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Topology in Liquid Crystal Phases

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The Role of Topology in Materials

Part of the book series: Springer Series in Solid-State Sciences ((SSSOL,volume 189))

Abstract

Liquid crystals exhibit a rich set of phenomena with a geometric and topological flavour. We provide a survey of this in the nematic, smectic and cholesteric mesophases. Starting with Schlieren textures in nematics, we introduce the topological methods used to identify and classify point and line defects as well as non-singular textures. Particular attention is given to the characterisation of disclination loops and the use of the Pontryagin–Thom construction as a method of visualising complex three-dimensional textures. We illustrate these with examples drawn from recent experiments. Smectics have non-uniform ground states, which adds geometrical constraints to their topological properties. We describe the implications this has for two-dimensional smectics with the aid of a three-dimensional visualisation in terms of ‘height’ functions. Next, we describe geometrical methods for analysing line fields in terms of their curvatures and the degeneracies in them, which we call umbilics. These methods are applied to identify defects in the pitch axis in cholesterics. Finally, we provide a brief overview of the topological characterisation of knotted defect lines in nematics as well as an explicit method for constructing a director field with arbitrary knotted disclination line.

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Authors and Affiliations

  1. Department of Physics and Centre for Complexity Science, University of Warwick, Coventry, CV4 7AL, UK

    Gareth P. Alexander

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  1. Gareth P. Alexander
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Correspondence to Gareth P. Alexander .

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Editors and Affiliations

  1. Department of Physics and Astronomy, Western Kentucky University, Bowling Green, Kentucky, USA

    Sanju Gupta

  2. Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico, USA

    Avadh Saxena

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Alexander, G.P. (2018). Topology in Liquid Crystal Phases. In: Gupta, S., Saxena, A. (eds) The Role of Topology in Materials. Springer Series in Solid-State Sciences, vol 189. Springer, Cham. https://doi.org/10.1007/978-3-319-76596-9_9

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