# Applied Mathematics and Statistics Research Unit

## Applications in liquid crystals

Liquid crystals are anisotropic fluids made up of elongated molecules which have an average molecular axis that aligns along a common direction in space which is usually denoted by the unit vector n, called the director. Smectic liquid crystals are layered materials with a well-defined interlayer distance. In equilibrium, smectic liquid crystals form equidistant parallel layers in which; the director is parallel to the layer normal a (smectic A) [1,2] or the director makes an angle $\theta$ to the layer normal (smectic C) [2]. Due to their natural affinity to align at high speeds with electric and magnetic fields, liquid crystals are used throughout the world in displays such as calculators, dashboards, monitors and televisions. Evidence also suggests that smectic liquid crystals are similar to biological lamellar systems (such as lipids), and hence the mathematical modelling of smectics takes similar forms to that of some biological membranes. Furthermore, liquid crystals can also be used as particle and wave sensors due to their optical properties.

Due to their molecular structure, smectic liquid crystals are fascinating materials to study and often provide unexpected results when considered in physical regimes. Current and future work in modelling smectic liquid crystals has included Couette flow [4], Poiseuille flow [5], investigations into anchoring (boundary conditions) strengths [6], investigations into cylindrically layered liquid crystals and their stability [7,8], acoustic waves in liquid crystals [9] and nonlinear modeling of layer normal and director orientation under applied strong anchoring [10].

PhD student, Ayad Al Sallo, is currently researching the effect of molecular anchoring on planar and cylindrically layered smectic liquid crystals. This project is a continuation of previous research [3,10] and will illustrate the significant differences between planar and cylindrical layered liquid crystals.

References:

[1] P. G. de Gennes and J. Prost The Physics of Liquid Crystals, Oxford University Press, Oxford, second edition, 1993.

[2] I. W. Stewart, The Static and Dynamic Continuum Theory of Liquid Crystals, Taylor and Francis, London and New York, 2004.

[3] I. W. Stewart, The alignment of smectic A liquid crystals with director tilt on the boundaries, J. Phys. A: Math. Theor., 40:5297, 2007.

[4] A. J. Walker and I. W. Stewart, Couette flow of a smectic A liquid crystal, J. Phys.: Condens. Matter, 21(155101), 2009.

[5] A. J. Walker and I. W. Stewart., Poiseuille flow of a smectic A liquid crystal, Int. J. Eng. Sci., 2010.

[6] A. J. Walker and I. W. Stewart, Layer undulations in a smectic {C} liquid crystal with weak anchoring. J. Phys. A: Math. Theor., 40:11849—11861, 2007.

[7] A. J. Walker and I. W. Stewart, Periodic disturbances in cylindrically layered smectic A, Mol. Cryst. Liq. Cryst., 478:788—799, 2007.

[8] A. J. Walker and I. W. Stewart, Wave Induced Perturbations in Cylindrically Layered Smectic A Liquid Crystals, Zeitschrift fur angewandte Mathematik und Physik (Journal of Applied Mathematics and Physics), 63 (2012), 357–371.

[9] A. J. Walker and A. J. Stewart, Acoustic waves in compressible planar layered smectic liquid crystals, J. Phys.: Condens. Matter, 22:325106, 2010.

[10] A. J. Walker, The alignment of cylindrically layered smectic A liquid crystals with director tilt on the boundaries, J. Phys. A: Math. Theor., 41:385205, 2008.