# First-Principles Calculations of Structural and Mechanical Properties of Cu–Ni Alloys

^{1}

^{2}

^{3}

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## Abstract

**:**

## 1. Introduction

_{4}alloy has a high Young’s modulus with lower elastic anisotropy, which is rationally recommended for improving the stiffness of Al–Cu–Zn alloys based on first-principles calculations [13]. It can be seen that the first-principles calculation is one of the most important tools to compute mechanical properties, aiming to improve stiffness. Cu–Ni-based alloys have high elasticity and high electrical conductivity [14,15,16], with the addition of Co improving the conductivity [14] and the addition of Sn enhancing the mechanical stability of alloys [15,16]. The high stiffness and corrosion protection found in experiments with Cu–Ni alloys has brought them wide attention. Those features of the mechanical properties of Cu–Ni alloys can also be obtained from calculations, but have been less reported previously.

## 2. Computational Details

^{−1}. In geometry optimization processing, the detailed convergence tolerance for atom is evinced by setting energy to 5.0 × 10

^{−6}eV/atom, max. force to 0.01 eV/Å, max. stress to 0.02 GPa, and max. displacement to 5.0 × 10

^{−5}nm, where the max. iteration is 100 to ensure the energy convergence. The Broyden–Fletcher–Goldfarb–Shanno (BFGS) method is applied to mathematical algorithm for convergence calculation [19].

_{0.5}Ni

_{0.5}, Ni

_{0.92}Cu

_{0.08}, and Cu

_{0.95}Ni

_{0.05}with same space group $Fm\overline{3}m$. Red points for k-point and green dash for Brillouin edge are marked, respectively. The vectors demonstrate the direction in reciprocal space by $\overrightarrow{\mathrm{g}}1$, $\overrightarrow{\mathrm{g}}2$, $\overrightarrow{\mathrm{g}}3$. The results of geometry optimization are listed in Table 1. It can be seen that the large Cu concentration has the larger lattice parameters, and the small atom percentage of Ni concentration has the smaller lattice parameters, which are consistent with experiments. Lattice parameters error is less than 2% compared with XRD data [2,9,10]. Figure 2 shows the simulated XRD patterns of Cu–Ni alloys based on pseudo-Voigt peak shape profile [20], where the peak broadening accounting for instrument broadening is described by Caglioti equation [21] by the U = 0.05, V = 0.002, and W = 0.002; the [UVW] is the direction of incident of the electron beam. It can be seen that the XRD patterns of Cu–Ni alloys are similar; the peak for (111) and (200) reflections shift towards larger angles as the Ni contents increase. The peak position and peak tendency for Cu–Ni alloys are in great agreement with experiments [2,9,10], conforming to the observed particle shapes and crystallite sizes. Hence, these structures are considered to further compute the mechanical properties with GGA-PBE functional.

## 3. Mechanical Properties

_{m}, Young’s modulus E, ratio B/G, and Poisson’s ratio $v$. In the cubic system, the equation can be simple:

_{U}, and Poisson’s ratio v are expressed [13]:

_{11}represents the deformation resistance in the x-axis for Cu–Ni alloys. The calculated elastic constant for C

_{11}reduces with the increasing Cu content, indicating that the deformation resistance in the x-axis for Cu–Ni is becoming weak. Similarly, C

_{44}represents the deformation resistance in the x,y-axis becoming weak, while C

_{12}represents the deformation resistance in the y-axis to stress in the x-axis decreasing first, then increasing. The Cu content in the crystal has a great effect on deformation resistance in the x-axis. In Figure 3b, the popular moduli for materials are shown in different Cu content. Bulk modulus has a similar tendency as C

_{12,}showing the total deformation resistance of alloys. Shear modulus G and Young’s modulus E decrease in Cu-rich Cu content, predicting the fluctuation of hardness.

_{0.95}Ni

_{0.05}and Ni-rich Cu

_{0.05}Ni

_{0.85}alloys, they are greatly ductile with the B/G > 1.75; meanwhile, the Cu-rich alloys have the higher ductility. On the other hand, the Cu

_{0.5}Ni

_{0.5}alloys are brittle due to the B/G = 1.67. It shows that the right ratio for the binary compound of Cu–Ni systems is beneficial to improve the brittleness of pure metal. For the blue curve, the anisotropic index has the same tendency, where the Cu

_{0.5}Ni

_{0.5}alloy with the smallest numerical index presented the lowest anisotropy. The low anisotropy with higher Young’s modulus shows that the Cu

_{0.5}Ni

_{0.5}alloy has a great stiffness.

_{0.5}Ni

_{0.5}alloy is brittle. In Figure 4b, the average velocity decreases with the increasing Cu content; the significant drop can be found in the Cu

_{0.5}Ni

_{0.5}position. It shows that the Cu

_{0.5}Ni

_{0.5}alloys have a good elastic strength by loading elastic wave velocity, almost comparable with pure copper.

## 4. Conclusions

- Based on DFT, we calculated elastic constants, bulk modulus, shear modulus, Young’s modulus, anisotropic index A
_{U}, Poisson’s ratio v, average velocity, and B/G in this paper. It improves and supports the results for the experiment. - Cu-rich and Ni-rich Cu–Ni alloys are ductile; the Ni-rich alloy has the highest uniaxial deformation resistance due to having the largest Young’s modulus.
- Cu
_{0.5}Ni_{0.5}as the most suitable binary compound is predicted to have great stiffness in the Cu–Ni system, due to the brittleness and low anisotropy.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Primitive cell of Cu–Ni; the blue atom corresponds to Cu and Ni; the contents of atom are Cu

_{0.5}Ni

_{0.5}, Ni

_{0.92}Cu

_{0.08}, and Cu

_{0.95}Ni

_{0.05}. Red line and point refer to integration path and k-point in Brillouin zone. $\overrightarrow{\mathrm{g}}1$, $\overrightarrow{\mathrm{g}}2$, $\overrightarrow{\mathrm{g}}3$ are reciprocal vector.

**Figure 2.**Simulated XRD patterns of Cu–Ni alloy crystal in the 2θ = 42–54° range. The dashed lines indicate the position of the (111) and (200) fcc reflection for pure copper and pure nickel, the copper (Cu) is used for anode type, λ = 0.1540562 nm.

**Figure 3.**Cu content-dependent mechanical properties: (

**a**) elastic constants; (

**b**) bulk modulus B, Shear modulus G, and Young’s modulus E.

**Figure 4.**Cu content-dependent brittleness and stiffness: (

**a**) criterion for ductile and brittle B/G, anisotropic index A

_{U}, and Poisson’s ratio v; (

**b**) average velocity v

_{m}.

Composition | Lattice Parameter a (nm) | Density (g·cm^{−3}) | Volume (nm^{3}) |
---|---|---|---|

Cu_{0.08}Ni_{0.92} | 0.35302 | 8.9223 | 0.010999 |

Cu_{0.5}Ni_{0.5} | 0.35557 | 9.0315 | 0.011239 |

Cu_{0.95}Ni_{0.05} | 3.6174 | 8.8827 | 0.011834 |

Cu_{0.97}Ni_{0.03} | 0.35836 [2] | ||

Cu_{0.46}Ni_{0.54} | 0.3560 2 [2] | ||

Cu_{0.55}N_{0.45} | 0.3561 [9] | ||

Cu_{0.13}Ni_{0.87} | 0.3534 [9] | ||

Cu_{100}Ni_{0} | 0.36148 [10] | ||

Cu_{0}Ni_{100} | 0.35232 [10] |

**Table 2.**Calculated mechanical properties of Cu–Ni alloys, ${C}_{\rho \sigma}$ is elastic constant, B is bulk modulus, G is shear modulus, v

_{m}is average velocity, E is Young’s modulus, and $v$ is Poisson’s ratio.

C_{11} | C_{12} | C_{44} | B | G | E | A_{U} | $\mathit{v}$ | v_{m} | B/G | |
---|---|---|---|---|---|---|---|---|---|---|

Cu_{0.05}Ni_{0.95} | 241.1 | 163.0 | 110.0 | 189.0 | 72.7 | 193.3 | 1.405 | 0.330 | 3145.7 | 2.60 |

Cu_{0.5}Ni_{0.5} | 165.2 | 89.4 | 102.2 | 114.6 | 68.7 | 171.7 | 1.278 | 0.250 | 3002.2 | 1.67 |

Cu_{0.95}Ni_{0.05} | 154.8 | 137.9 | 71.4 | 143.5 | 32.1 | 89.6 | 7.884 | 0.396 | 2011.8 | 4.47 |

Cu_{0.97}Ni_{0.03} | 111 [2] | |||||||||

Cu_{0.46}Ni_{0.54} | 163 [2] | |||||||||

Cu_{0.55}N_{0.45} | 172 [9] | |||||||||

Cu_{0.13}Ni_{0.87} | 195 [9] | |||||||||

Cu_{100}Ni_{0} | 158 [10] |

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**MDPI and ACS Style**

Wei, Y.; Niu, B.; Liu, Q.; Liu, Z.; Jiang, C.
First-Principles Calculations of Structural and Mechanical Properties of Cu–Ni Alloys. *Crystals* **2023**, *13*, 43.
https://doi.org/10.3390/cryst13010043

**AMA Style**

Wei Y, Niu B, Liu Q, Liu Z, Jiang C.
First-Principles Calculations of Structural and Mechanical Properties of Cu–Ni Alloys. *Crystals*. 2023; 13(1):43.
https://doi.org/10.3390/cryst13010043

**Chicago/Turabian Style**

Wei, Yun, Ben Niu, Qijun Liu, Zhengtang Liu, and Chenglu Jiang.
2023. "First-Principles Calculations of Structural and Mechanical Properties of Cu–Ni Alloys" *Crystals* 13, no. 1: 43.
https://doi.org/10.3390/cryst13010043