Skip to main content
SpringerLink
Log in
Book cover

Mathematical Thermodynamics of Complex Fluids pp 1–46Cite as

Liquid Crystals and Their Defects

  • Chapter
  • First Online:

Part of the book series: Lecture Notes in Mathematics ((LNMCIME,volume 2200))

Abstract

These lectures describe some classical models of liquid crystals, the relations between them, and the different ways in which these models describe defects.

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

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   49.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Notes

  1. 1.

    Here we consider only the shape of M as being important. More generally we could require the invariance of a vector u = u(x), x ∈ M, of additional molecular variables (such as mass or charge density), defining corresponding isotropy groups \(\tilde{G}_{M} =\{ \mathbf{R} \in O(3): \mathbf{R}M = M,\mathbf{u}(\mathbf{R}\mathbf{x}) = \mathbf{u}(\mathbf{x})\mbox{ for all }\mathbf{x} \in M\},\tilde{G}_{M}^{+} =\{ \mathbf{R} \in SO(3): \mathbf{R}M = M,\mathbf{u}(\mathbf{R}\mathbf{x}) = \mathbf{u}(\mathbf{x})\mbox{ for all }\mathbf{x} \in M\}\).

  2. 2.

    For example, in the case of the ellipsoid of revolution \(M =\{ \mathbf{x} = (x_{1},x_{2},x_{3}): \frac{x_{1}^{2}} {a^{2}} + \frac{x_{2}^{2}+x_{3}^{2}} {b^{2}} <1\}\), with semimajor axes a > 0, b > 0, a ≠ b, if \(\hat{\mathbf{R}}M = M\) then \(\hat{\mathbf{R}}\partial M = \partial M\), and since \(\vert \pm \hat{\mathbf{R}}a\mathbf{e}_{1}\vert = a\) and the only points of ∂M distant a from 0 are ± a e 1 we have that \(\hat{\mathbf{R}}\mathbf{e}_{1} = \pm \mathbf{e}_{1}\). Conversely, if \(\hat{\mathbf{R}}\mathbf{e}_{1} = \pm \mathbf{e}_{1}\) then it is easily checked that \(\hat{\mathbf{R}}M = M\).

  3. 3.

    Similarly, for a sixth order polynomial ψ B is a linear combination of 1, tr Q 2, tr Q 3, tr Q 2tr Q 3, (tr Q 2)3, (tr Q 3)2; see, for example, [47].

  4. 4.

    Since the L i are not dimensionless, some care is required in interpreting what it means for them to be small (see Gartland [44]).

  5. 5.

    A related, and even harder, open problem is that of proving that minimizers \(\mathbf{y}^{{\ast}}:\varOmega \rightarrow \mathbb{R}^{3}\) of the elastic energy I(y) = ∫ Ω W(∇y(x)) d x in nonlinear elasticity under the non-interpenetration hypothesis W(A) → ∞ as detA → 0+ satisfy det∇y ∗(x) ≥ δ > 0 a.e. in Ω.

  6. 6.

    This can be verified by separately estimating ∇n in neighbourhoods of the points where it is not smooth, namely x = 0, points on a cube edge, and corners of the cube.

References

  1. L. Ambrosio, E.G. Virga, A boundary value problem for nematic liquid crystals with a variable degree of orientation. Arch. Ration. Mech. Anal. 114(4), 335–347 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  2. L. Ambrosio, N. Fusco, D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems. Oxford Mathematical Monographs (Oxford University Press, Oxford, 2000)

    MATH  Google Scholar 

  3. D.R. Anderson, D.E. Carlson, E. Fried, A continuum-mechanical theory for nematic elastomers. J. Elast. 56(1), 33–58 (1999/2000)

    Google Scholar 

  4. J.M. Ball, Mathematics of liquid crystals. Mol. Cryst. Liq. Cryst. 647, 1–27 (2017)

    Article  MATH  Google Scholar 

  5. J.M. Ball, S.J. Bedford, Surface discontinuities of the director in liquid crystal theory (in preparation)

    Google Scholar 

  6. J.M. Ball, S.J. Bedford, Discontinuous order parameters in liquid crystal theories. Mol. Cryst. Liq. Cryst. 612(1), 1–23 (2015)

    Article  Google Scholar 

  7. J.M. Ball, A. Majumdar, Passage from the mean-field Maier-Saupe to the continuum Landau - de Gennes theory for nematic liquid crystals (in preparation)

    Google Scholar 

  8. J.M. Ball, A. Majumdar, Nematic liquid crystals: from Maier-Saupe to a continuum theory. Mol. Cryst. Liq. Cryst. 525, 1–11 (2010)

    Article  Google Scholar 

  9. J.M. Ball, V.J. Mizel, Singular minimizers for regular one-dimensional problems in the calculus of variations. Bull. Am. Math. Soc. 11, 143–146 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  10. J.M. Ball, V.J. Mizel, One-dimensional variational problems whose minimizers do not satisfy the Euler-Lagrange equations. Arch. Ration. Mech. Anal. 90, 325–388 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  11. J.M. Ball, A. Zarnescu, Orientability and energy minimization in liquid crystal models. Arch. Ration. Mech. Anal. 202, 493–535 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  12. J.M. Ball, A. Zarnescu, Partial regularity and smooth topology-preserving approximations of rough domains. Calc. Var. Partial Differ. Equ. 56(1), 13 (2017)

    Google Scholar 

  13. R. Barberi, F. Ciuchi, G.E. Durand, M. Iovane, D. Sikharulidze, A.M. Sonnet, E.G. Virga, Electric field induced order reconstruction in a nematic cell. Eur. Phys. J. E 13, 61–71 (2004)

    Article  Google Scholar 

  14. G. Barbero, R. Barberi, Critical thickness of a hybrid aligned nematic liquid crystal cell. J. Phys. 44, 609–616 (1983)

    Article  Google Scholar 

  15. P. Bauman, D. Phillips, Regularity and the behavior of eigenvalues for minimizers of a constrained Q-tensor energy for liquid crystals. Calc. Var. Partial Differ. Equ. 55(4), Paper No. 81, 22 (2016)

    Google Scholar 

  16. P. Bauman, J. Park, D. Phillips, Analysis of nematic liquid crystals with disclination lines. Arch. Ration. Mech. Anal. 205(3), 795–826 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  17. S.J. Bedford, Calculus of variations and its application to liquid crystals. Ph.D. thesis, Mathematical Institute, University of Oxford (2015)

    Google Scholar 

  18. S.J. Bedford, Function spaces for liquid crystals. Arch. Ration. Mech. Anal. 219(2), 937–984 (2016)

    Article  MATH  MathSciNet  Google Scholar 

  19. D.W. Berreman, S. Meiboom, Tensor representation of Oseen-Frank strain energy in uniaxial cholesterics. Phys. Rev. A 30(4), 1955 (1984)

    Google Scholar 

  20. F. Bethuel, D. Chiron, Some questions related to the lifting problem in Sobolev spaces, in Perspectives in Nonlinear Partial Differential Equations. Contemporary Mathematics (American Mathematical Society, Providence, RI, 2007), pp. 125–152

    Google Scholar 

  21. K. Bhattacharya, Microstructure of Martensite (Oxford University Press, Oxford, 2003)

    MATH  Google Scholar 

  22. P. Biscari, G.G. Peroli, A hierarchy of defects in biaxial nematics. Commun. Math. Phys. 186(2), 381–392 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  23. F. Bisi, E.C. Gartland, R. Rosso, E.G. Virga, Order reconstruction in frustrated nematic twist cells. Phys. Rev. E 68, 021707 (2003)

    Article  Google Scholar 

  24. P. Bladon, E.M. Terentjev, M. Warner, Transitions and instabilities in liquid crystal elastomers. Phys. Rev. E 47, R3838–3839 (1993)

    Article  Google Scholar 

  25. B. Bourdin, G.A. Francfort, J.-J. Marigo, The variational approach to fracture. J. Elast. 91, 5–148 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  26. J. Bourgain, H. Brezis, P. Mironescu, Lifting in Sobolev spaces. J. Anal. Math. 80, 37–86 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  27. H. Brezis, J.-M. Coron, E.H. Lieb, Harmonic maps with defects. Commun. Math. Phys. 107(4), 649–705 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  28. M.C. Calderer, C.A. Garavito Garzón, B. Yan, A Landau–de Gennes theory of liquid crystal elastomers. Discrete Contin. Dyn. Syst. Ser. S 8(2), 283–302 (2015)

    MATH  MathSciNet  Google Scholar 

  29. G. Canevari, Line defects in the small elastic constant limit of a three-dimensional Landau-de Gennes model. Arch. Ration. Mech. Anal. 223, 1–86 (2016)

    MathSciNet  Google Scholar 

  30. G. Carbone, G. Lombardo, R. Barberi, Mechanically induced biaxial transition in a nanoconfined nematic liquid crystal with a topological defect. Phys. Rev. Lett. 103, 167801 (2009)

    Article  Google Scholar 

  31. R. Cohen, M. Taylor, Weak stability of the map x∕ | x | for liquid crystal functionals. Commun. Partial Differ. Equ. 15(5), 675–692 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  32. D. Coursault, B.H. Ibrahim, L. Pelliser, B. Zappone, A. de Martino, E. Lacaze, B. Gallas, Modeling the optical properties of self-organized arrays of liquid crystal defects. Opt. Express 22, 023182 (2014)

    Article  Google Scholar 

  33. D. Coursault, B. Zappone, A. Coati, A. Boulaoued, L. Pelliser, D. Limagne, N. Boudet, B.H. Ibrahim, A. De Martino, M. Alba et al., Self-organized arrays of dislocations in thin smectic liquid crystal films. Soft Matter 12(3), 678–688 (2016)

    Article  Google Scholar 

  34. T.A. Davis, E.C. Gartland Jr., Finite element analysis of the Landau-de Gennes minimization problem for liquid crystals. SIAM J. Numer. Anal. 35(1), 336–362 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  35. P.G. de Gennes, Short range order effects in the isotropic phase of nematics and cholesterics. Mol. Cryst. Liq. Cryst. 12(3), 193–214 (1971)

    Article  Google Scholar 

  36. E. de Miguel, E. Martín del Rio, J.T. Brown, M.P. Allen, Effect of the attractive interactions on the phase behavior of the Gay–Berne liquid crystal model. J. Chem. Phys. 105(10), 4234–4249 (1996)

    Article  Google Scholar 

  37. A. DeSimone, G. Dolzmann, Macroscopic response of nematic elastomers via relaxation of a class of SO(3)-invariant energies. Arch. Ration. Mech. Anal. 161(3), 181–204 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  38. H.M. Edwards, Galois Theory. Graduate Texts in Mathematics, vol. 101 (Springer, New York, 1984)

    Google Scholar 

  39. J.L. Ericksen, Inequalities in liquid crystal theory. Phys. Fluids (1958–1988) 9(6), 1205–1207 (1966)

    Google Scholar 

  40. J.L. Ericksen, Liquid crystals with variable degree of orientation. Arch. Ration. Mech. Anal. 113(2), 97–120 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  41. L.C. Evans, O. Kneuss, H. Tran, Partial regularity for minimizers of singular energy functionals, with application to liquid crystal models. Trans. Am. Math. Soc. 368(5), 3389–3413 (2016)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  43. G.A. Francfort, J.-J. Marigo, Revisiting brittle fracture as an energy minimization problem. J. Mech. Phys. Solids 46, 1319–1342 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  44. E.C. Gartland, Scalings and limits of the Landau-de Gennes model for liquid crystals: a comment on some recent analytical papers (2015). arXiv:1512.08164

    Google Scholar 

  45. E.C. Gartland, S. Mkaddem, On the local instability of radial hedgehog configurations in nematic liquid crystals under Landau-de Gennes free-energy models. Phys. Rev. E. 59, 563–567 (1999)

    Article  Google Scholar 

  46. J. Gay, B. Berne, Modification of the overlap potential to mimic a linear site–site potential. J. Chem. Phys. 74(6), 3316–3319 (1981)

    Article  Google Scholar 

  47. E.F. Gramsbergen, L. Longa, W.H. de Jeu, Landau theory of the nematic-isotropic phase transition. Phys. Rep. 135(4), 195–257 (1986)

    Article  Google Scholar 

  48. R. Hardt, F.-H. Lin, Mappings minimizing the L p norm of the gradient. Commun. Pure Appl. Math. 40(5), 555–588 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  49. R. Hardt, D. Kinderlehrer, F.-H. Lin, Stable defects of minimizers of constrained variational principles. Ann. Inst. H. Poincaré Anal. Non Linéaire 5(4), 297–322 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  50. F. Hélein, Minima de la fonctionnelle énergie libre des cristaux liquides. C. R. Acad. Sci. Paris Sér. I Math. 305(12), 565–568 (1987)

    MATH  MathSciNet  Google Scholar 

  51. D. Henao, A. Majumdar, Symmetry of uniaxial global Landau-de Gennes minimizers in the theory of nematic liquid crystals. SIAM J. Math. Anal. 44(5), 3217–3241 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  52. D. Henao, A. Majumdar, A. Pisante, Uniaxial versus biaxial character of nematic equilibria in three dimensions. Calc. Var. Partial Differ. Equ. 56(2), 55 (2017)

    Google Scholar 

  53. R. Ignat, L. Nguyen, V. Slastikov, A. Zarnescu, Stability of the melting hedgehog in the Landau–de Gennes theory of nematic liquid crystals. Arch. Ration. Mech. Anal. 215(2), 633–673 (2015)

    Article  MATH  MathSciNet  Google Scholar 

  54. J. Katriel, G.F. Kventsel, G.R. Luckhurst, T.J. Sluckin, Free energies in the Landau and molecular field approaches. Liq. Cryst. 1, 337–355 (1986)

    Article  Google Scholar 

  55. D. Kinderlehrer, B. Ou, Second variation of liquid crystal energy at x∕ | x |. Proc. R. Soc. Lond. Ser. A 437(1900), 475–487 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  56. M. Kléman, Defects in liquid crystals. Rep. Prog. Phys. 52(5), 555–654 (1989)

    Article  MathSciNet  Google Scholar 

  57. S. Kralj, E.G. Virga, Universal fine structure of nematic hedgehogs. J. Phys. A 34(4), 829–838 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  58. I. Kundler, H. Finkelmann, Strain-induced director reorientation in nematic liquid single crystal elastomers. Macromol. Rapid Commun. 16, 679–686 (1995)

    Article  Google Scholar 

  59. E. Lacaze, J.-P. Michel, M. Alba, M. Goldmann, Planar anchoring and surface melting in the smectic-A phase. Phys. Rev. E 76, 041702 (2007)

    Article  Google Scholar 

  60. X. Lamy, Some properties of the nematic radial hedgehog in the Landau–de Gennes theory. J. Math. Anal. Appl. 397(2), 586–594 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  61. X. Lamy, Bifurcation analysis in a frustrated nematic cell. J. Nonlinear Sci. 24, 1197–1230 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  62. F.-H. Lin, A remark on the map x∕ | x |. C. R. Acad. Sci. Paris Sér. I Math. 305(12), 529–531 (1987)

    MATH  MathSciNet  Google Scholar 

  63. H. Liu, H. Zhang, P. Zhang, 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)

    Article  MATH  MathSciNet  Google Scholar 

  64. L. Longa, D. Monselesan, H. Trebin, An extension of the Landau-Ginzburg-de Gennes theory for liquid crystals. Liq. Cryst. 2, 769–796 (1987)

    Article  Google Scholar 

  65. G.R. Luckhurst, P.S.J. Simmonds, Computer simulation studies of anisotropic systems. Mol. Phys. 80(2), 233–252 (1993)

    Article  Google Scholar 

  66. A. Majumdar, The radial-hedgehog solution in Landau-de Gennes’ theory for nematic liquid crystals. Eur. J. Appl. Math. 23(1), 61–97 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  67. A. Majumdar, A. Zarnescu, Landau-De Gennes theory of nematic liquid crystals: the Oseen-Frank limit and beyond. Arch. Ration. Mech. Anal. 196(1), 227–280 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  68. L.R. Mead, N. Papanicolaou, Maximum entropy in the problem of moments. J. Math. Phys. 25(8), 2404–2417 (1984)

    Article  MathSciNet  Google Scholar 

  69. N.D. Mermin, The topological theory of defects in ordered media. Rev. Mod. Phys. 51, 591–648 (1979)

    Article  MathSciNet  Google Scholar 

  70. J.-P. Michel, E. Lacaze, M. Alba, M. de Boissieu, M. Gailhanou, M. Goldmann, Optical gratings formed in thin smectic films frustrated on a single crystalline substrate. Phys. Rev. E 70, 011709 (2004)

    Article  Google Scholar 

  71. J.-P. Michel, E. Lacaze, M. Goldmann, M. Gailhanou, M. de Boissieu, M. Alba, Structure of smectic defect cores: X-ray study of 8CB liquid crystal ultrathin films. Phys. Rev. Lett. 96, 027803 (2006)

    Article  Google Scholar 

  72. N. Mottram, C. Newton, An introduction to Q-tensor theory (2014). arXiv:1409.3542

    Google Scholar 

  73. L. Nguyen, A. Zarnescu, Refined approximation for minimizers of a Landau-de Gennes energy functional. Calc. Var. Partial Differ. Equ. 47(1–2), 383–432 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  74. B. Ou, Uniqueness of x∕ | x | as a stable configuration in liquid crystals. J. Geom. Anal. 2(2), 183–194 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  75. M.R. Pakzad, T. Rivière, Weak density of smooth maps for the Dirichlet energy between manifolds. Geom. Funct. Anal. 13(1), 223–257 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  76. P. Palffy-Muhoray, E.C. Gartland, J.R. Kelly, A new configurational transition in inhomogeneous nematics. Liq. Cryst. 16, 713–718 (1994)

    Article  Google Scholar 

  77. A. Pizzirusso, R. Berardi, L. Muccioli, M. Riccia, C. Zannoni, Predicting surface anchoring: molecular organization across a thin film of 5CB liquid crystal on silicon. Chem. Sci. 3, 573–579 (2012)

    Article  Google Scholar 

  78. A. Poniewierski, T. Sluckin, On the free energy density of non-uniform nematics. Mol. Phys. 55(5), 1113–1127 (1985)

    Article  Google Scholar 

  79. A. Rapini, M. Papoular, Distorsion d’une lamelle nématique sous champ magnétique conditions d’ancrage aux parois. Le Journal de Physique Colloques 30(C4), C4–54 (1969)

    Article  Google Scholar 

  80. M. Ricci, M. Mazzeo, R. Berardi, P. Pasini, C. Zannoni, A molecular level simulation of a twisted nematic cell. Faraday Discuss. 144, 171–185 (2010)

    Article  Google Scholar 

  81. R.T. Rockafellar, Convex Analysis (Princeton University Press, Princeton, NJ, 1970)

    Book  MATH  Google Scholar 

  82. K. Schiele, S. Trimper, On the elastic constants of a nematic liquid crystal. Phys. Status Solidi (b) 118(1), 267–274 (1983)

    Google Scholar 

  83. R. Schoen, K. Uhlenbeck, A regularity theory for harmonic maps. J. Differ. Geom. 17(2), 307–335 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  84. N. Schopohl, T.J. Sluckin, Defect core structure in nematic liquid crystals. Phys. Rev. Lett. 59(22), 2582 (1987)

    Google Scholar 

  85. T.J. Sluckin, D.A. Dunmur, H. Stegemeyer, Crystals That Flow. Liquid Crystals Series (Taylor & Francis, London, 2004)

    Google Scholar 

  86. G.F. Smith, On isotropic functions of symmetric tensors, skew-symmetric tensors and vectors. Int. J. Eng. Sci. 9, 899–916 (1971)

    Article  MATH  MathSciNet  Google Scholar 

  87. I.W. Stewart, The Static and Dynamic Theory of Liquid Crystals (Taylor and Francis, London, 2004)

    Google Scholar 

  88. V. Šverák, X. Yan, A singular minimizer of a smooth strongly convex functional in three dimensions. Calc. Var. Partial Differ. Equ. 10, 213–221 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  89. J.M. Taylor, Maximum entropy methods as the bridge between microscopic and macroscopic theory. J. Stat. Phys. 164(6), 1429–1459 (2016)

    Article  MATH  MathSciNet  Google Scholar 

  90. C. Truesdell, W. Noll, The non-linear field theories of mechanics, in Handbuch der Physik, ed. by S. Flügge, vol. III/3 (Springer, Berlin, 1965)

    Google Scholar 

  91. E.G. Virga, Variational Theories for Liquid Crystals. Applied Mathematics and Mathematical Computation, vol. 8 (Chapman & Hall, London, 1994)

    Google Scholar 

  92. M.A.C. Vollmer, Critical points and bifurcations of the three-dimensional Onsager model for liquid crystals. Arch. Ration. Mech. Anal. 226(2), 851–922 (2017)

    Article  MATH  MathSciNet  Google Scholar 

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

    Google Scholar 

  94. C. Zannoni, Molecular design and computer simulations of novel mesophases. J. Mater. Chem. 11, 2637–2646 (2001)

    Article  Google Scholar 

  95. B. Zappone, E. Lacaze, Surface-frustrated periodic textures of smectic-A liquid crystals on crystalline surfaces. Phys. Rev. E 78, 061704 (2008)

    Article  Google Scholar 

  96. B. Zappone, E. Lacaze, H. Hayeb, M. Goldmann, N. Boudet, P. Barois, M. Alba, Self-ordered arrays of linear defects and virtual singularities in thin smectic-A films. Soft Matter 7, 1161–1167 (2011)

    Article  Google Scholar 

  97. B. Zappone, C. Meyer, L. Bruno, E. Lacaze, Periodic lattices of frustrated focal conic defect domains in smectic liquid crystal films. Soft Matter 8, 4318–4326 (2012)

    Article  Google Scholar 

  98. S. Zhang, I.A. Kinloch, A.H. Windle, Mesogenicity drives fractionation in lyotropic aqueous suspensions of multiwall carbon nanotubes. Nano Lett. 6(3), 568–572 (2006). PMID: 16522064

    Article  Google Scholar 

Download references

Acknowledgements

This research was supported by EPSRC (GRlJ03466, the Science and Innovation award to the Oxford Centre for Nonlinear PDE EP/E035027/1, and EP/J014494/1), the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007–2013) / ERC grant agreement no 291053 and by a Royal Society Wolfson Research Merit Award. I offer warm thanks to Elisabetta Rocca and Eduard Feireisl for organizing such an interesting programme, to the other lecturers and participants for the lively interaction, and to Elvira Mascolo and the CIME staff for the smooth and friendly organization in a beautiful location.

I am indebted to my collaborators Apala Majumdar, Arghir Zarnescu and Stephen Bedford for many discussions related to the material in these notes, and to Apala Majumdar, Epifanio Virga, Claudio Zannoni and Arghir Zarnescu for kindly reading the notes and pointing out various errors and infelicitudes.

Author information

Authors and Affiliations

  1. Oxford Centre for Nonlinear PDE, Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, OX2 6GG, Oxford, UK

    John M. Ball

Authors
  1. John M. Ball
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to John M. Ball .

Editor information

Editors and Affiliations

  1. Institute of Mathematics AS CR , Prague, Czech Republic

    Eduard Feireisl

  2. Dipartimento di Matematica, University of Pavia, Pavia, Italy

    Elisabetta Rocca

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing AG

About this chapter

Cite this chapter

Ball, J.M. (2017). Liquid Crystals and Their Defects. In: Feireisl, E., Rocca, E. (eds) Mathematical Thermodynamics of Complex Fluids. Lecture Notes in Mathematics(), vol 2200. Springer, Cham. https://doi.org/10.1007/978-3-319-67600-5_1

Download citation

Publish with us

Policies and ethics

Search

Navigation