Dynamic theories of smectic 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 (smectic A) [1,2] or the director makes an angle θ 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 bi-layer 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.

Nematic liquid crystals, which have no layering structure, have been well described since the papers by Ericksen [3,4] and Leslie [5,6]. The resulting theory from these pieces of work is often referred to as the Ericksen-Leslie dynamic theory for nematics and is one of the most widely used and successful theories employed to model dynamic phenomena in nematics. There have been some proposed dynamic theories to describe smectic liquid crystals, notably the Smectic C (SmC) dynamic theory of Leslie, Stewart and Nakagawa [7] (commonly known as the LSN dynamic theory for Smectic C) and the dynamic theories for Smectic A (SmA) by Stewart [8], Auernhammer, Brand and Pleiner [9,10,11], E [12] and others.

One of the more recent developments in smectics is the acknowledgment of the importance of the smectic layers and the permeation phenomenon. In smectics, molecules may move (permeate) from one layer to another, which can be useful in the relaxation of flow or instabilities in dynamic situations. This phenomena has been accounted for in the recent SmA papers mentioned above along with de Gennes and Prost [1], in which compressibility in SmA, an aspect which is often missing in many other smectic theories, is considered.

At present, there is no definitive dynamic theory for smectic C liquid crystals which incorporates all aspects required for insightful and accurate modelling. Many of the previous attempts at dynamic theories have been isothermal, have not included permeation or have not included a full asymmetric viscous stress tensor. It is the construction of definitive smectic theories which can model the various phases of different types smectic liquid crystals that forms the bulk of this research. Once constructed, these theories will be useful in the construction of dynamic theories for other smart materials.

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] J. L. Ericksen, Anisotropic fluids, Arch. Rat. Mech. Anal., 4:231—237, 1960.

[4] J. L. Ericksen, Conservation laws for liquid crystals, Trans. Soc. Rheol., 5:23—34, 1961.

[5] F. M. Leslie, Some constitutive equations for anisotropic fluids, Q. Jl. Mech. Appl. Math., 19:357—370, 1966.

[6] F. M. Leslie, Some constitutive equations for liquid crystals, Arch. Rat. Mech. Anal., 28:265—283, 1968.

[7] F. M. Leslie et al., A continuum theory for smectic C liquid crystals, Mol. Cryst. Liq. Cryst., 198:443—454, 1991.

[8] I. W. Stewart, Dynamic theory for smectic A liquid crystals, Continuum Mech. Thermodyn., 18:343—360, 2007.

[9] G. K. Auernhammer et al., The undulation instability in layered systems under shear flow — a simple model, Rheol. Acta, 39:215—222, 2000.

[10] G. K. Auernhammer et al., Shear-induced instabilities in layered liquids, Phys. Rev. E, 66, 2002.

[11] G. K. Auernhammer et al., Erratum: Shear-induced instabilities in layered liquids, Phys. Rev. E, 71, 2005.

[12] W. E , Nonlinear continuum theory of smectic-A liquid crystals, Arch. Rat. Mech. Anal., 137:159—175, 1997.