Elasticity and Excitations of Minimal Crystals

  • R. Bruinsma
Conference paper
Part of the Springer Proceedings in Physics book series (SPPHY, volume 66)


The elastic properties of the unusual crystals encountered in surfactantrich solutions are investigated. Triply-periodic minimal surfaces provide a convenient frame-work for the understanding of such materials but, as is shown, degeneracy leads to vanishing elastic coefficients in the framework of the classical Helfrich energy. This degeneracy is lifted by higher-order corrections and by finite temperature effects. We show that, as a result, thermodynamic stability can be achieved at low levels of dilution but that with increasing dilution the P surface inevitably melts. The degeneracy also leads to an unusual collective excitation spectrum which has a smectic-like undulation dispersion, except at very long wavelengths where it becomes sound-like. The elastic moduli are found to have the same dependence on temperature and concentration as those of tethered stacked membranes and the shear moduli have a temperature and material independent ratio.


Minimal Surface Membrane Thickness Elastic Coefficient Unit Cell Size Periodic Bilayer 
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  1. 1.
    V. Luzatti in Biological Membranes edited by D. Chapman, Vol. 1, p.71 (Academic Press, New York, 1968); P. Mariani, V. Luzatti and H.J. Delacroix, J. Mol. Biol. 204, 165 (1988); K. Fontell, Liquid Crystals and Plastic Crystals edited by G.W. Gray and P. Winson, Vol. 2, 80 (Ellis Harwood, Chicester, 1974); K. Larsson, Z. Phys. Chem. 56 173 (1973); P. Kelucheff and B. Cabane, J. Phys. 48, 1571 (1987).Google Scholar
  2. 2.
    E.L. Thomas, D.B. Alward, D.J. Kinning, D.C. Martin, D.L. Handlin, L.J. Fetters, Macromolecules, 19, 2197 (1986); H. Hasegawa, K. Tanaka, K. Yamasalii, T. Hashimoto, Macromolecuies20, 1651 (1987); E.L. Thomas, J.R. Reffner and J. Bellare, J. Phys. (Paris), C7-363 (1990).CrossRefGoogle Scholar
  3. 3.
    Y. Bouligand, J. Phys. (Paris, C7-35 (1990).Google Scholar
  4. 4.
    See for instance: Physics of Amphiphilic Layers edited by J. Meunier, D. Langevin and N. Boccard (Springer, Berlin, 1987).Google Scholar
  5. 5.
    See for instance: A. Tardieu, Thesis, orsay (1972); D. Anderson, Thesis, University of Minnesota (1986); J. Charvolin and J.F. Sadoc, J. Phys. (Paris) 48 1559 (1987); S.T. Hyde, J. Phys. Chem. 93 1458 (1989).Google Scholar
  6. 6.
    P.B. Canham, J. Theor. Biol. 26 61 (1970); W. Helfrich, Z. Naturforsch. 28c, 693 (1973); H.J. Deuling and W. Helfrich, J. Physique 37 1335 (1976); E.A. Evans, Biophys. J 30 265 (1980); J.T. Jenkins, J. Math. Biology 4 149 (1977); M.A. Peterson, Mol. Cryst. Liq. Cryst. 127 257 (1985) and J. Math. Phys. 26 711 (1985).CrossRefGoogle Scholar
  7. 7.
    See for instance: J.C.C. Nitsche, Lectures on Minimal Surfaces, Vol.1 (Cambridge University Press, 1989); D.M. Anderson, H.T. Davis, J.C.C. Nitsche, Adv. Chem. Phys. 77 P. 337 (Interscience, New York, 1990); A.T. Fomenko, The Plateau Problem I & II (Gordon Breach, New York, 1989).Google Scholar
  8. 8.
    For a table of minimal surfaces and their symmetry see: E. Koch and W. Fischer, Acta Cryst. A46 33 (1990) and references therein.Google Scholar
  9. 9.
    P. Barois, D. Erdam, and S.T. Hyde, J. Phys. (Paris), C7–25 (1990).Google Scholar
  10. 10.
    S. Lidin and K. Hyde, J. Phys. (Paris), 48, 1585 (1987). The degeneracy is also known for the D, H and CLP Schwartz surfaces and the G, S ′ S″ Schoen surfaces.CrossRefGoogle Scholar
  11. 11.
    H. Karcher, Manus Math. 62 83 (1988).CrossRefGoogle Scholar
  12. 12.
    D. A. Hoffman, J. Phys. C7 197, (1990).Google Scholar
  13. 13.
    M. Mitov, Comptes Rendus de l’Academie Bulgare des Sciences 31 513 (1970).Google Scholar
  14. 14.
    W. Helfrich, Liquid Crystals 6, 1647 (1989).CrossRefGoogle Scholar
  15. 15.
    W. Helfrich and H. Rennschuh, J. Phys. (Paris), C7–189 (1990)Google Scholar
  16. 16.
    For a review: Statistical Mechanics of Membranes and Surfaces, (Jerusalem Winter School), edited by D.N. Nelson, T. Piran and S. Weinberg (World Scientific, Singapore, 1989).Google Scholar
  17. 17.
    See: C.R. Safinya et al., Phys. Rev. Lett. 57 2718 (1986), and references therein.CrossRefGoogle Scholar
  18. 18.
    D.A. Huse and S.J. Leibler, J. Phys. (Paris) 49 605 (1988).CrossRefGoogle Scholar
  19. 19.
    G. Porte, J. Marignan, P. Bassereau, and R. May, J. Phys. (Paris) 49, 511 (1988); G. Porte, J. Appel, P. Bassereau, and J. Marignan, J. Phys. (Paris) 50 1335 (1990).CrossRefGoogle Scholar
  20. 20.
    D.M. Anderson, J. Phys. (Paris), 51 C7-1 (1990).Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1992

Authors and Affiliations

  • R. Bruinsma
    • 1
  1. 1.Physics DepartmentUniversity of CaliforniaLos AngelesUSA

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