Machon, Thomas (2016) Aspects of geometry and topology in liquid crystalline phases. PhD thesis, University of Warwick.

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Abstract

Liquid crystals exhibit a rich set of phenomena with a geometric and topological flavour. In this thesis the study nematic and cholesteric phases of liquid crystals from the perspective of geometry and topology. We show that a global extension of the homotopy theory of defects in three dimensional nematics allows one to associate a topological ‘internal state’ to knotted and linked defects lines, such as those made experimentally. This internal state reflects the way in which the liquid crystal ‘wraps around’ the defects in the system and can be associated to baby Skyrmion tubes and relative orientations. Moving on from this we give a geometric analysis of orientational order, showing that there are a natural set of lines, realised as geometric singularities, the generalisation of the umbilical points of a surface. These lines can be identified with a number of structures seen experimentally in chiral systems such as double twist cylinders, the centres of Skyrmions and λ lines. We describe the theoretical properties of these lines, in particular how they reflect the topology of the underlying orientation by furnishing a representation of (four times) the Euler class of the orthogonal 2-plane bundle. We then combine these two theoretical structures to give a description of the cholesteric phase. In particular, our description allows all structures to be derived from the orientational order, and we show how previous theories and interesting structures such as τ lines fit into this scheme. We extend this theory by making the connection between cholesteric liquid crystals and contact structures, where we show that one can obtain additional topological distinctions between field configurations. We discuss topological aspects of colloidal inclusions and show that non-orientable colloids can precipitate the creation of knotted defect lines in silico. Finally,we show how one can use Milnor fibrations to give explicit formulae for director fields that contain knotted defect lines.

Item Type:	Thesis (PhD)
Subjects:	Q Science > QA Mathematics
Library of Congress Subject Headings (LCSH):	Liquid crystals
Official Date:	March 2016
Dates:	Date Event March 2016 Submitted
Institution:	University of Warwick
Theses Department:	Centre for Complexity Science
Thesis Type:	PhD
Publication Status:	Unpublished
Supervisor(s)/Advisor:	Alexander, Gareth P. ; Ravnik, Miha
Extent:	xvi, 151 leaves : illustrations
Language:	eng

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