.75Cu0.25Ti, Ni2MnCu0.25Ti0.75, Ni2Mn1.25Cu0.25Ti0.5, Ni1.5Co0.5MnTi, and Ni1.5Co0.5Mn1.25Cu0.25Ti0.5 alloys are obtained in the present work and the results are listed in Table 3. Guan et al. [37] calculated the elastic constants of the cubic austenite of the Ni2MnTi, Ni2MnIn, and Ni2MnGa alloys, and Xiong et al. [38] calculated the elastic constants of the cubic austenite of the (Ni2MnTi)0.89B0.11 alloys, these data are also listed in Table 3 for comparison.
Table
3.
Elastic constants (C11, C12, and C44), elastic modulus (bulk modulus B, shear modulus G, and Young’s modulus Y), Cauchy pressure Pc, B/G ratio, and Poisson’s ratio ν of Ni2Mn0.75Cu0.25Ti, Ni2MnCu0.25Ti0.75, Ni2Mn1.25Cu0.25Ti0.5, Ni1.5Co0.5MnTi, Ni1.5Co0.5Mn1.25Cu0.25Ti0.5, (Ni2MnTi)0.89B0.11, Ni2MnTi, Ni2MnGa, and Ni2MnIn alloys. Elastic properties of Ni2MnTi, Ni2MnGa, and Ni2MnIn alloy are cited from Ref. [37], elastic properties of (Ni2MnTi)0.89B0.11 are cited from Ref. [38]
The alloy can be classified as ductile material when the Pugh ratio B/G > 1.75 and Cauchy pressure Pc = C12 − C44 > 0. Compared with Ni2MnIn and Ni2MnGa alloys, Ni2Mn0.75Cu0.25Ti, Ni2MnCu0.25Ti0.75, Ni2Mn1.25Cu0.25Ti0.5, and Ni1.5Co0.5MnTi alloys possess smaller G and Y values. The results show that Ni–Mn–Cu–Ti and Ni–Co–Mn–Ti alloys have weaker shear resistance and normal stress resistance, but Ni–Co–Mn–Ti alloy possesses greater incompressibility. However, the Ni–Mn–Cu–Ti and Ni–Co–Mn–Ti alloys have a larger value of Pc, B/G, and ν than those of the Ni2MnIn and Ni2MnGa alloys, which is the essence for the enhancement of ductility and plasticity of the Ni–Mn–Ti-based alloys [18].
For the B, Pc, B/G, and ν, the Ni–Mn–Cu–Ti alloy is larger than the Ni1.5Co0.5Mn1.25Cu0.25Ti0.5 alloy and the Ni2MnTi alloy is larger than the Ni1.5Co0.5MnTi alloy, which indicates that the Co doping reduces the incompressibility and ductility of the Ni–Mn–(Cu)–Ti alloy. While the Ni1.5Co0.5MnTi alloy is larger than the Ni1.5Co0.5Mn1.25Cu0.25Ti0.5 alloy and the Ni2MnTi alloy is larger than the Ni–Mn–Cu–Ti alloy, which suggests that the Cu doping also reduces the incompressibility and ductility in the Ni–(Co)–Mn–Ti alloy.
The larger the Pc, B/G, and ν, the better the toughness. The Pc, B/G, and ν for the Ni1.5Co0.5MnTi alloy are larger than that of the Ni–Mn–Cu–Ti alloy, the toughness of the Ni1.5Co0.5MnTi alloy is better than that of the Ni–Mn–Cu–Ti alloy. Besides, doping Co in ternary Ni2MnTi alloy presents magnetocaloric effect. Therefore, Co doping is beneficial for the alloy to obtain both magnetocaloric effect and has little effect on toughness [21,39].
With the increase of Mn content and the decrease of Ti content, the G and Y values increase, but B/G decreases, which indicates that the increasing Mn and decreasing Ti content can improve the shear resistance and normal stress resistance, but reduce the toughness in the Ni–Mn–Cu–Ti alloy.
Three-dimensional Young’s modulus for the single-crystal Ni2MnTi, Ni1.5Cu0.5MnTi, Ni2Mn0.75Cu0.25Ti, Ni1.5Co0.5MnTi, and Ni1.5Co0.5Mn1.25Cu0.25Ti0.5 alloys is shown in Fig. 3. In Fig. 3, the anisotropy of the Ni1.5Co0.5MnTi alloy is the strongest, which means that doping Co can increase the anisotropy of the Ni–Mn–Ti-based alloys. In addition, the anisotropy of Ni2MnTi alloy can be reduced by doping Cu.
Fig.
3.
Three-dimensional Young’s modulus for single-crystal (a) Ni1.5Cu0.5MnTi, (b) Ni2Mn0.75Cu0.25Ti, (c) Ni2MnTi, (d) Ni1.5Co0.5MnTi, and (e) Ni1.5Co0.5Mn1.25Cu0.25Ti0.5 alloys.
The total and atomic magnetic moments of the Ni2Mn1.5−xCuxTi0.5 and Ni2−yCoyMn1.5−xCuxTi0.5 alloy systems are shown in Fig. 4. According to Fig. 4(a), the magnetic moment for the austenite and martensite is almost unchanged in the Ni2Mn1.5−xCuxTi0.5 and Ni2−yCoyMn1.5−xCuxTi0.5 (y < 0.625) series. However, the austenite total magnetic moment suddenly increases in the Ni2−yCoyMn1.5−xCuxTi0.5 (y = 0.625) series and gradually reduces with the increase of Cu content, which is mainly due to that the paramagnetic Cu substitutes ferromagnetic Mn, while the magnetism is mainly contributed by Mn atoms [40–44].
Fig.
4.
(a) Total magnetic moments of austenite and martensite and (b) atomic magnetic moments of Mn in austenite for Ni2Mn1.5−xCuxTi0.5 (x = 0.125, 0.25, 0.375 and 0.5) and Ni2−yCoyMn1.5−xCuxTi0.5 [(x = 0.125, y = 0.125, 0.25, 0.375, 0.5) and (x = 0.125, 0.25, 0.375, y = 0.625)] series.
Besides, the Mn moments are arranged in antiparallel with high symmetry in the Ni2Mn1.5−xCuxTi0.5 and Ni2−yCoyMn1.5−xCuxTi0.5 (y < 0.625) series, while parallel alignment in the austenite of the Ni2−yCoyMn1.5−xCuxTi0.5 (y = 0.625) series, leading to a sudden increase in the total magnetic moment. The above results are the same as that of Eform.
3.3
Electronic structures
The electronic density of states (DOS) of the Ni2−yCoyMn1.5−xCuxTi0.5 (x = 0.125, 0.25, 0.375, y = 0, 0.625) series was studied in detail in order to study the essence of the mechanical and magnetic properties.
The total DOS of the martensite and austenite for the Ni2Mn1.5−xCuxTi0.5 and Ni1.375Co0.625Mn1.5−xCuxTi0.5 series is shown in Fig. 5. The total DOS of the AFA and AFM phases have very high symmetry for the Cu1, Cu2, and Cu3 alloys according to Fig. 5(a), (c), and (e), which illustrates that their total magnetic moments are almost zero. However, for the Cu1Co5, Cu2Co5, and Cu3Co5 alloys as seen in Fig. 5(b), (d), and (f), the total DOS for the AFM phase has a high symmetry, while that of the FA phase possesses a relatively low symmetry. Such electronic structures provide explanation for the magnetic change of the austenite with Co doping.
Fig.
5.
Total electronic density of states of austenite and martensite for (a, c, e) Ni2Mn1.5−xCuxTi0.5 (x = 0.125, 0.25, 0.375) and (b, d, f) Ni1.375Co0.625Mn1.5−xCuxTi0.5 (x = 0.125, 0.25, 0.375) series.
The partial DOS of the Ni2Mn1.5−xCuxTi0.5 and Ni1.375Co0.625Mn1.5−xCuxTi0.5 (x = 0.125, 0.25, 0.375) series is displayed in Fig. 6. Fig. 6(a)–(c) shows that the up-spin and down-spin parts of the Ni, Mn, and Cu 3d DOS have hybridization effect under the Fermi level (EF). Meanwhile, the 3d Ni and Mn partial DOS exist hybridization effect within the energy range of −3.4 to −0.7 eV. The Cu and Mn partial DOS have strong hybridization effect around −3.2 and −1.5 eV. Strong d–d hybridization is closely related to the enhancement of ductility [18,37]. The partial DOS is symmetric and opposite, which indicates that Ni, Ti, and Cu magnetic moments are close to zero, in addition, the MnMn and MnTi partial DOS cancel each other out, so the total magnetic moment approaches zero.
Fig.
6.
Partial density of states of austenite for (a, b, c) Ni2Mn1.5−xCuxTi0.5 (x = 0.125, 0.25, 0.375) and (d, e, f) Ni1.375Co0.625Mn1.5−xCuxTi0.5 (x = 0.125, 0.25, 0.375) series.
The situation is different for the Co doping case, the up-spin and down-spin Ni, Mn, Ti, Co, and Cu partial 3d DOS are asymmetric, as shown in Fig. 6(d)–(f). Based on the electronic structures, it reasonably explains that the austenite is antiferromagnetic in the Ni2Mn1.5−xCuxTi0.5 series, but ferromagnetic state in the Ni1.375Co0.625Mn1.5−xCuxTi0.5 series.
4.
Conclusions
The first-principles calculations for the martensitic transformation, mechanical and magnetic properties for Ni2Mn1.5−xCuxTi0.5 (x = 0.125, 0.25, 0.375, 0.5) and Ni2−yCoyMn1.5−xCuxTi0.5 [(x = 0.125, y = 0.125, 0.25, 0.375, 0.5) and (x = 0.125, 0.25, 0.375, y = 0.625)] alloys are investigated in this paper. The main results are as follows.
For the formation energy, the martensite is smaller than the austenite, all the investigated Ni–(Co)–Mn–Cu–Ti alloys will undergo martensitic transformation. The martensite and austenite for the Ni2Mn1.5−xCuxTi0.5 and Ni2−yCoyMn1.5−xCuxTi0.5 (y < 0.625) alloys always exist in an antiferromagnetic state and the total magnetic moments are almost zero, but the austenite of the Ni2−yCoyMn1.5−xCuxTi0.5 (y = 0.625) alloys exists in ferromagnetic state. Cu doping reduces thermal hysteresis and anisotropy for the Ni–(Co)–Mn–Ti alloy. Increasing Mn and decreasing Ti content in the Ni–Mn–Cu–Ti alloy can improve the shear resistance and normal stress resistance, but reduce the toughness. The Ni–Mn–Cu–Ti and Ni–Co–Mn–Ti alloys have a relatively larger value of Pc, B/G, and ν. The ductility of the Ni1.5Co0.5MnTi alloy is stronger than that of the Ni–Mn–Cu–Ti alloy, while the ductility of the Ni–Co–Mn–Cu–Ti alloy is inferior to that of the Ni–Mn–Cu–Ti and Ni–Co–Mn–Ti alloys. Besides, Co, Cu doping, and Cu–Co co-doping can reduce the incompressibility and ductility of the Ni–Mn–Ti-based alloys.
Acknowledgements
This work was financially supported by the National Natural Science Foundation of China (No. 51771044), the Natural Science Foundation of Hebei Province (No. E2019501061), the Performance subsidy fund for Key Laboratory of Dielectric and Electrolyte Functional Material Hebei Province (No. 22567627H), the Fundamental Research Funds for the Central Universities (No. N2223025), the State Key Lab of Advanced Metals and Materials (No. 2022-Z02), and Programme of Introducing Talents of Discipline Innovation to Universities 2.0 (the 111 Project of China 2.0, No. BP0719037). This work was carried out at Shanxi Supercomputing Center of China, and the calculations were performed on TianHe-2.
Conflict of Interest
The authors declare no potential conflict of interest.
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Table
2.
Equilibrium lattice constants, volume, and volume change of austenite and martensite for Ni2Mn1.5−xCuxTi0.5 (x = 0, 0.125, 0.25, 0.375, 0.5) and Ni2−yCoyMn1.5−xCuxTi0.5 [(x = 0.125, y = 0.125, 0.25, 0.375, 0.5) and (x = 0.125, 0.25, 0.375, y = 0.625)] systems
Component
AFA
Non-modulated AFM
a / nm
Cell volume / nm3
a / nm
b / nm
c / nm
Cell volume / nm3
∆V / nm3
Cu0
0.58483
0.200027
0.89960
0.41761
0.51238
0.192492
0.007535
Cu1
0.58397
0.199146
0.86000
0.43085
0.52817
0.195703
0.003443
Cu2
0.58353
0.198696
0.84016
0.44325
0.51792
0.192874
0.005822
Cu3
0.58258
0.197727
0.87623
0.42103
0.52404
0.193328
0.004399
Cu4
0.58013
0.195243
0.86075
0.42927
0.52416
0.193674
0.001569
Cu1Co1
0.58369
0.198860
0.81653
0.40860
0.59234
0.197625
0.001235
Cu1Co2
0.58127
0.196396
0.74669
0.37374
0.69185
0.193073
0.003323
Cu1Co3
0.58123
0.196356
0.74698
0.37448
0.68892
0.192711
0.003645
Cu1Co4
0.58029
0.195405
0.74651
0.37483
0.68688
0.192199
0.003206
Component
FA
Non-modulated AFM
a / nm
Cell volume / nm3
a / nm
b / nm
c / nm
Cell volume / nm3
∆V / nm3
Cu1Co5
0.58309
0.198247
0.74685
0.37615
0.68292
0.191851
0.006396
Cu2Co5
0.58229
0.197432
0.75242
0.37926
0.67376
0.192266
0.005166
Cu3Co5
0.58108
0.196204
0.75695
0.38176
0.66582
0.192404
0.003800
Notes: a, b, and c are the side lengths of the cell along the x, y, and z axes, respectively.
Table
3.
Elastic constants (C11, C12, and C44), elastic modulus (bulk modulus B, shear modulus G, and Young’s modulus Y), Cauchy pressure Pc, B/G ratio, and Poisson’s ratio ν of Ni2Mn0.75Cu0.25Ti, Ni2MnCu0.25Ti0.75, Ni2Mn1.25Cu0.25Ti0.5, Ni1.5Co0.5MnTi, Ni1.5Co0.5Mn1.25Cu0.25Ti0.5, (Ni2MnTi)0.89B0.11, Ni2MnTi, Ni2MnGa, and Ni2MnIn alloys. Elastic properties of Ni2MnTi, Ni2MnGa, and Ni2MnIn alloy are cited from Ref. [37], elastic properties of (Ni2MnTi)0.89B0.11 are cited from Ref. [38]
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