Variational theories of liquid crystals: Materials science at many length scales

DATES: 18, 20, 21 and 22 March 2019 (4 sessions)
TIME: 10:00 - 12:30 (a total of 10 hours)
LOCATION: BCAM Seminar room

"Liquid crystals" are a broad class of materials that sit between the solid/fluid dichotomy. To the naked eye they typically appear as fluids ("Liquid"), but are structured over molecular length scales ("Crystals"). Their obvious exposure comes from their use in LCD (Liquid Crystal Display) screens, where their softness as fluids makes them easy to manipulate, while their structure over length scales comparable to the wavelength of visible light means they have useful optical properties. However such structured fluids appear everywhere, particularly in biological systems.

This course is aimed to be a broad, self-contained overview of the static/equilibrium theories of liquid crystals, typically phrased as problems in the calculus of variations (free energy minimisation). It is their very nature that liquid crystals exhibit a variety of behaviour at different length scales, and this is reflected in the wide variety of models to describe different features. This course will focus on three regimes. The first is the density functional theory, which is a statistical mechanical description appropriate at smaller length scales. Secondly, the Oseen-Frank model, which is in the realm of continuum mechanics. Thirdly, the coarsest description is the topological theory of defects, where only the singularities of such systems are considered. Though interesting in their own right, these descriptions are not independent, and we will explore their links as we proceed.


PROGRAMME:

1. A background in liquid crystals and their applications.
2. Mathematical preliminaries.
3. Density functional theory of liquid crystals: Bulk and elastic properties from pairwise interactions and asymptotics.
4. The Oseen-Frank continuum theory of nematic liquid crystals: Simple examples and deeper issues.
5. The topological theory of defects: Topological defect structures in the bulk and at surfaces.


REFERENCES:
[1] J. M. Ball. Mathematics and liquid crystals. Molec. Cryst. Liq. Cryst. 647(1):1-27, 2017. Proceedings of the 26th International Liquid Crystals Conference (ILCC 2016).
[2] P. G. De Gennes and J. Prost. The Physics of Liquid Crystals. International series of monographs on physics. Clarendon Press, 1993.
[3] N. D. Mermin. The topological theory of defects in ordered media. Rev. Modern Phys., 51(3):591-648, 1979.
[4] E. G. Virga. Variational Theories for Liquid Crystals, volume 8 of Applied Mathematics and Mathematical Computation. Chapman & Hall, London, 1994.

*Registration is free, but mandatory before March 15th: So as to inscribe go to https://bit.ly/2Ablov0 and fill the registration form. Student grants are available. Please, let us know if you need support for travel and accommodation expenses when you fill the form.

• Variational Theories Of Liquid Crystals
• Topological Defects In Variational Models From Materials Science
• BCAMCourse_2019_03_18_v2
• Examples