Skip to main content
SpringerLink
Log in

Curvature of the elastic deformations in a nematic sample

  • Regular Article
  • Published:
The European Physical Journal E Aims and scope Submit manuscript

Abstract.

In this work we study the geometry of the elastic deformations of the uniaxial nematic liquid crystals at the bulk. We will show that, at this region of the sample, the elastic terms of the free energy can be separated as the sum of two kinds of elastic deformations, the first is proportional to the Gaussian curvature obtained from the director field of a three-dimensional nematic sample and the second is composed by those terms that cannot be expressed as resulting from this curvature. To achieve these results we will construct the metric of an unixial nematic sample using the fact that the director gives the direction of the anisotropy of the system. With this approach we will give analytical and geometrical arguments to show that the elastic terms determined by \(K_{22}\), \(K_{13}\) and \(K_{24}\) are contained in a curvature term, while the terms fixed by the splay elastic term, \(K_{11}\), and the bend elastic term, \(K_{33}\), are not. The novelty here is that while \(K_{13}\) and \(K_{24}\) do not contribute the bulk elastic energy of a nematic sample, they have an important contribution to the curvature of the system.

Graphical abstract

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. P.G. de Gennes, J. Prost, The Physics of Liquid Crystals, 2nd edition (Clarendon Press, Oxford, 1993)

  2. D.A. Dunmur, Liquid Crystals: Fundamentals (World Scientific Publishing Co, 2002)

  3. C. Oldano, G. Barbero, J. Phys. (Paris) Lett. 46, 451 (1985)

    Article  Google Scholar 

  4. C. Oldano, G. Barbero, Phys. Lett. A 110, 213 (1985)

    Article  ADS  Google Scholar 

  5. C. Oldano, G. Barbero, J. Phys. (Paris) Lett. 46, L-451 (1985)

    Article  Google Scholar 

  6. C. Oldano, G. Barbero, Nuovo Cimento D 6, 479 (1985)

    Article  ADS  Google Scholar 

  7. H.P. Hinov, Mol. Cryst. Liq. Cryst. 148, 197 (1987)

    Article  Google Scholar 

  8. G. Barbero, A. Strigazzi, Liq. Cryst. 5, 693 (1989)

    Article  Google Scholar 

  9. C. Oldano, G. Barbero, Mol. Cryst. Liq. Cryst. 170, 99 (1989)

    Google Scholar 

  10. G. Barbero, A. Sparavigna, A. Strigazzi, Nuovo Cimento D 12, 1259 (1990)

    Article  ADS  Google Scholar 

  11. H.P. Hinov, Mol. Cryst. Liq. Cryst. 178, 53 (1990)

    Google Scholar 

  12. V.M. Pergamenshchik, Phys. Rev. E 48, 1256 (1993)

    ADS  MathSciNet  Google Scholar 

  13. S. Faetti, Phys. Rev. E 49, 4192 (1994)

    Article  ADS  Google Scholar 

  14. S. Stallinga, G. Vertogen, Phys. Rev. E 53, 1692 (1996)

    Article  ADS  Google Scholar 

  15. S. Faetti, Phys. Lett. A 255, 165 (1999)

    Article  ADS  Google Scholar 

  16. G. Barbero, L.R. Evangelista, An Elementary Course on the Continuum Theory for Nematic Liquid Crystals (World Scientific, 2001)

  17. F.C. Frank, Discuss. Faraday Soc. 25, 19 (1958)

    Article  Google Scholar 

  18. C.W. Oseen, Trans. Faraday Soc. 29, 883 (1993)

    Article  Google Scholar 

  19. H. Zocher, Trans. Faraday Soc. 29, 945 (1993)

    Article  Google Scholar 

  20. J. Nehring, A. Saupe, J. Chem. Phys. 56, 5527 (1972)

    Article  ADS  Google Scholar 

  21. M. Faetti, S. Faetti, Phys. Rev. E 57, 6741 (1998)

    Article  ADS  Google Scholar 

  22. G.P. Chen, H. Takezoe, A. Fukuda, Liq. Cryst. 5, 341 (1989)

    Article  Google Scholar 

  23. W.H. de Jeu, W.A.P. Claassen, M.J. Spruijt, Mol. Cryst. Liq. Cryst. 37, 269 (1976)

    Article  Google Scholar 

  24. N.V. Madhusudana, P.P. Karat, S. Chandrasekhar, Pramana Suppl. 1, 225 (1975)

    Google Scholar 

  25. H. Gruler, T.J. Sheffer, G. Meier, Z. Naturforsch. 27a, 966 (1972)

    ADS  Google Scholar 

  26. S. Scharkowski, H. Schmiedel, R. Srannarius, E. Weishuhn, Z. Naturforsch. 45a, 942 (1990)

    Google Scholar 

  27. M.P. Allen, D. Frenkel, Phys. Rev. A 37, R1813 (1988)

    Article  ADS  Google Scholar 

  28. M.A. Osipov, S. Hess, Liq. Cryst. 16, 845 (1994)

    Article  Google Scholar 

  29. J. Nehring, A. Saupe, J. Chem. Phys. 54, 337 (1971)

    Article  ADS  Google Scholar 

  30. J. Nehring, A. Saupe, J. Chem. Phys. 56, 5527 (1972)

    Article  ADS  Google Scholar 

  31. G. Vertogen, S.D. Flapper, C. Dullemond, J. Chem. Phys. 76, 616 (1982)

    Article  ADS  Google Scholar 

  32. G. Vertogen, Phys. Lett. A 89, 448 (1983)

    Article  ADS  Google Scholar 

  33. G. Vertogen, Physica A 117, 227 (1983)

    Article  ADS  Google Scholar 

  34. M.A. Osipov, S. Hess, Mol. Phys. 78, 1191 (1993)

    Article  ADS  Google Scholar 

  35. M.A. Osipov, S. Hess, Chem. Phys. 99, 4181 (1993)

    ADS  Google Scholar 

  36. S. Weinberg, Gravitation and Cosmology (John Wiley & Sons, New York, 1972)

  37. G. Napoli, L. Vergori, Phys. Rev. Lett. 108, 207803 (2012)

    Article  ADS  Google Scholar 

  38. C. Mostajeran, Phys. Rev. E 91, 062405 (2015)

    Article  ADS  MathSciNet  Google Scholar 

  39. C. Satiro, F. Moraes, Eur. Phys. J. E 20, 173 (2006)

    Article  Google Scholar 

  40. D. Baalss, S. Hess, Phys. Rev. Lett. 57, 86 (1986)

    Article  ADS  Google Scholar 

  41. M. Simões, M. Pazetti, EPL 92, 14001 (2010)

    Article  ADS  Google Scholar 

  42. F. Serra, Liq. Cryst. 43, 1920 (2016)

    Article  Google Scholar 

  43. D. Jesenek, S. Kralj, R. Rosso, E.G. Virga, Soft Matter 11, 2434 (2015)

    Article  ADS  Google Scholar 

  44. M.J. Stephen, J.P. Straley, Rev. Mod. Phys. 46, 617 (1974)

    Article  ADS  Google Scholar 

  45. L.R. Evangelista, I. Hibler, H. Mukai, Phys. Rev. E 58, 3245 (1998)

    Article  ADS  Google Scholar 

  46. G. Barbero, R. Barberi, in Physics of Liquid Crystalline Materials, edited by I.C. Khoo, F. Simoni (Gordon and Breach, New York, 1993)

  47. G. Barbero, L.R. Evangelista, Phys. Rev. E 56, 6189 (1997)

    Article  ADS  Google Scholar 

  48. G. Barbero, L.R. Evangelista, M. Giocondo, S. Ponti, J. Phys. II 4, 1519 (1994)

    Google Scholar 

  49. David Hilbert, Stephan Cohn-Vossen, Geometry and the Imagination, 2nd edition (Chelsea, New York, 1952) ISBN 978-0-8284-1087-8

  50. B.F. Schutz, Geometrical Methods of Mathematical Physics (Cambridge University Press, 1980)

  51. L.D. Landau, E.M. Lifshitz, Statistical Physics (Pergamon, London, 1959)

  52. D. Baalss, S. Hess, Z. Naturforsch. A: Phys. Sci. 43, 662 (1988)

    Article  ADS  Google Scholar 

  53. H. Sollich, D. Baalss, S. Hess, Mol. Cryst. Liq. Cryst. 168, 189 (1989)

    Google Scholar 

  54. S. Hess, J.F. Schwarzl, D.J. Baalss, J. Phys.: Condens. Matter 2, SA279 (1990) (Supplement)

    ADS  Google Scholar 

  55. H. Ehrentraut, S. Hess, Phys. Rev. E 51, 2203 (1995)

    Article  ADS  Google Scholar 

  56. M. Simões et al., Phys. Rev. E 75, 061710 (2007)

    Article  ADS  MathSciNet  Google Scholar 

  57. M. Simões et al., Phys. Rev. E 77, 041709 (2008)

    Article  ADS  Google Scholar 

  58. M. Simões et al., Mol. Cryst. Liq. Cryst. 576, 53 (2013)

    Article  Google Scholar 

  59. M. Simões et al., J. Chem. Phys. 137, 204905 (2012)

    Article  ADS  Google Scholar 

  60. M. Simões et al., Liq. Cryst. 38, 61 (2011)

    Article  Google Scholar 

  61. M. Simões et al., Phys. Rev. 83, 051702 (2011)

    Google Scholar 

  62. M. Simões et al., Physica A 389, 4000 (2010)

    Article  ADS  Google Scholar 

  63. M. Simões et al., Europhys. Lett. 92, 14001 (2010)

    Article  ADS  Google Scholar 

  64. M. Simões et al., Phys. Rev. E 80, 061701 (2009)

    Article  ADS  Google Scholar 

  65. M. Simões et al., Physica A 388, 3307 (2009)

    Article  ADS  Google Scholar 

  66. https://sites.google.com/site/nematiccurvature/curvature.pdf

  67. J.H. Heinbockel, Introduction to Tensor Calculus and Continuum Mechanics (Department of Mathematics and Statistics Old Dominion University, USA)

Download references

Author information

Authors and Affiliations

  1. Universidade Estadual de Londrina, Departamento de Fısica, Campus Universitário, 86051-990, Londrina (PR), Brazil

    M. Simões, W. Bertolino & T. Davincy

Authors
  1. M. Simões
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. W. Bertolino
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. T. Davincy
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to M. Simões.

Additional information

Publisher’s Note

The EPJ Publishers remain neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Simões, M., Bertolino, W. & Davincy, T. Curvature of the elastic deformations in a nematic sample. Eur. Phys. J. E 42, 59 (2019). https://doi.org/10.1140/epje/i2019-11817-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1140/epje/i2019-11817-8

Keywords

Search

Navigation