Introduction to Q-tensor theory

Abstract

This paper aims to provide an introduction to a basic form of the $\mathbf{Q}$-tensor approach to modelling liquid crystals, which has seen increased interest in recent years. The increase in interest in this type of modelling approach has been driven by investigations into the fundamental nature of defects and new applications of liquid crystals such as bistable displays and colloidal systems for which a description of defects and disorder is essential. The work in this paper is not new research, rather it is an introductory guide for anyone wishing to model a system using such a theory. A more complete mathematical description of this theory, including a description of flow effects, can be found in numerous sources but the books by Virga and Sonnet and Virga are recommended. More information can be obtained from the plethora of papers using such approaches, although a general introduction for the novice is lacking. The first few sections of this paper will detail the development of the $\mathbf{Q}$-tensor approach for nematic liquid crystalline systems and construct the free energy and governing equations for the mesoscopic dependent variables. A number of device surface treatments are considered and theoretical boundary conditions are specified for each instance. Finally, an example of a real device is demonstrated.

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• Boundary bubbling analysis of approximate harmonic maps under either weak or strong anchoring conditions in dimension two
• Discontinuous order parameters in liquid crystal theories
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• Liquid crystal defects in the Landau-de Gennes theory in two dimensions-beyond the one-constant approximation
• Higher dimensional Ginzburg-Landau equations under weak anchoring boundary conditions
• Effective models for nematic liquid crystals composites with ferromagnetic inclusions
• Remarks on Uniaxial Solutions in the Landau-de Gennes Theory
• Nematic equilibria on a two-dimensional annulus: defects and energies
• From the Q-tensor flow for the liquid crystal to the Harmonic map flow
• Dynamic Cubic Instability in a 2D Q-tensor Model for Liquid Crystals