Phase-field simulation of sintering process of ceramic composite fuel

Liao Yu-Xuan; Shen Wen-Long; Wu Xue-Zhi; La Yong-Xiao; Liu Wen-Bo

doi:10.7498/aps.73.20241112

Abstract

Due to the limitation of existing experimental techniques, it is difficult to observe the evolution of microstructure in the sintering process in real time, resulting in a lack of in-depth understanding of the sintering mechanism of two-phase composite fuels. Therefore, it is greatly important to carry out theoretical simulation studies in the sintering process of composite fuels. In this work, a phase-field model of the two-phase sintering process of ceramic composite fuel is established, and the sintering process of UN-U₃Si₂ composite fuel is simulated by using this method. The simulation results show that during the formation of sintering neck, the surface deformation of the grains with higher surface energy is significant. The size of the final equilibrium dihedral angle formed by the two-phase double grains depends on the ratio of the grain boundary energy to the surface energy of the two phases. The phenomenon of large grains swallowing small grains does not occur between the two unequal double grains. Subsequently, the pore shrinkage and the properties of the trident grain boundary among the two-phase three grains are investigated in the sintering process. It is found that the angle of the trident grain boundary formed by the two-phase three grains deviates from 120°. The high-energy barrier at the grain boundary hinders the diffusion of the pore vacancies along the grain boundary, resulting in a slow shrinkage rate of the pore vacancies at the trident grain boundary. In addition, the simulation results of the microstructure evolution of two-phase polycrystalline sintered tissue with different volume fraction ratios show that the grain boundary diffusion plays a major role in the two-phase sintering process. The grain growth of the phase with a higher volume fraction is dominant, and there exists a hindrance to the migration of grain boundaries between two-phase grains. The phenomenon of grain migration exists between grains of the same phase.

Keywords:

Authors and contacts

1.
Institute of Nuclear Science and Technology, Xian Jiaotong University, Xi’an 710049, China
2.
China institute of atomic energy, Beijing 102413, China
3.
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 (NSAF Joint Fund) (Grant No. U2130105) and the Innovative Scientific Program of China National Nuclear Corporation (CNNC).

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References

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[2]	Kurata M 2016 Nucl. Eng. Technol. 48 26 Google Scholar
[3]	Leenaers A, Van den Berghe S, Koonen E, Jacquet P, Jarousse C, Guigon B, Ballagny A, Sannen L 2004 J. Nucl. Mater. 327 121 Google Scholar
[4]	Zinkle S J, Terrani K A, Gehin J C, Ott L J, Snead L L 2014 J. Nucl. Mater. 448 374 Google Scholar
[5]	Terrani K A, Wang D, Ott L J, Montgomery R O 2014 J. Nucl. Mater. 448 512 Google Scholar
[6]	Johnson K D, Raftery A M, Lopes D A, Wallenius J 2016 J. Nucl. Mater. 477 18 Google Scholar
[7]	Watkins J K, Gonzales A, Wagner A R, Sooby E S, Jaques B J 2021 J. Nucl. Mater. 553 153048 Google Scholar
[8]	Wood E S, White J T, Nelson A T 2017 J. Nucl. Mater 484 245 Google Scholar
[9]	Ortega L H, Blamer B J, Evans J A, McDeavitt S M 2016 J. Nucl. Mater. 471 116 Google Scholar
[10]	White J T, Travis A W, Dunwoody J T, Nelson A T 2017 J. Nucl. Mater. 495 463 Google Scholar
[11]	Lopes D A, Uygur S, Johnson K D 2017 J. Nucl. Sci. Technol. 54 405 Google Scholar
[12]	刘续希, 高士森, 喇永孝, 玉栋梁, 柳文波 2024 物理学报 73 148201 Google Scholar Liu X X, Gao S S, La Y X, Yu D L, Liu W B 2024 Acta Phys. Sin. 73 148201 Google Scholar
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[15]	刘明治, 张瑞杰, 方伟, 章书周, 曲选辉 2012 金属学报 48 1207 Google Scholar Liu M Z, Zhang R J, Fang W, Zhang S Z, Qu X H 2012 Acta. Metall. Sin. 48 1207 Google Scholar
[16]	Kumar V, Fang Z Z, Fife P C 2010 Mater. Sci. Eng. A 528 254 Google Scholar
[17]	Biswas S, Schwen D, Wang H, Okuniewski M, Tomar V 2018 Comput. Mater. Sci. 148 307 Google Scholar
[18]	Du L F, Yang S M, Zhu X W, Jiang J, Hui Q, Du H L 2018 J. Mater. Sci. 53 9567 Google Scholar
[19]	Hötzer J, Seiz M, Kellner M, Rheinheimer W, Nestler B 2019 Acta Mater. 164 184 Google Scholar
[20]	Wang Y U 2006 Acta Mater. 54 953 Google Scholar
[21]	Fan D, Chen L Q 1997 Acta Mater. 45 611 Google Scholar
[22]	Moelans N, Blanpain B, Wollants P 2008 Phys. Rev. B 78 024113 Google Scholar
[23]	Ahmed K, Yablinsky C A, Schulte A, Allen T, El-Azab A 2013 Modell. Simul. Mater. Sci. Eng. 21 065005 Google Scholar
[24]	Cahn J W 1961 Acta Metall. 9 795 Google Scholar
[25]	Allen S M, Cahn J W 1979 Acta Metall. 27 1085 Google Scholar
[26]	Biner S B 2017 Programming Phase-Field Modeling (Switzerland: Springer International Publishing) p18
[27]	Holt J B, Almassy M Y 1969 J. Am. Ceram. Soc. 52 631 Google Scholar
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[34]	Rahaman M N 1995 Ceramic Processing and Sintering (New York: Marcel Dekker) p446
[35]	Riedel H, Svoboda J 1993 Acta Metall. Mater. 41 1929 Google Scholar
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[37]	Ahmed K, Allen T, El-Azab A 2016 J. Mater. Sci. 51 1261 Google Scholar
[38]	Yadav V, Vanherpe L, Moelans N 2016 Comput. Mater. Sci. 125 297 Google Scholar

Cited By

图 1 相场模型示意图 (ρ, ϕ_α, ϕ_β —浓度场变量; η₁—晶粒1的取向场变量; η₂—晶粒2的取向场变量)

Figure 1. Schematic of phase field model (ρ, ϕ_α, ϕ_β —concentration field variable; η₁—orientation field variable of grain 1; η₂—orientation field variable of grain 2.)

DownLoad: Full-Size Img PowerPoint

图 2 两个不同相等大圆形晶粒演化的相场模拟　(a) 6 × 10⁴ $ \Delta t $; (b) 50 × 10⁴ $ \Delta t $; (c) 200 × 10⁴ $ \Delta t $; (d) 500 × 10⁴$ \Delta t $

Figure 2. Phase-field simulation of the evolution of two equal-sized circular grains with different phases: (a) 6 × 10⁴ $ \Delta t $; (b) 50 × 10⁴ $ \Delta t $; (c) 200 × 10⁴ $ \Delta t $; (d) 500 × 10⁴ $ \Delta t $.

DownLoad: Full-Size Img PowerPoint

图 3 不同相晶粒的烧结颈对数增长曲线 (l — 颈部长度, t — 时间步)

Figure 3. Logarithmic growth curves of sintering neck of different phase grains (l —neck length, t — time step).

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图 4 演化时间500 × 10⁴ $ \Delta t $((a)—(c))与1500 × 10⁴ $ \Delta t $((d)—(f))下的平衡二面角　(a), (d) 2个UN晶粒; (b), (e) 2个U₃Si₂晶粒; (c), (f) 1个UN晶粒和1个U₃Si₂晶粒

Figure 4. Equilibrium dihedral angles at evolution times 500 × 10⁴ $ \Delta t $((a)–(c)) and 1500 × 10⁴ $ \Delta t $((d)–(f)): (a), (d) Two UN grains; (b), (e) two U₃Si₂ grains; (c), (f) one UN grain and one U₃Si₂ grain.

DownLoad: Full-Size Img PowerPoint

图 5 两个不同相不等大圆形晶粒演化的相场模拟　(a) 2 × 10⁴ $ \Delta t $; (b) 50 × 10⁴ $ \Delta t $; (c) 100 × 10⁴ $ \Delta t $; (d) 500 × 10⁴ $ \Delta t $.

Figure 5. Phase-field simulation of the evolution of two unequal-sized circular grains with different phases: (a) 2 × 10⁴ $ \Delta t $; (b) 50 × 10⁴ $ \Delta t $; (c) 100 × 10⁴ $ \Delta t $; (d) 500 × 10⁴ $ \Delta t $.

DownLoad: Full-Size Img PowerPoint

图 6 晶粒面积与烧结颈尺寸的演化曲线

Figure 6. Evolution curves of grain area and sintering neck sizes.

DownLoad: Full-Size Img PowerPoint

图 7 四种不同相晶粒个数比例下3个晶粒烧结的三叉晶界　(a) 3个UN晶粒; (b) 2个UN晶粒和 1个U₃Si₂晶粒; (c) 1个UN晶粒和2个U₃Si₂晶粒; (d) 3个U₃Si₂晶粒

Figure 7. Trident grain boundaries of the sintering of three grains with four different phase grain number ratios: (a) Three UN grains; (b) two UN grains and one U₃Si₂ grain; (c) one UN grain and two U₃Si₂ grains; (d) three U₃Si₂ grains.

DownLoad: Full-Size Img PowerPoint

图 8 三叉晶界处气孔率的演化曲线

Figure 8. Evolution curves of the porosity at triple grain boundaries.

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图 9 不同体积分数比的两相多晶组织初始形貌　(a) 80% α-20% β; (b) 70% α-30% β; (c) 60% α-40% β; (d) 50% α-50% β

Figure 9. Initial morphology of two-phase polycrystalline structures with different volume fraction ratios: (a) 80% α-20% β; (b) 70% α-30% β; (c) 60% α-40% β; (d) 50% α-50% β.

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图 10 不同体积分数比的两相多晶组织最终形貌　(a) 80% α-20% β; (b) 70% α-30% β; (c) 60% α-40% β; (d) 50% α-50% β

Figure 10. Final morphology of two-phase polycrystalline structures with different volume fraction ratios: (a) 80% α-20% β; (b) 70% α-30% β; (c) 60% α-40% β; (d) 50% α-50% β.

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表 1 UN的物理参数^[27–29]

Table 1. Physical parameters of UN at 1823 K^[27–29]

Parameter	Value	Unit	Ref.
$ D_\alpha ^{\text{s}} $	7.5 × 10^–12	m²·s^–1	[27]
$ D_\alpha ^{{\text{gb}}} $	$ 0.01D_\alpha ^{\text{s}} $	m²·s^–1	[28]
$ \gamma _{\text{s}}^\alpha $	1.6	J·m^–2	[29]
$ \gamma _{{\text{gb}}}^\alpha $	0.8	J·m^–2	[29]
δ	6	nm	[29]

DownLoad: CSV

表 2 U₃Si₂的物理参数^[30–32]

Table 2. Physical parameters of U₃Si₂ at 1823 K^[30–32]

Parameter	Value	Unit	Ref.
$ D_\beta ^{\text{s}} $	100$ D_\beta ^{{\text{gb}}} $	m²·s^–1
$ D_\beta ^{{\text{gb}}} $	6.5923 × 10^–10	m²·s^–1	[30]
$ \gamma _{\text{s}}^\beta $	2.0	J·m^–2	[31]
$ \gamma _{{\text{gb}}}^\beta $	1.3	J·m^–2	[32]
δ	6	nm	[32]
Note: $ D_\alpha ^{\text{s}} $, $ D_\beta ^{\text{s}} $— surface diffusivity; $ D_\alpha ^{{\text{gb}}} $, $ D_\beta ^{{\text{gb}}} $— grain-boundary diffusivity; $ \gamma _{\text{s}}^\alpha $, $ \gamma _{\text{s}}^\beta $— surface energy; $ \gamma _{{\text{gb}}}^\alpha $, $ \gamma _{{\text{gb}}}^\beta $— grain-boundary energy; δ — diffuse interface width.

DownLoad: CSV

表 3 模拟中的无量纲参数表

Table 3. Non-dimensional parameters used in simulation.

Parameter	Value	Parameter	Value
$ \tilde A({\varphi _\alpha } = 1, {\text{ }}{\varphi _\beta } = 0) $	17	$ {\tilde \kappa _\eta } $	6.75
$ \tilde A({\varphi _\alpha } = 0, {\text{ }}{\varphi _\beta } = 1) $	11.5	$\tilde {\boldsymbol{M}}({\varphi _\alpha } = 1, {\text{ }}{\varphi _\beta } = 0)$	6817.5
$ \tilde B $	1	$\tilde {\boldsymbol{M}}({\varphi _\alpha } = 0, {\varphi _\beta } = 1)$	6817.5
$ {\tilde \kappa _\rho }({\varphi _\alpha } = 1, {\varphi _\beta } = 0) $	20.25	$ \tilde L $	1
$ {\tilde \kappa _\rho }({\varphi _\alpha } = 0, {\text{ }}{\varphi _\beta } = 1) $	14	$ \Delta x = \Delta y $	1
$ {\tilde \kappa _\phi }({\varphi _\alpha } = 1, {\text{ }}{\varphi _\beta } = 0) $	20.25	$ \Delta t $	2 × 10^–5
$ {\tilde \kappa _\phi }({\varphi _\alpha } = 0, {\text{ }}{\varphi _\beta } = 1) $	14
Note: $\tilde A$, $\tilde B$, $ {\tilde \kappa _\rho } $, $ {\tilde \kappa _\phi } $, $ {\tilde \kappa _\eta } $— non-dimensional parameters of free energy function; $ \tilde {\boldsymbol{M}} $— non-dimensional mobility; $\tilde L$—non-dimensional Allen-Cahn mobility; ∆x, ∆y — space scale; ∆t — time scale.

DownLoad: CSV

[1]	Kim H G, Yang J H, Kim W J, Koo Y H 2016 Nucl. Eng. Technol. 48 1 Google Scholar
[2]	Kurata M 2016 Nucl. Eng. Technol. 48 26 Google Scholar
[3]	Leenaers A, Van den Berghe S, Koonen E, Jacquet P, Jarousse C, Guigon B, Ballagny A, Sannen L 2004 J. Nucl. Mater. 327 121 Google Scholar
[4]	Zinkle S J, Terrani K A, Gehin J C, Ott L J, Snead L L 2014 J. Nucl. Mater. 448 374 Google Scholar
[5]	Terrani K A, Wang D, Ott L J, Montgomery R O 2014 J. Nucl. Mater. 448 512 Google Scholar
[6]	Johnson K D, Raftery A M, Lopes D A, Wallenius J 2016 J. Nucl. Mater. 477 18 Google Scholar
[7]	Watkins J K, Gonzales A, Wagner A R, Sooby E S, Jaques B J 2021 J. Nucl. Mater. 553 153048 Google Scholar
[8]	Wood E S, White J T, Nelson A T 2017 J. Nucl. Mater 484 245 Google Scholar
[9]	Ortega L H, Blamer B J, Evans J A, McDeavitt S M 2016 J. Nucl. Mater. 471 116 Google Scholar
[10]	White J T, Travis A W, Dunwoody J T, Nelson A T 2017 J. Nucl. Mater. 495 463 Google Scholar
[11]	Lopes D A, Uygur S, Johnson K D 2017 J. Nucl. Sci. Technol. 54 405 Google Scholar
[12]	刘续希, 高士森, 喇永孝, 玉栋梁, 柳文波 2024 物理学报 73 148201 Google Scholar Liu X X, Gao S S, La Y X, Yu D L, Liu W B 2024 Acta Phys. Sin. 73 148201 Google Scholar
[13]	刘东昆, 王庆宇, 张田, 周羽, 王翔 2024 物理学报 73 066102 Google Scholar Liu D K, Wang Q Y, Zhang T, Zhou Y, Wang X 2024 Acta Phys. Sin. 73 066102 Google Scholar
[14]	王凯乐, 杨文奎, 史新成, 侯华, 赵宇宏 2023 物理学报 72 076102 Google Scholar Wang K L, Yang W K, Shi X C, Hou H, Zhao Y H 2023 Acta Phys. Sin. 72 076102 Google Scholar
[15]	刘明治, 张瑞杰, 方伟, 章书周, 曲选辉 2012 金属学报 48 1207 Google Scholar Liu M Z, Zhang R J, Fang W, Zhang S Z, Qu X H 2012 Acta. Metall. Sin. 48 1207 Google Scholar
[16]	Kumar V, Fang Z Z, Fife P C 2010 Mater. Sci. Eng. A 528 254 Google Scholar
[17]	Biswas S, Schwen D, Wang H, Okuniewski M, Tomar V 2018 Comput. Mater. Sci. 148 307 Google Scholar
[18]	Du L F, Yang S M, Zhu X W, Jiang J, Hui Q, Du H L 2018 J. Mater. Sci. 53 9567 Google Scholar
[19]	Hötzer J, Seiz M, Kellner M, Rheinheimer W, Nestler B 2019 Acta Mater. 164 184 Google Scholar
[20]	Wang Y U 2006 Acta Mater. 54 953 Google Scholar
[21]	Fan D, Chen L Q 1997 Acta Mater. 45 611 Google Scholar
[22]	Moelans N, Blanpain B, Wollants P 2008 Phys. Rev. B 78 024113 Google Scholar
[23]	Ahmed K, Yablinsky C A, Schulte A, Allen T, El-Azab A 2013 Modell. Simul. Mater. Sci. Eng. 21 065005 Google Scholar
[24]	Cahn J W 1961 Acta Metall. 9 795 Google Scholar
[25]	Allen S M, Cahn J W 1979 Acta Metall. 27 1085 Google Scholar
[26]	Biner S B 2017 Programming Phase-Field Modeling (Switzerland: Springer International Publishing) p18
[27]	Holt J B, Almassy M Y 1969 J. Am. Ceram. Soc. 52 631 Google Scholar
[28]	戚晓勇, 柳文波, 何宗倍, 王一帆, 恽迪 2023 金属学报 59 1513 Google Scholar Qi X Y, Liu W B, He Z B, Wang Y F, Yun D 2023 Acta Metall. Sin. 59 1513 Google Scholar
[29]	Bocharov D, Gryaznov D, Zhukovskii Y F, Kotomin E A 2013 J. Nucl. Mater. 435 102 Google Scholar
[30]	Cooper M W D, Gamble K A, Capolungo L, Matthews C, Andersson D A, Beeler B, Stanek C R, Metzger K 2021 J. Nucl. Mater. 555 153129 Google Scholar
[31]	Beeler B, Baskes M, Andersson D, Cooper M W D, Zhang Y F 2019 J. Nucl. Mater. 514 290 Google Scholar
[32]	Cheniour A, Tonks M R, Gong B, Yao T K, He L F, Harp J M, Beeler B, Zhang Y F, Lian J 2020 J. Nucl. Mater 532 152069 Google Scholar
[33]	Chockalingam K, Kouznetsova V G, van der Sluis O, Geers M G D 2016 Comput. Methods Appl. Mech. Eng. 312 492 Google Scholar
[34]	Rahaman M N 1995 Ceramic Processing and Sintering (New York: Marcel Dekker) p446
[35]	Riedel H, Svoboda J 1993 Acta Metall. Mater. 41 1929 Google Scholar
[36]	孙正阳, 杨超, 柳文波 2020 金属学报 56 1295 Google Scholar Sun Z Y, Yang C, Liu W B 2020 Acta Metall. Sin. 56 1295 Google Scholar
[37]	Ahmed K, Allen T, El-Azab A 2016 J. Mater. Sci. 51 1261 Google Scholar
[38]	Yadav V, Vanherpe L, Moelans N 2016 Comput. Mater. Sci. 125 297 Google Scholar

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