Design of Cu–Cr Alloys with High Strength and High Ductility Based on First-Principles Calculations

Abstract

Designing a material to realize the simultaneous improvement in strength and ductility is very meaningful to its industrial application. Here, the first-principles calculations based on density functional theory (DFT) were adopted to investigate the stability, elastic properties and Debye temperature of binary Cu–Cr alloys; and the effect of micro-alloying elements on their mechanical properties, including the bulk modulus B, shear modulus G, Yong’s modulus E and Poisson’s ratio σ, was discussed. The elastic constants show that all the studied binary Cu–Cr alloys are mechanically stable, and the Cu–0.7Cr alloy has a combination of good strength and ductility. Moreover, the addition of Ag, Sn, Nb, Ti and Zr can improve the basic properties of Cu–0.7Cr alloy, and the Cu–0.7Cr–1.1Sn possess a large strength combined with improved ductility and strong covalent bonds due to the large Debye temperature. Additionally, the introduction of Y and In further improves the mechanical properties (strength and ductility) of the excellent Cu–0.7Cr–1.1Sn alloy. Our studied results can provide guidance for the theoretical design and experimental improvement of Cu-based alloys.

1. Introduction

Due to their excellent strength, plasticity and conductivity, the Cu–Cr alloys have been widely used in the railway transportation industries [1], electrode materials [2] and even ultra-large-scale integrated circuit lead frames [3,4,5], but the industrial application of binary Cu–Cr alloy is limited because of the poor softening resistance at high temperatures, which is caused by the unstable and coarsening Cr precipitate of Cu matrix during aging treatment [3,6]. To further improve the mechanical properties of binary Cu–Cr alloys, the micro-alloying elements are added to the Cu matrix [7,8]. For instance, Xiao et al. [9] found that the introduction of trace Ca and Sr could obviously improve the softening resistance of Cu–0.57Cr alloy; the high-density dislocations and fine Cr precipitates resulted in the high strengths of Cu–Cr–Ca and Cu–Cr–Sr alloys. Additionally, adding 0.28 wt% Ti into Cu–0.45Cr alloy can inhibit its recrystallization and increase the softening temperature of this alloy [10]. Moreover, Sn addition can realize the simultaneous improvement in yield strength and electrical conductivity of Cu–0.67Cr alloy [11]. Apart from those micro-alloying elements, the introduction of Nb, Mg, Zr and Hf was also demonstrated to improve the comprehensive properties of binary Cu–Cr alloys [12,13,14,15].

Currently, many researchers also employ the experimental method to study the feasibility and effectiveness of adding noble metal and rare earth (RE) into Cu–Cr alloys. For example, Xie et al. [4] reported that a small amount of Ag could enhance the solution strengthening and obviously restrict the chain precipitation of Cr precipitates for the Cu–Cr alloys and revealed that the Cu–0.89Cr alloy containing 0.44 wt% Ag had the outstanding mechanical-electrical properties. Jie et al. [5] investigated the effects of various Y content on the mechanical properties of Cu–0.9Cr and Cu–1.45Cr alloys and found that the addition of Y could effectively refine the eutectic structure, inhibit the coarsening of Cr precipitates and increase the dislocation density, which results in the good balance of tensile strength, hardness and electrical conductivity. Moreover, the Y was also found to obviously improve the high-temperature performance of the Cu–Cr–Zr alloy, and the minor Sc, Er had a similar positive effect on the Cu–Cr–Zr alloy [16]. Additionally, the addition of Yb restricted the coarsening of Cr-rich precipitates in Cu–Cr–Yb alloy, which makes this alloy possess better softening resistance, higher strength and electrical conductivity in comparison with Cu–Cr alloy [17]. All those studies show that the microalloying method is an effective way to realize the improvement of strength and ductility.

Nowadays, the first-principles calculations based on the density functional theory (DFT) method are also an effective way to design high-performance materials, such as Fe-based alloy [18], Al-based alloy [19] and high entropy alloys [20,21]. Compared with the experimental method, the DFT computations possess the advantages of low cost, convenience and high accuracy and thus have been extensively used in the evaluation and screening of materials. For example, Vitos et al. [18] employed quantum mechanical calculations to conduct the design of austenitic stainless steels, they found that the Fe₅₈Cr₁₈Ni₂₄ alloy had the optimal combination of hardness, ductility and corrosion resistance, and its basic properties would be further enhanced by the addition of osmium and iridium. With the help of the DFT method, Wang et al. [22] reported that the Nd and Mn had great strengthening potentials for the Mg alloy, and the local atomic ordering strategy could be used to design the high-strength Mg alloys. Additionally, the high-performance high entropy alloys can be effectively screened by calculating elastic and mechanical properties [20]. However, there are few pieces of literature about the theoretical design of high-strength Cu–Cr alloys based on the DFT method.

In this work, the first-principles calculations were employed to study the lattice constant, elastic constant and mechanical properties of different Cu–Cr alloys. The virtual crystal approximation (VCA) approach [23] was used to construct all the Cu-based alloys models. The stability and elastic properties of binary Cu–Cr alloys were firstly calculated and analyzed. After the screening of the excellent binary Cu–Cr alloys, the different micro-alloying elements, including Ag, Sn, Nb, Ti, Zr, In and rare earth (Y, Sc, Yb), were added to this outstanding class of Cu alloys, and their elastic constants and moduli were systematically discussed. Finally, the ternary and quaternary Cu–Cr alloys with high strength and ductility were obtained. Our work can provide guidance for the theoretical design and experimental improvement of Cu-based alloys.

2. Calculation Method and Details

All the calculations in this work were performed by using the Cambridge Serial Total Energy Package (CASTEP) Code [24,25] based on the density functional theory (DFT). The ultra-soft pseudopotential scheme [26] was used to describe the interactions of ionic-core and valence-electrons. The exchange–correlation energy was treated by the Perdew–Burke–Ernzerh of (PBE) of generalized gradient approximation (GGA) functional [27]. The cutoff energy of plane waves was set as 500 eV for the geometry optimization and elastic properties calculations, and the k-points meshes of 14 × 14 × 14 sampled in the first irreducible Brillouin zone [28] were chosen for high accuracy. The convergence thresholds of the total energy tolerance, maximum force tolerance and maximal displacement were 5.0 × 10⁻⁶ eV/atom, 0.01 eV/Å, and 5.0 × 10⁻⁴ Å, respectively. The Broyden–Flecher–Goldfarb–Shanno (BFGS) method was used for geometry optimization. All the Cu–Cr alloy models were built by the virtual crystal approximate (VCA) method, which was proved to be accurate enough for the disorder solid solution [23,29,30,31]. The calculated lattice constant of fcc-Cu in this work is 3.630 Å, which matches well with experimental data (3.636 Å) and other calculated results [32,33], indicating that our parameters are reasonable.

3. Results and Discussion

3.1. Elastic Properties of Binary Cu–Cr Alloy

The elastic constants C_ij of a compound or alloy can be used to evaluate its mechanical property, which can be obtained by the stress–strain method based on the generalized Hooke’s law. For the face-centered cubic crystal, the mechanical stability follows the following conditions:

C₁₁ > 0, C₄₄ > 0, C₁₁ − C₁₂ > 0, C₁₁ + 2C₁₂ > 0

(1)

our calculated C₁₁, C₁₂ and C₄₄ of fcc-Cu are 185.16 GPa, 113.61 GPa and 79.87 GPa, which are consistent with the calculated results of Zhan [32]. Moreover, all the studied pure Cu and Cu–Cr alloys are demonstrated to meet the stability conditions. According to the elastic constants C_ij, the elastic properties of Cu–Cr alloys, including bulk modulus (B), shear modulus (G), Young’s modulus (E) and Poisson’s ratio (σ), can be determined by the Voigt–Reuss–Hill approximation:

$$\begin{gathered} B_H{=B}_v{=B}_R=\frac{1}{3}{(C}_{11}{+2C}_{12}) G_v=\frac{1}{5}{(C}_{11}{-C}_{12}{+3C}_{44}) G_R=\frac{{5(C}_{11}{-C}_{12}{)C}_{44}}{{4C}_{44}{+3(C}_{11}{-C}_{12})} G_H=\frac{1}{2}{(G}_v{+G}_R) \\ E=\frac{{9B}_H{G}_H}{{3(B}_H{+G}_H)} \mathsf{\sigma }=\frac{{3B}_H{-2G}_H}{{2(3B}_H{+G}_H)} \end{gathered}$$

(2)

where the B_V and B_R are the bulk modulus calculated by Voigt and Reuss method, respectively. The G_V and G_R are the shear modulus calculated by Voigt and Reuss method, respectively.

It is well known that the solubility of Cr atom in Cu is less than 1.28 wt%; thus, the Cu-based alloys containing 0~1.3% Cr with the interval of 0.2% are taken into account, and the calculated elastic properties are summarized in

Table 1. The calculated elastic constant (C₁₁, C₁₂ and C₄₄, GPa), Zener’s elastic anisotropy constant (A), bulk modulus (B, GPa), shear modulus (G, GPa), B/G, Young’s modulus (E, GPa) and Poisson’s ratio (σ) of different Cu–Cr alloy, where the A = 2C₄₄/(C₁₁−C₁₂).

Composition (wt%)	C11	C12	C44	A	B	G	B/G	E	σ
Pure Cu	185.16	113.61	79.87	2.23	137.46	57.86	2.38	152.23	0.32
Cu–0.1Cr	185.56	113.33	80.88	2.24	137.41	58.52	2.35	153.74	0.31
Cu–0.3Cr	182.73	113.50	83.14	2.40	136.58	58.50	2.33	153.58	0.31
Cu–0.5Cr	176.98	117.87	83.69	2.83	137.57	55.17	2.49	146.00	0.32
Cu–0.7Cr	167.57	122.21	81.23	3.58	137.33	48.89	2.81	131.11	0.34
Cu–0.9Cr	160.45	127.30	76.46	4.61	138.35	41.89	3.30	114.14	0.36
Cu–1.1Cr	153.27	127.88	71.18	5.61	136.34	36.41	3.74	100.31	0.38
Cu–1.3Cr	150.85	129.63	66.07	6.23	136.70	32.63	4.19	90.69	0.39

. Our calculated B (137.46 GPa) and G (57.86 GPa) of pure Cu agree well with the previously reported results [34]. The B of polycrystalline alloy is often in direct proportion to the cohesive energy [35], which represents the resistance of the material to bond rupture [36], while the G of polycrystalline alloy represents the opposition of the material to plastic deformation, which can be used to evaluate the mechanical hardness of alloys in the annealed state [37,38]. Moreover, the B/G ratio is closely related to the ductility of alloys, the large (small) B/G ratios mean the ductile (brittle) alloys [36]. Figure 1 gives the calculated B, G and B/G of various binary Cu–Cr alloys; one can see that the shear modulus gradually decreases with the increasing Cr content, and the B/G exhibits the opposite tendency, but the bulk modulus irregularly fluctuates. After comprehensive consideration, the Cu–0.7Cr alloy has a relatively large B corresponding to the large G and large B/G. Therefore, the Cu alloy containing about 0.7% Cr should have a combination of high strength, large hardness and good ductility.

The Poisson ratio σ can be employed to estimate the degree of the covalent bond and predict the ductile or brittle of a material [39]; when the σ is larger than 0.26, the material has better ductility. From Table 1, the σ values of all the alloys are changed from 0.31 to 0.39, suggesting they exhibit good ductility characteristics. The stiffness of a material can be described by the Young’s modulus, and the E greatly decreases with the increment of Cr content due to the very large decrease in shear modulus. Moreover, the Zener’s elastic anisotropy constant A is in the range of 2.24~6.32. When the Cr content is more than 0.7, the A of Cu–Cr alloys is comparable and even larger than austenite stainless steel AISI 304 [18].

The Debye temperature (Θ_D) is closely related to many physical properties, such as the melting temperature [40], specific heat [41] and strength of covalent bonds in solids [42]. The higher Θ_D indicates the stronger chemical bonding. The Debye temperature (Θ_D) can be calculated by [43]:

$${\Theta }_{\mathrm{D}}=\frac{h}{k_B}{\left[\frac{3\mathrm{n}}{4\pi }\left(\frac{N_{\mathrm{A}}\rho }{M}\right)\right]}^{\frac{1}{3}}{v}_m$$

(3)

$$v_m=[\frac{1}{3}\left(\frac{2}{v_t^3}+\frac{1}{v_l^3}\right){]}^{-\frac{1}{3}}$$

(4)

$$v_{\mathrm{l}}=\sqrt{\frac{(B+\frac{4}{3}G)}{\rho } }{v}_{\mathrm{t}}=\sqrt{\frac{G}{\rho }}$$

(5)

where h, k_B and N_A are the Planck constant, Boltzmann constant and Avogadro constant, respectively. n, M and ρ are the total atom numbers per formula, molecular weight per formula and theoretical density, respectively. ν_l and ν_t are the longitudinal sound velocity and transverse sound velocity. According to the above equations, the calculated result is Figure 2. It can be observed that the Debye temperature and theoretical density of Cu–Cr alloys gradually decreased with the increment of Cr content, indicating that the high Cr content can decrease the melting temperature and weaken the chemical bonds of Cu–Cr alloys. In particular, the chemical bond strength of Cu–Cr alloys was dramatically decreased when the Cr content was larger than 0.7%. Combining the mechanical properties and Debye temperature, the Cu–0.7Cr alloy is chosen to be further studied in the next sections.

3.2. Mechanical Properties of Ternary Cu–Cr Alloy

To improve the performance of Cu–0.7Cr alloy, the effects of the third component on its mechanical properties are investigated. The common transition metals (X, X = Ag, Sn, Nb, Ti and Zr) in the industrial application are taken into account. For each transition metal, 0~1.3% X with the interval of 0.2% is added into the Cu–0.7Cr alloy due to the small solubility of those elements in Cu. The calculated lattice parameters, B, G, and B/G, of different ternary Cu–Cr alloys are shown in Figure 3 and Figure 4. The Sn can obviously decrease the lattice parameters of Cu–0.7Cr due to the small atomic radius of Sn, while the other alloying elements have a negligible effect on it. When the Ag and Sn are introduced into the Cu–0.7Cr alloy, the B, G and B/G are synchronously improved, and the higher Ag and Sn concentrations lead to more excellent mechanical properties, as displayed in Figure 4a,b. Moreover, the bulk modulus and shear modulus of Cu–Cr–Sn alloy is much larger than those of C–Cr–Ag alloy for the same contents, which indicates that the Cu–Cr–Sn alloys have the larger strength and mechanical hardness in comparison with Cu–Cr–Ag alloys. However, the B/G is in the range of 2.65~2.78 and 2.80~2.83 for the Cu–Cr–Sn and Cu–Cr–Ag alloys. Evidently, the Cu–Cr–Ag alloys exhibit better ductility due to the larger B/G. Hence, the Cu–0.7Cr–1.1Sn and Cu–0.7Cr–1.1Ag, which have high strength and toughness, should be the optimal compositions for the two alloys.

When the Nb, Ti and Zr are added into the Cu–0.7Cr alloy, the B, G, and B/G undergo great variations. With the increment of Nb, Ti and Zr contents, the shear modulus is gradually reduced, and the B/G shows the opposite trend, as shown in Figure 4c–e. This suggests that the large Nb, Ti and Zr concentrations are beneficial to improving the ductility of Cu–0.7Cr alloy. In addition, the bulk modulus of Cu–0.7Cr alloy exhibits the upward tendency with the increasing Nb, Ti and Zr, and the B of Cu–Cr–X (X = Nb, Ti and Zr) alloys is comparable for the same X content. By comparing the B, G, and B/G, we can find that the optimal compositions are Cu–0.7Cr–0.7Nb, Cr–0.7Cr–0.7Ti and Cu–0.7Cr–0.5Zr for the ternary Cu–Cr–Nb, Cu–Cr–Ti and Cu–Cr–Zr alloys, respectively.

In order to conduct the screening of excellent Cu-based alloys, the elastic and mechanical properties of the optimal ternary Cu–Cr–X alloys are further compared and discussed, as listed in

Table 2. The calculated bulk modulus (B), shear modulus (G), B/G, Poisson’s ratio (σ), elastic anisotropy constant (A) and Debye temperature (Θ_D) of the optimal ternary Cu–Cr–X alloys.

Composition (wt%)	B, GPa	G, GPa	B/G	σ	A	ΘD, K
Cu–0.7Cr–1.1Sn	180.76	65.06	2.78	0.34	2.37	491.41
Cu–0.7Cr–1.1Ag	140.40	49.67	2.83	0.34	3.79	433.00
Cu–0.7Cr–0.7Nb	138.62	42.13	3.29	0.36	4.66	398.82
Cr–0.7Cr–0.7Ti	138.55	41.38	3.35	0.36	4.77	396.05
Cu–0.7Cr–0.5Zr	138.00	45.92	3.01	0.35	4.12	414.95

and Figure 5. One can see that the bulk modulus (180.76 GPa) and shear modulus (65.06 GPa) of Cu–0.7Cr–1.1Sn are much larger than those of the other alloys, suggesting that the Cu–0.7Cr–1.1Sn alloy has the largest strength and mechanical hardness. While the Cu–0.7Cr–1.1Sn alloy has a relatively small B/G value and A (Figure 5), which is much smaller than that of Cr–0.7Cr–0.7Ti. Additionally, the five ternary alloys have the comparable Poisson’s ratio σ (higher than 0.26), and thus, they all exhibit the feature of good ductility. Moreover, the Debye temperature Θ_D follows the order: Cu–0.7Cr–1.1Sn > Cu–0.7Cr–1.1Ag > Cu–0.7Cr–0.5Zr > Cu–0.7Cr–0.7Nb > Cr–0.7Cr–0.7Ti, Therefore, we can conclude that the covalent bonds in Cu–0.7Cr–1.1Sn is stronger than other alloys. The Θ_D of these alloys is smaller than that of pure Cu (574.65 K); this is because the addition of alloy elements increases the molecular weight of pure copper, as illustrated in Equation (2). Based on the above discussion, should has the optimum compositions with good mechanical properties is the Cu–0.7Cr–1.1Sn alloy, followed by the Cu–0.7Cr–1.1Ag alloy.

3.3. Mechanical Properties of Quaternary Cu-Cr Alloy

Based on the above discussion, the optimum ternary composition with good performance is the Cu–0.7Cr–1.1Sn. In this section, the fourth alloying elements are introduced into this alloy to further improve its mechanical properties. The calculated B, G and B/G ratios of Cu–0.7Cr–1.1Sn alloy with different contents of In, Y, Sc and Yb are shown in Figure 6. One can see that Sc, Y and In can increase the bulk modulus of Cu–0.7Cr–1.1Sn alloy, while the Yb can slightly decrease its bulk modulus (Figure 6a). Whereas the addition of all the alloying elements can result in the increment of shear modulus for the Cu–0.7Cr–1.1Sn alloy, as depicted in Figure 6b. This implies that these alloying elements can further improve the strength and mechanical hardness, which is consistent with the experimental observation that a small amount of Y, Sc and Yb significantly improves the strength of Cu–Cr alloy [5,16,17]. In particular, the enhancement effects of Y are much more significant than that of the other elements for both B and G, and the larger Y content leads to the greater enhancement. However, most of the B/G of Cu–0.7Cr–1.1Sn alloy is decreased after the introduction of these alloying elements, meaning that its ductility is weakened to a different degree. By contrast, adding about 0.1% Y or 0.7% In can improve the ductility of the Cu–0.7Cr–1.1Sn alloy, as shown in Figure 6c, which matches well with the experimental results of Wang [5]. Therefore, the overall effect of Y or In on the Cu–0.7Cr–1.1Sn alloys realizes the goal of simultaneously increasing the bulk and shear modulus as well as B/G.

4. Conclusions

In this work, the DFT method was employed to investigate the stability, elastic properties and Debye temperature of binary Cu–Cr alloys, and the effect of different alloying elements on the mechanical properties, including C_ij, B, G, E and σ, of Cu-based alloys, was discussed. The results show that all the binary Cu–Cr alloys exhibit ductility characteristics, and the Cu–0.7Cr alloy has a combination of high strength, large hardness and good ductility due to the relatively large B, G and B/G. Moreover, the addition of Ag, Sn, Nb, Ti and Zr can improve the strength and toughness of Cu–0.7Cr alloy, and the optimal compositions are Cu–0.7Cr–0.7Nb, Cr–0.7Cr–0.7Ti and Cu–0.7Cr–0.5Zr for the ternary Cu–Cr–Nb, Cu–Cr–Ti and Cu–Cr–Zr alloys, respectively. Among these ternary Cu–Cr alloys, the Cu–0.7Cr–1.1Sn has the largest strength, mechanical hardness and covalent bonds. Furthermore, the overall effect of Y or In on the Cu–0.7Cr–1.1Sn alloys realizes the goal of simultaneously increasing the bulk and shear modulus. Thus, the Cu–0.7Cr–1.1Sn alloys containing a certain Y or In should have excellent mechanical properties. Those results illustrate that the theoretical modeling of Cu alloys based on DFT computations can be an effective method in modern copper alloy design.