# Large-Scale Multi-Phase-Field Simulation of 2D Subgrain Growth

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## Abstract

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## 1. Introduction

## 2. Methodology

#### 2.1. Microstructure of Deformed State

#### 2.2. Phase-Field Modeling and Implementation

#### 2.3. Generation of Synthetic Microstructures

## 3. Results

#### 3.1. Initial Microstructures

#### 3.1.1. Baseline Conditions

#### 3.1.2. Result of Phase-Field Simulation for the Baseline Conditions

#### 3.1.3. Modification from Baseline Conditions

#### 3.2. Role of Initial Subgrain Size Distribution

#### 3.3. Role of Initial Disorientation Distribution

## 4. Discussion

#### 4.1. Role of Initial Subgrain Size Distribution

#### 4.2. Role of Initial Subgrain Disorientation Distribution

## 5. Summary and Conclusions

- Simulations conducted on microstructures with different subgrain size distributions (but the same disorientation distribution) show that regardless of the initial subgrain size distribution, a self-similar regime is achieved after an initial transition. However, the self-similar state is not the same for different initial size distributions. As the initial subgrain size distribution gets wider, it reaches a wider subgrain size distribution during the steady-state growth regime but with a smaller average subgrain size as compared with narrower initial size distributions.
- The effect of disorientation distribution on the evolution of subgrains is more pronounced compared with that of the subgrain size distribution. By increasing the width of the disorientation distribution, a larger increase in the average subgrain size is observed. When the cases of 50% narrower and 50% wider disorientation distributions are compared with the baseline, it is found that the average subgrain size is 8 μm smaller and 5 μm larger than the baseline, respectively, after 7.5 s of simulation time. The significant effect of the width of disorientation distribution can primarily be related to the associated mobility distributions.

## Supplementary Materials

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**A schematic representation of the approach taken in the current work for the message parsing interface (MPI) communication. Red points are buffer nodes for MPI communications, other colors are used for simulation nodes. Different colors are used for different field levels.

**Figure 2.**(

**a**) Subdivision of a large 2D domain into 256 divisions for a faster microstructure generation (256 times faster than undivided domain); (

**b**) an example of microstructure that is generated after doing tessellation in a divided domain; (

**c**) final step where the seeds and their weights are input to Neper to anneal out the straight boundaries between parent cells. Note: the colors in (

**b**,

**c**) represent subgrain IDs within each distinct subdomain, and they were used to improve the visualization of boundaries between subdomains before and after annealing.

**Figure 3.**(

**a**) an EBSD IPF map of the grain structure for the Al–Mn–Si–Fe alloy homogenized at 375 °C for 24h and extruded at 350 °C; (

**b**) a grain with <001>||ED orientation; (

**c**) a grain with <111>||ED orientation.

**Figure 4.**(

**a**) The disorientation distribution within <001>||ED and <111>||ED; (

**b**) subgrain size distribution within grains with <001>||ED and <111>||ED.

**Figure 5.**The microstructures at the end of the simulation for (

**a**) <111>||ED and (

**b**) <001>||ED, and (

**c**) the average grain size vs. simulation time for the two different initial microstructures.

**Figure 6.**Different simulations to study the role of (

**a**) initial subgrain size distribution and (

**b**) disorientation distribution on growth. See Supplementary Figures S1 and S2 for the initial conditions for <001>||ED grain and initial generated microstructures.

**Figure 7.**The subgrain structure at the end of the simulation for different initial microstructures, i.e., subgrain size distributions: (

**a**) 50% narrower, (

**b**) baseline, and (

**c**) 50% wider (note: Figure S3 in the Supplementary Materials shows the results for <001>||ED).

**Figure 8.**(

**a**) Evolution of the equivalent area average 2D diameter of <111>||ED microstructures with different initial subgrain size distribution; (

**b**) evolution of the rate of change in the equivalent area average diameter of the same microstructures (note: Figure S4 in the Supplementary Materials shows the results for <001>||ED).

**Figure 9.**Histogram of evolved normalized diameter for microstructures with different initial standard deviation and same initial average diameter in different time steps: (

**a**) 50% narrower, (

**b**) baseline, and (

**c**) 50% wider standard deviation (note: Figure S5 in the Supplementary Materials shows the results for <001>||ED).

**Figure 10.**Evolution of (

**a**) the total length of high-angle boundaries and (

**b**) the fraction length of the high-angle boundaries of subgrains <111>||ED (note: Figure S6 in Supplementary Materials shows the results for the same analysis in the case of <001>||ED).

**Figure 11.**The evolved microstructure of a grain in <111>||ED fiber with the same subgrain size as baseline and disorientation distribution of different half-width angles (

**a**) 3.45°, (

**b**) 6.9° (baseline), and (

**c**) 8.62° at 7.5 s and (

**d**) evolution of diameters for different half-width angles (note: Figure S7 in Supplementary Materials shows the results for <001>||ED).

**Figure 12.**The evolution of average disorientation in the different microstructures of different disorientation distributions with the same subgrain size in a grain with <111>||ED (note: Figure S8 in Supplementary Materials shows the results for <001>||ED).

**Figure 13.**Evolution of average subgrain area as a function of time for a grain with <111>||ED orientation (note: Figure S9 in Supplementary Materials shows the results for the same analysis for <001>||ED).

**Figure 14.**Fraction of shrinking subgrains as a function of time for <111>||ED grain. Note: Figure S10 in Supplementary Materials shows the results for the same analysis for <001>||ED).

**Figure 15.**(

**a**) Comparison of initial disorientation distribution in <111>||ED with halfwidth of 10.35ΰ with a random texture; (

**b**) comparison of subgrain average diameter evolution for microstructure with <111>||ED orientation and normal grain growth.

**Figure 16.**Normalized effective mobility of the initial microstructure for different initial disorientation distributions in <111>||ED grain.

**Table 1.**Key parameters for the initial microstructures synthesized to study the role of subgrain size distribution (see Table S1 in the Supplementary Materials for <001>||ED).

Orientation | 50% Narrower | Baseline | 50% Wider |
---|---|---|---|

<111>||ED | $\mu =2.06\mathsf{\mu}\mathrm{m},\sigma =0.71\mathsf{\mu}\mathrm{m}$ | $\mu =2.06\mathsf{\mu}\mathrm{m},\sigma =1.42\mathsf{\mu}\mathrm{m}$ | $\mu =2.06\mathsf{\mu}\mathrm{m},\sigma =2.13\mathsf{\mu}\mathrm{m}$ |

**Table 2.**Half-width angle in degrees for ODF calculation of the initial microstructures synthesized to study the role of disorientation distribution (See Table S2 in Supplementary Materials for <001>||ED).

Orientation | 50% Narrower | 25% Narrower | Baseline | 25% Wider | 50% Wider |
---|---|---|---|---|---|

<111>||ED | 3.45 | 5.17 | 6.9 | 8.62 | 10.35 |

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**MDPI and ACS Style**

Khajezade, A.; Poole, W.J.; Greenwood, M.; Militzer, M.
Large-Scale Multi-Phase-Field Simulation of 2D Subgrain Growth. *Metals* **2024**, *14*, 584.
https://doi.org/10.3390/met14050584

**AMA Style**

Khajezade A, Poole WJ, Greenwood M, Militzer M.
Large-Scale Multi-Phase-Field Simulation of 2D Subgrain Growth. *Metals*. 2024; 14(5):584.
https://doi.org/10.3390/met14050584

**Chicago/Turabian Style**

Khajezade, Ali, Warren J. Poole, Michael Greenwood, and Matthias Militzer.
2024. "Large-Scale Multi-Phase-Field Simulation of 2D Subgrain Growth" *Metals* 14, no. 5: 584.
https://doi.org/10.3390/met14050584