# Liquid crystal

Schlieren texture of Liquid Crystal nematic phase

Liquid crystals are substances that exhibit a phase of matter that has properties between those of a conventional liquid, and those of a solid crystal. For instance, a liquid crystal (LC) may flow like a liquid, but have the molecules in the liquid arranged and/or oriented in a crystal-like way. There are many different types of LC phases, which can be distinguished based on their different optical properties (such as birefringence). When viewed under a microscope using a polarized light source, different liquid crystal phases will appear to have a distinct texture. Each 'patch' in the texture corresponds to a domain where the LC molecules are oriented in a different direction. Within a domain, however, the molecules are well ordered. Liquid crystal materials may not always be in an LC phase (just as water is not always in the liquid phase: it may also be found in the solid or gas phase). Liquid crystals can be divided into thermotropic and lyotropic LCs. Thermotropic LCs exhibit a phase transition into the LC phase as temperature is changed, whereas lyotropic LCs exhibit phase transitions as a function of concentration of the mesogen in a solvent (typically water) as well as temperature.

## Mesogens

Molecules that exhibit liquid crystal phases are called mesogens. For a molecule to display an LC phase, it must generally be fairly rigid and anisotropic (i.e. longer in one direction than another). Most mesogens fall into the 'rigid-rod' class (calamitic mesogens), which orient based on their long axis. Disk-like (discotic) mesogens are also known, and these orient in the direction of their short axis. In addition to molecules, polymers and colloidal suspensions can also form LC phases. For instance, micrometre-sized objects (such as anisotropic colloids, latex particles, clay platelets, and even some viruses, such as the tobacco mosaic virus) can organize themselves in liquid crystal phases. These more exotic mesogens generally fall into the category of lyotropic liquid crystals.

## Liquid crystal phases

The various LC phases (called mesophases) can be characterized by the type of ordering that is present. One can distinguish positional order (whether or not molecules are arranged in any sort of ordered lattice) and orientational order (whether or not molecules are mostly pointing in the same direction), and moreover order can be either short-range (only between molecules close to each other) or long-range (extending to larger, sometimes macroscopic, dimensions). Most thermotropic LCs will have an isotropic phase at high temperature. That is, heating will eventually drive them into a conventional liquid phase characterized by random and isotropic molecular ordering (little to no long-range order), and fluid-like flow behaviour. Under other conditions (for instance, lower temperature), an LC might inhabit one or more phases with significant anisotropic orientational structure and long-range orientational order while still having an ability to flow.

The ordering of liquid crystalline phases is extensive on the molecular scale. This order extends up to the entire domain size, which may be on the order of micrometres, but usually does not extend to the macroscopic scale as often occurs in classical crystalline solids. However, some techniques (such as the use of boundaries or an applied electric field) can be used to enforce a single ordered domain in a macroscopic liquid crystal sample. The ordering in a liquid crystal might extend along only one dimension, with the material being essentially disordered in the other two directions.

### Thermotropic liquid crystals

Thermotropic phases are those that occur in a certain temperature range. If the temperature is raised too high, thermal motion will destroy the delicate cooperative ordering of the LC phase, pushing the material into a conventional isotropic liquid phase. At too low a temperature, most LC materials will form a conventional (though anisotropic) crystal. Many thermotropic LCs exhibit a variety of phases as temperature is changed. For instance, a particular mesogen may exhibit various smectic and nematic (and finally isotropic) phases as temperature is increased.

#### Nematic phase

One of the most common LC phases is the nematic, where the molecules have no positional order, but they do have long-range orientational order. Thus, the molecules flow and their centre of mass positions are randomly distributed as in a liquid, but they all point in the same direction (within each domain). Most nematics are uniaxial: they have one axis that is longer and preferred, with the other two being equivalent (can be approximated as cylinders). Some liquid crystals are biaxial nematics, meaning that in addition to orienting their long axis, they also orient along a secondary axis.

Liquid crystals are a phase of matter whose order is intermediate between that of a liquid and that of a crystal. The molecules are typically rod-shaped organic moieties about 25 angstroms (2.5 nanometers) in length and their ordering is a function of temperature. The nematic phase, for example, is characterized by the orientational order of the constituent molecules. The molecular orientation (and hence the material's optical properties) can be controlled with applied electric fields. Nematics are (still) the most commonly used phase in liquid crystal displays (LCDs), with many such devices using the twisted nematic geometry. The smectic phases, which are found at lower temperatures than the nematic, form well-defined layers that can slide over one another like soap. The smectics are thus positionally ordered along one direction. In the Smectic A phase, the molecules are oriented along the layer normal, while in the Smectic C phase they are tilted away from the layer normal. These phases, which are liquid-like within the layers, are illustrated below. There is a very large number of different smectic phases, all characterized by different types and degrees of positional and orientational order.

#### Chiral phases

The chiral nematic phase exhibits chirality (handedness). This phase is often called the cholesteric phase because it was first observed for cholesterol derivatives. Only chiral molecules (i.e.: those that lack inversion symmetry) can give rise to such a phase. This phase exhibits a twisting of the molecules along the director, with the molecular axis perpendicular to the director. The finite twist angle between adjacent molecules is due to their asymmetric packing, which results in longer-range chiral order. In the smectic C* phase, the molecules orient roughly along the director, with a finite tilt angle, and a twist relative to other mesogens. This results in, again, a spiral twisting of molecular axis along the director.

The chiral pitch refers to the distance (along the director) over which the mesogens undergo a full 360° twist (but note that the structure repeats itself every half-pitch, since the positive and negative directions along the director are equivalent). The pitch may be varied by adjusting temperature or adding other molecules to the LC fluid. For many types of liquid crystals, the pitch is on the same order as the wavelength of visible light. This causes these systems to exhibit unique optical properties, such as selective reflection. These properties are exploited in a number of optical applications.

#### Discotic phases

Disk-shaped mesogens can orient themselves in a layer-like fashion known as the discotic nematic phase. If the disks pack into stacks, the phase is called a discotic columnar. The columns themselves may be organized into rectangular or hexagonal arrays. Chiral discotic phases, similar to the chiral nematic phase, are also known.

### Lyotropic liquid crystals

A lyotropic liquid crystal consists of two or more components that exhibit liquid-crystalline properties in certain concentration ranges. In the lyotropic phases, solvent molecules fill the space around the compounds to provide fluidity to the system. In contrast to thermotropic liquid crystals, these lyotropics have another degree of freedom of concentration that enables them to induce a variety of different phases.

A compound which has two immiscible hydrophilic and hydrophobic parts within the same molecule is called an amphiphilic molecule. Many amphiphilic molecules show lyotropic liquid-crystalline phase sequences depending on the volume balances between the hydrophilic part and hydrophobic part. These structures are formed through the micro-phase segregation of two incompatible components on a nanometer scale. Soap is an everyday example of a lyotropic liquid crystal.

The content of water or other solvent molecules changes the self-assembled structures. At very low amphiphile concentration, the molecules will be dispersed randomly without any ordering. At slightly higher (but still low) concentration, amphiphilic molecules will spontaneously assemble into micelles or vesicles. This is done so as to 'hide' the hydrophobic tail of the amphiphile inside the micelle core, exposing a hydrophilic (water-soluble) surface to aqueous solution. These spherical objects do not order themselves in solution, however. At higher concentration, the assemblies will become ordered. A typical phase is a hexagonal columnar phase, where the amphiphiles form long cylinders (again with a hydrophilic surface) that arrange themselves into a roughly hexagonal lattice. This is called the middle soap phase. At still higher concentration, a lamellar phase (neat soap phase) may form, wherein extended sheets of amphiphiles are separated by thin layers of water. For some systems, a cubic (also called viscous isotropic) phase may exist between the hexagonal and lamellar phases, wherein spheres are formed that create a dense cubic lattice. These spheres may also be connected to one another, forming a bicontinuous cubic phase.

The objects created by amphiphiles are usually spherical (as in the case of micelles), but may also be disc-like (bicelles), rod-like, or biaxial (all three micelle axes are distinct). These anisotropic self-assembled nano-structures can then order themselves in much the same way as liquid crystals do, forming large-scale versions of all the thermotropic phases (such as a nematic phase of rod-shaped micelles).

For some systems, at high concentration, inverse phases are observed. That is, one may generate an inverse hexagonal columnar phase (columns of water encapsulated by amphiphiles) or an inverse micellar phase (a bulk liquid crystal sample with spherical water cavities).

A generic progression of phases, going from low to high amphiphile concentration, is:

• Discontinuous cubic phase (micellar phase)
• Hexagonal columnar phase (middle phase)
• Bicontinuous cubic phase
• Lamellar phase
• Bicontinuous cubic phase
• Reverse hexagonal columnar phase
• Inverse cubic phase (Inverse micellar phase)

Even within the same phases, their self-assembled structures are tunable by the concentration: for example, in lamellar phases, the layer distances increase with the solvent volume. Since lyotropic liquid crystals rely on a subtle balance of intermolecular interactions, it is more difficult to analyze their structures and properties than those of thermotropic liquid crystals.

Similar phases and characteristics can be observed in immiscible diblock copolymers.

### Metallotropic liquid crystals

Liquid crystal phases can also be based on low-melting inorganic phases like ZnCl2 that have a structure formed of linked tetrahedra and easily form glasses. The addition of long chain soaplike molecules leads to a series of new phases that show a variety of liquid crystalline behaviour both as a function of the inorganic-organic composition ratio and of temperature. This class of materials has been named metallotropic J.D. Martin et al.

## Biological liquid crystals

Lyotropic liquid-crystalline nanostructures are abundant in living systems. Accordingly, lyotropic liquid crystals attract particular attention in the field of biomimetic chemistry. In particular, biological membranes and cell membranes are a form of liquid crystal. Their constituent rod-like molecules (e.g., phospholipids) are organized perpendicularly to the membrane surface, yet the membrane is fluid and elastic. The constituent molecules can flow in-plane quite easily, but tend not to leave the membrane, and can flip from one side of the membrane to the other with some difficulty. These liquid crystal membrane phases can also host important proteins such as receptors freely "floating" inside, or partly outside, the membrane.

Many other biological structures exhibit LC behaviour. For instance, the concentrated protein solution that is extruded by a spider to generate silk is, in fact, a liquid crystal phase. The precise ordering of molecules in silk is critical to its renowned strength. DNA and many polypeptides can also form LC phases. Since biological mesogens are usually chiral, chirality often plays a role in these phases.

## Theoretical treatment of liquid crystals

Microscopic theoretical treatment of fluid phases can become quite involved, owing to the high material density, which means that strong interactions, hard-core repulsions, and many-body correlations cannot be ignored. In the case of liquid crystals, anisotropy in all of these interactions further complicate analysis. There are a number of fairly simple theories, however, that can at least predict the general behaviour of the phase transitions in liquid crystal systems.

### Order parameter

The description of liquid crystals involves an analysis of order. To make this quantitative, an orientational order parameter is usually defined based on the average of the second Legendre polynomial:

$S = \langle P_2(\cos \theta) \rangle = \left \langle \frac{3 \cos^2 \theta-1}{2} \right \rangle$

where θ is the angle between the mesogen molecule axis and the local director (which is the 'preferred direction' in a liquid crystal sample). This definition is convenient, since for a completely random and isotropic sample, S=0, whereas for a perfectly aligned sample S=1. For a typical liquid crystal sample, S is on the order of 0.3 to 0.8, and generally decreases as the temperature is raised. In particular, a sharp drop of the order parameter to 0 is observed when one undergoes a phase transition from an LC phase into the isotropic phase. The order parameter can be measured experimentally in a number of ways. For instance, diamagnetism, birefringence, Raman scattering, and NMR can also be used to determine S.

One could also characterize the order of a liquid crystal using other even Legendre polynomials (all the odd polynomials average to zero since the director can point in either of two antiparallel directions). These higher-order averages are more difficult to measure, but can yield additional information about molecular ordering.

### Onsager hard-rod model

A very simple model which predicts lyotropic phase transitions is the hard-rod model proposed by Lars Onsager. This theory considers the volume excluded from the center-of-mass of one idealized cylinder as it approaches another. Specifically, if the cylinders are oriented parallel to one another, there is very little volume that is excluded from the center-of-mass of the approaching cylinder (it can come quite close to the other cylinder). If, however, the cylinders are at some angle to one another, then there is a large volume surrounding the cylinder where the approaching cylinder's centre-of-mass cannot enter (due to the hard-rod repulsion between the two idealized objects). Thus, this angular arrangement sees a decrease in the net positional entropy of the approaching cylinder (there are fewer states available to it).

The fundamental insight here is that that while parallel arrangements of anisotropic objects leads to a decrease in orientational entropy, there is an increase in positional entropy. Thus in some case greater positional order will be entropically favorable. This theory thus predicts that a solution of rod-shaped objects will undergo a phase transition, at sufficient concentration, into a nematic phase. Recently this theory is used to observe the phase transition between nematic and smectic-A at very high concentration also Hanif et al.. Although this model is conceptually helpful, its mathematical formulation makes several assumptions that limit its applicability to real systems.

### Maier-Saupe mean field theory

This statistical theory includes contributions from an attractive intermolecular potential. The anisotropic attraction stabilizes parallel alignment of neighboring molecules, and the theory then considers a mean-field average of the interaction. Solved self-consistently, this theory predicts thermotropic phase transitions, consistent with experiment.

### Elastic continuum theory

In this formalism, a liquid crystal material is treated as a continuum; molecular details are entirely ignored. Rather, this theory considers perturbations to a presumed oriented sample. One can identify three types of distortions that could occur in an oriented sample: (1) twists of the material, where neighboring molecules are forced to be angled with respect to one another, rather than aligned; (2) splay of the material, where bending occurs perpendicular to the director; and (3) bend of the material, where the distortion is parallel to the director and mesogen axis. All three of these types of distortions incur an energy penalty. They are defects that often occur near domain walls or boundaries of the enclosing container. The response of the material can then be decomposed into terms based on the elastic constants corresponding to the three types of distortions.

## Effect of chirality

As already described, chiral mesogens usually give rise to chiral mesophases. For molecular mesogens, this means that the molecule must possess an asymmetric carbon atom. An additional requirement is that the system not be racemic: a mixture of right- and left-handed versions of the mesogen will cancel the chiral effect. Due to the cooperative nature of liquid crystal ordering, however, a small amount of chiral dopant in an otherwise achiral mesophase is often enough to select out one domain handedness, making the system overall chiral.

Chiral phases usually have a helical twisting of the mesogens. If the pitch of this twist is on the order of the wavelength of visible light, then interesting optical interference effects will be observed. The chiral twisting that occurs in chiral LC phases also makes the system respond differently to right- and left-handed circularly polarized light. These materials can thus be used as polarization filters.

It is possible for chiral mesogens to produce essentially achiral mesophases. For instance, in certain ranges of concentration and molecular weight, DNA will form an achiral line hexatic phase. A curious recent observation is of the formation of chiral mesophases from achiral mesogens. Specifically, bent-core molecules (sometimes called banana liquid crystals) have been shown to form liquid crystal phases that are chiral. In any particular sample, various domains will have opposite handedness, but within any given domain, strong chiral ordering will be present. The appearance mechanism of this macroscopic chirality is not yet entirely clear. It appears that the molecules stack in layers and orient themselves in a tilted fashion inside the layers. These liquid crystals phases are ferroelectric and anti- ferroelectric, both of which are of interest for applications.

## Applications of liquid crystals

Liquid crystals find wide use in liquid crystal displays, which rely on the optical properties of certain liquid crystalline molecules in the presence or absence of an electric field. In a typical device, a liquid crystal layer sits between two polarizers that are crossed (oriented at 90° to one another). The liquid crystal is chosen so that its relaxed phase is a twisted one. This twisted phase reorients light that has passed through the first polarizer, allowing it to be transmitted through the second polarizer and reflected back to the observer. The device thus appears clear. When an electric field is applied to the LC layer, all the mesogens align (and are no longer twisting). In this aligned state, the mesogens do not reorient light, so the light polarized at the first polarizer is absorbed at the second polarizer, and the entire device appears dark. In this way, the electric field can be used to make a pixel switch between clear or dark on command. Color LCD systems use the same technique, with colour filters used to generate red, green, and blue pixels. Similar principles can be used to make other liquid crystal based optical devices.

Thermotropic chiral LCs whose pitch varies strongly with temperature can be used as crude thermometers, since the color of the material will change as the pitch is changed. Liquid crystal color transitions are used on many aquarium and pool thermometers. Other liquid crystal materials change colour when stretched or stressed. Thus, liquid crystal sheets are often used in industry to look for hot spots, map heat flow, measure stress distribution patterns, and so on. Liquid crystal in fluid form is used to detect electrically generated hot spots for failure analysis in the semiconductor industry. Liquid crystal memory units with extensive capacity were used in Space Shuttle navigation equipment.

It is also worth noting that many common fluids are in fact liquid crystals. Soap, for instance, is a liquid crystal, and forms a variety of LC phases depending on its concentration in water.