Cluster dynamics of proton irradiated RPV steel at room temperature

The life of reactor pressure vessels in nuclear power plants will be extended to 60 years, while the irradiation data of materials from commercial and experimental reactors are rare and expensive. There is an urgent need to solve how to accurately predict the changes in microstructure and mechanical properties of high-dose irradiated materials of RPV. The cluster dynamics method can be used to establish the relationship between nanostructure and macro properties and to study its evolution over a long period of years. Based on the average rate field theory, a cluster dynamics model was developed for the evolution of irradiated microstructure of RPV steel, which considers both homogeneous and heterogeneous nucleation of solute clusters. The model was successfully used to simulate the influence of the proton irradiation-induced cluster size and number density of RPV model steel with a low copper content at room temperature.


Introduction
The ductile-brittle transition temperature of reactor pressure vessel (RPV) steels will increase, and the upper shelf energy will decrease in the neutron irradiation environment during the reactor operation.The embrittlement process of RPV steel endangers its structural integrity and operation safety [1] .This mechanical property degradation is partly due to the formation of nanometer-size solute and point-defect (PD) clusters.Many breakthroughs have been made in the study of the radiation damage mechanism of RPV steel using many research methods, such as neutron irradiation experiments, irradiation experiments, or theoretical simulations of energy-carrying ions [2] .However, it is a crucial issue to deeply understand their formation and to form a recognized mechanism.The image of the synergistic evolution of solute clusters or matrix damage is still very blurred.The relative contribution of the two to embrittlement has not been clarified.The existing radiation embrittlement prediction model of RPV materials could not clearly express the contribution of these two types of irradiation damage features.For ultra-pure RPV steel, the equivalence law of proton irradiation and neutron irradiation damage needs to be revealed by multi-scale simulation [3] .

AMCE-2023
A homogeneous precipitation mechanism is at the origin of the experimentally observed copperenriched clusters in the irradiated high copper content RPV steels [4] .But for the low copper content RPV steels, the formation of solute-rich clusters under cascade damage conditions occurs via heterogeneous precipitation on PD clusters [5][6] .In this paper, the microstructure evolution of irradiated RPV steel will be studied.The irradiation damage cluster dynamics simulation method is used combined with the accelerator proton irradiation experiment at room temperature.The research results provide a reference basis for the evaluation of the irradiation properties of domestic RPV steel.

Cluster-dynamics model
2.1 Model description 2.1.1Monomer concentration evolution.Isolated vacancies, interstitial atoms, and Cu atoms can be collectively referred to as monomers.In the process of irradiation, the concentration of vacancy, interstitial atom, and Cu atom will evolve with time.Its evolution equation can be expressed as: where loop N , void N , and crp N are the number of monomers contained in each dislocation loop, cavity, and Cu-rich phase, respectively; , ic dc N and , vc dc N are respectively the number of interstitial atoms and the number of vacancies contained in each cluster of endogenous interstitial atoms in the offsite cascade.loop R , void R , and crp R are the radii of each dislocation ring, cavity, and Cu-rich phase, respectively.The unit is the number of monomers.

Parameters
To use this model, we need to know the input parameters.When solving the average rate field partial differential equation, it is very important to select the material parameters as accurately as possible.This is conducive to obtaining reliable prediction results.Some of them are intrinsic to the material, such as the sample dislocation density or the grain size.Others are energetic and kinetics parameters, like the formation or migration energies of species, controlling the rate of capture and emission of point defects by clusters.The parameters used to simulate proton irradiation RPV steel are reported in Table 1.Most of the parameters come from literature data [7][8][9] .

Multidimensional ill conditioned differential equation algorithm
Equations ( 1) ~ ( 9) are 9-dimensional ill-conditioned rigid differential equations.If the size distribution of various clusters is considered, the number of equations will reach about 10 9 .In this study, the size distribution of clusters is not considered temporarily, and only the average size of clusters is calculated.The algorithm used is the Runge Kutta method.Based on the cluster dynamics model, a long-time microstructure evolution simulation program is developed.

Cluster evolution
Figure 1 shows the results of the simulation case, that is, the dynamic simulation results of A508-3 steel with a low copper content of 0.06wt.%irradiated by proton irradiation at room temperature, including 3 × 5 array diagrams.In Figure 1, the 5 diagrams in the first row show the variation of content, size, and hardening components of interstitial defects or dislocation loops with irradiation fluence.In Figure 1, the 5 diagrams in the second row show the variation of content, size, and hardening components of vacancy defects or voids with irradiation fluence.In Figure 1, the 5 diagrams in the last row show the variation of content, size, and hardening components of copper or copper-rich phase with irradiation fluence.
The size of dislocation loops and voids first increases rapidly and slowly up to 0.4 dpa.At 0.4 dpa, The diameter of dislocation loops is about 3 nm, while the diameter of voids is about 0.5 nm.The density of loops and voids continuously increases up to 0.4 dpa.The diameter of the copper-rich cluster increases to about 0.5 nm with a relatively low density of 10 22 m -3 up to 0.4 dpa.

Cluster hardening contribution
The low copper RPV steels were proton irradiated at room temperature.The irradiation-induced nanoclusters are mainly dislocations loops and voids with relatively minor copper-rich clusters, as shown in Figure 2. Therefore irradiation-induced dislocation loops and cavities are the main contributors to hardening.The dislocation loops maybe play a leading role, while the voids maybe play a secondary role in irradiation hardening.In addition, cavity effect gradually increases with the increase of proton fluence.

Conclusion
Based on the average rate field theory, a cluster dynamics model was developed for the evolution of irradiated microstructure of RPV steel, which considers both homogeneous and heterogeneous nucleation of solute clusters.The model was successfully used to simulate the influence of the proton irradiation-induced cluster size and number density of RPV model steel with a low copper content at room temperature.The size of dislocation loops and voids first increases rapidly and then slowly up to 0.4 dpa.At 0.4 dpa, the diameter of dislocation loops is about 3 nm, while the diameter of voids is about 0.5 nm.The density of loops and voids continuously increases up to 0.4 dpa.The diameter of the copperrich cluster increases to about 0.5 nm with a relatively low density of 10 22 m -3 up to 0.4 dpa.The dislocation loops maybe play a leading role, while the voids maybe play a secondary role in irradiation hardening with a minor role of solute cluster.
are the vacancy concentration on the surface of the cavity and the Cu atom concentration at the interface of the Cu-rich phase, respectively; P is the damage rate, that is, the fluence rate, in dpa/s; Cluster size evolution.During irradiation, the sizes of dislocation loops, vacancies, and Cu-rich phases will evolve with time.The evolution equation can be expressed as: r  is the probability of recombination of Frenkel point defects during offsite cascade cooling, and r 1   represents the damage efficiency.ic,dc  is the percentage of interstitial atoms in the cluster of endogenous interstitial atoms in the offsite cascade.vc,dc  is the percentage of vacancies in the cluster of endogenous vacancies in the offsite cascade; Z is recombination or capture reaction constants, and the lower corners i , v , Cu , ic , vc , CRP , and d represent interstitial atoms, vacancies, Cu atoms, dislocation rings, cavities, Cu rich phases, and dislocation lines in the matrix respectively.i

Table 1 .
The main input parameters for cluster dynamics simulating proton irradiation and neutron irradiation RPV steel.