Calculation of Critical Nucleus Size and Minimum Energy Path of Cu-Riched Precipitates During Radiation in Fe-Cu Alloy Using String Method
LIU Xuxi1, LIU Wenbo1(), LI Boyan2, HE Xinfu3, YANG Zhaoxi1, YUN Di1
1.School of Nuclear Science and Technology, Xi'an Jiaotong University, Xi'an 710049, China 2.School of Materials Science and Engineering, Tsinghua University, Beijing 100084, China 3.China Institute of Atomic Energy, Beijing 102413, China
Cite this article:
LIU Xuxi, LIU Wenbo, LI Boyan, HE Xinfu, YANG Zhaoxi, YUN Di. Calculation of Critical Nucleus Size and Minimum Energy Path of Cu-Riched Precipitates During Radiation in Fe-Cu Alloy Using String Method. Acta Metall Sin, 2022, 58(7): 943-955.
As a pressure containment shell that supports all components in the nuclear reactor, reactor pressure vessel (RPV) is an irreplaceable core component during the whole life of nuclear power plant. Cu-riched particles precipitated in the early stage of radiation have significant effects on the mechanical property (such as radiation hardening and embrittlement) changes during the application of RPV steel. However, the Cu-riched precipitate with extremely small size (smaller than 2 nm) cannot be detected by the conventional experimental method, such as scanning electron microscope and transmission electron microscope. Hence, it is essential to calculate the critical nucleus size of Cu-riched precipitate under radiation in RPV steel. In this study, based on the constrained string method and phase-field theory, the critical nucleus size and minimum energy path of Cu-riched precipitate in Fe-Cu alloy under irradiation were calculated, and the minimum energy path, critical nucleus radius, and vacancy concentration distribution were also studied. The calculated results showed that both temperature and Cu concentration have a great influence on the energy path and critical nucleus cluster size of Cu-riched particles in Fe-Cu binary alloy. Temperature is the main factor influencing the energy path direction of the nucleus, while Cu concentration is the main factor influencing the growth rate of the nucleus radius. With the increase of temperature, the Cu concentration in the nucleus increases, while the time needed for the Cu-riched particles to reach its critical nucleus size decreases, and the energy barrier needed to be crossed also decreases. The distribution of Cu concentration also has a great influence on the distribution of vacancy during radiation. The vacancy concentration in the Cu-riched cluster is lower than that in the Fe-Cu matrix. The vacancy concentration decreased as the Cu concentration increased. The calculated results are consistent with the experimental results.
Fund: National Natural Science Foundation of China(U1830124);National Natural Science Foundation of China(11705137);China Postdoctoral Science Foundation(2019M663738);Innovative Scientific Program of China National Nuclear Corporation
About author: LIU Wenbo, associate professor, Tel: (029)82668948, E-mail: liuwenbo@xjtu.edu.cn
Fig.1 Schematics of saddle points in the nucleus evolutionary correction process (a) energy path saddle point diagram(b) energy path height contrast diagram
Element
α1
α2
α3
α4
β
γ
L
Cu
6242.2
122.833
-23.5902
-0.004761
-5.919
70000
38022.8
Fe
1225.7
124.134
-23.5143
-0.004397
-5.892
77358.5
38022.8
Table 1 Gibbs free energy parameters of Fe-Cu alloy[18,28,30]
Parameter
Value
Unit
κCu
5 × 10-15
J·m2·mol-1
κV
1 × 10-15
J·m2·mol-1
εCu
122.833
εV
124.134
Table 2 Interfacial energy correlation coefficients and eigenstrain coefficients of Fe-Cu alloy[28,29]
T / K
DFe / (m2·s-1)
DCu / (m2·s-1)
MFe / (mol·m·s·kg-1)
MCu / (mol·m·s·kg-1)
550
7.2 × 10-32
5.0 × 10-29
6.7 × 10-17
4.6 × 10-14
650
4.2 × 10-27
8.0 × 10-25
3.9 × 10-12
7.4 × 10-10
Table 3 Values of diffusion coefficients and mobility at different temperatures[29,35]
Fig.2 Energy paths of Fe-0.3%Cu alloy at 650 K and radiation intensity of 0.01 dpa/s (EP2 for energy path after 2-step-revolution, EP50 for energy path after 50-step-revolution. The abscissa represents the number of nucleus time states, the same in the figures below)
Fig.3 Cu concentration distributions and top views of cluster morphologies of critical nucleus (insets) of Fe-0.3%Cu alloy at 650 K and radiation intensity of 0.01dpa/s (a) cluster after 2-step-revolution (b) cluster after 50-step-revolution
Fig.4 Minimum energy path curves of Fe-0.3%Cu alloy after 50-step-revolution at 550 K (a), 600 K (b), 650 K (c), and 700 K (d)
Fig.5 Cu concentration distributions and top views of cluster morphologies of critical nucleus (insets) of Fe-0.3%Cu alloy after 50-step-revolution at 550 K (a), 600 K (b), 650 K (c), and 700 K (d)
T / K
RC / nm
CC / %
550
1.722
45.4355
600
1.762
47.8542
650
1.794
50.2122
700
1.821
52.5147
Table 4 Calculation results of critical nucleus radius (RC) and average Cu concentration in critical nucleus clusters (CC) of Fe-0.3%Cu alloy after 50-step-revolution at different temperatures
Fig.6 1D Cu concentration distribution curves in critical nucleus cluster of Fe-0.3%Cu alloy after 50-step-revolution at different temperatures
Fig.7 Minimum energy path curves of Fe-Cu alloy after 50-step-revolution with Cu concentrations of 0.05% (a), 0.1% (b), 0.3% (c), and 0.5% (d) at 650 K
Fig.8 Cu concentration distributions and top views of cluster morphologies of critical nucleus (insets) of Fe-Cu alloy after 50-step-revolution with Cu concentrations of 0.05% (a), 0.1% (b), 0.3% (c), and 0.5% (d) at 650 K
Alloy
RC / nm
CC / %
Fe-0.05Cu
1.190
26.9862
Fe-0.1Cu
1.292
35.3291
Fe-0.3Cu
1.762
47.8542
Fe-0.5Cu
2.014
52.8991
Table 5 Calculation results of RC and CC for different Fe-Cu alloys at 650 K after 50-step-revolution
Fig.9 1D Cu concentration distribution curves in critical nucleus clusters of Fe-Cu alloy with different Cu concentrations at 650 K after 50-step-revolution
Fig.10 Changes of cluster radius for Fe-Cu alloy after 50-step-revolution (a) 650 K for different Cu concentrations (b) 0.3%Cu concentration at different temperatures
Fig.11 Distributions of the increase of vacancy concentration at critical nucleus cluster state after 50-step-revolution at 650 K (a) Fe-0.3%Cu, 0.01 dpa/s (b) Fe-0.3%Cu, 0.05 dpa/s (c) Fe-0.5%Cu, 0.01 dpa/s (d) Fe-0.5%Cu, 0.05 dpa/s
Fig.12 Vacancy concentration distributions after 50-step-revolution at the different time (t) of nucleus of Fe-0.3%Cu alloy at 650 K and 0.01 dpa/s (a) t = 30 step (b) t = 60 step (c) t = 90 step (d) t = 120 step
t / step
RC / nm
Min.CV
Max.CV - Min.CV
Sum.CV
30
1.558
0.00102
8.63 × 10-5
17.149
60
1.763
0.00182
3.89 × 10-4
34.381
90
1.884
0.00229
8.92 × 10-4
51.535
120
2.039
0.00280
1.65 × 10-3
69.190
Table 6 Calculation results of partial parameters of critical nucleus clusters after 50-step-revolution at different t
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