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Multi-Objective Optimization of the Nanocavities Diffusion in Irradiated Metals^{ †}

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## Abstract

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## 1. Introduction

## 2. Problem Formulation

**annealing**temperatures, ${T}_{j}$, the real temperatures at which the samples are put during the experiment. The others are the

**diffusion**temperatures, ${\theta}_{i}$, inputs of the model and hypothetical temperatures at which nanocavities of various sizes start their diffusion. The link between these two series of temperatures is that if one

**diffusion**temperature is close to one

**annealing**temperature, it is likely that the related nanocavity family diffuses and modifies the sample microstructure. However, there is no established method to define the

**diffusion**temperature.

- The dependence of the objectives from one temperature to the next one: ${O}_{d,s}^{j}$ depends on ${O}_{d,s}^{j-1}$. If the simulation is far from the observation at ${T}_{j}$, there is little chance to get it correct at ${T}_{j}+\delta T$.
- An unexpected interdependence of $density$ and $size$ objectives. At each temperature stage, we observed that the optimization of $density$, ${O}_{d}^{j}$ tends to disfavor $size$, ${O}_{s}^{j}$.

## 3. Experimental Data and Numerical Model

#### 3.1. Collision Cascade

#### 3.2. Experimental Data

#### 3.3. Numerical Model

## 4. Results

#### 4.1. Pareto Front in the Objective Space

#### 4.2. Projection or Mapping of the Objective Function

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Experimental data: measured nanocavity total densities and mean sizes as a function of temperature, i.e., at the end of the irradiation stages and each of the 8 annealing stages. The lines indicate the 0.9, 0.5 and 0.1 contours of our designed likelihood function. (

**b**) Data attach function which describes how strongly the optimization should converge close to the likelihood maximum.

**Figure 2.**(

**a**) High density of small nanocavities visible in the TEM micrographs in under-focused beam conditions of tungsten samples after irradiation. (

**b**) Visualization of the defects (SIAs clusters and nanocavities) in the simulation box at the end of the irradiation.

**Figure 3.**Pareto front of the first objectives (i.e., the ones corresponding to the irradiation stage) of size and density at ${T}_{1}$, the irradiation temperature. The point colors correspond to the temperature of diffusion of the smallest nanocavities, ${\theta}_{1}$, and the point sizes correspond to the nanocavity mean sizes at the end of the irradiation, ${s}_{1}$.

**Figure 4.**Projection of the Pareto optimal solutions in the space of parameters $\theta \left(s\right)$ (

**a**) $density$ objective, (

**b**) $size$ objective.

**Figure 5.**(

**a**) Pareto front and Pareto optimal solution projected in the parameter space, diffusion temperature, ${\theta}_{i}$, as a function of the nanocavity size. Stars on the y axis indicate the annealing temperatures, ${T}_{j}$. (

**b**) Comparison of our results with literature.

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## Share and Cite

**MDPI and ACS Style**

De Backer, A.; Souidi, A.; Hodille, E.A.; Autissier, E.; Genevois, C.; Haddad, F.; Della Noce, A.; Domain, C.; Becquart, C.S.; Barthe, M.F. Multi-Objective Optimization of the Nanocavities Diffusion in Irradiated Metals. *Phys. Sci. Forum* **2022**, *5*, 41.
https://doi.org/10.3390/psf2022005041

**AMA Style**

De Backer A, Souidi A, Hodille EA, Autissier E, Genevois C, Haddad F, Della Noce A, Domain C, Becquart CS, Barthe MF. Multi-Objective Optimization of the Nanocavities Diffusion in Irradiated Metals. *Physical Sciences Forum*. 2022; 5(1):41.
https://doi.org/10.3390/psf2022005041

**Chicago/Turabian Style**

De Backer, Andrée, Abdelkader Souidi, Etienne A. Hodille, Emmanuel Autissier, Cécile Genevois, Farah Haddad, Antonin Della Noce, Christophe Domain, Charlotte S. Becquart, and Marie France Barthe. 2022. "Multi-Objective Optimization of the Nanocavities Diffusion in Irradiated Metals" *Physical Sciences Forum* 5, no. 1: 41.
https://doi.org/10.3390/psf2022005041