Skip to main content

Modelling of grain boundary dynamics using amplitude equations


We discuss the modelling of grain boundary dynamics within an amplitude equations description, which is derived from classical density functional theory or the phase field crystal model. The relation between the conditions for periodicity of the system and coincidence site lattices at grain boundaries is investigated. Within the amplitude equations framework, we recover predictions of the geometrical model by Cahn and Taylor for coupled grain boundary motion, and find both \({\langle100\rangle}\) and \({\langle110\rangle}\) coupling. No spontaneous transition between these modes occurs due to restrictions related to the rotational invariance of the amplitude equations. Grain rotation due to coupled motion is also in agreement with theoretical predictions. Whereas linear elasticity is correctly captured by the amplitude equations model, open questions remain for the case of nonlinear deformations.

This is a preview of subscription content, access via your institution.


  1. 1

    Adland A., Karma A., Spatschek R., Buta D., Asta M.: Phase-field-crystal study of grain boundary premelting and shearing in bcc iron. Phys. Rev. B 87, 024110 (2013)

    ADS  Article  Google Scholar 

  2. 2

    Adland A., Xu Y., Karma A.: Unified theoretical framework for poltcrystalline pattern evolution. Phys. Rev. Lett. 110, 265504 (2013)

    ADS  Article  Google Scholar 

  3. 3

    Alexander A., McTague J.: Should all crystals be bcc? Landau theory of solidification and crystal nucleation. Phys. Rev. Lett. 41, 702 (1978)

    ADS  Article  Google Scholar 

  4. 4

    Bhogireddy V.S.P.K., Hüter C., Neugebauer J., Steinbach I., Karma A., Spatschek R.: Phase-field modeling of grain-boundary premelting using obstacle potentials. Phys. Rev. E 90, 012401 (2014)

    ADS  Article  Google Scholar 

  5. 5

    Boettinger W.J., Warren J., Beckermann C., Karma A.: Phase-field simulation of solidification. Annu. Rev. Mater. Res. 32, 163 (2002)

    Article  Google Scholar 

  6. 6

    Bollmann W.: The basic concepts of the O-lattice theory. Surf. Sci. 31, 1–11 (1972)

    ADS  Article  Google Scholar 

  7. 7

    Boussinot G., Hüter C., Brener E.A.: Growth of a two-phase finger in eutectics systems. Phys. Rev. E 83, 020601 (2011)

    ADS  Article  Google Scholar 

  8. 8

    Brener E.A., Boussinot G., Hüter C., Fleck M., Pilipenko D., Spatschek R., Temkin D.E.: Pattern formation during diffusional transformations in the presence of triple junctions and elastic effects. J. Phys. Condens. Matter 21, 464106 (2009)

    ADS  Article  Google Scholar 

  9. 9

    Brener E.A., Marchenko V.I., Müller-Krumbhaar H., Spatschek R.: Coarsening kinetics with elastic effects. Phys. Rev. Lett. 84, 4914 (2000)

    ADS  Article  Google Scholar 

  10. 10

    Cahn J., Hilliard J.: Free energy of a nonuniform system 1: interfacial free energy. J. Chem. Phys. 28, 258 (1958)

    ADS  Article  Google Scholar 

  11. 11

    Cahn J., Hilliard J.: Free energy of a nonuniform system 3: nucleation in a two component incompressible fluid. J. Chem. Phys. 31, 688 (1959)

    ADS  Article  Google Scholar 

  12. 12

    Cahn J.W.: Theory of crystal growth and interface motion in crystalline materials. Acta Metall. 8, 554 (1960)

    Article  Google Scholar 

  13. 13

    Cahn J.W., Mishin Y., Suzuki A.: Coupling grain boundary motion to shear deformations. Acta Mater. 54, 4953 (2006)

    Article  Google Scholar 

  14. 14

    Cahn J.W., Taylor J.E.: A unified approach to motion of grain boundaries, relative tangential translation along grain boundaries, and grain rotation. Acta Mat. 52, 4887 (2004)

    Article  Google Scholar 

  15. 15

    Chan P.Y., Goldenfeld N.: Nonlinear elasticity of the phase-field crystal model from the renormalization group. Phys. Rev. E 80, 065105 (R) (2009)

    ADS  Article  Google Scholar 

  16. 16

    Chen L.: Phase-field models for microstructure evolution. Annu. Rev. Mater. Res. 32, 113 (2002)

    Article  Google Scholar 

  17. 17

    Cross M.C., Hohenberg P.C.: Pattern formation outside of equilibrium. Rev. Mod. Phys. 65, 851–1112 (1993). doi:10.1103/RevModPhys.65.851

    ADS  Article  Google Scholar 

  18. 18

    Dreyer W., Mueller W.: A study of the coarsening in tin/lead solders. Int. J. Solids Struct. 37, 3841 (2000)

    Article  MATH  Google Scholar 

  19. 19

    Elder K.R., Grant M.: Modeling elastic and plastic deformations in nonequilibrium processing using phase field crystals. Phys. Rev. E 70, 051605 (2004)

    ADS  Article  Google Scholar 

  20. 20

    Elder K.R., Katakowski M., Haataja M., Grant M.: Modeling elasticity in crystal growth. Phys. Rev. Lett. 88, 245701 (2002). doi:10.1103/PhysRevLett.88.245701

    ADS  Article  Google Scholar 

  21. 21

    Emmerich H., Löwen H., Wittkowski R., Gruhn T., Tóth G.I., Tegze G., Gránásy L.: Phase-field-crystal models for condensed matter dynamics on atomic length and diffusive time scales: an overview. Adv. Phys. 61, 665 (2012)

    ADS  Article  Google Scholar 

  22. 22

    Fix, G.J.: Phase field methods for free boundary problems. In: Free Boundary Problems: Theory and Applications, vol. 79, p. 580. Pitman Research Notes in Mathematics Series, Boston (1983)

  23. 23

    Graham R.: Systematic derivation of a rotationally covariant extension of the two-dimensional newell-whitehead-segel equation. Phys. Rev. Lett. 76, 2185–2187 (1996). doi:10.1103/PhysRevLett.76.2185

    ADS  Article  Google Scholar 

  24. 24

    Graham R.: Erratum: Systematic derivation of a rotationally covariant extension of the two-dimensional newell-whitehead-segel equation. Phys. Rev. Lett. 80, 3888–3888 (1998). doi:10.1103/PhysRevLett.80.3888

    ADS  Article  Google Scholar 

  25. 25

    Grasselli M., Wu H.: Erratum: Systematic derivation of a rotationally covariant extension of the two-dimensional newell-whitehead-segel equation. Phys. Rev. Lett. 80, 3888 (1998)

    ADS  Article  Google Scholar 

  26. 26

    Gunaratne G.H., Ouyang Q., Swinney H.L.: Pattern formation in the presence of symmetries. Phys. Rev. E 50, 2802–2820 (1994). doi:10.1103/PhysRevE.50.28

    ADS  MathSciNet  Article  MATH  Google Scholar 

  27. 27

    Harrowell P., Oxtoby D.W.: A molecular theory of crystal nucleation from the melt. J. Chem. Phys. 80(4), 1639–1646 (1984). doi:10.1063/1.446864

    ADS  Article  Google Scholar 

  28. 28

    Haymet A.D.J., Oxtoby D.W.: A molecular theory for the solid–liquid interface. J. Chem. Phys. 74(4), 2559–2565 (1981). doi:10.1063/1.441326

    ADS  Article  Google Scholar 

  29. 29

    Hillert, M.: A theory of nucleation for solid solutions. Master’s thesis, Cambridge, MA (1956)

  30. 30

    Hüter C., Boussinot G., Brener E.A., Temkin D.E.: Solidification along the interface between demixed liquids in monotectic systems. Phys. Rev. E 83, 050601 (2011)

    Article  Google Scholar 

  31. 31

    Hüter, C., G.Boussinot, Brener, E.A., Spatschek, R.: Solidification in syntectic and monotectic systems. Phys. Rev. E (2012)

  32. 32

    Hüter, C., Nguyen, C.-D., Spatschek, R.P., Neugebauer, J.: Scale bridging between atomistic and mesoscale modelling: applications of amplitude equation descriptions. Model. Simul. Mater. Sci. Eng. 22(3), 034001 (2014). doi:10.1088/0965-0393/22/3/034001

  33. 33

    Hüter C., Twiste F., Brener E.A., Neugebauer J., Spatschek R.: Influence of short-range forces on melting along grain boundaries. Phys. Rev. B 89, 224104 (2014)

    ADS  Article  Google Scholar 

  34. 34

    Karma, A. et al.: Phase-field methods. In: Buschow, K. (ed.) Encyclopedia of Materials Science and Technology, pp. 6873. Elsevier, Oxford (2001)

  35. 35

    Kerr, W., Killough, M., Saxena, A., Swart, J., Bishop, A.R.: Role of elastic role of elastic compatibility in martensitic texture evolution. Phase Transitions 69 (1999)

  36. 36

    Khachaturyan A.G.: Theory of Structural Transformation in Solids. Wiley, London (1983)

    Google Scholar 

  37. 37

    Laird B.B., McCoy J.D., Haymet A.D.J.: Density functional theory of freezing—analysis of crystal density. J. Chem. Phys. 87(9), 5449–5456 (1987). doi:10.1063/1.453663

    ADS  Article  Google Scholar 

  38. 38

    Landau L.: On the theory of phase transitions. Zh. Eksp. Teor. Fiz. 7, 19 (1937)

    Google Scholar 

  39. 39

    Langer J.S.: Directions in Condensed Matter. World Scientific, Singapore (1986)

    Google Scholar 

  40. 40

    Langer, J.S.: Lectures on the theory of pattern formation. In: Chance and Matter, p. 629. Amsterdam: North Holland (1986)

  41. 41

    Provatas N., Elder K.: Introduction, in Phase-Field Methods in Materials Science and Engineering. Wiley-VCH, Weinheim, Germany (2010)

    Book  Google Scholar 

  42. 42

    Radhakrishnan, B., Gorti, S., Nicholson, D.M., Dantzig, J.: Comparison of phase field crystal and molecular dynamics: simulations for a shrinking grain. J. Phys. Conf. Ser. 402, 012043 (2012)

  43. 43

    Rubin G., Khachaturyan A.G.: Three-dimensional model of precipitation of ordered intermetallics. Acta Mater. 47, 1995 (1999)

    Article  Google Scholar 

  44. 44

    Shen Y., Oxtoby D.: Density functional theory of crystal growth: Lennard–Jones fluids. J. Chem. Phys. 104(11), 4233–4242 (1996). doi:10.1063/1.471234

    ADS  Article  Google Scholar 

  45. 45

    Shen Y., Oxtoby D.: Nucleation of Lennard–Jones fluids: a density functional approach. J. Chem. Phys. 105(15), 6517–6524 (1996). doi:10.1063/1.472461

    ADS  Article  Google Scholar 

  46. 46

    Singh, Y.: Density-functional theory of freezing and properties of the ordered phase. Physics Reports 207(6), 351–444 (1991). doi:10.1016/0370-1573(91)90097-6.

  47. 47

    Spatschek R., Adland A., Karma A.: Structural short-range forces between solid–melt interfaces. Phys. Rev. B 97, 024109 (2013)

    ADS  Article  Google Scholar 

  48. 48

    Spatschek R., Brener E., Karma A.: Phase field modeling of crack propagation. Philos. Mag. 91, 75 (2011)

    ADS  Article  Google Scholar 

  49. 49

    Spatschek R., Karma A.: Amplitude equations for polycrystalline materials with interaction between composition and stress. Phys. Rev. B 81, 214201 (2010)

    ADS  Article  Google Scholar 

  50. 50

    Spatschek R., Müller-Gugenberger C., Brener E., Nestler B.: Phase field modeling of fracture and stress-induced phase transitions. Phys. Rev. E 75, 066111 (2007)

    ADS  MathSciNet  Article  Google Scholar 

  51. 51

    Stefanovic P., Haataja M., Provatas N.: Phase field crystal study of deformation and plasticity in nanocrystalline materials. Phys. Rev. E 80, 046107 (2009)

    ADS  Article  Google Scholar 

  52. 52

    Steinbach, I.: Phase-field models in materials science. Model. Simul. Mater. Sci. Eng. 17, 073001 (2009)

  53. 53

    Wang N., Spatschek R., Karma A.: Multi-phase-field analysis of short-range forces between diffuse interfaces. Phys. Rev. E 81, 051601 (2010). doi:10.1103/PhysRevE.81.051601

    ADS  Article  Google Scholar 

  54. 54

    Wang Y., Banerjee D., Su C.C., Khachaturyan A.G.: Field kinetic model and computer simulation of precipitation of Ll(2) ordered intermetallics from fcc solid solution. Acta Mater. 46, 2983 (1998)

    Article  Google Scholar 

  55. 55

    Wang Y., Khachaturyan A.G.: Multi-scale phase field approach to martensitic transformation. Mater. Sci. Eng. A 438(440), 55–63 (2006)

    Article  Google Scholar 

  56. 56

    Wang Y., Li J.: Phase field modeling of defects and deformation. Acta Mater. 58, 1212 (2010)

    Article  Google Scholar 

  57. 57

    Wu K.A., Adland A., Karma A.: Phase-field-crystal model for fcc ordering. Phys. Rev. E 81, 061601 (2010)

    ADS  Article  Google Scholar 

  58. 58

    Wu K.A., Karma A.: Phase-field crystal modeling of equilibrium bcc–liquid interfaces. Phys. Rev. B 76, 184107 (2007)

    ADS  Article  Google Scholar 

  59. 59

    Wu K.A., Karma A., Hoyt J.J., Asta M.: Ginzburg–Landau theory of crystalline anisotropy for bcc–liquid interfaces. Phys. Rev. B 73, 094101 (2006)

    ADS  Article  Google Scholar 

  60. 60

    Wu K.A., Vorhees P.: Phase field crystal simulations of nanocrystalline grain growth in two dimensions. Acta Mater. 60, 407 (2012)

    Article  Google Scholar 

Download references

Author information



Corresponding author

Correspondence to Claas Hüter.

Additional information

Communicated by Ralf Müller.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Hüter, C., Neugebauer, J., Boussinot, G. et al. Modelling of grain boundary dynamics using amplitude equations. Continuum Mech. Thermodyn. 29, 895–911 (2017).

Download citation


  • Amplitude equations
  • Grain rotation
  • Coupled motion
  • Nonlinear elasticity