# Effect of Cu-Rich Phase Growth on Creep Deformation of Fe-Cr-Ni-Cu Medium-Entropy Alloy: A Phase Field Study

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## Abstract

**:**

## 1. Introduction

_{ij}and B = B

_{ij}over one and two indices as $A\cdot B={A}_{ij}{B}_{jk}$ and $A:B={A}_{ij}{B}_{ij}$. The transpose of A is A

^{T}, and I is the unit tensor.

## 2. Phase Field Model

#### 2.1. Free Energy Formulation of Fe-Cr-Ni-Cu Alloy System

_{m}is the mole volume of the alloy, F

^{c}is the mole Gibbs energy of the Fe-Cr-Ni-Cu alloy, which can be obtained by the CALPHAD method, F

^{e}is the elastic energy density, and F

^{▽}is gradient energy density.

^{c}can be expressed as [27]:

_{i}is the concentration obeying constraint $\sum {c}_{i}=1$ and G

_{i}is the Gibbs free energy of the pure element i with the FCC crystal structure. The subscripts i = 1, 2, 3, and 4 correspond to Fe, Cr, Ni, and Cu, respectively.

^{E}G is the excess free energy corresponding to the heat of mixing, and

^{mg}G is the magnetic contribution to the Gibbs free energy. R and T are the gas constant and the absolute temperature (K), respectively.

^{E}G for a quaternary system is described as [28]:

_{i,j}and L

_{i,j,k}are the binary and ternary interaction parameters, respectively.

^{mg}G is formulated as [29]

_{c}is a dimensionless temperature, T

_{c}is Curie temperature, p is a material constant, and p = 0.28 for the FCC phase. If β or T

_{c}is negative, they should be revised as −β/3 and −T

_{c}/3.

_{i}is the interfacial energy.

_{e}is the elastic strain. According to Hook’s law

_{y}can be expressed as [34]

_{y}

_{0}is the initial yield stress, and ${\overline{\epsilon}}_{p}$ is the equivalent plastic strain. Following the Prandtl–Reuss plastic flow rule [35]

#### 2.2. Kinetic Equation

_{ij}is the matrix of Onsager coefficients and given by Darken’s equation [4]

_{ij}is the Kronecker delta and D

_{i}is the self-diffusion coefficient

_{0i}is a frequency factor and Q

_{0i}is the activation energy.

_{y}, and hardening modulus H) are assumed to be the mix of the matrix P

_{m}and Cu-rich phase P

_{Cu}[37]

## 3. Simulation Data

_{i}is the boundary flux, k

_{i}is the diffusion coefficient, and c

_{0i}is the initial concentration of element i. The lower left corner is fixed in all directions and the lower right corner is in the y direction to avoid rigid body motion. In Case 1, a single Cu-rich phase nucleus is prescribed at the center of the model, while in Cases 2~4, two nuclei with a distance of l

_{0}were located on the vertical center line as the initial condition, and l

_{0}= 15, 22.5, and 30 nm, respectively. The conditions of the four simulated cases are shown in Table 1.

## 4. Simulation Results

_{0}due to the overlapping of the diffusion zone [45]. It should be noticed that the radius of Cu-rich particles in Case 4 is even larger than in Case 1. This is probably because the interface of the Cu-rich phase in Case 4 is very close to the boundary of the simulated domain, and the consumption of Cu atoms in the matrix around the Cu-rich particles can be easily supplied from the boundary. For comparison, the Cu concentration at 9 nm above the interface (defined as c

_{Cu}= 0.5) of the upper Cu-rich particles in Case 4 (located on the upper boundary of the simulated domain) is 0.024; however, it is only about 0.011 at the same location in Case 1 (located in the matrix). The evolution of the Cu-rich phase radius and mean axial strain $\overline{\epsilon}$ (defined as the mean axial elongation divided by the initial length of the simulated domain) in the four simulated cases are shown in Figure 3.

_{c}= 11 nm. Another notable phenomenon is that when there are two Cu-rich phase particles in the simulated grain, the creep strain increases with the increase in the distance between the two Cu-rich particles in same creep time; in contrast, the creep strain decreases with the increase in the distance between the two Cu-rich phase particles for same Cu-rich phase size.

## 5. Discussion

#### 5.1. Influence of Cu-Rich Phase Growth on Creep Strain

_{c}= 11 nm. Zhou et al. [43] reported a very similar critical size of r

_{c}= 13 nm at the same temperature in the Sanicro 25 alloy, which has roughly the same chemical composition as the simulated alloy.

#### 5.2. Influence of the Distance between Two Cu-Rich Particles on Creep Strain

## 6. Conclusions

- (1)
- Creep strain of the simulated grain was intensified with the coarsening of Cu-rich particles. When the Cu-rich precipitates were relatively fine (~<11 nm), the plastic strain tended to shear the Cu-rich phase, and the size of the Cu-rich particle has a slight influence on the creep strain at this stage. However, for coarse Cu-rich precipitates (~>11 nm), the plastic strain will bypass them due to the enhancing stress concentration around the interface, and the creep strain is increased rapidly with the growth of Cu-rich particles.
- (2)
- The coarsening of Cu-rich particles will be retarded by the adjacent particles due to the overlapping of the diffusion zone, and hence the creep strain was reduced when creep occurred for the same time. The retard effect will vanish when the distance is sufficiently long (~>60 nm).
- (3)
- When the size of the Cu-rich particles is identical, the creep strain will be mitigated with elongation of the distance between the two Cu-rich particles, because a more homogeneous stress distribution is generated.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Schematic of the simulation model and boundary conditions for simulated case 1 (

**a**) and cases 2–4 (

**b**).

**Figure 2.**Morphology evolution of Cu precipitates (

**a**,

**c**,

**e**,

**g**) and concentration distributions along L1 line (

**b**,

**d**,

**f**,

**h**) in Case 1 (

**a**,

**b**), Case 2 (

**c**,

**d**), Case 3 (

**e**,

**f**) and Case 4 (

**g**,

**h**).

**Figure 3.**Evolution of Cu-rich phase radius and mean axial strain as a function of creep time and the experimental data reported by Zhou et al. (

**a**) and variation of mean axial strain with radius of the Cu-rich phases in the four simulated cases (

**b**). Adapted with permission from Ref. [43]. Copyright 2020, Elsevier.

**Figure 4.**Distribution of equivalent stress (

**a**–

**c**) and equivalent plastic strain (

**d**–

**f**) after creep for 500 h (

**a**,

**d**), 1000 h (

**b**,

**e**), and 1500 h (

**c**,

**f**) in Case 1.

**Figure 5.**Distribution of equivalent stress (

**a**–

**c**) and equivalent plastic strain (

**d**–

**f**) when r = 12 nm in Case 2 (

**a**,

**d**), Case 3 (

**b**,

**e**), and Case 4.

**Figure 6.**Effect of distance between two Cu-rich phase particles on equivalent stress distribution along L2 line.

Case | Number of Nuclei | Distance between Nuclei/nm |
---|---|---|

Case 1 | 1 | / |

Case 2 | 2 | 15 |

Case 3 | 2 | 22.5 |

Case 4 | 2 | 30 |

Parameters | Values |
---|---|

E_{m} | 144.8 GPa [38] |

ν_{m} | 0.3 |

σ_{y}_{0m} | 220 MPa [38] |

H_{m} | 2 GPa [38] |

E_{Cu} | 95 GPa [39] |

ν_{Cu} | 0.3 |

σ_{y}_{0Cu} | 45 MPa [39] |

H_{Cu} | 50 MPa [39] |

D_{Cu} | D_{0Cu} = 4.16 × 10^{−4} m^{2}/s, Q_{0Cu} = 306.2 kJ/mol [40] |

D_{Cr} | D_{0Cr} = 2.29 × 10^{−15} m^{2}/s, Q_{0Cr} = 74.4 kJ/mol [41] |

D_{Fe} | D_{0Fe} = 1.0 × 10^{−5} m^{2}/s, Q_{0Fe} = 260.0 kJ/mol [42] |

D_{Ni} | D_{0Ni} = 1.7 × 10^{−5} m^{2}/s, Q_{0Ni} = 272.0 kJ/mol [42] |

γ | 8.1 mJ/m^{2} [43] |

Δ | 1 nm |

a_{m} | 0.3639 nm [43] |

δ_{Cr} | 6.1 × 10^{−3} [44] |

δ_{Ni} | 4.75 × 10^{−4} [25] |

δ_{Cu} | 3.29 × 10^{−2} [25] |

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**MDPI and ACS Style**

Gao, J.; Hu, L.; Ma, N.; Fang, X.; Xu, Z.; He, Y.
Effect of Cu-Rich Phase Growth on Creep Deformation of Fe-Cr-Ni-Cu Medium-Entropy Alloy: A Phase Field Study. *Metals* **2023**, *13*, 1219.
https://doi.org/10.3390/met13071219

**AMA Style**

Gao J, Hu L, Ma N, Fang X, Xu Z, He Y.
Effect of Cu-Rich Phase Growth on Creep Deformation of Fe-Cr-Ni-Cu Medium-Entropy Alloy: A Phase Field Study. *Metals*. 2023; 13(7):1219.
https://doi.org/10.3390/met13071219

**Chicago/Turabian Style**

Gao, Jianbing, Lei Hu, Ninshu Ma, Xudong Fang, Zhenlin Xu, and Yizhu He.
2023. "Effect of Cu-Rich Phase Growth on Creep Deformation of Fe-Cr-Ni-Cu Medium-Entropy Alloy: A Phase Field Study" *Metals* 13, no. 7: 1219.
https://doi.org/10.3390/met13071219