Introduction to Q-tensor theory

TL;DR

This paper provides an accessible introduction to the Q-tensor theory for modeling liquid crystals, focusing on its mathematical formulation, boundary conditions, and applications in devices, aimed at newcomers to the field.

Contribution

It offers a comprehensive beginner-friendly overview of the Q-tensor approach, including derivation of equations and boundary conditions, with practical device examples.

Findings

Development of Q-tensor model for nematic liquid crystals

Formulation of free energy and governing equations

Application to real device boundary conditions

Abstract

This paper aims to provide an introduction to a basic form of the ${\bf Q}$-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 ${\bf Q}$-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.