Phase-field simulation of high-temperature corrosion of binary Zr-2.5Sn alloy

Liu Xu-Xi; Gao Shi-Sen; La Yong-Xiao; Yu Dong-Liang; Liu Wen-Bo

doi:10.7498/aps.73.20240393

Abstract

Due to the small neutron absorption cross section and excellent thermal creep performance, zirconium alloy is one of the most important cladding materials for fuel rods in commercial fission reactors. However, quantitative analysis of the effects of temperature and grain boundaries on the corrosion microstructure evolution of zirconium alloys is still needed. The establishing of a phase field simulation for the corrosion process of polycrystalline zirconium alloy and the systematical investigating of the thermodynamic influence are both very important. In this study, the phase field model of the corrosion process in zirconium alloys is developed by combining corrosion electrochemistry through calculating the interfacial energy at the metal-oxide and oxide-fluid boundaries. Then the model is used to investigate the uniform corrosion behavior on the surface of Zr-2.5Sn alloy, which demonstrates that the corrosion kinetic curve follows a cubic rule. Subsequently, the influence of temperature on the corrosion thickening curve of zirconium alloy is examined, and good agreement between simulation and experimental results is achieved. It is observed that during early stage of oxide layer formation, there is a high growth rate with minimal temperature dependence; however, as the oxide layer thickness increases, temperature becomes a significant factor affecting its growth rate, with higher temperatures resulting in faster corrosion rates. Furthermore, the effect of polycrystalline zirconium alloy matrices on corrosion rate is investigated, revealing that the grain boundaries accelerate oxide layer thickening due to enhanced oxygen diffusion rates. At metal-oxide interface, O^2– bands are formed in areas with higher O^2– concentration along these grain boundaries towards the metal matrix, which mainly influences oxidation-corrosion rate during the initial oxidation stage.

Keywords:

Authors and contacts

1.
Institute of Nuclear Science and Technology, Xi’an Jiaotong University, Xi’an 710049, China
2.
Shaanxi Key Laboratory of Advanced Nuclear Energy and Technology, Xi’an Jiaotong University, Xi’an 710049, China

Corresponding author: Liu Wen-Bo, liuwenbo@xjtu.edu.cn

Funds: Project supported by the Joint Fund of the National Natural Science Foundation of China and the China Academy of Engineering Physics (Grant No. U2130105) and the China National Nuclear Corporation Limited Leading Innovation Research Project, China.

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References

[1]	Kai J J, Huang W I, Chou H Y 1990 J. Nucl. Mater. 170 193 Google Scholar
[2]	Zhao W Z, Quan Q W, Zhang S M, Yang X Y, Wen H Y, Wu Y C, Liu X B, Cao X Z, Wang B Y 2023 Radiat. Phys. Chem. 209 110986 Google Scholar
[3]	Jones R H, Simonen E P 1994 Mater. Sci. Eng. 176 211 Google Scholar
[4]	Hu J, Liu J L, Lozano P S, Grovenor C R M, Christensen M, Wolf W, Wimmer E, Mader E V 2019 Acta Mater. 180 105 Google Scholar
[5]	Cann C D, So C B, Styles R C, Coleman C E 1993 J. Nucl. Mater. 205 267 Google Scholar
[6]	Wei T G, Dai X, Long C S, Sun C, Long S J, Zheng J Y, Wang P F, Jia Y Z, Zhang J S 2021 Corros. Sci. 192 109808 Google Scholar
[7]	Tian Z, Peng J C, Lin X D, Hu Y Y, Yao M Y, Xie Y P, Liang X, Zhou B X 2024 Corros. Sci. 202 111937 Google Scholar
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[9]	Jiang G Y, Xu D H, Yang W P, Liu L, Zhi Y W, Yang J Q 2022 Prog. Nucl. Energy 154 104490 Google Scholar
[10]	Yuan R, Xie Y P, Li T, Xu C H, Yao M Y, Xu J X, Guo H B, Zhou B X 2021 Acta Mater. 209 116804 Google Scholar
[11]	Mcdeavitt S M, Billings G W, Indacochea J E 2002 J. Mater. Sci. 37 3765 Google Scholar
[12]	Jerlerud P R, Toffolon M C, Joubert J M, Sundman B 2008 CALPHAD 32 593 Google Scholar
[13]	Asle Zaeem M, El Kadiri H 2014 Comput. Mater. Sci. 89 122 Google Scholar
[14]	张更, 王巧, 沙立婷, 李亚捷, 王达, 施思齐 2020 69 226401 Google Scholar Zhang G, Wang Q, Sha L T, Li Y J, Wang D, Shi S Q 2020 Acta Phys. Sin 69 226401 Google Scholar
[15]	Chen L Q, Zhao Y 2022 Prog. Mater. Sci. 124 100868 Google Scholar
[16]	Chen L Q 2002 Annu. Rev. Mater. Res. 32 113 Google Scholar
[17]	刘续希, 柳文波, 李博岩, 贺新福, 杨朝曦, 恽迪 2022 金属学报 58 943 Liu X X, Liu W B, Li B Y, He X F, Yang Z X, Yun D 2022 Acta Metal. Sin 58 943
[18]	Mai W, Soghrati S 2017 Corros. Sci. 125 87 Google Scholar
[19]	Fang X R, Pan Z C, Ma R J, Chen A R 2023 Comuput. Methods Appl. Mech. Engrg. 414 116196 Google Scholar
[20]	冯力, 王智平, 路阳, 朱昌盛 2008 57 1084 Google Scholar Feng L, Wang Z P, Lu Y, Zhu C S 2008 Acta Phys. Sin 57 1084 Google Scholar
[21]	Yang C, Huang H B, Liu W B, Wang J S, Wang J, Jafri H M, Liu Y, Han G M, Song H F, Chen L Q 2021 Adv. Theor. Simul. 4 2000162 Google Scholar
[22]	Zhou F Y, Qiu K J, Bian D, Zheng Y F, Lin J P 2014 J. Mater. Sci. Technol. 30 299 Google Scholar
[23]	Dinsdale A T 1991 CALPHAD 15 317 Google Scholar
[24]	Aricó S F, Gribaudo L M, 1999 Scripta Mater. 41 159 Google Scholar
[25]	Wang C, Zinkevich M, Aldinger F 2004 CALPHAD 28 281 Google Scholar
[26]	Yin T, Lee J, Moosavi K E, Jung I H 2021 Ceram. Int. 47 29267 Google Scholar
[27]	Isomäki I, Hämäläinen M, Gierlotka W, Onderka B, Fitzner K 2006 J. Alloys Compd. 422 173 Google Scholar
[28]	Fang X R, Pan Z C, Chen A R, Tian H, Ma R J 2023 Eng. Fract. Mech. 281 109131 Google Scholar
[29]	Allen S M, Cahn J W 1979 Acta Metall. 27 1085 Google Scholar
[30]	Larché F C, Cahn J W 1985 Acta Metall. 33 331 Google Scholar
[31]	Cahn J W 1961 Acta Metall. 9 795 Google Scholar
[32]	Cox B, Pemsler J P 1968 J. Nucl. Mater. 28 73 Google Scholar
[33]	Ritchie I G, Atrens A 1977 J. Nucl. Mater. 67 254 Google Scholar
[34]	Keneshea F J, Douglass D L 1971 Oxid. Met. 3 1 Google Scholar
[35]	姜彦博, 柳文波, 孙志鹏, 喇永孝, 恽迪 2022 71 026103 Google Scholar Jiang Y B, Liu W B, Sun Z P, La Y X, Yun D 2022 Acta Phys. Sin 71 026103 Google Scholar
[36]	Qiu K, Wang R, Peng C, Lu X, Wang N 2015 CALPHAD 48 175 Google Scholar
[37]	Fisher E, Renken C 1964 Phys. Rev. 135 A482 Google Scholar
[38]	Fogaing E Y, Lorgouilloux Y, Huger M, Gault C P 2006 J. Mater. Sci. 41 7663 Google Scholar
[39]	Mirgorodsky A, Smirnov M, Quintard P 1997 Phys. Rev. B 55 19 Google Scholar
[40]	刘建章 2007 核结构材料 (北京: 化学工业出版社) 第89页 Liu J Z 2007 Nuclear Structure Material (Beijing: Chemical Industry Press) p89
[41]	Fan D, Chen L Q 1997 Acta Mater. 45 611 Google Scholar
[42]	Bi Y B, Zhang X L, Lu L, Xu Z F, Xie Z G, Chen B B, Liang Z X, Sun Z Q, Luo Z 2023 J. Mater. Res. Technol. 26 5888 Google Scholar
[43]	Gwinner B, Bataillon C, Chelagemdib R, Gruet N, Lorentz V, Puga B 2023 Electrochim. Acta 470 143334 Google Scholar
[44]	Zhang M H, Liu S D, Jiang J Y, Wei W C 2023 T Nonferr Metal. Soc. 33 1963 Google Scholar
[45]	Liu J L, Yu H B, Karamched P, Hu J, He G Z, Goran D, Hughes G M, Wilkinson A J, Sergio L P, Grovenor C R 2019 Acta Mater. 179 328 Google Scholar

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图 1 (a) 金属表面腐蚀过程及组织分布示意图; (b) 腐蚀过程相场序参量η及界面分布示意图

Figure 1. (a) Schematic diagram of metal surface corrosion process and microstructure distribution; (b) schematic diagram of phase field parameter η and interface distribution during metal surface corrosion.

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图 2 均匀腐蚀形貌演变的相场模拟结果

Figure 2. Microstructure evolution of uniform corrosion obtained by phase-field simulation.

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图 3 Zr合金均匀腐蚀氧化层增厚曲线及拟合结果

Figure 3. Kinetic curves of uniform corrosion.

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图 4 不同温度下Zr合金均匀腐蚀氧化层增长模拟结果

Figure 4. Simulated results of uniform corrosion at different temperatures.

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图 5 不同温度下Zr-2.5Sn合金表面氧化层增厚曲线

Figure 5. The corrosion kinetic curves of Zr-2.5Sn at different temperatures.

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图 6 不同温度下沿试样深度方向的O^2–分布

Figure 6. The distribution of O^2– at different temperatures.

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图 7 相场模型采用的多晶Zr-2.5Sn合金初始组织

Figure 7. The initial microstructure used by phase-field simulation.

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图 8 Zr-2.5Sn在673 K腐蚀不同时间后的相场模拟结果　(a) 腐蚀形貌; (b) O^2–分布

Figure 8. Corrosion of Zr-2.5Sn at 673 K with different times obtained by phase-field simulation: (a) Morphology of the crystalline after corrosion; (b) distribution of the O^2– concentration.

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图 9 不同温度下Zr-2.5Sn腐蚀25 d后的相场模拟结果　(a) 腐蚀形貌; (b) O^2–分布

Figure 9. Corrosion of Zr-2.5Sn after 25 days with different temperatures obtained by phase-field simulation: (a) Morphology of the crystalline; (b) distribution of the O^2– concentration.

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图 10 不同温度下多晶Zr-2.5Sn的腐蚀增厚曲线及与实验结果^[9]的对比

Figure 10. The corrosion kinetic curves of polycrystalline Zr-2.5Sn alloy and the experimental results ^[9].

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表 1 Zr-Sn合金自由能参数

Table 1. Free energy parameters of Zr-Sn alloys.

Parameter	Value	Reference
E_Zr/(J·m^–3)	$-7827.6 + 125.65T - 24.1618T\ln T - 4.38{\rm e}^{-3} T^3 + 34971T^{-1} $	[23]
E_Sn/(J·m^–3)	$2524.7 + 4.00T - 8.26T \lnT - 16.81{\rm e}^{-3} T^2 + 2.62 {\rm e}^{-6}T^3 - 1.08{\rm e}^6 T^{-1} $	[23]
E_O/(J·m^–3)	$-3480.9 - 25.50T - 11.13T\ln T - 5.10 {\rm e}^{-3}T^2 + 0.66 {\rm e}^{-6}T^3 - 38365T^{-1} $	[23]
A_ZrSn	$-148022.5+19.41T+(173681.9-22T)(c_{\rm Zr}-c_{\rm Sn}) + 104271.96(c_{\rm Zr}-c_{Sn})^2 $	[24]
A_ZrO	$-37876.66+17.2915T - 4471.4(c_{\rm Zr}-c_{\rm O}) $	[25]
A_SnO	140878 – 23.9326T	[26, 35]
$g_2^0$	$-1117869+420.3T - 69.6T\ln T - 0.003766T^2 + 702910T^{-1} +4.59 {\rm e}^{-21}T^7 $	[25]
$ g_3^0 $	0
k₂/(J·m^–3)	2.5 × 10⁵	[13]
k₃/(J·m^–3)	2.5 × 10⁵	[13]

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表 2 各温度下不同相中O^2–扩散系数 ^[33,34]

Table 2. Diffusion coefficients of O^2– in different phases at different temperatures ^[33,34].

Temperature/K	D₁/(m²·s^–1)	D₂/(m²·s^–1)	D₃/(m²·s^–1)	D_GB/(m²·s^–1)
633	3.99×10^–17	1.61×10^–17	2.98×10^–17	1.46×10^–16
653	1.26×10^–16	5.18×10^–17	8.77×10^–17	8.11×10^–16
673	3.09×10^–16	1.11×10^–16	2.25×10^–16	3.76×10^–15
693	1.04×10^–15	3.69×10^–16	7.64×10^–16	2.82×10^–14

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表 3 锆合金基体及氧化层的弹性模量分量

Table 3. Elastic modulus of Zr alloy and oxide layer.

Parameter	Value	Ref.	Parameter	Value	Ref.
$ {C}_{11}^{1} $	106.4 GPa	[37]	$ {C}_{11}^{2} $	395 GPa	[38, 39]
$ {C}_{12}^{1} $	84 GPa	[37]	$ {C}_{12}^{2} $	26 GPa	[38, 39]
$ {C}_{13}^{1} $	656 GPa	[37]	$ {C}_{13}^{2} $	105 GPa	[38, 39]
$ {C}_{66}^{1} $	10.5 GPa	[37]	$ {C}_{66}^{2} $	56 GPa	[38, 39]

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map

[1]	Kai J J, Huang W I, Chou H Y 1990 J. Nucl. Mater. 170 193 Google Scholar
[2]	Zhao W Z, Quan Q W, Zhang S M, Yang X Y, Wen H Y, Wu Y C, Liu X B, Cao X Z, Wang B Y 2023 Radiat. Phys. Chem. 209 110986 Google Scholar
[3]	Jones R H, Simonen E P 1994 Mater. Sci. Eng. 176 211 Google Scholar
[4]	Hu J, Liu J L, Lozano P S, Grovenor C R M, Christensen M, Wolf W, Wimmer E, Mader E V 2019 Acta Mater. 180 105 Google Scholar
[5]	Cann C D, So C B, Styles R C, Coleman C E 1993 J. Nucl. Mater. 205 267 Google Scholar
[6]	Wei T G, Dai X, Long C S, Sun C, Long S J, Zheng J Y, Wang P F, Jia Y Z, Zhang J S 2021 Corros. Sci. 192 109808 Google Scholar
[7]	Tian Z, Peng J C, Lin X D, Hu Y Y, Yao M Y, Xie Y P, Liang X, Zhou B X 2024 Corros. Sci. 202 111937 Google Scholar
[8]	Ferreirós P A, Polack E C S, Lanzani L A, Alonso P R, Quirós P D, Mieza J I, Rubiolo G H 2021 J. Nucl. Mater. 553 153039 Google Scholar
[9]	Jiang G Y, Xu D H, Yang W P, Liu L, Zhi Y W, Yang J Q 2022 Prog. Nucl. Energy 154 104490 Google Scholar
[10]	Yuan R, Xie Y P, Li T, Xu C H, Yao M Y, Xu J X, Guo H B, Zhou B X 2021 Acta Mater. 209 116804 Google Scholar
[11]	Mcdeavitt S M, Billings G W, Indacochea J E 2002 J. Mater. Sci. 37 3765 Google Scholar
[12]	Jerlerud P R, Toffolon M C, Joubert J M, Sundman B 2008 CALPHAD 32 593 Google Scholar
[13]	Asle Zaeem M, El Kadiri H 2014 Comput. Mater. Sci. 89 122 Google Scholar
[14]	张更, 王巧, 沙立婷, 李亚捷, 王达, 施思齐 2020 69 226401 Google Scholar Zhang G, Wang Q, Sha L T, Li Y J, Wang D, Shi S Q 2020 Acta Phys. Sin 69 226401 Google Scholar
[15]	Chen L Q, Zhao Y 2022 Prog. Mater. Sci. 124 100868 Google Scholar
[16]	Chen L Q 2002 Annu. Rev. Mater. Res. 32 113 Google Scholar
[17]	刘续希, 柳文波, 李博岩, 贺新福, 杨朝曦, 恽迪 2022 金属学报 58 943 Liu X X, Liu W B, Li B Y, He X F, Yang Z X, Yun D 2022 Acta Metal. Sin 58 943
[18]	Mai W, Soghrati S 2017 Corros. Sci. 125 87 Google Scholar
[19]	Fang X R, Pan Z C, Ma R J, Chen A R 2023 Comuput. Methods Appl. Mech. Engrg. 414 116196 Google Scholar
[20]	冯力, 王智平, 路阳, 朱昌盛 2008 57 1084 Google Scholar Feng L, Wang Z P, Lu Y, Zhu C S 2008 Acta Phys. Sin 57 1084 Google Scholar
[21]	Yang C, Huang H B, Liu W B, Wang J S, Wang J, Jafri H M, Liu Y, Han G M, Song H F, Chen L Q 2021 Adv. Theor. Simul. 4 2000162 Google Scholar
[22]	Zhou F Y, Qiu K J, Bian D, Zheng Y F, Lin J P 2014 J. Mater. Sci. Technol. 30 299 Google Scholar
[23]	Dinsdale A T 1991 CALPHAD 15 317 Google Scholar
[24]	Aricó S F, Gribaudo L M, 1999 Scripta Mater. 41 159 Google Scholar
[25]	Wang C, Zinkevich M, Aldinger F 2004 CALPHAD 28 281 Google Scholar
[26]	Yin T, Lee J, Moosavi K E, Jung I H 2021 Ceram. Int. 47 29267 Google Scholar
[27]	Isomäki I, Hämäläinen M, Gierlotka W, Onderka B, Fitzner K 2006 J. Alloys Compd. 422 173 Google Scholar
[28]	Fang X R, Pan Z C, Chen A R, Tian H, Ma R J 2023 Eng. Fract. Mech. 281 109131 Google Scholar
[29]	Allen S M, Cahn J W 1979 Acta Metall. 27 1085 Google Scholar
[30]	Larché F C, Cahn J W 1985 Acta Metall. 33 331 Google Scholar
[31]	Cahn J W 1961 Acta Metall. 9 795 Google Scholar
[32]	Cox B, Pemsler J P 1968 J. Nucl. Mater. 28 73 Google Scholar
[33]	Ritchie I G, Atrens A 1977 J. Nucl. Mater. 67 254 Google Scholar
[34]	Keneshea F J, Douglass D L 1971 Oxid. Met. 3 1 Google Scholar
[35]	姜彦博, 柳文波, 孙志鹏, 喇永孝, 恽迪 2022 71 026103 Google Scholar Jiang Y B, Liu W B, Sun Z P, La Y X, Yun D 2022 Acta Phys. Sin 71 026103 Google Scholar
[36]	Qiu K, Wang R, Peng C, Lu X, Wang N 2015 CALPHAD 48 175 Google Scholar
[37]	Fisher E, Renken C 1964 Phys. Rev. 135 A482 Google Scholar
[38]	Fogaing E Y, Lorgouilloux Y, Huger M, Gault C P 2006 J. Mater. Sci. 41 7663 Google Scholar
[39]	Mirgorodsky A, Smirnov M, Quintard P 1997 Phys. Rev. B 55 19 Google Scholar
[40]	刘建章 2007 核结构材料 (北京: 化学工业出版社) 第89页 Liu J Z 2007 Nuclear Structure Material (Beijing: Chemical Industry Press) p89
[41]	Fan D, Chen L Q 1997 Acta Mater. 45 611 Google Scholar
[42]	Bi Y B, Zhang X L, Lu L, Xu Z F, Xie Z G, Chen B B, Liang Z X, Sun Z Q, Luo Z 2023 J. Mater. Res. Technol. 26 5888 Google Scholar
[43]	Gwinner B, Bataillon C, Chelagemdib R, Gruet N, Lorentz V, Puga B 2023 Electrochim. Acta 470 143334 Google Scholar
[44]	Zhang M H, Liu S D, Jiang J Y, Wei W C 2023 T Nonferr Metal. Soc. 33 1963 Google Scholar
[45]	Liu J L, Yu H B, Karamched P, Hu J, He G Z, Goran D, Hughes G M, Wilkinson A J, Sergio L P, Grovenor C R 2019 Acta Mater. 179 328 Google Scholar

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