Skip to main content

Critical Points and Bifurcations of the Three-Dimensional Onsager Model for Liquid Crystals

Abstract

We study the bifurcation diagram of the Onsager free-energy functional for liquid crystals with orientation parameter on the sphere. In particular, we concentrate on the bifurcations from the isotropic solution for a general class of two-body interaction potentials including the Onsager kernel. Reformulating the problem as a non-linear eigenvalue problem for the kernel operator, we prove that spherical harmonics are the corresponding eigenfunctions and we present a direct relationship between the coefficients of the Taylor expansion of this class of interaction potentials and their eigenvalues. We find explicit expressions for all bifurcation points corresponding to bifurcations from the isotropic state of the Onsager free-energy functional equipped with the Onsager interaction potential. A substantial amount of our analysis is based on the use of spherical harmonics and a special algorithm for computing expansions of products of spherical harmonics in terms of spherical harmonics is presented. Using a Lyapunov–Schmidt reduction, we derive a bifurcation equation depending on five state variables. The dimension of this state space is further reduced to two dimensions by using the rotational symmetry of the problem and the invariant theory of groups. On the basis of these results, we show that the first bifurcation from the isotropic state of the Onsager interaction potential is a transcritical bifurcation and that the corresponding solution is uniaxial. In addition, we prove some global properties of the bifurcation diagram such as the fact that the trivial solution is the unique local minimiser if the bifurcation parameter is high, that it is not a local minimiser if the bifurcation parameter is small, the boundedness of all equilibria of the functional and that the bifurcation branches are either unbounded or that they meet another bifurcation branch.

References

  1. Abramovich, Y.A., Aliprantis, C.D.: An Invitation to Operator Theory, Vol. 50. American Mathematical Society, Providence, 2002

  2. Abud M., Sartori G.: The geometry of spontaneous symmetry breaking. Ann. Phys. 150, 307–372 (1983)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  3. Ambrosetti, A., Prodi, G.: A primer of nonlinear analysis. Cambridge Studies in Advanced Mathematics, Vol. 34. Cambridge University Press, Cambridge, 1993

  4. Atkinson, K., Han, W.: Spherical harmonics and approximations on the unit sphere: an introduction. Lecture Notes in Mathematics, Vol. 2044. Springer, Heidelberg, 2012. doi:10.1007/978-3-642-25983-8

  5. Bailey, W.N.: Cambridge Tracts in Mathematics and Mathematical Physics, No. 32, Generalized Hypergeometric Series. Stechert-Hafner Service Agency, New York, 1964

  6. Bröcker, T., Tom Dieck, T.: Representations of compact Lie groups. Graduate Texts in Mathematics, Vol.98. Springer, New York, 1985

  7. Chen W., Li C., Wang G.: On the stationary solutions of the 2D Doi–Onsager model. Nonlinear Anal. Theor. 73(8), 2410–2425 (2010)

    MathSciNet  Article  MATH  Google Scholar 

  8. Chen, Y.C.: Singularity theory and nonlinear bifurcation analysis. (Eds. Fu Y.B. and Ogden R.W.) Nonlinear Elasticity: Theory and Applications, London Mathematical Society Lecture Note Series, Vol. 283. Cambridge University Press, Cambridge, 305–344, 2001

  9. Chossat, P., Lauterbach, R.: Methods in equivariant bifurcations and dynamical systems. Advanced Series in Nonlinear Dynamics, Vol. 15. World Scientific Publishing Co. Inc., River Edge, 2000

  10. Constantin P., Kevrekidis I.G., Titi E.S.: Asymptotic states of a Smoluchowski equation. Arch. Ration. Mech. Anal. 174(3), 365–384 (2004)

    MathSciNet  Article  MATH  Google Scholar 

  11. Elliott, C.: Theory of PDEs. Lecture Notes. University of Warwick, Coventry

  12. Fatkullin I., Slastikov V.: Critical points of the Onsager functional on a sphere. Nonlinearity 18(6), 2565–2580 (2005)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  13. Freiser M.J.: Ordered states of a nematic liquid. Phys. Rev. Lett. 24(2), 1041–1043 (1970)

    ADS  Article  Google Scholar 

  14. Friedrich, T.: On types of non-integrable geometries. Proceedings of the 22nd Winter School “Geometry and Physics” (Srní, 2002), Vol. 71, 99–113, 2003

  15. Garrett, P.: Lecture notes on harmonic analysis on spheres 1, 2011. http://www.math.umn.edu/~garrett/m/mfms/notes_c/spheres_I.pdf

  16. Garrett, P.: Lecture notes on harmonic analysis on spheres 2, 2011. http://www.math.umn.edu/~garrett/m/mfms/notes_c/spheres_II.pdf

  17. Hussain, S.A.: Properties of spherical harmonics, 2014. https://sahussaintu.files.wordpress.com/2014/03/spherical_harmonics.pdf

  18. Kayser R.F., Raveché H.J.: Bifurcations in Onsager’s model of isotropic-nematic transition. Phys. Rev. A 17, 2067–2072 (1978)

    ADS  Article  Google Scholar 

  19. Liu H., Zhang H., Zhang P.: Axial symmetry and classification of stationary solutions of Doi–Onsager equation on the sphere with Maier–Saupe potential. Commun. Math. Sci. 3(2), 201–218 (2005)

    MathSciNet  Article  MATH  Google Scholar 

  20. Lucia M., Vukadinovic J.: Exact multiplicity of nematic states for an Onsager model. Nonlinearity 23(12), 3157–3185 (2010)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  21. Maier W., Saupe A.: Eine einfache molekulare Theorie des nematischen kristallinflüssigen Zustandes. Zeitschrift für Naturforschung 13, 564 (1958)

    ADS  Google Scholar 

  22. Marinucci, D., Peccati, G.: Random fields on the sphere. London Mathematical Society Lecture Note Series, Vol. 389. Cambridge University Press, Cambridge, 2011

  23. Niksirat, M.A., Yu, X.: On stationary solutions of the 2D Doi-Onsager model. ArXiv e-prints 2015

  24. Olver, F.W.J., Lozier, D.W., Boisvert, R.F., Clark, C.W.: NIST Handbook of Mathematical Functions. Cambridge University Press, Cambridge, 2010

  25. Onsager, L.: The Effects of Shape on the Interaction of Collodial Particles. Sterling Chemistry Laboratory, Yale University, New Haven, 1949

  26. Palais R.S.: The principle of symmetric criticality. Commun. Math. Phys. 69, 19–30 (1979)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  27. Rabinowitz P.H.: Some global results for nonlinear eigenvalue problems. J. Funct. Anal. 7, 487–513 (1971)

    MathSciNet  Article  MATH  Google Scholar 

  28. Schwarz G.W.: Smooth functions invariant under the action of a compact Lie group. Topology 14, 63–68 (1975)

    MathSciNet  Article  MATH  Google Scholar 

  29. Truesdell, C., Noll, W.: The Non-Linear Field Theories of Mechanics, 3rd edn. Springer, New York, 2004

  30. Vollmer, M.A.C.: Critical points and bifurcations of the three-dimensional Onsager model for liquid crystals, 2015. http://arxiv.org/abs/1509.02469

  31. Wachsmuth, J.: Suspensions of rod-like molecules: the istropic–nematic phase transition and flow alignment in 2-d. Master’s thesis, University of Bonn 2006. Unpublished

  32. Wang H., Zhou H.: Multiple branches of ordered states of polymer ensembles with the Onsager excluded volume potential. Phys. Lett. Sect. A 372(19), 3423–3428 (2008)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  33. Wang H., Zhou H.: Phase diagram of nematic polymer monolayers with the Onsager interaction potential. J. Comput. Theor. Nanosci. 7(4), 738–755 (2010)

    MathSciNet  Article  Google Scholar 

  34. Weisstein, E.W.: Associated Legendre polynomials. http://mathworld.wolfram.com/AssociatedLegendrePolynomial.html. From MathWorld-A Wolfram Web Resource

  35. Weisstein, E.W.: Spherical harmonics. http://mathworld.wolfram.com/SphericalHarmonic.html. From MathWorld-A Wolfram Web Resource

  36. Weisstein, E.W.: Spherical harmonics. http://mathworld.wolfram.com/LegendrePolynomial.html. From MathWorld-A Wolfram Web Resource

  37. Weisstein, E.W.: CRC Concise Encyclopedia of Mathematics, 2nd edn. Chapman Hall/CRC, Boca Raton, 2002

  38. Whittaker, E.T., Watson, G.N.: A Course of Modern Analysis. Cambridge University Press, Cambridge, 1996

  39. Wiggins, S.: Introduction to applied nonlinear dynamical systems and chaos. In:Marsden, J.E., Sirovich, L.,Antman, S.S. (eds) Texts in Applied Mathematics, Vol. 2. Springer, New York, 1990

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Michaela A. C. Vollmer.

Additional information

Communicated by E. G. Virga

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Vollmer, M.A.C. Critical Points and Bifurcations of the Three-Dimensional Onsager Model for Liquid Crystals. Arch Rational Mech Anal 226, 851–922 (2017). https://doi.org/10.1007/s00205-017-1146-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00205-017-1146-8