# Comparison of Grain-Growth Mean-Field Models Regarding Predicted Grain Size Distributions

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Mean-Field Models

#### 2.1. Burke and Turnbull Model

#### 2.2. Hillert Model

#### 2.3. Abbruzzese et al. Model

#### 2.4. Maire et al. Model

## 3. Input Data for Mean-Field Modeling

#### 3.1. Material-Dependent Model Parameters Acquisition

#### Experimental Data

#### 3.2. Use of Saltykov Algorithm to Obtain a 3D GSD

#### 3.2.1. GB Mobility Parameter Identification

#### A First Approximation Using the Classical B&T Law

#### Refined Identification

#### Model-Dependence of Reduced Mobility

## 4. Results and Discussion

#### 4.1. Numerical Parameters

#### 4.1.1. Convergence Study Concerning the Number of Grain Classes Introduced in the Model

#### 4.1.2. Different Spatial Dimensions Considered to Define the Contact Probability

#### Description of the Spatial Dimensions

#### Impact on the Distribution Results

#### 4.1.3. Impact of the Selection Order of Grain Classes on the Neighborhood Construction

#### 4.2. Comparison of Mean-Field Models Using Different Initial Microstructures

#### 4.2.1. Comparison of Mean-Field Models with a Monomodal Initial Microstructure

#### 4.2.2. Comparison of Mean-Field Models on Bimodal Initial Microstructure

#### Impact of the Selection Order of Neighborhood Construction on the Distribution Prediction

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Statistical neighborhood construction of the 2D GG model of Abbruzzese et al.: (

**a**) the statistical medium, and (

**b**) the concept of the contact probability ${p}_{j}$.

**Figure 5.**In the background: IPF Z maps of 316L microstructures of (

**a**) the as-received material, (

**b**) after 2 h at 1000 °C, (

**c**) after 2 h at 1050 °C, and (

**d**) after 2 h at 1100 °C. The black lines denote the grain boundaries. In the foreground: the corresponding 3D GSD after an inverse Saltykov transformation.

**Figure 6.**Inverse Saltykov method illustrated on the 2D GSD of a sample heat-treated at 1050 °C for 5 h. The blue histogram corresponds to the 2D GSD and the orange discrete distribution corresponds to the 3D results obtained through the inverse Saltykov algorithm.

**Figure 7.**Use of the Burke and Turnbull law to obtain the first value of ${\left({M}_{GB}{\gamma}_{GB}\right)}_{ini}$ for 316L at (

**a**) 1000 °C, (

**b**) 1050 °C, and (

**c**) 1100 °C.

**Figure 8.**Curve fitting of the simulation points (obtained with ${\left({M}_{GB}{\gamma}_{GB}\right)}_{ini}$) with respect to the experimental points by minimizing L${}^{2}$ error for the Maire et al. and Hillert models at 1050 °C for 316L.

**Figure 9.**(

**a**) Histogram of the ${L}^{2}$ (1 h) error method for analyzing the convergence of simulations at 1100 °C, (

**b**) convergence study of the initial number of grain classes by computing the ${L}^{2}$ (1 h) error at 1100 °C for the Hillert, Abbruzzese et al., and Maire et al. models on 316L, and (

**c**) the same convergence study computing the ${L}^{2}$ (1 h) error for the three models using the final number of grain classes as the x-axis.

**Figure 10.**Description of the different contact probabilities ${p}_{(i,j)}^{m}$ with neighbor classes of the first class ($i=0$) in the microstructure at t = 0 s.

**Figure 11.**Comparison of the impact on the distribution of the ${p}_{(i,j)}^{m}$ with $m\in \u301a0,3\u301b$ for an annealing time of 2 h at different temperatures for 316L: (

**a**) 1000 °C. (

**b**) 1050 °C. (

**c**) 1100 °C.

**Figure 12.**Comparison of different selection orders for the neighborhood construction using the Maire et al. model. The test case selected here is a 1h annealing time at 1100 °C. (

**a**) MGS evolution with respect to time, GSD at the end of the heat treatment (t = 1 h) considering (

**b**) the frequency of occurrence and (

**c**) volume fraction.

**Figure 13.**Specific neighborhood construction considering (

**a**) ascending and (

**b**) descending selection order in the GSD.

**Figure 14.**Comparisons of GSDs in terms of the (

**a**,

**c**,

**e**,

**g**) frequency of occurrence and (

**b**,

**d**,

**f**,

**h**) volume fraction for the Hillert, Abbruzzese et al., and Maire et al. models with experimental data at 1100 °C for (

**a**,

**b**) 1 h, (

**c**,

**d**) 2 h, (

**e**,

**f**) 3 h, and (

**g**,

**h**) 5 h of thermal treatment on 316L.

**Figure 15.**Comparison of ${L}^{2}\left(t\right)$ errors computed from GSDs for the three models and heat treatments from 1 h to 5 h at 1100 °C in terms of the (

**a**) frequency of occurrence and (

**b**) volume fraction.

**Figure 16.**Comparison of the GG predictions at 1100 °C, starting from an initial bimodal distribution with a neighborhood construction selected in ascending order for the Maire et al. model: (

**a**) Initial bimodal distribution, (

**b**) MGS evolution over time, and GSDs at (

**c**) 5 min, (

**d**) 10 min, and (

**e**) 2 h.

**Figure 17.**Comparison of GG predictions at 1100 °C, starting from an initial bimodal distribution with a neighborhood construction selected in descending order for the Maire et al. model: (

**a**) MGS evolution over time, and GSDs at (

**b**) 5 min, (

**c**) 10 min, and (

**d**) 2 h.

**Table 1.**Heat treatment campaign conditions, with orange indicating the values used only for calibration, red indicating the values used for both calibration and validation, and green indicating the values used only for validation.

1000 °C | 1050 °C | 1100 °C |
---|---|---|

30 min | 30 min | 30 min |

1 h | 1 h | 1 h |

2 h | 2 h | 2 h |

3 h | 3 h | 3 h |

5 h | 5 h | 5 h |

**Table 2.**Summary of post-processing details for the various isothermal treatment temperatures and holding times, including the EBSD step size (h), the dimensions in each direction (${L}_{x}\times {L}_{y}$) of the analyzed areas of the sample, and the number of grains represented in the EBSD images, considering all the grains and without taking into account the twins boundaries (#G).

T = 1000 °C | T = 1050 °C | T = 1100 °C | |||||||
---|---|---|---|---|---|---|---|---|---|

$\mathit{t}$ | $\mathit{h}$ ($\mathsf{\mu}$m) | ${\mathit{L}}_{\mathit{x}}$ × ${\mathit{L}}_{\mathit{y}}$ (mm × mm) | #G | $\mathit{h}$ ($\mathsf{\mu}$m) | ${\mathit{L}}_{\mathit{x}}$ × ${\mathit{L}}_{\mathit{y}}$ (mm × mm) | #G | $\mathit{h}$ ($\mathsf{\mu}$m) | ${\mathit{L}}_{\mathit{x}}$ × ${\mathit{L}}_{\mathit{y}}$ (mm × mm) | #G |

Initial | 1.49 | 1.1 × 0.85 | 980 | 1.49 | 1.1 × 0.85 | 980 | 1.49 | 1.1 × 0.85 | 980 |

30 min | 2.5 | 2 × 1.4 | 2654 | 1.13 | 1 × 0.7 | 534 | 3.3 | 3.7 × 2.8 | 3509 |

1 h | 2.5 | 2 × 1.4 | 2078 | 3 | 3 × 2.2 | 1964 | 3.3 | 3.7 × 2.8 | 3590 |

2 h | 1.13 | 1 × 0.7 | 456 | 3 | 3 × 2.2 | 1154 | 3.77 | 3.7 × 2.8 | 2208 |

3 h | 1.13 | 1 × 0.7 | 468 | 1.13 | 1 × 0.7 | 300 | 3.77 | 3.7 × 2.8 | 2263 |

5 h | 1.13 | 1 × 0.7 | 243 | 1.13 | 1 × 0.7 | 133 | 3.77 | 3.7 × 2.8 | 2304 |

**Table 3.**Initial reduced mobility values ${\left({M}_{GB}{\gamma}_{GB}\right)}_{ini}$ for the Maire et al. model in ascending order of the initial GSD for the investigated temperatures for 316L.

Temperature | 1000 °C | 1050 °C | 1100 °C |
---|---|---|---|

${M}_{GB}{\gamma}_{GB}$ (m${}^{2}$s${}^{-1}$) | 2.30 × 10${}^{-15}$ | 1.08 × 10${}^{-13}$ | 1.10 × 10${}^{-13}$ |

Model | Hillert | Abbruzzese | Maire |
---|---|---|---|

${M}_{GB}{\gamma}_{GB}$ (m${}^{2}$s${}^{-1}$) | 1.08 × 10${}^{-13}$ | 1.27 × 10${}^{-13}$ | 1.10 × 10${}^{-13}$ |

**Table 5.**Identified reduced mobility values for the Maire et al. model for different selection orders at 1100 °C for 316L.

Sorting Order | Ascending | Descending | Shuffle |
---|---|---|---|

${M}_{GB}{\gamma}_{GB}$ (m${}^{2}$s${}^{-1}$) | 2.19 × 10${}^{-13}$ | 5.00 × 10${}^{-13}$ | 2.30 × 10${}^{-13}$ |

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**MDPI and ACS Style**

Roth, M.; Flipon, B.; Bozzolo, N.; Bernacki, M.
Comparison of Grain-Growth Mean-Field Models Regarding Predicted Grain Size Distributions. *Materials* **2023**, *16*, 6761.
https://doi.org/10.3390/ma16206761

**AMA Style**

Roth M, Flipon B, Bozzolo N, Bernacki M.
Comparison of Grain-Growth Mean-Field Models Regarding Predicted Grain Size Distributions. *Materials*. 2023; 16(20):6761.
https://doi.org/10.3390/ma16206761

**Chicago/Turabian Style**

Roth, Marion, Baptiste Flipon, Nathalie Bozzolo, and Marc Bernacki.
2023. "Comparison of Grain-Growth Mean-Field Models Regarding Predicted Grain Size Distributions" *Materials* 16, no. 20: 6761.
https://doi.org/10.3390/ma16206761