# Predicting the Irradiation Swelling of Austenitic and Ferritic/Martensitic Steels, Based on the Coupled Model of Machine Learning and Rate Theory

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Machine Learning of Steady-State Swelling Onset Dose

#### 2.2. Rate Theory of the Swelling Behavior after Incubation Period

## 3. Results

#### 3.1. Prediction of the Onset Dose of Swelling

#### 3.2. Simulation of Irradiation Swelling

#### 3.2.1. Irradiation Swelling of Austenitic Steels

^{14}~1.8 × 10

^{16}cm

^{−3}and the void concentration ${C}_{\mathrm{v}}$ is in the range of 3.1 × 10

^{14}~2.8 × 10

^{16}cm

^{−3}. This data range is more close to the experimental results [34] compared with the values in the previous rate theory [3]. The formulas of ${N}_{\mathrm{l}}$ and ${C}_{\mathrm{v},}$ with their temperatures, are shown in Table 3. The results of the swelling rates are different at different temperatures; for example, they are about 0.4%/dpa at 427 °C and about 0.9%/dpa at 510 °C. The major reason that influences the swelling rate under different temperatures is the diffusion coefficient of the point defects. The swelling could not be simulated properly at high temperatures using the previous rate theory model [2]; the main reasons for this are the improper description of the vacancy loop evolution and cascade effects, as the point defect flux ${Z}_{\mathrm{i},\mathrm{v}}{D}_{\mathrm{i},\mathrm{v}}{c}_{\mathrm{i},\mathrm{v}}$ at high temperatures would cause the accumulating rate to become negative. In this work, the effect of the vacancy loop has been ignored while the focus is on the evolution of interstitial loops. The cascade efficiency of point defects production has been used to calculate the value of the surviving point defects after the cascade process. By using this modified ML-RT model, the neutron swelling at high temperatures (>525 °C) can be predicted, as shown in Figure 4a.

#### 3.2.2. Irradiation Swelling of F/M Steels

#### 3.2.3. Prediction of Irradiation Swelling in CLAM Steel

## 4. Discussion

#### 4.1. Steady-State Swelling Onset Dose

#### 4.2. Cascade Efficiency

#### 4.3. Point Defect Diffusion

#### 4.4. Sink Strength

^{10}cm

^{−2}and 1.1 × 10

^{11}cm

^{−2}and the dislocation loop densities are 2.5 × 10

^{15}cm

^{−2}and 4.5 × 10

^{15}cm

^{−2}for 316 and Fe-9Cr, respectively [27]. This density difference results in about 3% lower swelling in Fe-9Cr than that in AISI 316. In general, the more non-neutral sinks there are, the greater the amount of swelling the material has. However, this density discrepancy of dislocations in AISI 316 and Fe-9Cr is not sufficient to cause the much lower swelling seen in Fe-9Cr.

#### 4.5. Helium Effects

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**The mean absolute error of the different ensemble methods (decision tree regression (DTR), support vector regression (SVR), random forest regression (RFR), gradient boosting regression (GBR), and k-nearest neighbor regression (KNR)), training set, and testing set.

**Figure 2.**The correlation coefficient (R) and the root mean square error (RMSE), using different ensemble methods on the testing set.

**Figure 3.**Comparison of the predicted and experimental results for the onset dose of swelling using the RFR method.

**Figure 4.**Neutron irradiation swelling of austenitic AISI 316 steel: (a) swelling as a function of temperature with the irradiation dose of 120 dpa, (b) swelling as a function of the dose at 510 °C, where the black squares represent experimental data from [29] and the red line corresponds to the results of theoretical calculations.

**Figure 6.**Predicted irradiation swelling of CLAM steel as a function of the dose under neutron irradiation (with low helium production rate in fission reactor) at 400 °C and compared with the experimental data of F/M steels, data from [32,33]. Due to the production of H and He in the fusion reactor, the swelling in the fusion reactor may be higher than in the predictions.

**Figure 7.**(

**a**) The diffusion coefficients of vacancy, (

**b**) the diffusion coefficients of SIA in F/M steels and austenitic steels under different temperatures.

**Figure 8.**(

**a**) The swelling of AISI 316 with different E

_{mv}or E

_{mi}. The black line with triangles: E

_{mv}= 1.4 eV and E

_{mi}= 0.85 eV; the red line with circles: E

_{mv}= 1.4 eV and E

_{mi}= 1.4 eV; the blue line with squares: E

_{mv}= 1.2 eV and E

_{mi}= 0.85 eV. (

**b**) The swelling plots as a function of doses at 450 °C with the loop biases ${Z}_{\mathrm{il}}$ equal to 1.05, 1.15, and 1.20, respectively.

Variables | Min | Max | Variables | Min | Max |
---|---|---|---|---|---|

Fe/(wt %) | 8.0 | 97.0 | V/(wt %) | 0 | 2 |

Cr/(wt %) | 3 | 24.7 | W/(wt %) | 0 | 2.1 |

Ni/(wt %) | 0.0 | 74.9 | Temperature/(K) | 500 | 1013 |

Si/(wt %) | 0.0 | 1.3 | Dose rate/(dpa/s) | 8 × 10^{−9} | 0.06 |

Mn/(wt %) | 0.0 | 15 | Dislocation density/(m^{−2}) | 5 × 10^{13} | 8.5 × 10^{15} |

Mo/(wt %) | 0.0 | 2.8 | Cascade efficiency | 0.01 | 0.3 |

Ta/(wt %) | 0.0 | 0.36 | Dose(dpa) | 0.2 | 120 |

Material Parameters | AISI 316 | Fe-9Cr |
---|---|---|

V-formation energy (eV) | 1.8 [21] | 1.9 [25] |

SIA-formation energy (eV) | 1.8 [21] | 4.1 [25] |

V-migration energy (eV) | 1.4 [24] | 1.1 [26] |

SIA-migration energy (eV) | 0.85 [24] | 0.2 [26] |

Dv0 (cm^{2}·s^{−1}) | 1.29 × 10^{−2} [21] | 4.5 × 10^{−3} [25] |

Di0 (cm^{2}·s^{−1}) | 1.29 × 10^{−2} [21] | 3.0 × 10^{−5} [25] |

Recombination coefficient | 5.69 × 10^{26} | 5.48 × 10^{27} |

Recombination radius (cm) | 1.27 × 10^{−7} | 1.1 × 10^{−7} |

Dislocation density (cm^{−2}) | 1.5 × 10^{10} [27] | 1.1 × 10^{11} [27] |

Dislocation bias ${Z}_{id}$ | 1.20 [24] | 1.05 [28] |

Loop bias ${Z}_{il}$ | 1.20 [24] | 1.05 [28] |

Loop initial radius (cm) | 10^{−7} | 10^{−7} |

Burger’s vector (cm) | 2.0 × 10^{−8} | 2.86 × 10^{−8} |

Lattice parameter (cm) | 3.64 × 10^{−8} | 2.8 × 10^{−8} |

Poisson’s ratio | 0.264 | 0.3 |

Parameters | AISI 316 [29] | Fe-9Cr [30] | JLF-1 [31] | CLAM [32,33] |
---|---|---|---|---|

Dose rate (dpa/s) | 10^{−6} | 10^{−6} | 10^{−6} | 10^{−6} |

Cascade efficiency | 0.2 | 0.25 | 0.25 | 0.25 |

Interstitial loop density (cm ^{−3}) | 2.2 × 10^{4} exp (1.7/${\mathrm{k}}_{\mathrm{B}}T$) | 6.5 × 10^{16} at 425 °C | 1.8 × 10^{3} exp (1.8/${\mathrm{k}}_{\mathrm{B}}T$) | 1.1 × 10^{16} |

Void concentration (cm^{−3}) | 3.0 × 10^{16} exp {−[(1/${\mathrm{k}}_{\mathrm{B}}T$ × 1.2]^{2}} | 8.7 × 10^{15} at 425 °C | 7.45 × 10^{13} at 390 °C2.2 × 10 ^{14} at 430 °C | 5.0 × 10^{14} |

Irradiation type | neutron | neutron | neutron | neutron |

Neutron fluence (n m ^{−2}) | 10 × 10^{26}–25 × 10^{26} | 19 × 10^{26} | 5.8 × 10^{26} | 7.7 × 10^{26} |

T (°C) | 427–593 | 420 | 390–460 | 400 |

**Table 4.**Experimental data of neutron swelling in austenitic AISI 316 steel under different irradiation doses or at different temperatures from [29].

Dose/(dpa) | Swelling/% 510 °C | T/(°C) | Dose/(dpa) | Swelling/% |
---|---|---|---|---|

123.3 | 71.1 | 427 | 87.8 | 16.9 |

141.3 | 87.5 | 482 | 112.5 | 46.4 |

89.7 | 41.5 | 510 | 123.3 | 51.6 |

71.8 | 23.6 | 538 | 118.1 | 41.5 |

56.1 | 11.5 | 593 | 127.2 | 29.6 |

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**MDPI and ACS Style**

Zhu, X.; Li, X.; Zheng, M.
Predicting the Irradiation Swelling of Austenitic and Ferritic/Martensitic Steels, Based on the Coupled Model of Machine Learning and Rate Theory. *Metals* **2022**, *12*, 651.
https://doi.org/10.3390/met12040651

**AMA Style**

Zhu X, Li X, Zheng M.
Predicting the Irradiation Swelling of Austenitic and Ferritic/Martensitic Steels, Based on the Coupled Model of Machine Learning and Rate Theory. *Metals*. 2022; 12(4):651.
https://doi.org/10.3390/met12040651

**Chicago/Turabian Style**

Zhu, Xiaohan, Xiaochen Li, and Mingjie Zheng.
2022. "Predicting the Irradiation Swelling of Austenitic and Ferritic/Martensitic Steels, Based on the Coupled Model of Machine Learning and Rate Theory" *Metals* 12, no. 4: 651.
https://doi.org/10.3390/met12040651