REFERENCES
1. Janssens, K. G. F.; Raabe, D.; Kozeschnik, E.; Miodownik, M.; Nestler, B. Computational materials engineering: an introduction to microstructure evolution. Burlington: Elsevier Academic Press; 2007. Available from: https://www.sciencedirect.com/book/9780123694683/computational-materials-engineering [Last accessed on 25 Apr 2025]
2. Bulatov, V. V.; Cai, W. Computer simulations of dislocations. New York: Oxford University Press; 2006, pp. 241.
3. Ginzburg, V. L.; Landau, L. D. 73 - On the theory of superconductivity. In: Collected Papers of L.D. Landau; 1965, pp. 546-68.
4. Cahn, J. W.; Hilliard, J. E. Free energy of a nonuniform system. I. interfacial free energy. J. Chem. Phys. 1958, 28, 258-67.
5. Collins, J. B.; Levine, H. Diffuse interface model of diffusion-limited crystal growth. Phys. Rev. B. Condens. Matter. 1985, 31, 6119-22.
6. Emmerich, H. The diffuse interface approach in materials science: thermodynamic concepts and applications of phase-field models. New York: Springer; 2003.
7. Chen, L.; Moelans, N. Phase-field method of materials microstructures and properties. MRS. Bull. 2024, 49, 551-5.
8. Zhao, Y. Understanding and design of metallic alloys guided by phase-field simulations. NPJ. Comput. Mater. 2023, 9, 1038.
9. Zhuang, X.; Zhou, S.; Huynh, G.; Areias, P.; Rabczuk, T. Phase field modeling and computer implementation: a review. Eng. Fract. Mech. 2022, 262, 108234.
10. Tourret, D.; Liu, H.; Llorca, J. Phase-field modeling of microstructure evolution: recent applications, perspectives and challenges. Prog. Mater. Sci. 2022, 123, 100810.
11. Tonks, M. R.; Aagesen, L. K. The phase field method: mesoscale simulation aiding material discovery. Annu. Rev. Mater. Res. 2019, 49, 79-102.
12. Emmerich, H. Advances of and by phase-field modelling in condensed-matter physics. Adv. Phys. 2008, 57, 1-87.
13. Boettinger, W. J.; Warren, J. A.; Beckermann, C.; Karma, A. Phase-field simulation of solidification. Annu. Rev. Mater. Res. 2002, 32, 163-94.
14. Chen, L. Phase-field models for microstructure evolution. Annu. Rev. Mater. Res. 2002, 32, 113-40.
16. Wang, F.; Altschuh, P.; Matz, A. M.; et al. Phase-field study on the growth of magnesium silicide occasioned by reactive diffusion on the surface of Si-foams. Acta. Mater. 2019, 170, 138-54.
17. Uddagiri, M.; Tegeler, M.; Steinbach, I. Interface stabilization and propagation in phase field models of solidification: resolving the issue of large driving forces. Model. Simul. Mater. Sci. Eng. 2024, 32, 065034.
18. Cai, Y.; Wang, F.; Czerny, A.; Seifert, H. J.; Nestler, B. Phase-field investigation on the microstructural evolution of eutectic transformation and four-phase reaction in Mo-Si-Ti system. Acta. Mater. 2023, 258, 119178.
19. Chen, L. Q.; Yang, W. Computer simulation of the domain dynamics of a quenched system with a large number of nonconserved order parameters: the grain-growth kinetics. Phys. Rev. B. Condens. Matter. 1994, 50, 15752-6.
20. Kobayashi, R.; Warren, J. A.; Craig, C. W. A continuum model of grain boundaries. Phys. D. 2000, 140, 141-50.
21. Iii C, Chen L. Computer simulation of 3-D grain growth using a phase-field model. Acta. Mater. 2002, 50, 3059-75.
22. Moelans, N.; Blanpain, B.; Wollants, P. Quantitative analysis of grain boundary properties in a generalized phase field model for grain growth in anisotropic systems. Phys. Rev. B. 2008, 78, 024113.
23. Gránásy, L.; Tóth, G. I.; Warren, J. A.; et al. Phase-field modeling of crystal nucleation in undercooled liquids - A review. Prog. Mater. Sci. 2019, 106, 100569.
24. Wang, Y.; Chen, L.; Khachaturyan, A. Kinetics of strain-induced morphological transformation in cubic alloys with a miscibility gap. Acta. Metall. Mater. 1993, 41, 279-96.
25. Zhu, J.; Wang, T.; Ardell, A.; Zhou, S.; Liu, Z.; Chen, L. Three-dimensional phase-field simulations of coarsening kinetics of γ′ particles in binary Ni-Al alloys. Acta. Mater. 2004, 52, 2837-45.
26. Liu, H.; Gao, Y.; Liu, J.; Zhu, Y.; Wang, Y.; Nie, J. A simulation study of the shape of β′ precipitates in Mg-Y and Mg-Gd alloys. Acta. Mater. 2013, 61, 453-66.
27. Zhao, Y.; Liu, K.; Zhang, H.; et al. Dislocation motion in plastic deformation of nano polycrystalline metal materials: a phase field crystal method study. Adv. Compos. Hybrid. Mater. 2022, 5, 2546-56.
28. Tian, X. L.; Zhao, Y. H.; Peng, D. W.; Guo, Q. W.; Guo, Z.; Hou, H. Phase-field crystal simulation of evolution of liquid pools in grain boundary pre-melting regions. Trans. Nonfer. Metal. Soc. China. 2021, 31, 1175-88.
29. Clayton, J.; Knap, J. Phase-field analysis of fracture-induced twinning in single crystals. Acta. Mater. 2013, 61, 5341-53.
30. Ansari, T. Q.; Xiao, Z.; Hu, S.; Li, Y.; Luo, J.; Shi, S. Phase-field model of pitting corrosion kinetics in metallic materials. NPJ. Comput. Mater. 2018, 4, 89.
31. Kovacevic, S.; Ali, W.; Martínez-Pañeda, E.; LLorca, J. Phase-field modeling of pitting and mechanically-assisted corrosion of Mg alloys for biomedical applications. Acta. Biomater. 2023, 164, 641-58.
32. Wang, Q.; Zhang, G.; Li, Y.; Hong, Z.; Wang, D.; Shi, S. Application of phase-field method in rechargeable batteries. NPJ. Comput. Mater. 2020, 6, 445.
33. Wang, Y.; Khachaturyan, A. G. Multi-scale phase field approach to martensitic transformations. Mater. Sci. Eng. A. 2006, 438-40, 55-63.
34. Wang, Y.; Khachaturyan, A. Three-dimensional field model and computer modeling of martensitic transformations. Acta. Mater. 1997, 45, 759-73.
35. Levitas, V. I.; Preston, D. L.; Lee, D. W. Three-dimensional Landau theory for multivariant stress-induced martensitic phase transformations. III. Alternative potentials, critical nuclei, kink solutions, and dislocation theory. Phys. Rev. B. 2003, 68, 134201.
36. Mamivand, M.; Zaeem, M. A.; El, K. H. A review on phase field modeling of martensitic phase transformation. Comput. Mater. Sci. 2013, 77, 304-11.
37. Wang, J.; Wang, B.; Chen, L. Understanding, predicting, and designing ferroelectric domain structures and switching guided by the phase-field method. Annu. Rev. Mater. Res. 2019, 49, 127-52.
38. Guo, C.; Huang, H. Design of super-elastic freestanding ferroelectric thin films guided by phase-field simulations. Microstructures 2022, 2, 2022021.
39. Wang, J.; Zhang, J. A real-space phase field model for the domain evolution of ferromagnetic materials. Int. J. Solids. Struct. 2013, 50, 3597-609.
40. Huang, Y. Y.; Jin, Y. M. Phase field modeling of magnetization processes in growth twinned Terfenol-D crystals. Appl. Phys. Lett. 2008, 93, 142504.
41. Shen, Z. H.; Wang, J. J.; Jiang, J. Y.; et al. Phase-field modeling and machine learning of electric-thermal-mechanical breakdown of polymer-based dielectrics. Nat. Commun. 2019, 10, 1843.
42. Bui, T. Q.; Hu, X. A review of phase-field models, fundamentals and their applications to composite laminates. Eng. Fract. Mech. 2021, 248, 107705.
43. Chen, L.; Zhao, Y. From classical thermodynamics to phase-field method. Prog. Mater. Sci. 2022, 124, 100868.
44. Wang, J.; Li, X. K.; Liu, C.; Shi, Y. N. Phase field simulations of microstructure evolution. Chin. J. Solid. Mechan. 2016, 37, 1-33.
45. Cahn, J. W. Free energy of a nonuniform system. II. Thermodynamic basis. J. Chem. Phys. 1959, 30, 1121-4.
46. Cahn, J. W.; Hilliard, J. E. Free energy of a nonuniform system. III. nucleation in a two-component incompressible fluid. J. Chem. Phys. 1959, 31, 688-99.
47. Cahn, J. W.; Allen, S. M. A microscopic theory for domain wall motion and its experimental verification in Fe-Al alloy domain growth kinetics. J. Phys. Colloques. 1977, 38, C7-51.
48. Allen, S. M.; Cahn, J. W. A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening. Acta. Metall. 1979, 27, 1085-95.
49. Clark, A. E.; Wun-Fogle, M. Modern magnetostrictive materials: classical and nonclassical alloys. Proc. SPIE. 2002, 4699, 421.
50. Deng, Z.; Dapino, M. J. Review of magnetostrictive vibration energy harvesters. Smart. Mater. Struct. 2017, 26, 103001.
51. Jiles, D. Recent advances and future directions in magnetic materials. Acta. Mater. 2003, 51, 5907-39.
52. Olabi, A.; Grunwald, A. Design and application of magnetostrictive materials. Mater. Des. 2008, 29, 469-83.
53. Klinger, T.; Pfutzner, H.; Schonhuber, P.; Hoffmann, K.; Bachl, N. Magnetostrictive amorphous sensor for biomedical monitoring. IEEE. Trans. Magn. 1992, 28, 2400-2.
54. Hart, S.; Bucio, R.; Dapino, M. Magnetostrictive actuation of a bone loading composite for accelerated tissue formation. Smart. Mater. Res. 2012, 2012, 1-7.
55. Clark, A. E. Chapter 7 Magnetostrictive rare earth-Fe2 compounds. Amsterdam: North-Holland Publishing Co.; 1980, pp. 531-89.
56. Zhang, J. X.; Chen, L. Q. Phase-field microelasticity theory and micromagnetic simulations of domain structures in giant magnetostrictive materials. Acta. Mater. 2005, 53, 2845-55.
57. Hubert, A.; Schafer, R. Magnetic domains: the analysis of magnetic microstructures. Berlin: Springe; 1998.
59. Hu, C. C.; Shi, Y. G.; Shi, D. N.; Tang, S. L.; Fan, J. Y.; Du, Y. W. Anisotropy compensation and magnetostrictive properties in
60. O’Handley, R. C. Modern magnetic materials: principles and applications. New York: Wiley; 1999. Available from: https://www.wiley.com/en-us/Modern+Magnetic+Materials%3A+Principles+and+Applications-p-9780471155669 [Last accessed on 25 Apr 2025]
61. Khachaturyan, A. G. Theory of structural transformation in solids. New York: Wiley; 2013. Available from: https://store.doverpublications.com/products/9780486783444?srsltid=AfmBOoqp5K0c7IvoyBkllCIrPmbaCngBtMhhWcV_MvzTq79RDwfwl65Y [Last accessed on 25 Apr 2025]
62. Artemev, A.; Jin, Y.; Khachaturyan, A. Three-dimensional phase field model of proper martensitic transformation. Acta. Mater. 2001, 49, 1165-77.
63. Wang, X.; Garcı́a-Cervera, C. J.; E, W. A gauss-seidel projection method for micromagnetics simulations. J. Comput. Phys. 2001, 171, 357-72.
64. Chen, L.; Shen, J. Applications of semi-implicit Fourier-spectral method to phase field equations. Comput. Phys. Commun. 1998, 108, 147-58.
65. Hu, C.; Zhang, Z.; Yang, T.; et al. Phase field simulation of grain size effects on the phase coexistence and magnetostrictive behavior near the ferromagnetic morphotropic phase boundary. Appl. Phys. Lett. 2019, 115, 162402.
66. Liu, J.; Jiang, C.; Xu, H. Giant magnetostrictive materials. Sci. China. Technol. Sci. 2012, 55, 1319-26.
67. Ren, W.; Zhang, Z. Progress in bulk MgCu2-type rare-earth iron magnetostrictive compounds. Chin. Phys. B. 2013, 22, 077507.
68. Atulasimha, J.; Flatau, A. B. A review of magnetostrictive iron-gallium alloys. Smart. Mater. Struct. 2011, 20, 043001.
69. Gou, J.; Ma, T.; Qiao, R.; Yang, T.; Liu, F.; Ren, X. Dynamic precipitation and the resultant magnetostriction enhancement in [001]-oriented Fe-Ga alloys. Acta. Mater. 2021, 206, 116631.
71. Yang, S.; Bao, H.; Zhou, C.; et al. Large magnetostriction from morphotropic phase boundary in ferromagnets. Phys. Rev. Lett. 2010, 104, 197201.
72. Bergstrom, R. J.; Wuttig, M.; Cullen, J.; et al. Morphotropic phase boundaries in ferromagnets: Tb1-xDyxFe2 alloys. Phys. Rev. Lett. 2013, 111, 017203.
73. Hunter, D.; Osborn, W.; Wang, K.; et al. Giant magnetostriction in annealed Co1-xFex thin-films. Nat. Commun. 2011, 2, 518.
74. Wang, B. L.; Jin, Y. M. Magnetization and magnetostriction of Terfenol-D near spin reorientation boundary. J. Appl. Phys. 2012, 111, 103908.
75. Zhou, C.; Ren, S.; Bao, H.; et al. Inverse effect of morphotropic phase boundary on the magnetostriction of ferromagnetic
76. Zhang, D.; Ma, X.; Yang, S.; Song, X. Role of the electronic structure in the morphotropic phase boundary of Tb1-xDyxCo2 studied by first-principle calculation. J. Alloys. Compd. 2016, 689, 1083-7.
77. Ma, T.; Liu, X.; Gou, J.; et al. Sign-changed-magnetostriction effect of morphotropic phase boundary in pseudobinary DyCo2-DyFe2 laves compounds. Phys. Rev. Mater. 2019, 3, 034411.
78. Hu, C.; Zhang, Z.; Cheng, X.; Huang, H.; Shi, Y.; Chen, L. Ultrasensitive magnetostrictive responses at the pre-transitional rhombohedral side of ferromagnetic morphotropic phase boundary. J. Mater. Sci. 2021, 56, 1713-29.
79. Wang, B.; Busbridge, S.; Li, Y.; Wu, G.; Piercy, A. Magnetostriction and magnetization process of Tb0.27Dy0.73Fe2 single crystal. J. Magn. Magn. Mater. 2000, 218, 198-202.
80. Zhao, Y.; Jiang, C.; Zhang, H.; Xu, H. Magnetostriction of <110> oriented crystals in the TbDyFe alloy. J. Alloys. Compd. 2003, 354, 263-8.
81. Jiles, D.; Thoelke, J. Theoretical modelling of the effects of anisotropy and stress on the magnetization and magnetostriction of
82. Desimone, A.; James, R. D. A theory of magnetostriction oriented towards applications. J. Appl. Phys. 1997, 81, 5706-8.
83. Zhao, X. G.; Lord, D. G. Effect of demagnetization fields on the magnetization processes in Terfenol-D. J. Magn. Magn. Mater. 1999, 195, 699-707.
84. Armstrong, W. D. An incremental theory of magneto-elastic hysteresis in pseudo-cubic ferro-magnetostrictive alloys. J. Magn. Magn. Mater. 2003, 263, 208-18.
85. Xu; YX; Cai, T. T.; Hu, C. C.; et al. Magnetic-field driven domain wall evolution in rhombohedral magnetostrictive single crystals: a phase-field simulation. Microstructures 2024, 4, 2024052.
86. Xu, Y.; Wu, Y.; Hu, C.; et al. Domain engineering in ferromagnetic morphotropic phase boundary with enhanced and non-hysteretic magnetostriction: a phase-field simulation. Scripta. Mater. 2024, 242, 115916.
87. Hu, C.; Yang, T.; Huang, H.; et al. Phase-field simulation of domain structures and magnetostrictive response in Tb1-xDyxFe2 alloys near morphotropic phase boundary. Appl. Phys. Lett. 2016, 108, 141908.
88. Ma, T.; Liu, X.; Pan, X.; et al. Local rhombohedral symmetry in Tb0.3Dy0.7Fe2 near the morphotropic phase boundary. Appl. Phys. Lett. 2014, 105, 192407.
89. Hu, C.; Zhang, Z.; Cai, T.; et al. Room-temperature ultrasensitive magnetoelastic responses near the magnetic-ordering tricritical region. J. Appl. Phys. 2021, 130, 063901.
90. Ke, X.; Zhou, C.; Tian, B.; et al. Direct evidence of magnetization rotation at the ferromagnetic morphotropic phase boundary in
91. Dai, M.; Demirel, M. F.; Liang, Y.; Hu, J. Graph neural networks for an accurate and interpretable prediction of the properties of polycrystalline materials. NPJ. Comput. Mater. 2021, 7, 574.
92. Sun, S.; Gong, Q.; Ni, Y.; Yi, M. A micromagnetic-mechanically coupled phase-field model for fracture and fatigue of magnetostrictive alloys. J. Mech. Phys. Solids. 2024, 191, 105767.
93. Rezaei, Y.; Jafari, M.; Jamshidian, M. Phase-field modeling of magnetic field-induced grain growth in polycrystalline metals. Comput. Mater. Sci. 2021, 200, 110786.
94. Guruswamy, S.; Srisukhumbowornchai, N.; Clark, A.; Restorff, J.; Wun-Fogle, M. Strong, ductile, and low-field-magnetostrictive alloys based on Fe-Ga. Scr. Mater. 2000, 43, 239-44.
95. Petculescu, G.; Wu, R.; Mcqueeney, R. Chapter three - Magnetoelasticity of bcc Fe-Ga Alloys. Elsevier; 2012, pp. 123-226.
96. Li, X.; Bao, X.; Yu, X.; Gao, X. Magnetostriction enhancement of Fe73Ga27 alloy by magnetic field annealing. Scr. Mater. 2018, 147, 64-8.
97. Gou, J.; Ma, T.; Liu, X.; et al. Large and sensitive magnetostriction in ferromagnetic composites with nanodispersive precipitates. NPG. Asia. Mater. 2021, 13, 276.
98. Yi, M.; Xu, B. X. A constraint-free phase field model for ferromagnetic domain evolution. Proc. R. Soc. A. 2014, 470, 20140517.
99. He, Y.; Ke, X.; Jiang, C.; et al. Interaction of trace rare-earth dopants and nanoheterogeneities induces giant magnetostriction in Fe-Ga alloys. Adv. Funct. Mater. 2018, 28, 1800858.
100. Yan, K.; Xu, Y.; Niu, J.; et al. Unraveling the origin of local chemical ordering in Fe-based solid-solutions. Acta. Mater. 2024, 264, 119583.
101. Zhang, J. X.; Chen, L. Q. Phase-field model for ferromagnetic shape-memory alloys. Philos. Mag. Lett. 2005, 85, 533-41.
102. Ullakko, K.; Huang, J. K.; Kantner, C.; O’handley, R. C.; Kokorin, V. V. Large magnetic-field-induced strains in Ni2MnGa single crystals. Appl. Phys. Lett. 1996, 69, 1966-8.
103. Sozinov, A.; Likhachev, A. A.; Lanska, N.; Ullakko, K. Giant magnetic-field-induced strain in NiMnGa seven-layered martensitic phase. Appl. Phys. Lett. 2002, 80, 1746-8.
104. Lai, Y. W.; Scheerbaum, N.; Hinz, D.; et al. Absence of magnetic domain wall motion during magnetic field induced twin boundary motion in bulk magnetic shape memory alloys. Appl. Phys. Lett. 2007, 90, 192504.
105. Chen, H.; Wang, Y. D.; Nie, Z.; et al. Unprecedented non-hysteretic superelasticity of [001]-oriented NiCoFeGa single crystals. Nat. Mater. 2020, 19, 712-8.
106. Jin, Y. M. Effects of twin boundary mobility on domain microstructure evolution in magnetic shape memory alloys: Phase field simulation. Appl. Phys. Lett. 2009, 94, 062508.
107. Jin, Y. M. Domain microstructure evolution in magnetic shape memory alloys: phase-field model and simulation. Acta. Mater. 2009, 57, 2488-95.
108. Wu, P.; Ma, X.; Zhang, J.; Chen, L. Phase-field simulations of magnetic field-induced strain in Ni2MnGa ferromagnetic shape memory alloy. Philos. Mag. 2011, 91, 2102-16.
109. Peng, Q.; Huang, J.; Chen, M.; Sun, Q. Phase-field simulation of magnetic hysteresis and mechanically induced remanent magnetization rotation in Ni-Mn-Ga ferromagnetic shape memory alloy. Scr. Mater. 2017, 127, 49-53.
110. Peng, Q.; He, Y.; Moumni, Z. A phase-field model on the hysteretic magneto-mechanical behaviors of ferromagnetic shape memory alloy. Acta. Mater. 2015, 88, 13-24.
111. Sun, Y.; Luo, J.; Zhu, J. Phase field study of the microstructure evolution and thermomechanical properties of polycrystalline shape memory alloys: grain size effect and rate effect. Comput. Mater. Sci. 2018, 145, 252-62.
112. Wu, H. H.; Pramanick, A.; Ke, Y. B.; Wang, X. Real-space phase field investigation of evolving magnetic domains and twin structures in a ferromagnetic shape memory alloy. J. Appl. Phys. 2016, 120, 183904.
113. Xu, C.; Huang, Y.; Liang, Y.; Wu, P. Phase field simulations of microstructures in porous ferromagnetic shape memory alloy Ni2MnGa. Metals 2023, 13, 1572.
114. Xu, Y.; Hu, C.; Liu, L.; et al. A nano-embryonic mechanism for superelasticity, elastic softening, invar and elinvar effects in defected pre-transitional materials. Acta. Mater. 2019, 171, 240-52.
115. Rao, W.; Xu, Y.; Hu, C.; Khachaturyan, A. G. Magnetoelastic equilibrium and super-magnetostriction in highly defected pre-transitional materials. Acta. Mater. 2020, 188, 539-50.
116. Nan, C. W. Magnetoelectric effect in composites of piezoelectric and piezomagnetic phases. Phys. Rev. B. Condens. Matter. 1994, 50, 6082-8.
117. Srinivas, S.; Li, J. Y. The effective magnetoelectric coefficients of polycrystalline multiferroic composites. Acta. Mater. 2005, 53, 4135-42.
118. Ramesh, R.; Spaldin, N. A. Multiferroics: progress and prospects in thin films. Nat. Mater. 2007, 6, 21-9.
119. Ni, Y.; Khachaturyan, A. G. Phase field approach for strain-induced magnetoelectric effect in multiferroic composites. J. Appl. Phys. 2007, 102, 113506.
120. Zhang, J. X.; Li, Y. L.; Schlom, D. G.; et al. Phase-field model for epitaxial ferroelectric and magnetic nanocomposite thin films. Appl. Phys. Lett. 2007, 90, 052909.
121. Wu, P.; Ma, X.; Zhang, J.; Chen, L. Phase-field model of multiferroic composites: Domain structures of ferroelectric particles embedded in a ferromagnetic matrix. Philos. Mag. 2010, 90, 125-40.
122. Hu, J.; Sheng, G.; Zhang, J. X.; Nan, C. W.; Chen, L. Q. Phase-field simulation of strain-induced domain switching in magnetic thin films. Appl. Phys. Lett. 2011, 98, 112505.
