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Liquid Crystals and Their Defects

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Mathematical Thermodynamics of Complex Fluids

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.

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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.

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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.

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    John M. Ball

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    Eduard Feireisl

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

    Elisabetta Rocca

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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

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