Crystalline Motion of Interfaces Between Patterns

Abstract

We consider the dynamical problem of an antiferromagnetic spin system on a two-dimensional square lattice \(\varepsilon \mathbb {Z}^2\) with nearest-neighbour and next-to-nearest neighbour interactions. The key features of the model include the interaction between spatial scale \(\varepsilon \) and time scale \(\tau \), and the incorporation of interfacial boundaries separating regions with microstructures. By employing a discrete-time variational scheme, a limit continuous-time evolution is obtained for a crystal in \(\mathbb {R}^2\) which evolves according to some motion by crystalline curvatures. In the case of anti-phase boundaries between striped patterns, a striking phenomenon is the appearance of some “non-local” curvature dependence velocity law reflecting the creation of some defect structure on the interface at the discrete level.

This is a preview of subscription content, log in to check access.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21
Fig. 22
Fig. 23
Fig. 24
Fig. 25
Fig. 26
Fig. 27
Fig. 28
Fig. 29
Fig. 30

Notes

  1. 1.

    A bisectric direction has slope \(\pm 1\). A bisectric segment is a segment along the bisectric direction.

  2. 2.

    Note that there is a difference in the definition of the number of layers moving in for the bisectric segments between the \(e_1\)-\(e_2\)-interface in Sect. 4 and the current \(e_1\)-\(-e_1\)-interface. In the former case, the number is measured along the normal bisectric direction while here it is measured along the horizontal direction. This is simply for arithmetic convenience.

References

  1. 1.

    Alicandro, R., Braides, A., Cicalese, M.: Phase and anti-phase boundaries in binary discrete systems: a variational viewpoint. Netw. Heterog. Media 1, 85–107 (2006)

    MathSciNet  Article  MATH  Google Scholar 

  2. 2.

    Allen, S., Cahn, J.: A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening. Acta Metall. 27, 1084–1095 (1979)

    Article  Google Scholar 

  3. 3.

    Almgren, F., Taylor, J.E.: Flat flow is motion by crystalline curvature for curves with crystalline energies. J. Differ. Geom. 42, 1–22 (1995)

    MathSciNet  MATH  Google Scholar 

  4. 4.

    Almgren, F., Taylor, J.E., Wang, L.: Curvature driven flows: a variational approach. SIAM J. Control Optim. 50, 387–438 (1983)

    MathSciNet  MATH  Google Scholar 

  5. 5.

    Ambrosio, L., Gigli, N., Savaré, G.: Gradient Flows in Metric Spaces and in the Space of Probability Measures. Lectures in Mathematics ETH. Zürich. Birkhhäuser, Basel (2008)

  6. 6.

    Ansini, N., Braides, A., Chiadò Piat, V.: Gradient theory of phase transitions in composite media. Proc. Royal Soc. Edin 133A, 265–296 (2003)

    MathSciNet  Article  MATH  Google Scholar 

  7. 7.

    Braides, A.: \(\Gamma \)-convergence for Beginners. Oxford University Press, Oxford (2002)

    Google Scholar 

  8. 8.

    Braides, A.: Local Minimization, Variational Evolution and Gamma-Convergence. Lecture Notes in Mathematics, vol. 2094. Springer Verlag, Berlin (2014)

  9. 9.

    Braides, A., Cicalese, M.: Interfaces, modulated phases and textures in lattice systems (to appear in Arch. Ration. Mech. Anal)

  10. 10.

    Braides, A., Gelli, M.S., Novaga, M.: Motion and pinning of discrete interfaces. Arch. Ration. Mech. Anal. 95, 469–498 (2010)

    MathSciNet  Article  MATH  Google Scholar 

  11. 11.

    Braides, A., Scilla, G.: Motion of discrete interfaces in periodic media. Interfaces Free Bound. 15, 451–476 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  12. 12.

    Braides, A., Solci, M.: Motion of discrete interfaces through mushy layers. J. Nonlinear Sci. 26, 1031–1053 (2016)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  13. 13.

    Caffarelli, L.A., de la Llave, R.: Interfaces of ground states in ising models with periodic coefficients. J. Stat. Phys. 118, 687–719 (2005)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  14. 14.

    Cahn, J.W., Van Vleck, E.S.: Quadrijunctions do not stop two-dimensional grain growth. Scipta Mater. 34, 909–912 (1996)

    Article  Google Scholar 

  15. 15.

    Cahn, J.W., Van Vleck, E.S.: On the co-existence and stability of trijunctions and quadrijunctions in a simple model. Acta Mater. 47(18), 4627–4639 (1999)

    Article  Google Scholar 

  16. 16.

    Chen, X.: Generation and propagation of interfaces for reaction-diffusion equations. J. Diff. Equ. 96, 116–141 (1992)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  17. 17.

    De Masi, A., Ferrai, P.A., Lebowitz, J.L.: Rigorous derivation of reaction-diffusion equations with fluctuations. Phys. Rev. Lett. 55(19), 1947–1949 (1985)

    ADS  MathSciNet  Article  Google Scholar 

  18. 18.

    De Masi, A., Ferrai, P.A., Presutti, E.: Reaction-diffusion equations for interacting particle systems. J. Stat. Phys. 44(3/4), 589–644 (1986)

    ADS  MathSciNet  Article  Google Scholar 

  19. 19.

    De Masi, A., Orlandi, E., Presutti, E., Triolo, L.: Glauber evolution with the Kac potentials. I. Mesoscopic and macroscopic limits, interface dynamics. Nonlinearity 7(3), 633–696 (1994)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  20. 20.

    de Mottoni, P., Schatzman, M.: Geometric evolution of developed interfaces. Trans. Am. Math. Soc. 347, 1533–1589 (1995)

    Article  MATH  Google Scholar 

  21. 21.

    Evans, L.C., Soner, H.M., Souganidis, P.E.: Phase transitions and generalized motion by mean curvature. Commun. Pure Appl. Math. 45, 1097–1123 (1992)

    MathSciNet  Article  MATH  Google Scholar 

  22. 22.

    Giacomin, G., Lebowitz, J.L.: Phase segregation dynamics in particle systems with long range interactions. I. Macroscopic limits. J. Stat. Phys. 87(1–2), 37–61 (1997)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  23. 23.

    Giacomin, G., Lebowitz, J.L.: Phase segregation dynamics in particle systems with long range interactions. II. Interface motion. SIAM J. Appl. Math. 58(6), 1707–1729 (1998)

    MathSciNet  Article  MATH  Google Scholar 

  24. 24.

    Giuliani, A., Lieb, E.H., Seiringer, R.: Formation of stripes and slabs near the ferromagnetic transition. Commun. Math. Phys. 331, 333–350 (2014)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  25. 25.

    Giuliani, A., Seiringer, R.: Periodic striped ground states in Ising models with competing interactions. arXiv:1509.00057 (to appear in Comm. Math. Phys)

  26. 26.

    Glauber, R.J.: Time-dependent statistics of the Ising model. J. Math. Phys. 4(2), 294–307 (1963)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  27. 27.

    Ilmanen, T.: Convergence of the Allen–Cahn equation to Brakke’s motion by mean curvature. J. Differ. Geom. 38, 417–461 (1993)

    MathSciNet  MATH  Google Scholar 

  28. 28.

    Katsoulakis, M.K., Souganidis, P.E.: Interacting particle systems and generalized evolution of fronts. Arch. Ration. Mech. Anal. 127(2), 133–157 (1994)

    MathSciNet  Article  MATH  Google Scholar 

  29. 29.

    Katsoulakis, M.K., Souganidis, P.E.: Generalized motion by mean curvature as a macroscopic limit of stochastic Ising models with long range interactions and Glauber dynamics. Commun. Math. Phys. 169(1), 61–97 (1995)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  30. 30.

    Katsoulakis, M.K., Souganidis, P.E.: Stochastic Ising models and anisotropic front propagation. J. Stat. Phys. 87(1–2), 63–89 (1997)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  31. 31.

    Kawasaki, K.: Diffusion constant near the critical point for time-dependent ising models. I. Phys. Rev. 145(1), 224–230 (1966)

    ADS  MathSciNet  Article  Google Scholar 

  32. 32.

    Kawasaki, K.: Diffusion constant near the critical point for time-dependent ising models. II. Phys. Rev. 148(1), 375–381 (1966)

    ADS  MathSciNet  Article  Google Scholar 

  33. 33.

    Kinderlehrer, D., Liu, C.: Evolution of grain boundaries. Math. Models Methods Appl. Sci. 11, 713–729 (2001)

    MathSciNet  Article  MATH  Google Scholar 

  34. 34.

    Luckhaus, S., Sturzenhecker, T.: Implicit time discretization for the mean curvature flow. Calc. Var. 3, 253–271 (1995)

    MathSciNet  Article  MATH  Google Scholar 

  35. 35.

    Magni, A., Mantegazza, C., Novaga, M.: Motion by curvature of planar networks II. Ann. Sc. Norm. Super. Pisa Cl. Sci. 15, 117–144 (2016)

    MathSciNet  MATH  Google Scholar 

  36. 36.

    Mantegazza, C., Novaga, M., Tortorelli, V.M.: Motion by curvature of planar networks. Ann. Sc. Norm. Super. Pisa Cl. Sci. 3, 235–324 (2004)

    MathSciNet  MATH  Google Scholar 

  37. 37.

    Modica, L.: The gradient theory of phase transitions and the minimal interface criterion. Arch. Ration. Mech. Anal. 98, 123–142 (1987)

    MathSciNet  Article  MATH  Google Scholar 

  38. 38.

    Sandier, E., Serfaty, S.: Gamma-convergence of gradient flows with applications to Ginzburg–Landau. Commun. Pure Appl. Math. 57(12), 1627–1672 (2004)

    MathSciNet  Article  MATH  Google Scholar 

  39. 39.

    Sternberg, P.: The effect of a singular perturbation on nonconvex variational problems. Arch. Ration. Mech. Anal. 101, 209–260 (1988)

    MathSciNet  Article  MATH  Google Scholar 

  40. 40.

    Taylor, J.E.: Motion of curves by crystalline curvature, including triple junctions and boundary points. Proc. Symp. Pure Math. Differ. Geom. 51(part 1), 417–438 (1993)

    MathSciNet  Article  MATH  Google Scholar 

  41. 41.

    Taylor, J.E., Cahn, J., Handwerker, C.: Geometric models of crystal growth. Acta Metall. Mater. 40, 1443–1474 (1992)

    ADS  Article  Google Scholar 

  42. 42.

    Taylor, J.E., Cahn, J., Handwerker, C.: Mean curvature and weighted mean curvature. Acta Metall. Mater. 40, 1475–1485 (1992)

    Article  Google Scholar 

  43. 43.

    Taylor, J.E.: A variational approach to crystalline triple-junction motion. J. Stat. Phys. 95(5–6), 1221–1244 (1999)

    ADS  MathSciNet  Article  MATH  Google Scholar 

Download references

Acknowledgments

The authors would like to thank the hospitality of the Institute for Mathematics and Its Applications (IMA), Minnesota, it was where this project started. The second author was supported by the DFG Collaborative Research Center TRR 109 Discretization in Geometry and Dynamics. The third author also appreciates the hosting by the Dipartimento di Matematica, Università degli Studi di Roma Tor Vergata of his several visits which facilitated the completion of this project.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Nung Kwan Yip.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Braides, A., Cicalese, M. & Yip, N.K. Crystalline Motion of Interfaces Between Patterns. J Stat Phys 165, 274–319 (2016). https://doi.org/10.1007/s10955-016-1609-6

Download citation

Keywords

  • Antiferromagnetic spin system
  • Anti-phase boundaries
  • Microstructures
  • Defects
  • Interface motion
  • Crystalline curvature motion