Cite this Article
Peng Guang-Wei, Gan Xue-Ping, Li Zhou, Zhou Ke-Chao. First-principles study of the (CuxNi1 − x)3Sn precipitations with different structures in Cu–Ni–Sn alloys. Chinese Physics B, 2018, 27(8): 086302  
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First-principles study of the (CuxNi1 − x)3Sn precipitations with different structures in Cu–Ni–Sn alloys
Peng Guang-Wei1, 2, Gan Xue-Ping1, †, Li Zhou3, Zhou Ke-Chao1
State Key Laboratory of Powder Metallurgy, Central South University, Changsha 410083, China
Hunan Automotive Engineering Vocational College, Zhuzhou 412001, China
School of Materials Science and Engineering, Central South University, Changsha 410083, China

 

† Corresponding author. E-mail: ganxueping@csu.edu.cn

Project supported by the National Key Research and Development Program of China (Grant No. 2016YFB0301402), the Project of Innovation-Driven Plan in Central South University, and the State Key Laboratory of Powder Metallurgy, Central South University, Changsha, China.

Abstract

The structural parameters, the formation energies, and the elastic and thermodynamic properties of the (CuxNi1 − x)3Sn phase with different structures are studied by the virtual crystal approximation (VCA) and super-cell (SC) methods. The lattice constants, formation energies, and elastic constants obtained by SC and VCA are generally consistent with each other. It can be inferred that the VCA method is suitable for (CuxNi1 − x)3Sn ordered phase calculation. The calculated results show that the equilibrium structures of Cu3Sn and Ni3Sn are D0a and D019 respectively. (CuxNi1 − x)3Sn_D03 with various components are the metastable phase at temperature of 0 K, just as D022 and L12. With the temperature increase, the free energy of the D03 is lower than those of D022 and L12, and D022 and L12 eventually turn into D03 in the aging process. The (CuxNi1 − x)3Sn_D022 is first precipitated in a solid solution because its structure and cell volume are most similar to those of a solid solution matrix. The L12 and the D022 possess better mechanical stability than the D03. Also, they may play a more important role in the strengthening of Cu–Ni–Sn alloys. This study is valuable for further research on Cu–Ni–Sn alloys.

PACS: 63.20.dk;71.20.Lp;71.15.Nc
Keyword:(CuxNi1 − x)3Sn;first-principle;virtual crystal approximation;super-cell
1. Introduction

The Cu–Ni–Sn alloy is a significant material of interest in science and technology due to its high strength, elasticity, and good corrosion and wear resistance.[1,2] Many studies have been performed to understand the precipitation hardening mechanism of the Cu–Ni–Sn alloy.[3,4] It has been proved that (CuxNi1 − x)3Sn can be formed into five different structures: D022(tI8), L12 (cP4), D03(cF16), D0a (oP8), and D019 (hP8),[5–7] as shown in Fig. 1.

Fig. 1. (color online) Lattice structures of (CuxNi1 − x)3Sn: (a) L12, (b) D022, (c)D03, (d) D0a, and (e) D019.

Although the equilibrium structures of Cu3Sn and Ni3Sn are D0a and D019 respectively, D0a and D019 can be hardly precipitated from a face-centered cubic (fcc) solid solution, in which the atoms do not have enough diffusion capacity to overcome the structural difference in the solid phase transition. The D022, L12, and D03 are the main strengthening phases precipitated from the supersaturated solid solution of the Cu–Ni–Sn alloy, as indicated by the experimental time-temperature-transformation (TTT) curves of the Cu–15Ni–8Sn alloy in Fig. 2.[8] When the alloy is aged at a temperature range from about 300 °C to 500 °C, D022 is first precipitated from the α supersaturated solution. As the aging time goes by, D022 will be transformed into L12, and with the aging time further increasing, both the D022 and L12 will be transformed into D03. When the aging temperature increases to a certain extent, atoms obtain a greater diffusion capacity, and D03 will be precipitated from the α supersaturated solution directly.

Fig. 2. The time-temperature-transformation (TTT) diagram of the Cu–15Ni–8Sn alloy.[8]

Nevertheless, the compositions, structures, and properties of (CuxNi1 − x)3Sn with different ordered structures have not yet been fully understood due to the phase transition complexity of the Cu–Ni–Sn system. There has been no systematic study of the (CuxNi1 − x)3Sn with various structures. The compositions, structural stabilities, and mechanical properties of the D022, the L12, and the D03 need to be discussed. Some important fundamental issues arise as follows.

(i) It is generally accepted that Ni only strengthens the matrix by solid solution and does not affect the phase transformation, but some documents indicated that when the content of Sn is constant, the brittleness of the alloy decreases with the increase of the Ni content.[9,10] What role does the Ni play in the (CuxNi1 − x)3Sn phase? Is there any relationship between the properties of (CuxNi1 − x)3Sn and the x value?

(ii) As transient ordered phases, D022 and L12 during the aging of supersaturated solid solution were well established. It has been found that D022 ordering takes place prior to L12 and D03, and both L12 and D022 may co-exist in a Cu–15Ni–8Sn alloy.[11,12] What are the relations and differences between the two metastable phases?

In order to understand the effect of (CuxNi1 − x)3Sn precipitations in Cu–Ni–Sn alloys, it is necessary to gain more knowledge about the (CuxNi1 − x)3Sn with different structures and different Ni content. The first principles calculation is a most powerful method to investigate the phase stabilities and properties of these (CuxNi1 − x)3Sn phases.

In the present study, the lattice constants, elastic properties, and electronic and thermodynamic properties of the D022, L12, and D03 are investigated by first-principles calculations. By the use of calculations, certain conclusions drawn from the experiments may be well explained, which consequently assist in optimizing the Cu–Ni–Sn alloy properties in subsequent studies.

2. Calculation method

Since the x in (CuxNi1 − x)3Sn is a variable, neither the Ni content nor the occupation of Ni atoms is definite. Two different calculation methods were used in this work in order to obtain the results that can be compared.

The first method was the super-cell (SC). For a certain composition, the total energies of all possible structures after structural optimization with Ni atoms in different positions were calculated, and the structure with the lowest total energy was identified as the most stable one and taken to the further calculations, and the results are listed in Table 1. In order to simplify the SC structure and save computation time, the x values of (CuxNi1 − x)3Sn were set to be 1, 5/6, 2/3, 1/2, 1/3, 1/6, and 0, corresponding to the systems of Cu3Sn, Cu2.5Ni0.5Sn, Cu2NiSn, Cu1.5Ni1.5Sn, CuNi2Sn, Cu0.5Ni2.5Sn, and Ni3Sn respectively.

Table. 1.

(color online) SC structures identified for calculations (x = 5/6, 2/3, 1/2, 1/3, 1/6).

.

The second method was the virtual crystal approximation (VCA). Since the atomic radius and electronic structure of Ni are very similar to those of Cu, (CuxNi1 − x)3Sn phases could be treated as a disordered solid solution in the VCA framework, considering the atomic sites in terms of a mixture of atoms that consist of x Cu and (1 − x) Ni.[13] In this work, the Cu atoms were taken as mixed atoms with Ni in proportions of 0, 16.667, 33.333, 50, 66.667, 83.333, and 100 at.%, corresponding to the x values of 1, 5/6, 2/3, 1/2, 1/3, 1/6, and 0, respectively.

All calculations were made with the CASTEP program based on density functional theory (DFT). The PW91version of the generalized gradient approximation (GGA) was used in the calculations.[14] The atomic geometry optimizations were performed, with the cutoff energy (Ecut) of atomic wave function set to be 440 eV, the convergence tolerance energy 5.0 × 10−6 eV/atom, the Max force 0.01 eV/Å, Max stress 0.02 GPa, and Max displacement 5.0 × 10−4 Å. The Brillouin zone sample integrations used were based on the Monkhorst–Pack grids of 8 × 8 × 8-dimensional points for the L12 structure, 7 × 7 × 3-dimensional points for the D022 structure, 7 × 7 × 7-dimensional points for the D03 structure, 5 × 6 × 5-dimensional points for the D0a structure, and 5 × 5 × 6-dimensional points for the D019 structure, respectively. All the calculations of phonons properties were based on the “finite displacement” method with the super-cell defined by cutoff radius 5 Å and the dispersion separation was set to be 0.015 1/Å.

3. Results and discussion
3.1. Structural parameters

The calculated equilibrium lattice constants are listed in Table 2. It can be seen that the calculated values of lattice constants are slightly higher than the actual experimental data, which is consistent with the fact that the GGA method usually overestimates the lattice constants.[15] Therefore, the chosen calculation method in this work can be regarded as valid. In Table 1, the calculated lattice constants and the average volumes of atoms by VCA are smaller and closer to the experimental results than those by SC. It indicates that in the (Cu, Ni)3Sn phases, the Cu and Ni atoms tend to be irregularly distributed so that a more compact arrangement can be obtained.

Table 2.

Calculated equilibrium lattice constants, average volumes of atoms, and total energies of different structures.

.

The average volumes per atom as a function of composition x are presented in Fig. 3. The average volumes of (CuxNi1 − x)3Sn phases with L12, D022, D03, D0a, and D019 structures show negative deviation from Vegard’s law, and the volumes from the VCA deviate more from those from Vegard’s law than those from the SC. It implies that the attraction between Cu and Ni atoms is greater than that between the atoms of the same kind, and the VCA method is more suitable for the calculation of the alloy than the SC method.

Fig. 3. (color online) Calculated average volumes per atom of (CuxNi1 − x)3Sn for (a) L12, (b) D022, (c) D03, (d) D0a, and (e) D019.

The volume comparison among the five different structures is shown in Fig. 4. Although there is a difference between the calculated volumes from SC and VCA, the volumes of the five structures show roughly the same variation tendency. With the same x value, the volumes of the D022 are the smallest in all structures in the period of 1 ≥ x ≥ 1/2. It can therefore explain the case appearing in the experiment[8] that the first phase precipitated from the α supersaturated solid solutions at a lower temperature is the D022. Since at low temperature, the activity of atoms is limited, and the precipitation of D022 leads to the smallest matrix distortion.

Fig. 4. (color online) Calculated average volumes per atom of (CuxNi1 − x)3Sn (a) by SC method and (b) VCA method.
3.2. Formation energies

The calculations of formation energies are performed to compare the stabilities among the L12, D022, D03, D0a, and D019 structures.

For the ordered super-cell structure, the formation energy of the Cu–Ni–Sn alloy is defined as follows:[23]

where is the average formation energy per atom, Etot is the total energy of the unit cell, is the energy per atom of Cu with a face-centered cubic (fcc) structure, is the energy per atom of Ni with the fcc structure, and represents the energy per atom of β-Sn in the solid state.

Therefore, the formation energy of (CuxNi1 − x)3Sn super-cell structure in this work can be written as

As far as the disordered VCA structure is concerned, there is no doubt that equation (2) is unsuitable due to the unreasonable energy standard of the pure element, and the formation energy must be developed by considering the influence of mixture atoms on the formation energy.[24,25] Since the Cu and Ni atoms are treated as virtual mixed atoms, the and (1 − x) should be replaced by the total energy of CuxNi1 − x calculated by VCA. Therefore, the formation energy of (CuxNi1 − x)3Sn VCA structure can be written as

where is the energy of the mixed atom in the optimized α-CuxNi1 − x solution, calculated by the VCA method.

The calculated formation energies of the (CuxNi1 − x)3Sn alloy are shown in Fig. 5. It can be seen that the results obtained by the two methods are very close to each other, and the variation trend of the formation energies from the VCA is the same as that from the SC. The formation energies of all the structures decrease with Ni atom content increasing. Since the VCA is based on the disorder structure, while SC is based on ordered replacements of Ni in Cu sites, the difference between formation energies obtained by the VCA and SC method can help indicate the tendency of atomic arrangement in different structures. For L12, D0a, and D019 phases, the formation energies calculated by SC are more negative than those by VCA when 1 > x ≥ 2/3, which indicates that Cu and Ni atoms tend to be ordered. While the opposite scenario occurs when 1/2 ≥ x > 0 as shown in Figs. 5(a), 5(d), and 5(e).

Fig. 5. (color online) Calculated formation energies of (CuxNi1 − x)3Sn with different structures: (a) L12, (b) D022, (c) D03, (d) D0a, and (e) D019.

For the D03 phase, the formation energies calculated by SC are a little more negative than those by the VCA as shown in Fig. 5(c), indicating that the Cu and Ni atoms in D03 structure prefer to be arranged regularly. By comparing the formation energies among the D03 supercellular structures with different atomic arrangements, an interesting rule is found. The body-central positions are prone to be occupied by Cu atoms rather than Ni atoms, since this kind of atomic arrangement possesses a more compact structure with lower energy. For example, the energy-lowest structure of Cu2NiSn is D03, but the energy-lowest structure of CuNi2Sn is L21 as shown in Fig. 6. These are consistent with the experimental results that the (Ni,Cu)3Sn alloys have both D03 and L21 structures.[26] It may be explained that the sizes of Cu and Sn atoms are more close than those of Ni and Sn atoms, and the more compact structures with lower formation energies can be formed with Cu atoms at the body-central positions.

Fig. 6. (color online) Energy-lowest structures of (CuxNi1 − x)3− D03: (a) Cu2NiSn and (b) CuNi2Sn(L21).

The formation energies comparisons of the five different structures are shown in Fig. 7. When x = 1, the formation energies of L12, D022, and D03 are all positive, indicating that Cu3Sn compounds with these structures are very difficult to form. The D0a and D019 phases possess the negative formation energies, and the formation energy of D0a is lowest, so the equilibrium phase structure of Cu3Sn should be D0a. When x = 0, the formation energy of D019 is lowest, indicating that the equilibrium phase structure of Ni3Sn should be D019. These results are consistent with those in the literature.[27] However, because of the structural difference, it is difficult for D0a and D019 to precipitate directly from the fcc solid solution matrix. With a comprehensive analysis of structures and formation energies, the structure of the (CuxNi1 − x)3 first precipitated from the α supersaturated solid solution should be D022. As time goes by, the D022 phase with high Ni content will be transformed into L12.

Fig. 7. (color online) Calculated formation energies of (CuxNi1 − x)3 by (a) SC method and (b) VCA method.
3.3. Electronic structure

To confirm the validity of the VCA method and further reveal the underlying mechanisms of the structural stability for the (CuxNi1 − x)3Sn phases, the electronic total and partial densities of states (TDOSs and PDOSs) at equilibrium lattice constants for Cu2NiSn are calculated by the VCA and SC method, respectively, and the results are shown in Fig. 8. The Fermi energy level is set to be zero on the x axis. It can be seen that the Cu2NiSn compounds with L12, D022, and D03 structures exhibit metallic behaviors because the values of TDOS at the Fermi level are higher than 0. TDOSs of Cu are very similar to those of Ni in all the figures of various structures, which indicates that the Cu and Ni have similar electronic structures and properties in the (CuxNi1 − x)3Sn. The TDOSs are mainly dominated by Cu-d (the energy of PDOS peak is in a range from about −4.8 eV to −1.0 eV) or Ni-d ( the energy of PDOS peak is in a range from about −4.2 eV to 0 eV). As seen from the PDOSs of Figs. 8(a), 8(c), and 8(e), the Cu-d states show strong hybridizations with Ni-d states. It can be seen by comparing Fig. (a) with Fig. 8(b) that the peak of TDOS calculated by the VCA method is narrower and higher than that of TDOS calculated by the SC method, although their positions are roughly the same. It can be explained that in the calculation by the VCA, the PDOSs of mixed (Cu,Ni) atoms are the average of the respective PDOSs of Cu and Ni.

Fig. 8. (color online) TDOSs and PDOSs of (CuxNi1 − x)3Sn with different structures, with Fermi level set to zero.
3.4. Mechanical properties

Generally, the elastic constants are closely related to the mechanical behaviors of the materials and also provide information regarding bonding characteristics. The requirement of mechanical stability leads to the following restrictions of the elastic constants,[28] for cubic unit cells

for tetragonal crystals[29]

The bulk modulus (B) is the measure of unit cell incompressibility, and shear modulus (G) reflects material resistance to the shear strain. Young’s modulus (E) is the comprehensive index of the B and G, which reflects the material stiffness and an internal property of the material.[30] Poisson’s ratio (ν) is used to quantify the stability of the crystal against shear, and the larger the ν, the better the plasticity is. These polycrystalline structural properties are important for determining the mechanical properties of materials and can be calculated by using the Voigt–Reuss–Hill (VRH) approximation[31] from the following formulas:

Generally, the ductility of a material can be estimated by the Cauchy pressure value (C12–C44).[32] If the Cauchy value has a positive sign, the material is ductile and displays metallic characteristics, otherwise the material is brittle. Alternatively, the Pugh ratio (G/B) reflects the ductility of alloys.[33] The smaller the Pugh ratio, the more ductile the alloy will be, and the 0.57 ratio is the critical point of all brittle/ductile behaviors.

The mechanical properties of (CuxNi1 − x)3Sn, calculated by SC and VCA are shown in Table 3.

Table 3.

Calculated values of elastic constants Cij (in unit GPa), modulus (in unit GPa), and Poisson’s ratio of (CuxNi1 − x)3Sn.

.

As it is observed in Table 3, the elastic constants of all the L12 and D022 structures meet the restrictions of mechanical stability. The D03 structures with x ⩽ 0.5 are mechanically unstable since none of their elastic constants meets the restriction of the condition C11 > |C12|. The Cauchy pressures (C12–C44) of all structures are positive, and all the calculated Pugh ratios (G/B) are less than 0.57. These criteria indicate that D022, L12, and D03 structures with various compositions each possess a favorable ductility.

Figure 9 shows the calculated values of Young’s modulus (E) and Poisson’s ratio (ν) of different structures with different x values. With Ni content increasing, the E value of L12 and D022 increase while ν decreases, and D03 presents roughly the opposite scenario. It indicates that L12 and D022 play a more important role in the strengthening of Cu–Ni–Sn alloys than D03.

Fig. 9. (color online) Mechanical properties of different structures: (a) Young’s moduli and (b) Poisson’s ratios.
3.5. Thermodynamic properties

In order to predict the thermodynamic properties of (CuxNi1 − x)3Sn, phonon dispersions are calculated by the SC method. The phonon dispersion curves of (CuxNi1 − x)3Sn (x = 1, 2/3, 1/3, 0) with L12, D022, and D03 structures are shown in Fig. 10. It can be seen from the figure that there are no imaginary frequencies for all the structures, this indicates that they are stable phases.

Fig. 10. (color online) Phonon dispersion curves for (CuxNi1 − x)3 Sn.

Figure 11 shows the predicted thermodynamic properties of (CuxNi1 − x)3 Sn. It can be clearly found that each of the thermal entropies always has a higher value than enthalpies at all temperatures, resulting in the free energies decreasing with the rise of the temperature. However, the decreasing rates of free energies are not the same. In a range from x = 1 to x = 0, the free energies of the D03 are lower than those of D022 and L12, and the free energy of Cu2NiSn_D03 is the lowest in the free energies of all the compounds (CuxNi1 − x)3Sn as shown in Fig. 10(b). The free energies of the L12 are lower than those of D022 in all x range values except x = 1. This may explain why D022 and L12 eventually turn into D03 in the aging process.

Fig. 11. (color online) Thermodynamic properties calculated by super-cell method.
4. Conclusions

(i) For Cu–Ni–Sn alloys, VCA is a very effective calculation method. It can greatly reduce the amount of calculation and provide the high computational efficiency. The lattice constants, formation energies, and elastic constants obtained by SC and VCA are generally consistent with each other. The density of states shows that the Cu and Ni have similar electronic structures and properties in the (CuxNi1 − x)3Sn, and the TDOSs are mainly dominated by the electron valence of Cu(d) or Ni(d).

(ii) The (CuxNi1 − x)3 Sn_D022 is first precipitated in a solid solution because its structure and cell volume are most similar to those of a solid solution matrix. The L12 and the D022 possess better mechanical stability than the D03, and they may play a more important role in the strengthening of Cu–Ni–Sn alloys.

(iii) The calculated results show the equilibrium structures of Cu3Sn and Ni3Sn are D0a and D019 respectively. While (CuxNi1 − x)3 Sn_D03 with various components are metastable phases at temperature of 0 K, just as D022 and L12. Nevertheless, with the temperature increasing, the free energy of the D03 is lower than those of D022 and L12, and the D022 and L12 eventually turn into the D03 in the aging process.

(iv) A comparison among the formation energies of the D03 supercellular structures with different atomic arrangements shows that the body-central positions are prone to be occupied by Cu atoms rather than Ni atoms, since this kind of atomic arrangement possesses a more compact structure with lower energy. The energy-lowest structure of Cu2NiSn is D03, but the energy-lowest structure of CuNi2Sn is L21, which is in good consistence with the experimental results that the (Ni,Cu)3Sn alloys have both D03 and L21 structures.

(v) All the calculated results are in good agreement with the previous experimental data, and this study has been shown to be valuable for further research on the Cu–Ni–Sn alloys.

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