Advertisement

Archive for Rational Mechanics and Analysis

, Volume 223, Issue 2, pp 977–1017 | Cite as

Interfaces, Modulated Phases and Textures in Lattice Systems

  • Andrea Braides
  • Marco Cicalese
Article

Abstract

We introduce a class of n-dimensional (possibly inhomogeneous) spin-like lattice systems presenting modulated phases with possibly different textures. Such systems can be parameterized according to the number of ground states, and can be described by a phase-transition energy which we compute by means of variational techniques. Degeneracies due to frustration are also discussed.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

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(1), 85–107 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Alicandro, R., Braides, A., Cicalese, M.: Book, 2016 (in preparation)Google Scholar
  3. 3.
    Alicandro R., Cicalese M., Ruf M.: Domain formation in magnetic polymer composites: an approach via stochastic homogenization. Arch. Rat. Mech. Anal. 218(2), 945–984 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Alicandro R., Gelli M.S.: Local and non local continuum limits of Ising type energies for spin systems. SIAM J. Math. Anal. 48(2), 895–931 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Ambrosio L., Braides A.: Functionals defined on partitions in sets of finite perimeter, I: Integral representation and gamma-convergence. J. Math. Pure Appl. 69, 285–306 (1990)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Ambrosio L., Braides A.: Functionals defined on partitions of sets of finite perimeter, I: semicontinuity, relaxation and homogenization. J. Math. Pures Appl. 69, 307–333 (1990)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Bellettini G., Braides A., Riey G.: Variational approximation of anisotropic functionals on partitions. Ann. di Mat. Pura ed Appl. 184(1), 75–93 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Braides, A.: \({\Gamma}\)-convergence for beginners, volume 22 of Oxford Lecture Series in Mathematics and its Applications. Oxford University Press, Oxford, 2002Google Scholar
  9. 9.
    Braides, A.: Discrete-to-continuum variational methods for lattice systems. In: Proceedings International Congress of Mathematicians. Seoul, pp. 997–1015, 2014Google Scholar
  10. 10.
    Braides A.: An example of non-existence of plane-like minimizers for an almost-periodic Ising system. J. Stat. Phys. 157(2), 295–302 (2014)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Braides, A., Chiadò Piat V.: Integral representation results for functionals defined on \({SBV (\Omega; R^m)}\). J. de Math. Pures et Appl. 75(6), 595–626 (1996)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Braides A., ChiadoPiat V., Solci M.: Discrete double-porosity models for spin systems. Math. Mech. Complex Syst. 4, 79–102 (2016)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Braides A., Defranceschi A.: Homogenization of multiple integrals. Oxford University Press, Oxford (1998)zbMATHGoogle Scholar
  14. 14.
    Braides A., Garroni A., Palombaro M.P.: Interfacial energies of systems of chiral molecules. SIAM Multiscale Model Simul. 14, 1037–1062 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Braides A., Piatnitski A.: Variational problems with percolation: dilute spin systems at zero temperature. J. Stat. Phys. 149(5), 846–864 (2012)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Braides A., Piatnitski A.: Homogenization of surface and length energies for spin systems. J. Funct. Anal. 264(6), 1296–1328 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Caffarelli L.A., de la Llave R.: Planelike minimizers in periodic media. Commun. Pure Appl. Math. 54(12), 1403–1441 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Ciach, A., Hoye, J.S., Stell, G.: Microscopic model for microemulsions. J. Phys. A Math. General. 21(15), L777 (1988)Google Scholar
  19. 19.
    Coleman B.D., Marcus M., Mizel V.J.: On the thermodynamics of periodic phases. Arch. Ration. Mech. Anal. 117(4), 321–347 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    De Giorgi E., Letta G.: Une notion générale de convergence faible pour des fonctions croissantes d’ensemble. Ann. della Scuola Normale Superiore di Pisa-Classe di Sci. 4(1), 61–99 (1977)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Giuliani A., Lebowitz J.L., Lieb E.H.: Ising models with long-range antiferromagnetic and short-range ferromagnetic interactions. Phys. Rev. B 74(6), 064420 (2006)ADSCrossRefGoogle Scholar
  22. 22.
    Giuliani A., Lebowitz J.L., Lieb E.H.: Checkerboards, stripes, and corner energies in spin models with competing interactions. Phys. Rev. B 84(6), 064205 (2011)ADSCrossRefGoogle Scholar
  23. 23.
    Giuliani A., Lieb E.H., Seiringer R.: Realization of stripes and slabs in two and three dimensions. Phys. Rev. B 88(6), 064401 (2013)ADSCrossRefGoogle Scholar
  24. 24.
    Ohta T., Kawasaki K.: Equilibrium morphology of block copolymer melts. Macromolecules 19(10), 2621–2632 (1986)ADSCrossRefGoogle Scholar
  25. 25.
    Seul M., Andelman D.: Domain shapes and patterns: the phenomenology of modulated phases. Science 267(5197), 476–483 (1995)ADSCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Dipartimento di MatematicaUniversità di Roma ‘Tor Vergata’RomeItaly
  2. 2.Zentrum Mathematik-M7Technische Universität MünchenGarchingGermany

Personalised recommendations