# Phase-Field Simulation of Spinodal Decomposition in Mn-Cu Alloys

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Numerical Formulation

#### 2.1. Thermodynamic Equilibrium

_{N}, and T

_{N}is the Neel temperature. The values of $\beta $ and T

_{N}can be obtained experimentally, and their values would be negative when they are theoretically obtained. To obtain the real value of $\beta $ and T

_{N}, they must be divided by −3 for phases with fcc and hcp crystalline structures and −1 for phases with a bcc crystalline structure [28]. $g\left(\tau \right)$ is a function of the value of $\tau $ and given by:

#### 2.2. Elastic Strain Energy

#### 2.3. Phase-Field Model

#### 2.4. Simulation Conditions

## 3. Results

#### 3.1. Evolution of Calculated Profiles

#### 3.2. Microstructural Evolution

## 4. Discussion

#### 4.1. Growth Kinetics

#### 4.2. Contribution of the Spinodal Decomposition to Mn-Cu Alloys

## 5. Conclusions

- The calculated Cu concentration profile confirmed the presence of spinodal decomposition at the early stages of aging because of the increase in the amplitude of modulation composition.
- The morphology of the decomposed phases was interconnected and of irregular shape, designated as percolated in the Cahn–Hilliard theory of spinodal decomposition. The growth kinetics of spinodal decomposition was slow at the early stages of aging.
- The rate of growth of spinodal decomposition was faster for the Mn-20 and 30 at. %Cu alloys than for Mn-40 at. %Cu during aging at 500 °C because of its higher driving force.
- The growth kinetics of phase decomposition increased with the decrease in aging temperature because of a reduction in driving force.
- The size of the decomposed phases increased with the aging temperature. The coarsening of the decomposed phases was a diffusion-controlled process despite the nanometric size.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Composition-dependence of the free energy curves at 300 °C, 400 °C and 500 °C. Where c

_{γ}and c

_{γ’}are the equilibrium compositions of γ and γ’ phases respectively.

**Figure 2.**Calculated second-order partial derivative of free energy at 300 °C, 400 °C, and 500 °C corresponding to Figure 1.

**Figure 3.**Equilibrium phase diagram of Mn-Cu calculated with Thermo-Calc. The blue line and marks represent the calculated miscibility gap by Wang and Lui (2007).

**Figure 4.**Calculated concentration profiles of (

**a**) Mn-40 at. %Cu, (

**b**) Mn-30 at. %Cu, and (

**c**) Mn-20 at.%Cu alloys aged at 400 °C for different times.

**Figure 5.**Calculated concentration profiles of Mn-30 at. %Cu alloys aged at (

**a**) 300 °C, (

**b**) 400 °C, and (

**c**) 500 °C for different times.

**Figure 6.**Microstructural evolution of (

**a**) Mn-40 at. %Cu, (

**b**) Mn-30 at. %Cu, and (

**c**) Mn-20 at. %Cu alloys aged at 500 °C for different times.

**Figure 7.**Microstructural evolution of Mn-30 at. %Cu alloy aged at (

**a**) 300 °C, (

**b**) 400 °C, and (

**c**) 500 °C for different times.

**Figure 8.**Volume fraction of γ′ phase for (

**a**) Mn-20 at. %Cu, (

**b**) Mn-30 at. %Cu, and (

**c**) Mn-40 at. %Cu alloys aged at 500 °C. The green marks in (

**b**) represent the experimental results of Vitek and Warlimont (1976).

**Figure 9.**Plot of average equivalent radius of γ′ phase versus time for the Mn-30 at. %Cu alloy aged at 300, 400, and 500 °C. The green marks represent the experimental results of Vitek and Warlimont (1976).

**Figure 10.**Plot of interparticle spacing $\lambda $ in γ′ phase versus time for the Mn-30 at. %Cu alloy aged at 500 °C. The green marks represent the experimental results of Vitek and Warlimont (1976).

Parameter | FCC_A1 γ-Phase | Reference |
---|---|---|

$\xb0{G}_{Cu}^{\gamma}$ J/mol | −7770.458 + 130.485235 T − 24.112392 T ln(T) − 2.65684 × 10^{−3} T^{2} + 0.129223 × 10^{6} T^{3} + 52,478 T^{−1} | [33] |

$\xb0{G}_{Mn}^{\gamma}$ J/mol | −3439.3 + 131.884 T − 24.5177 T ln (T) − 6 × 10^{−3} T^{2} + 69,600 T^{−1} + ${G}_{mag}$ | |

${T}_{{N}_{CuMn}}^{\gamma}$ | 540/(−3) | |

${\beta}_{CuMn}^{\gamma}$ | 0.62/(−3) | |

$L{0}_{CuMn}^{\gamma}$ | 20,235.508 − 13.2437 T | [37] |

$L{1}_{CuMn}^{\gamma}$ | −12,154.853 + 2.9399 T | |

Diffusion coefficient m ^{2}s^{−1} | ${D}_{0Cu}^{\gamma}=4.3\times {10}^{-5}$ | [28,39] |

${D}_{0Mn}^{\gamma}=1.78\times {10}^{-5}$ | ||

Q J/mol | ${Q}_{Cu}^{\gamma}=2.80\times {10}^{5}$ | |

${Q}_{Mn}^{\gamma}=2.64\times {10}^{5}$ |

Parameter | FCC_A1 γ-Phase | Reference | |
---|---|---|---|

$\mathrm{Lattice}\mathrm{parameter}\left({a}_{i}\right)$ $i=Cu,Mn$ (nm) | 0.36074 0.38546 | [40] | |

Elastic constants $J\times {m}^{-3}$ Cu/Mn | ${C}_{11}=168.400\times {10}^{10}$ | ${C}_{11}=223.000\times {10}^{10}$ | [41] |

${C}_{12}=121.400\times {10}^{10}$ | ${C}_{12}=120.000\times {10}^{10}$ | ||

${C}_{44}=75.400\times {10}^{10}$ | ${C}_{44}=79.000\times {10}^{10}$ | ||

Lattice mismatch ${\epsilon}_{i}=\left({a}_{Mn}-{a}_{Cu}\right)/{a}_{Cu}$ | 0.06835 | [40] |

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**MDPI and ACS Style**

Sigala-García, D.A.; López-Hirata, V.M.; Saucedo-Muñoz, M.L.; Dorantes-Rosales, H.J.; Villegas-Cárdenas, J.D.
Phase-Field Simulation of Spinodal Decomposition in Mn-Cu Alloys. *Metals* **2022**, *12*, 1220.
https://doi.org/10.3390/met12071220

**AMA Style**

Sigala-García DA, López-Hirata VM, Saucedo-Muñoz ML, Dorantes-Rosales HJ, Villegas-Cárdenas JD.
Phase-Field Simulation of Spinodal Decomposition in Mn-Cu Alloys. *Metals*. 2022; 12(7):1220.
https://doi.org/10.3390/met12071220

**Chicago/Turabian Style**

Sigala-García, Darío A., Víctor M. López-Hirata, Maribel L. Saucedo-Muñoz, Héctor J. Dorantes-Rosales, and José D. Villegas-Cárdenas.
2022. "Phase-Field Simulation of Spinodal Decomposition in Mn-Cu Alloys" *Metals* 12, no. 7: 1220.
https://doi.org/10.3390/met12071220