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Nonlocal orientation field phase-field model for misorientation- and inclination-dependent grain boundaries

Xiao Han and Axel van de Walle

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Phys. Rev. B 113, 184114 – Published 11 May, 2026

DOI: https://doi.org/10.1103/24p4-5grt

Abstract

We propose to incorporate grain boundary (GB) anisotropy in phase-field modeling by extending the standard partial differential equations (PDEs) formulation to include a nonlocal functional of an orientation field. Regardless of the number of grains in the simulation, the model uses a single orientation field and incorporates grain misorientation and inclination information obtained from sampling the orientation field at optimized locations in the vicinity of the GB. The formalism enables simple and precise tuning of GB energy anisotropy while avoiding an extensive fitting procedure. The functional includes explicit GB functions to control the GB energy as a function of both misorientation and inclination. The model is validated by reproducing the linear grain growth rate, Wulff shapes with varying misorientations and anisotropic coefficients, and analytical equilibrium dihedral angles at triple junctions. Polycrystalline simulations further demonstrate grain growth, coalescence, triple junction behavior, and the influence of anisotropy on grain morphology.

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References (71)

  1. B. J. Alder and T. E. Wainwright, Studies in molecular dynamics. I. General method, J. Chem. Phys. 31, 459 (1959).
  2. S. L. Thomas, K. Chen, J. Han, P. K. Purohit, and D. J. Srolovitz, Reconciling grain growth and shear-coupled grain boundary migration, Nat. Commun. 8, 1764 (2017).
  3. K. R. Elder, M. Katakowski, M. Haataja, and M. Grant, Modeling elasticity in crystal growth, Phys. Rev. Lett. 88, 245701 (2002).
  4. K. R. Elder and M. Grant, Modeling elastic and plastic deformations in nonequilibrium processing using phase field crystals, Phys. Rev. E 70, 051605 (2004).
  5. R. J. Amodeo and N. M. Ghoniem, Dislocation dynamics. I. A proposed methodology for deformation micromechanics, Phys. Rev. B 41, 6958 (1990).
  6. R. J. Amodeo and N. M. Ghoniem, Dislocation dynamics. II. Applications to the formation of persistent slip bands, planar arrays, and dislocation cells, Phys. Rev. B 41, 6968 (1990).
  7. R. Hill and J. R. Rice, Constitutive analysis of elastic-plastic crystals at arbitrary strain, J. Mech. Phys. Solids 20, 401 (1972).
  8. F. Roters, P. Eisenlohr, L. Hantcherli, D. D. Tjahjanto, T. R. Bieler, and D. Raabe, Overview of constitutive laws, kinematics, homogenization and multiscale methods in crystal plasticity finite-element modeling: Theory, experiments, applications, Acta Mater. 58, 1152 (2010).
  9. N. Moelans, B. Blanpain, and P. Wollants, An introduction to phase-field modeling of microstructure evolution, Calphad 32, 268 (2008).
  10. L.-Q. Chen, Phase-field models for microstructure evolution, Annu. Rev. Mater. Res. 32, 113 (2002).
  11. I. Steinbach, Phase-field models in materials science, Model. Simul. Mater. Sci. Eng. 17, 073001 (2009).
  12. J. J. Eggleston, G. B. McFadden, and P. W. Voorhees, A phase-field model for highly anisotropic interfacial energy, Physica D 150, 91 (2001).
  13. J. J. Hoyt, M. Asta, and A. Karma, Atomistic and continuum modeling of dendritic solidification, Mater. Sci. Eng. R Rep. 41, 121 (2003).
  14. P. Staublin, A. Mukherjee, J. A. Warren, and P. W. Voorhees, Phase-field model for anisotropic grain growth, Acta Mater. 237, 118169 (2022).
  15. L. C. Yeo and J. L. Bair, Phase field modeling of anisotropic bicrystal grain growth using a spherical-Gaussian-based 5-D computational approach, J. Cryst. Growth 627, 127508 (2024).
  16. H. Salama, J. Kundin, O. Shchyglo, V. Mohles, K. Marquardt, and I. Steinbach, Role of inclination dependence of grain boundary energy on the microstructure evolution during grain growth, Acta Mater. 188, 641 (2020).
  17. N. C. Admal, J. Segurado, and J. Marian, A three-dimensional misorientation axis- and inclination-dependent Kobayashi-Warren-Carter grain boundary model, J. Mech. Phys. Solids 128, 32 (2019).
  18. T. Fujiwara, E. Miyoshi, and A. Yamanaka, Evaluating inclination-dependent anisotropic grain boundary energies: Bayesian data assimilation approach using molecular dynamics and phase-field simulations, Comput. Mater. Sci. 248, 113605 (2025).
  19. A. van de Walle, B. G. Chirranjeevi, S. Demers, Q.-J. Hong, A. Kowalski, L. Miljacic, G. S. Pomrehn, and P. Tiwary, Ab initio calculation of anisotropic interfacial excess free energies, Phys. Rev. B 89, 184101 (2014).
  20. D. Fan and L.-Q. Chen, Computer simulation of grain growth using a continuum field model, Acta Mater. 45, 611 (1997).
  21. N. Moelans, New phase-field model for polycrystalline systems with anisotropic grain boundary properties, Mater. Des. 217, 110592 (2022).
  22. I. Steinbach, F. Pezzolla, B. Nestler, M. Seeßelberg, R. Prieler, G. J. Schmitz, and J. L. L. Rezende, A phase field concept for multiphase systems, Physica D 94, 135 (1996).
  23. J. Eiken, B. Böttger, and I. Steinbach, Multiphase-field approach for multicomponent alloys with extrapolation scheme for numerical application, Phys. Rev. E 73, 066122 (2006).
  24. https://openphase.rub.de.
  25. M. Tegeler, O. Shchyglo, R. D. Kamachali, A. Monas, I. Steinbach, and G. Sutmann, Parallel multiphase field simulations with OpenPhase, Comput. Phys. Commun. 215, 173 (2017).
  26. A. Kazaryan, Y. Wang, S. A. Dregia, and B. R. Patton, Generalized phase-field model for computer simulation of grain growth in anisotropic systems, Phys. Rev. B 61, 14275 (2000).
  27. D. Schwen, L. K. Aagesen, J. W. Peterson, and M. R. Tonks, Rapid multiphase-field model development using a modular free energy based approach with automatic differentiation in MOOSE/MARMOT, Comput. Mater. Sci. 132, 36 (2017).
  28. https://mooseframework.inl.gov/source/postprocessors/GrainTracker.html.
  29. S. Y. Hu and L. Q. Chen, A phase-field model for evolving microstructures with strong elastic inhomogeneity, Acta Mater. 49, 1879 (2001).
  30. E. Shahnooshi, M. Jamshidian, M. Jafari, S. Ziaei-Rad, and T. Rabczuk, Phase field modeling of stressed grain growth: Effect of inclination and misorientation dependence of grain boundary energy, J. Cryst. Growth 518, 18 (2019).
  31. J. Wang, S.-Q. Shi, L.-Q. Chen, Y. Li, and T.-Y. Zhang, Phase field simulations of ferroelectric/ferroelastic polarization switching, Acta Mater. 52, 749 (2004).
  32. R. Kobayashi, J. A. Warren, and W. C. Carter, Vector-valued phase field model for crystallization and grain boundary formation, Physica D 119, 415 (1998).
  33. J. A. Warren, R. Kobayashi, and W. C. Carter, Modeling grain boundaries using a phase-field technique, J. Cryst. Growth 211, 18 (2000).
  34. R. Kobayashi, J. A. Warren, and W. C. Carter, A continuum model of grain boundaries, Physica D 140, 141 (2000).
  35. J. Kim, M. Jacobs, S. Osher, and N. C. Admal, A crystal symmetry-invariant Kobayashi–Warren–Carter grain boundary model and its implementation using a thresholding algorithm, Comput. Mater. Sci. 199, 110575 (2021).
  36. H. Henry, J. Mellenthin, and M. Plapp, Orientation-field model for polycrystalline solidification with a singular coupling between order and orientation, Phys. Rev. B 86, 054117 (2012).
  37. B. Korbuly, M. Plapp, H. Henry, J. A. Warren, L. Gránásy, and T. Pusztai, Topological defects in two-dimensional orientation-field models for grain growth, Phys. Rev. E 96, 052802 (2017).
  38. M. Allen and J. W. Cahn, A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening, Acta Metall. 27, 1085 (1979).
  39. S. M. Allen and J. W. Cahn, Ground state structures in ordered binary alloys with second neighbor interactions, Acta Metall. 20, 423 (1972).
  40. S. M. Allen and J. W. Cahn, A correction to the ground state of fcc binary ordered alloys with first and second neighbor pairwise interactions, Scr. Metall. 7, 1261 (1973).
  41. T. Hirouchi, T. Tsuru, and Y. Shibutani, Grain growth prediction with inclination dependence of 110 tilt grain boundary using multi-phase-field model with penalty for multiple junctions, Comput. Mater. Sci. 53, 474 (2012).
  42. R. A. Marks and A. M. Glaeser, Equilibrium and stability of triple junctions in anisotropic systems, Acta Mater. 60, 349 (2012).
  43. P. D. Lax, Stability of difference schemes, in The Courant–Friedrichs–Lewy (CFL) Condition: 80 Years After Its Discovery (Birkhäuser, Boston, 2013), Chap. 1, pp. 1–7.
  44. W. T. Read and W. Shockley, Dislocation models of crystal grain boundaries, Phys. Rev. 78, 275 (1950).
  45. W. Cai and W. D. Nix, Imperfections in Crystalline Solids (Cambridge University Press, Cambridge, 2016).
  46. H. Miura, M. Kato, and T. Mori, Temperature dependence of the energy of Cu 110 symmetrical tilt grain boundaries, J. Mater. Sci. Lett. 13, 46 (1994).
  47. V. V. Bulatov, B. W. Reed, and M. Kumar, Grain boundary energy function for fcc metals, Acta Mater. 65, 161 (2014).
  48. M. A. Tschopp, S. P. Coleman, and D. L. McDowell, Symmetric and asymmetric tilt grain boundary structure and energy in Cu and Al (and transferability to other fcc metals), Integr. Mater. Manuf. Innov. 4, 176 (2015).
  49. J. A. Warren, R. Kobayashi, A. E. Lobkovsky, and W. C. Carter, Extending phase field models of solidification to polycrystalline materials, Acta Mater. 51, 6035 (2003).
  50. W. W. Mullins, Two-dimensional motion of idealized grain boundaries, J. Appl. Phys. 27, 900 (1956).
  51. A. Dimokrati, Y. le Bouar, M. Benyoucef, and A. Finel, S-PFM model for ideal grain growth, Acta Mater. 201, 147 (2020).
  52. K. J. Laidler, The development of the Arrhenius equation, J. Chem. Educ. 61, 494 (1984).
  53. C. Herring, Surface tension as a motivation for sintering, in The Physics of Powder Metallurgy (McGraw-Hill, New York, 1951), Chap. 8, pp. 143–179.
  54. J. Pujol, Hamilton Rodrigues, Gauss, quaternions, and rotations: A historical reassessment, Commun. Math. Anal. 13, 1 (2012).
  55. R. Kobayashi and J. A. Warren, Modeling the formation and dynamics of polycrystals in 3D, Physica A 356, 127 (2005).
  56. D. Rowenhorst et al., Consistent representations of and conversions between 3D rotations, Model. Simul. Mater. Sci. Eng. 23, 083501 (2015).
  57. W. T. Thompson, Introduction to Space Dynamics (Dover Publications, New York, 1986).
  58. M. M. de Jong, J. Kacher, M. H. F. Sluiter, L. Qi, D. L. Olmsted, A. van de Walle, J. W. Morris, A. M. Minor, and M. D. Asta, Electronic origins of anomalous twinning in hexagonal close packed transition metals, Phys. Rev. Lett. 115, 065501 (2015).
  59. Q. Hong, J. Schroer, D. Hofmann, S. Curtarolo, M. Asta, and A. van de Walle, Theoretical prediction of melting temperature for a Mo-Ru-Ta-W HCP multi-principal element alloy, npj Comput. Mater. 7, 1 (2021).
  60. K. Thool, K. U. Yazar, V. Kavimani, A. Gupta, and S.-H. Choi, Microstructural and textural evolution in hexagonal close-packed metals: The case of zirconium, magnesium, and titanium, Crystals 14, 727 (2024).
  61. E. B. Tadmor, R. S. Elliott, J. P. Sethna, R. E. Miller, and C. A. Becker, The potential of atomistic simulations and the knowledgebase of interatomic models, JOM 63, 17 (2011).
  62. Z. T. Trautt, L. M. Hale, and C. A. Becker, Evaluating variability with atomistic simulations: The effect of potential and calculation methodology on the modeling of lattice and elastic constants, Model. Simul. Mater. Sci. Eng. 26, 055003 (2018).
  63. R. Jacobs, D. Morgan, S. Attarian, et al., A practical guide to machine learning interatomic potentials—Status and future, Curr. Opin. Solid State Mater. Sci. 35, 101214 (2025).
  64. I. S. Winter and T. Frolov, Quantifying and visualizing the microscopic degrees of freedom of grain boundaries in the Wigner–Seitz cell of the displacement-shift-complete lattice, Acta Mater. 291, 120968 (2025).
  65. T. Frolov, D. L. Olmsted, M. Asta, and Y. Mishin, Structural phase transformations in metallic grain boundaries, Nat. Commun. 4, 1899 (2013).
  66. A. Bhattacharya, Y.-F. Shen, C. M. Hefferan, S. F. Li, J. Lind, R. M. Suter, C. E. Krill III, and G. S. Rohrer, Grain boundary velocity and curvature are not correlated in Ni polycrystals, Science 374, 189 (2021).
  67. C. Qiu, D. J. Srolovitz, G. S. Rohrer, J. Han, and M. Salvalaglio, Why grain growth is not curvature flow, Proc. Natl. Acad. Sci. USA 122, e2500707122 (2025).
  68. https://mooseframework.inl.gov.
  69. D. Gaston, C. Newman, G. Hansen, and D. Lebrun-Grandié, MOOSE: A parallel computational framework for coupled systems of nonlinear equations, Nucl. Eng. Des. 239, 1768 (2009).
  70. T. Keller, D. Lewis, pdetwiler44, Y. Tan, fields4242, lauera, and Zhyrek, mesoscale/mmsp: Zenodo integration (5.2.1). Zenodo (2019), https://doi.org/10.5281/zenodo.2583258.
  71. https://github.com/hanxustc/phaseField.git.

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Phys. Rev. B 113, 184114– Published 11 May, 2026
  • Received 23 February 2026
  • Accepted 21 April 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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