A review: applications of the phase field method in predicting microstructure and property evolution of irradiated nuclear materials
Source abstract
Abstract
Abstract Complex microstructure changes occur in nuclear fuel and structural materials due to the extreme environments of intense irradiation and high temperature. This paper evaluates the role of the phase field method in predicting the microstructure evolution of irradiated nuclear materials and the impact on their mechanical, thermal, and magnetic properties. The paper starts with an overview of the important physical mechanisms of defect evolution and the significant gaps in simulating microstructure evolution in irradiated nuclear materials. Then, the phase field method is introduced as a powerful and predictive tool and its applications to microstructure and property evolution in irradiated nuclear materials are reviewed. The review shows that (1) Phase field models can correctly describe important phenomena such as spatial-dependent generation, migration, and recombination of defects, radiation-induced dissolution, the Soret effect, strong interfacial energy anisotropy, and elastic interaction; (2) The phase field method can qualitatively and quantitatively simulate two-dimensional and three-dimensional microstructure evolution, including radiation-induced segregation, second phase nucleation, void migration, void and gas bubble superlattice formation, interstitial loop evolution, hydrate formation, and grain growth, and (3) The Phase field method correctly predicts the relationships between microstructures and properties. The final section is dedicated to a discussion of the strengths and limitations of the phase field method, as applied to irradiation effects in nuclear materials.
Source: crossref · Quality: baseline
Research context
OpenAlex topics: Nuclear Materials and Properties · Fusion materials and technologies · Nuclear materials and radiation effects
SciVX discovery tags are terminology-based. OpenAlex topics are source-provided classifications.
References (100+)
Show source-provided references
1. Little, E. A. Void-swelling in irons and ferritic steels.1. Mechanisms of swelling suppression. J. Nucl. Mater. 87, 11–24 (1979).10.1016/0022-3115(79)90122-3 ↗
2. Little, E. A. & Stow, D. A. Void-swelling in irons and ferritic steels. 2. Experimental survey of materials irradiated in a fast-reactor. J. Nucl. Mater. 87, 25–39 (1979).10.1016/0022-3115(79)90123-5 ↗
3. Marwick, A. D. Segregation in irradiated alloys - inverse Kirkendall effect and effect of constitution on void swelling. J. Phys. F-Met. Phys. 8, 1849–1861 (1978).10.1088/0305-4608/8/9/008 ↗
4. Mansur, L. K. Void swelling in metals and alloys under irradiation - assessment of theory. Nucl. Technol. 40, 5–34 (1978).10.13182/NT78-2 ↗
5. Singh, B. N., Eldrup, M., Zinkle, S. J. & Golubov, S. I. On grain-size-dependent void swelling in pure copper irradiated with fission neutrons. Philos. Mag. A 82, 1137–1158 (2002).10.1080/01418610208240021 ↗
6. Terentyev, D., Haghighat, S. M. H. & Schaublin, R. Strengthening due to Cr-rich precipitates in Fe-Cr alloys: effect of temperature and precipitate composition. J. Appl. Phys. 107, 061806 (2010).10.1063/1.3340522 ↗
7. Garner, F. A., Toloczko, M. B. & Sencer, B. H. Comparison of swelling and irradiation creep behavior of fcc-austenitic and bcc-ferritic/martensitic alloys at high neutron exposure. J. Nucl. Mater. 276, 123–142 (2000).10.1016/S0022-3115(99)00225-1 ↗
8. Scott, P. A review of irradiation assisted stress-corrosion cracking. J. Nucl. Mater. 211, 101–122 (1994).10.1016/0022-3115(94)90360-3 ↗
9. Was, G. S. et al. Corrosion and stress corrosion cracking in supercritical water. J. Nucl. Mater. 371, 176–201 (2007).10.1016/j.jnucmat.2007.05.017 ↗
10. Tan, L., Ren, X., Sridharan, K. & Allen, T. R. Corrosion behavior of Ni-base alloys for advanced high temperature water-cooled nuclear plants. Corros. Sci. 50, 3056–3062 (2008).10.1016/j.corsci.2008.08.024 ↗
11. Okada, O., Nakata, K. & Kasahara, S. Effects of thermal sensitization on radiation-induced segregation in type 304 stainless steel irradiated with He-ions. J. Nucl. Mater. 265, 232–239 (1999).10.1016/S0022-3115(98)00733-8 ↗
12. Wilkes, P. Phase-stability under irradiation—review of theory and experiment. J. Nucl. Mater. 83, 166–175 (1979).10.1016/0022-3115(79)90602-0 ↗
13. Maydet, S. I. & Russell, K. C. Precipitate stability under irradiation—point-defect effects. J. Nucl. Mater. 64, 101–114 (1977).10.1016/0022-3115(77)90013-7 ↗
14. Turnbull, J. A. & Cornell, R. M. Observations demonstrating re-solution of gas from bubbles and sintering pores during irradiation of uo2 at a high temperature. J. Nucl. Mater. 37, 355–357 (1970).10.1016/0022-3115(70)90170-4 ↗
15. Olander, D. R. & Wongsawaeng, D. Re-solution of fission gas—a review: Part I. Intragranular bubbles. J. Nucl. Mater. 354, 94–109 (2006).10.1016/j.jnucmat.2006.03.010 ↗
16. Johnson, P. B. & Mazey, D. J. Gas-bubble superlattice formation in bcc metals. J. Nucl. Mater. 218, 273–288 (1995).10.1016/0022-3115(94)00674-1 ↗
17. Trinkaus, H. & Singh, B. N. Helium accumulation in metals during irradiation—where do we stand? J. Nucl. Mater. 323, 229–242 (2003).10.1016/j.jnucmat.2003.09.001 ↗
18. Zacharie, I. et al. Thermal treatment of uranium oxide irradiated in pressurized water reactor: swelling and release of fission gases. J. Nucl. Mater. 255, 85–91 (1998).10.1016/S0022-3115(98)00039-7 ↗
19. Dvoriashin, A. M. et al. Mechanical properties and microstructure of three Russian ferritic/martensitic steels irradiated in BN-350 reactor to 50 dpa at 490 degrees C. J. Nucl. Mater. 367, 92–96 (2007).10.1016/j.jnucmat.2007.03.161 ↗
20. Braski, D. N., Schroeder, H. & Ullmaier, H. Effect of tensile-stress on the growth of helium bubbles in an austenitic stainless-steel. J. Nucl. Mater. 83, 265–277 (1979).10.1016/0022-3115(79)90611-1 ↗
21. Zacharie, I. et al. Microstructural analysis and modelling of intergranular swelling of an irradiated UO2 fuel treated at high temperature. J. Nucl. Mater. 255, 92–104 (1998).10.1016/S0022-3115(98)00040-3 ↗
22. Trinkaus, H., Singh, B. N. & Foreman, A. J. E. Impact of glissile interstitial loop production in cascades on defect accumulation in the transient. J. Nucl. Mater. 206, 200–211 (1993).10.1016/0022-3115(93)90124-H ↗
23. Bacon, D. J., Gao, F. & Osetsky, Y. N. The primary damage state in fcc, bcc and hcp metals as seen in molecular dynamics simulations. J. Nucl. Mater. 276, 1–12 (2000).10.1016/S0022-3115(99)00165-8 ↗
24. Osetsky, Y. N., Bacon, D. J., Serra, A., Singh, B. N. & Golubov, S. I. Stability and mobility of defect clusters and dislocation loops in metals. J. Nucl. Mater. 276, 65–77 (2000).10.1016/S0022-3115(99)00170-1 ↗
25. Becquart, C. S. et al. Effect of displacement cascade structure and defect mobility on the growth of point defect clusters under irradiation. J. Nucl. Mater. 351, 39–46 (2006).10.1016/j.jnucmat.2006.02.022 ↗
26. Vortler, K., Bjorkas, C., Terentyev, D., Malerba, L. & Nordlund, K. The effect of Cr concentration on radiation damage in Fe-Cr alloys. J. Nucl. Mater. 382, 24–30 (2008).10.1016/j.jnucmat.2008.09.007 ↗
27. Domain, C. & Legris, A. Ab initio atomic-scale determination of point-defect structure in hcp zirconium. Philos. Mag. A 85, 569–575 (2005).10.1080/14786430412331334625 ↗
28. Klaver, T. P. C., Olsson, P. & Finnis, M. W. Interstitials in FeCr alloys studied by density functional theory. Phys. Rev. B 76, 214110 (2007).10.1103/PhysRevB.76.214110 ↗
29. Klaver, T. P. C., Hepburn, D. J. & Ackland, G. J. Defect and solute properties in dilute Fe-Cr-Ni austenitic alloys from first principles. Phys. Rev. B 85, 174111 (2012).10.1103/PhysRevB.85.174111 ↗
30. Dudarev, S. L. Density functional theory models for radiation damage. Ann. Rev. Mater. Res. 43, 35–61 (2013).10.1146/annurev-matsci-071312-121626 ↗
31. Soneda, N., Ishino, S., Takahashi, A. & Dohi, K. Modeling the microstructural evolution in bcc-Fe during irradiation using kinetic Monte Carlo computer simulation. J. Nucl. Mater. 323, 169–180 (2003).10.1016/j.jnucmat.2003.08.021 ↗
32. Caturla, M. J., Soneda, N., de la Rubia, T. D. & Fluss, M. Kinetic Monte Carlo simulations applied to irradiated materials: the effect of cascade damage in defect nucleation and growth. J. Nucl. Mater. 351, 78–87 (2006).10.1016/j.jnucmat.2006.02.019 ↗
33. Heinisch, H. L. Computer-simulation of high-energy displacement cascades. Radiat. Eff. Defects Solids 113, 53–73 (1990).10.1080/10420159008213055 ↗
34. Torre, J. D., Fu, C. C., Willaime, F., Barbu, A. & Bocquet, J. L. Resistivity recovery simulations of electron-irradiated iron: Kinetic Monte Carlo versus cluster dynamics. J. Nucl. Mater. 352, 42–49 (2006).10.1016/j.jnucmat.2006.02.040 ↗
35. Domain, C., Becquart, C. S. & Malerba, L. Simulation of radiation damage in Fe alloys: an object kinetic Monte Carlo approach. J. Nucl. Mater. 335, 121–145 (2004).10.1016/j.jnucmat.2004.07.037 ↗
36. Woo, C. H. & Singh, B. N. Production bias due to clustering of point-defects in irradiation-induced cascades. Philos. Mag. A 65, 889–912 (1992).10.1080/01418619208205596 ↗
37. Surh, M. P., Sturgeon, J. B. & Wolfer, W. G. Master equation and Fokker-Planck methods for void nucleation and growth in irradiation swelling. J. Nucl. Mater. 325, 44–52 (2004).10.1016/j.jnucmat.2003.10.013 ↗
38. Xu, D. H., Wirth, B. D., Li, M. M. & Kirk, M. A. Defect microstructural evolution in ion irradiated metallic nanofoils: Kinetic Monte Carlo simulation versus cluster dynamics modeling and in situ transmission electron microscopy experiments. Appl. Phys. Lett. 101, 101905 (2012).10.1063/1.4748980 ↗
39. Xu, D. H., Wirth, B. D., Li, M. M. & Kirk, M. A. Combining in situ transmission electron microscopy irradiation experiments with cluster dynamics modeling to study nanoscale defect agglomeration in structural metals. Acta Mater. 60, 4286–4302 (2012).10.1016/j.actamat.2012.03.055 ↗
40. Surh, M. P., Sturgeon, J. B. & Wolfer, W. G. Radiation swelling behavior and its dependence on temperature, dose rate and dislocation structure evolution. J. Nucl. Mater. 341, 235–236 (2005). vol 336, pg 217, 2004.10.1016/j.jnucmat.2005.03.002 ↗
41. Ortiz, C. J., Caturla, M. J., Fu, C. C. & Willaime, F. He diffusion in irradiated alpha-Fe: an ab-initio-based rate theory model. Phys. Rev. B 75, 100102 (2007).10.1103/PhysRevB.75.100102 ↗
42. Brailsford, A. D. & Bullough, R. The rate theory of swelling due to void growth in irradiated metals. J. Nucl. Mater. 44, 121–135 (1972).10.1016/0022-3115(72)90091-8 ↗
43. Singh, B. N. & Zinkle, S. J. Defect accumulation in pure fcc metals in the transient regime—a review. J. Nucl. Mater. 206, 212–229 (1993).10.1016/0022-3115(93)90125-I ↗
44. Fu, C. C., Dalla Torre, J., Willaime, F., Bocquet, J. L. & Barbu, A. Multiscale modelling of defect kinetics in irradiated iron. Nat. Mater. 4, 68–74 (2005).10.1038/nmat1286 ↗
45. Stan, M. Discovery and design of nuclear fuels. Mater. Today 12, 20–28 (2009).10.1016/S1369-7021(09)70295-0 ↗
46. Chen, L. Q. Phase-field models for microstructure evolution. Ann. Rev. Mater. Res. 32, 113–140 (2002).10.1146/annurev.matsci.32.112001.132041 ↗
47. Granasy, L., Pusztai, T. & Warren, J. A. Modelling polycrystalline solidification using phase field theory. J. Phys. -Condes. Matter. 16, R1205–R1235 (2004).10.1088/0953-8984/16/41/R01 ↗
48. Moelans, N., Blanpain, B. & Wollants, P. An introduction to phase-field modeling of microstructure evolution. Calphad 32, 268–294 (2008).10.1016/j.calphad.2007.11.003 ↗
49. Steinbach, I. Phase-field models in materials science. Modell. Simul. Mater. Sci. Eng. 17, 073001 (2009).10.1088/0965-0393/17/7/073001 ↗
50. Steinbach, I. Phase-field model for microstructure evolution at the mesoscopic scale. Ann. Rev. Mater. Res. 43, 89–107 (2013).10.1146/annurev-matsci-071312-121703 ↗
51. Wang, Y. Z. & Li, J. Phase field modeling of defects and deformation. Acta Mater. 58, 1212–1235 (2010).10.1016/j.actamat.2009.10.041 ↗
52. Mamivand, M., Zaeem, M. A. & El Kadiri, H. A review on phase field modeling of martensitic phase transformation. Comput. Mater. Sci. 77, 304–311 (2013).10.1016/j.commatsci.2013.04.059 ↗
53. Karma, A. & Rappel, W. J. Quantitative phase-field modeling of dendritic growth in two and three dimensions. Phys. Rev. E 57, 4323–4349 (1998).10.1103/PhysRevE.57.4323 ↗
54. Stan, M. et al. Models and simulations of nuclear fuel materials properties. J. Alloy. Compd. 444, 415–423 (2007).10.1016/j.jallcom.2007.01.102 ↗
55. Hu, S. et al. Phase-field modeling of gas bubbles and thermal conductivity evolution in nuclear fuels. J. Nucl. Mater. 392, 292–300 (2009).10.1016/j.jnucmat.2009.03.017 ↗
56. Hu, S. et al. Application of the phase-field method in predicting gas bubble microstructure evolution in nuclear fuels. Int. J. Mater. Res. 101, 515–522 (2010).10.3139/146.110304 ↗
57. Millett, P. C. & Tonks, M. Phase-field simulations of gas density within bubbles in metals under irradiation. Comput. Mater. Sci. 50, 2044–2050 (2011).10.1016/j.commatsci.2011.02.006 ↗
58. Millett, P. C., El-Azab, A. & Wolf, D. Phase-field simulation of irradiated metals Part II: gas bubble kinetics. Comput. Mater. Sci. 50, 960–970 (2011).10.1016/j.commatsci.2010.10.032 ↗
59. Millett, P. C., Tonks, M. & Biner, S. B. Mesoscale modeling of intergranular bubble percolation in nuclear fuels. J. Appl. Phys. 111, 083511 (2012).10.1063/1.3702872 ↗
60. Millett, P. C. et al. Phase-field simulation of intergranular bubble growth and percolation in bicrystals. J. Nucl. Mater. 425, 130–135 (2012).10.1016/j.jnucmat.2011.07.034 ↗
61. Li, Y., Hu, S., Montgomery, R., Gao, F. & Sun, X. Phase-field simulations of intragranular fission gas bubble evolution in UO2 under post-irradiation thermal annealing. Nucl. Instrum. Methods Phys. Res. B. 303, 62–67 (2013).10.1016/j.nimb.2012.11.028 ↗
62. Hu, S. et al. Formation mechanism of gas bubble superlattice in UMo metal fuels: phase-field modeling investigation. J. Nucl. Mater. 479, 202–215 (2016).10.1016/j.jnucmat.2016.07.012 ↗
63. Rokkam, S., El-Azab, A., Millett, P. & Wolf, D. Phase field modeling of void nucleation and growth in irradiated metals. Modell. Simul. Mater. Sci. Eng. 17, 064002 (2009).10.1088/0965-0393/17/6/064002 ↗
64. Millett, P. C., Rokkam, S., El-Azab, A., Tonks, M. & Wolf, D. Void nucleation and growth in irradiated polycrystalline metals: a phase-field model. Modell. Simul. Mater. Sci. Eng. 17, 064003 (2009).10.1088/0965-0393/17/6/064003 ↗
65. Li, Y. et al. Phase-field modeling of void evolution and swelling in materials under irradiation. Sci. China-Phys. Mech. Astron 54, 856–865 (2011).10.1007/s11433-011-4316-y ↗
66. Millett, P. C., El-Azab, A., Rokkam, S., Tonks, M. & Wolf, D. Phase-field simulation of irradiated metals Part I: Void kinetics. Comput. Mater. Sci. 50, 949–959 (2011).10.1016/j.commatsci.2010.10.034 ↗
67. Semenov, A. A. & Woo, C. H. Interfacial energy in phase-field emulation of void nucleation and growth. J. Nucl. Mater. 411, 144–149 (2011).10.1016/j.jnucmat.2011.01.100 ↗
68. Semenov, A. A. & Woo, C. H. Phase-field modeling of void formation and growth under irradiation. Acta Mater. 60, 6112–6119 (2012).10.1016/j.actamat.2012.07.049 ↗
69. Xiao, Z. H., Semenov, A. A., Woo, C. H. & Shi, S. Q. Single void dynamics in phase field modeling. J. Nucl. Mater. 439, 25–32 (2013).10.1016/j.jnucmat.2013.03.076 ↗
70. Wang, N., Rokkam, S., Hochrainer, T., Pernice, M. & El-Azab, A. Asymptotic and uncertainty analyses of a phase field model for void formation under irradiation. Comput. Mater. Sci. 89, 165–175 (2014).10.1016/j.commatsci.2014.03.045 ↗
71. El-Azab, A., Ahmed, K., Rokkam, S. & Hochrainer, T. Diffuse interface modeling of void growth in irradiated materials. Mathematical, thermodynamic and atomistic perspectives. Curr. Opin. Solid State Mater. Sci. 18, 90–98 (2014).10.1016/j.cossms.2014.01.002 ↗
72. Semenov, A. A. & Woo, C. H. Modeling void development in irradiated metals in the phase-field framework. J. Nucl. Mater. 454, 60–68 (2014).10.1016/j.jnucmat.2014.07.031 ↗
73. Hochrainer, T. & El-Azab, A. A sharp interface model for void growth in irradiated materials. Philos. Mag. 95, 948–972 (2015).10.1080/14786435.2015.1009516 ↗
74. Hu, S. & Henager, C. H. Jr Phase-field modeling of void lattice formation under irradiation. J. Nucl. Mater. 394, 155–159 (2009).10.1016/j.jnucmat.2009.09.002 ↗
75. Hu, S. Y. & Henager, C. H. Jr Phase-field simulation of void migration in a temperature gradient. Acta Mater. 58, 3230–3237 (2010).10.1016/j.actamat.2010.01.043 ↗
76. Li, Y. et al. Phase-field modeling of void migration and growth kinetics in materials under irradiation and temperature field. J. Nucl. Mater. 407, 119–125 (2010).10.1016/j.jnucmat.2010.09.048 ↗
77. Zhang, L. Z. et al. Phase-field modeling of temperature gradient driven pore migration coupling with thermal conduction. Comput. Mater. Sci. 56, 161–165 (2012).10.1016/j.commatsci.2012.01.002 ↗
78. Hu, S. et al. Evolution kinetics of interstitial loops in irradiated materials: a phase-field model. Modell. Simul. Mater. Sci. Eng. 20, 015011 (2012).10.1088/0965-0393/20/1/015011 ↗
79. Li, Y. et al. Computer simulations of interstitial loop growth kinetics in irradiated bcc Fe. J. Nucl. Mater. 427, 259–267 (2012).10.1016/j.jnucmat.2012.05.004 ↗
80. Rouchette, H., Thuinet, L., Legris, A., Ambard, A. & Domain, C. Quantitative phase field model for dislocation sink strength calculations. Comput. Mater. Sci. 88, 50–60 (2014).10.1016/j.commatsci.2014.02.011 ↗
81. Ma, X. Q., Shi, S. Q., Woo, C. H. & Chen, L. Q. Phase-field simulation of hydride precipitation in bi-crystalline zirconium. Scr. Mater. 47, 237–241 (2002).10.1016/S1359-6462(02)00131-8 ↗
82. Ma, X. Q., Shi, S. Q., Woo, C. H. & Chen, L. Q. Effect of applied load on nucleation and growth of gamma-hydrides in zirconium. Comput. Mater. Sci. 23, 283–290 (2002).10.1016/S0927-0256(01)00226-9 ↗
83. Ma, X. Q., Shi, S. Q., Woo, C. H. & Chen, L. Q. Simulation of gamma-hydride precipitation in bi-crystalline zirconium under uniformly applied load. Mater. Sci. Eng. A-Struct. Mater. Prop. Microstruct. Process. 334, 6–10 (2002).10.1016/S0921-5093(01)01770-1 ↗
84. Ma, X. Q., Shi, S. Q., Woo, C. H. & Chen, L. Q. The phase field model for hydrogen diffusion and gamma-hydride precipitation in zirconium under non-uniformly applied stress. Mech. Mater. 38, 3–10 (2006).10.1016/j.mechmat.2005.05.005 ↗
85. Guo, X. H., Shi, S. Q., Zhang, Q. M. & Ma, X. Q. An elastoplastic phase-field model for the evolution of hydride precipitation in zirconium. Part I: smooth specimen. J. Nucl. Mater. 378, 110–119 (2008).10.1016/j.jnucmat.2008.05.008 ↗
86. Guo, X. H., Shi, S. Q., Zhang, Q. M. & Ma, X. Q. An elastoplastic phase-field model for the evolution of hydride precipitation in zirconium. Part II: specimen with flaws. J. Nucl. Mater. 378, 120–125 (2008).10.1016/j.jnucmat.2008.05.006 ↗
87. Zhao, Z. et al. Characterization of zirconium hydrides and phase field approach to a mesoscopic-scale modeling of their precipitation. J. ASTM Int. 5, 31–51 (2008).
88. Li, Y. L., Hu, S. Y., Zhang, L. & Sun, X. Non-classical nuclei and growth kinetics of Cr precipitates in FeCr alloys during aging. Modelling Simul. Mater. Sci. Eng. 22, 025002 (2014).10.1088/0965-0393/22/2/025002 ↗
89. Badillo, A., Bellon, P. & Averback, R. S. A phase field model for segregation and precipitation induced by irradiation in alloys. Modelling Simul. Mater. Sci. Eng. 23, 035008 (2015).10.1088/0965-0393/23/3/035008 ↗
90. Biner, S. B., Rao, W. F. & Zhang, Y. F. The stability of precepitates and the role of lattice defects in Fe-1at% Cu-1at%Ni-1at%Mn alloy: a phase-field model study. J. Nucl. Mater. 468, 9–16 (2016).10.1016/j.jnucmat.2015.10.062 ↗
91. Millett, P. C., Wolf, D., Desai, T., Rokkam, S. & El-Azab, A. Phase-field simulation of thermal conductivity in porous polycrystalline microstructures. J. Appl. Phys. 104, 033512 (2008).10.1063/1.2964116 ↗
92. Ahmed, K., Pakarinen, J., Allen, T. & El-Azab, A. Phase field simulation of grain growth in porous uranium dioxide. J. Nucl. Mater. 446, 90–99 (2014).10.1016/j.jnucmat.2013.11.036 ↗
93. Tonks, M. R., Zhang, Y., Bai, X. & Millett, P. C. Demonstrating the temperature gradient impact on grain growth in uo2 using the phase field method. Mater. Res. Lett. 2, 23–28 (2014).10.1080/21663831.2013.849300 ↗
94. Mei, Z. G. et al. Grain growth in U-7Mo alloy: a combined first-principles and phase field study. J. Nucl. Mater. 473, 300–308 (2016).10.1016/j.jnucmat.2016.01.027 ↗
95. Liang, L. et al. Mesoscale model for fission-induced recrystallization in U-7Mo alloy. Comput. Mater. Sci. 124, 228–237 (2016).10.1016/j.commatsci.2016.07.033 ↗
96. Millett, P. C. & Tonks, M. Application of phase-field modeling to irradiation effects in materials. Curr. Opin. Solid State Mater. Sci 15, 125–133 (2011).10.1016/j.cossms.2010.10.002 ↗
97. Hu, S. Y., Burkes, D., Lavender, C. A. & Joshi, V. Effect of grain morphology on gas bubble swelling in UMo fuels—a 3D microstructure dependent booth model. J. Nucl. Mater. 480, 323–331 (2016).10.1016/j.jnucmat.2016.08.038 ↗
98. Chang, K., Lee, G. G. & Kwon, J. A phase-field modeling of void swelling in the Austenitic stainless steel. Radiat. Eff. Defects Solids 171, 242–251 (2016).10.1080/10420150.2016.1179304 ↗
99. Welland, M. J., Lewis, B. J. & Thompson, W. T. A comparison of Stefan and phase field modeling techniques for the simulation of melting nuclear fuel. J. Nucl. Mater. 376, 229–239 (2008).10.1016/j.jnucmat.2008.03.003 ↗
100. Welland, M. J., Thompson, W. T., Lewis, B. J. & Manara, D. Computer simulations of non-congruent melting of hyperstoichiometric uranium dioxide. J. Nucl. Mater. 385, 358–363 (2009).10.1016/j.jnucmat.2008.12.023 ↗