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Introduction

Liquid crystals are an intermediate phase of matter existing between the crystalline solid and amorphous liquids. They retain orientational order on melting from the solid form, but can flow readily like a liquid, and become totally disordered in the high temperature fluid phase. Liquid crystalline states arise in materials with molecules that possess geometric anisotropy, either
being rod or disc shaped. The ordering of these molecules gives rise to anisotropy of the electric and optical properties.

Liquid Crystal Phases

Liquid crystals can be broadly categorized as either lyotropic and thermotropic. Lyotropic liquid crystal molecules comprise a hydrophobic tail and a hydrophilic head and form liquid crystal phases when dissolved in a polar solvent. The molecules assemble so that the tails group together, exposing the hydrophilic heads to the solvent. A wide range of phases can be formed, for example the micellar or lamellar phase (molecules form bilayers separated by solvent). Phase transitions are induced by changes to the solvent concentration or to the temperature. For very specific combinations of molecules and solvent concentrations, micelles can form that are not spherical, which can give rise to liquid crystal phases that are similar to those seen in thermotropic liquid crystals.

Thermotropic liquid crystals typically are formed from pure compounds or mixtures of rod or disc shaped molecules and give rise to liquid crystal hases over a fixed temperature range. Generally speaking, there are three liquid crystal phases, known as the smectic, nematic and cholesteric phases.

Ordinary liquids generally possess some short-range order on a molecular scale but lack any macroscopic order. This is known as the isotropic phase. If a liquid crystal is heated sufficiently it will become isotropic. On cooling there will be a transition temperature, where various degrees of ordering of the fluid give rise to a liquid crystal phase, specifically the nematic phase.
Further cooling can gives rise to another liquid crystal phase, called the smectic phase, which not only posses orientational order but also a degree of positional ordering. When the temperature falls beneath the melting point the crystalline solid forms, where the molecules are rigidly bound in both space and orientation.

Nematic Phase

The simplest of the liquid crystal phases is the nematic phase. In this phase the molecules behave much like those of a liquid, being free to move and with no positional order, but differ in the respect that the molecules tend to align with one another. The director can change from point to point, and is a function of space. The word nematic is derived from the Greek nematos meaning thread like, referring to thread-like structures that can be
observed when these materials are viewed using a polarising microscope. This phase of liquid crystal is used in virtually all commercially available liquid crystal displays. Most nematic liquid crystals are uniaxial, meaning that the major of axis of the molecules tend to align with each other. Biaxial nematics also orient along a secondary axis.

Smectic Phase

Smectic liquid crystals are closest in structure to solid crystals and posses a high degree of order, with diffuse positional order as well as orientational order. The additional smectic order results in the formation of layered structures. There are several classes of a smectic material depending on the in layer ordering and the angle between the layer normal and the director (labelled A, B, C, and so on, according to the chronological sequence of
their detection). The most common smectic materials are smectic A, where the director is parallel to the layer normal, and smectic C, where the director is at an angle to the layer normal. Smectic materials show wax like properties, whereas nematic materials flow more freely making them more suitable for liquid crystal displays.

Chiral Phase

The chiral nematic phase replaces the nematic phase when the molecules lack inversion symmetry. The chiral component of the molecular interaction produces a helical twisted structure of pitch , which can be as short as nm [2]. The helical twist may be clockwise or anti-clockwise, depending on the molecular conformation.

The helical structure of the chiral nematic phase has the ability to selectively reflect light of a wavelength equal to that of the helical pitch length; however, in order for this to happen the material must be aligned so that the helical axes are parallel to the light propagation direction. A common commercial application for this type of phase is in thermochromic thermometer devices and other devices that change colour with temperature.The tilted smectic phases can also be chiral, where the director rotates about the cone between the layer normal and the tilt. The chiral version of the smectic A phase is referred to as the phase.

Properties of Liquid Crystals

As a result of the uniaxial symmetry, the dielectric constants differ in the value along the preferred axis ( $\varepsilon_\|$ ) and perpendicular to this axis ( $\varepsilon_\perp$ ). The dielectric anisotropy ( $\Delta\varepsilon$ ) is the difference between the parallel and perpendicular dielectric constants. When an electric field is applied to a liquid crystal, the induced
dipole moment of the molecules creates a net torque which tends to align the molecules along the direction of the electric field. This is true for molecules with a positive dielectric anisotropy.This uniaxial symmetry also gives rise to anisotropy in the refractive index. For this reason a glass flask filled with
nematic LC often appears as an opaque fluid, as light is scattered due to the fluctuations in refractive index. The ordinary refractive index no, is for light with electric field polarisation perpendicular to the director, and the extraordinary refractive index ne, is for light with electric field polarisation parallel to the director. The birefringence is the difference between these
two values. The viscosity of fluid is the internal resistance to flow, defined
as the ratio of the shearing stress to the rate of shear. A flow process induces a favoured director orientation, and conversely a field-induced reorientation of the director induces a flow effect. A detailed analysis of the liquid crystal requires 5 independent viscous coefficients, but usually the behaviour is approximated using a single rotational viscosity coefficient ( $\gamma_0$ ).

Phase Transitions and Orientational Order

A phase transition is the transformation of a thermodynamic system from one phase to another. At the transition, a small change in thermodynamic variable such as the temperature or pressure result an abrupt change in one or more physical properties. An example is the transition between solid, liquid and gaseous phases. Here, the transition can be characterised by an abrupt change in density.

When a liquid crystal is cooled from the isotropic state to the nematic there is a reduction in symmetry and an associated (abrupt) increase in order at the phase transition. This transition can be characterised as first-order, as it involves a latent heat; an amount of energy is absorbed or released during
the transformation between phases. First-order transitions are associated with "mixed-phase regimes"; in which some parts of the system have completed the transition and others have not. To quantify this behaviour it is necessary to introduce the concept of an order parameter.

If we define a unit vector to represent the long axis of each
molecule, described by a distribution function $f(\theta,\phi)$ , then the director is the statistical average of these unit vectors over a small volume element around a point. A measure of the fluctuation in these vectors about the director is defined by the order parameter S, which is usually based on the average of the second Legendre polynomial: S=<\frac{1}{2}\cos^2 \theta - 1> where $\theta$ is the angle between the long axis of an individual molecule and the director and the angular brackets denote a statistical average. Due to the head tail symmetry of the molecules the first order polynomial vanishes and cannot be used. Equivalently:

$S=\frac{1}{2}\int f(\theta) (3 \cos^2 \theta - 1)d\Omega.$

If $f(\theta)$ is zero everywhere except for at $\theta = 0$ and $\theta = \pi$ then the molecules are all aligned parallel to one another and , corresponding to a crystalline state. If the orientation is completely random $f(\theta)=1/2\pi$ , integrating the last equation leads to , the isotropic state. In the nematic phase, the order parameter takes intermediate value that typically ranges from to . It is possible for this equation to yield if $f(\theta)$ is peaked at $\theta = \pi/2$ , however this corresponds the unlikely situation where a collection of rods favour perpendicular alignment. The order parameter can be measured in a number of ways, for example by optical birefringence or Raman scattering.

This definition has the problem that the isotropic state is not the only molecular configuration that can give the state of minimum order parameter. If the molecules are arranged in a cone about the director with a specific angle, can equal zero [1,p.17]. Additional order parameters can be defined using higher order Legendre polynomials, which can remove this ambiguity, however, for the purposes of modelling it is convenient restrict the definition to second-order.

The majority of liquid crystal phases are uniaxial in the bulk, with a single degree of rotational symmetry about the director. The search for a liquid crystal that is biaxial in the bulk has been the subject of much research, and only recently have such phases been created, made possible by molecules possessing bent cores.

It is possible to have a uniaxial or biaxial arrangement of biaxial molecules. Biaxial molecules can be pictured as being in shape like a plank of wood. Both short and long axis of the molecules can posses orientational order, a biaxial arrangement. It is also possible that only the long axes exhibit orientational order and the short axes are randomly orientated, or vice versa.
This is a uniaxial arrangement.

Uniaxial molecules can also form uniaxial or biaxial arrangements. In the uniaxial arrangement the long axes of the molecules align with each other. The biaxial arrangement of uniaxial is the most difficult configuration to picture, but is important as it arises in proximity to disclinations. This state can be induced by an applied electric field or by constrained geometries, where there are two directions in which the long axes of the molecules tend to align.

The previous definition of order parameter was in terms of angle, but it useful to define the order in terms of an order tensor.

$Q = \frac{1}{2} S_1 (3 \hat{n}\otimes\hat{n} - I) + \frac{1}{2}S_2 (3 \hat{m}\otimes\hat{m} - I)$

Using the Spectral Theorem

$Q = \lambda_1 \textbf{e}_1\otimes \textbf{e}_1+ \lambda_2\textbf{e}_2\otimes \textbf{e}_2 +\lambda_3 \textbf{e}_3\otimes\textbf{e}_3,$

where $\lambda_1,\lambda_2$ and $\lambda_3$ are the eigenvalues of and $\textbf{e}_1,\textbf{e}_2$ and $\textbf{e}_3$ are the corresponding eigenvectors.

$\lambda_1 = S_1 - \frac{1}{2}S_2,$

$\lambda_2 = S_2 - \frac{1}{2}S_1,$

$\lambda_3 = -\frac{1}{2}(S_1+S_2).$

When two eigenvalues are equal this represents a uniaxial state, corresponding to one of three possible cases; only is nonzero, only only is nonzero or finally . When all three eigenvalues differ this represents a biaxial state. The isotropic state occurs when all the eigenvalues are identically zero.

[1] E. G. Virga, "Variational theories for liquid crystals", ser. Applied mathematics and mathematical computation. London: Chapman & Hall, 1994

[2] P. J. Collings and M. Hird, "Introduction to liquid crystals : chemistry and physics", ser. The liquid crystals book series. London: Taylor & Francis, 1997.

This page last modified 6 October, 2006 by r.james