Liquid crystal

The order of a liquid crystal could also be characterized by 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.[19]

A positional order parameter is also used to describe the ordering of a liquid crystal. It is characterized by the variation of the density of the center of mass of the liquid crystal molecules along a given vector. In the case of positional variation along the z-axis the density ${\displaystyle \rho (z)}$ is often given by:

${\displaystyle \rho (\mathbf {r} )=\rho (z)=\rho _{0}+\rho _{1}\cos(q_{s}z-\varphi )+\cdots \,}$

The complex positional order parameter is defined as ${\displaystyle \psi (\mathbf {r} )=\rho _{1}(\mathbf {r} )e^{i\varphi (\mathbf {r} )}}$ and ${\displaystyle \rho _{0}}$ the average density. Typically only the first two terms are kept and higher order terms are ignored since most phases can be described adequately using sinusoidal functions. For a perfect nematic ${\displaystyle \psi =0}$ and for a smectic phase ${\displaystyle \psi }$ will take on complex values. The complex nature of this order parameter allows for many parallels between nematic to smectic phase transitions and conductor to superconductor transitions.^[20]

Onsager hard-rod model

A 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 which the approaching cylinder's center-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).^[63][64]

The fundamental insight here is that, whilst parallel arrangements of anisotropic objects lead 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. Although this model is conceptually helpful, its mathematical formulation makes several assumptions that limit its applicability to real systems.^[64] An extension of Onsager Theory was proposed by Flory to account for non entropic effects.

Maier–Saupe mean field theory

This statistical theory, proposed by Alfred Saupe and Wilhelm Maier, includes contributions from an attractive intermolecular potential from an induced dipole moment between adjacent rod-like liquid crystal molecules. 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 nematic-isotropic phase transitions, consistent with experiment.^[65][66][67] Maier-Saupe mean field theory is extended to high molecular weight liquid crystals by incorporating the bending stiffness of the molecules and using the method of path integrals in polymer science.^[68]

McMillan's model

McMillan's model, proposed by William McMillan,^[69] is an extension of the Maier–Saupe mean field theory used to describe the phase transition of a liquid crystal from a nematic to a smectic A phase. It predicts that the phase transition can be either continuous or discontinuous depending on the strength of the short-range interaction between the molecules. As a result, it allows for a triple critical point where the nematic, isotropic, and smectic A phase meet. Although it predicts the existence of a triple critical point, it does not successfully predict its value. The model utilizes two order parameters that describe the orientational and positional order of the liquid crystal. The first is simply the average of the second Legendre polynomial and the second order parameter is given by:

${\displaystyle \sigma =\left\langle \cos \left({\frac {2\pi z_{i}}{d}}\right)\left({\frac {3}{2}}\cos ^{2}\left(\theta _{i}\right)-{\frac {1}{2}}\right)\right\rangle }$

The values z_i, θ_i, and d are the position of the molecule, the angle between the molecular axis and director, and the layer spacing. The postulated potential energy of a single molecule is given by:

${\displaystyle U_{i}(\theta _{i},z_{i})=-U_{0}\left(S+\alpha \sigma \cos \left({\frac {2\pi z_{i}}{d}}\right)\right)\left({\frac {3}{2}}\cos ^{2}\left(\theta _{i}\right)-{\frac {1}{2}}\right)}$

Here constant α quantifies the strength of the interaction between adjacent molecules. The potential is then used to derive the thermodynamic properties of the system assuming thermal equilibrium. It results in two self-consistency equations that must be solved numerically, the solutions of which are the three stable phases of the liquid crystal.^[23]

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. The distortions of the liquid crystal are commonly described by the

The ability of the director to align along an external field is caused by the electric nature of the molecules. Permanent electric dipoles result when one end of a molecule has a net positive charge while the other end has a net negative charge. When an external electric field is applied to the liquid crystal, the dipole molecules tend to orient themselves along the direction of the field.[72]

Even if a molecule does not form a permanent dipole, it can still be influenced by an electric field. In some cases, the field produces slight re-arrangement of electrons and protons in molecules such that an induced electric dipole results. While not as strong as permanent dipoles, orientation with the external field still occurs.

The response of any system to an external electrical field is

${\displaystyle D_{i}=\epsilon _{0}E_{i}+P_{i}}$

where ${\displaystyle E_{i}}$, ${\displaystyle D_{i}}$ and ${\displaystyle P_{i}}$ are the components of the electric field, electric displacement field and polarization density. The electric energy per volume stored in the system is

${\displaystyle f_{\text{elec}}=-{\frac {1}{2}}D_{i}E_{i}}$

(summation over the doubly appearing index ${\displaystyle i}$). In nematic liquid crystals, the polarization, and electric displacement both depend linearly on the direction of the electric field. The polarization should be even in the director since liquid crystals are invariants under reflexions of ${\displaystyle n}$. The most general form to express ${\displaystyle D}$ is

${\displaystyle D_{i}=\epsilon _{0}\epsilon _{\bot }E_{i}+\left(\epsilon _{\parallel }-\epsilon _{\bot }\right)n_{i}n_{j}E_{j}}$

(summation over the index ${\displaystyle j}$) with ${\displaystyle \epsilon _{\bot }}$ and ${\displaystyle \epsilon _{\parallel }}$ the electric permittivity parallel and perpendicular to the director ${\displaystyle n}$. Then density of energy is (ignoring the constant terms that do not contribute to the dynamics of the system)^[73]

${\displaystyle f_{\text{elec}}=-{\frac {1}{2}}\epsilon _{0}\left(\epsilon _{\parallel }-\epsilon _{\bot }\right)\left(E_{i}n_{i}\right)^{2}}$

(summation over ${\displaystyle i}$). If ${\displaystyle \epsilon _{\parallel }-\epsilon _{\bot }}$ is positive, then the minimum of the energy is achieved when ${\displaystyle E}$ and ${\displaystyle n}$ are parallel. This means that the system will favor aligning the liquid crystal with the externally applied electric field. If ${\displaystyle \epsilon _{\parallel }-\epsilon _{\bot }}$ is negative, then the minimum of the energy is achieved when ${\displaystyle E}$ and ${\displaystyle n}$ are perpendicular (in nematics the perpendicular orientation is degenerated, making possible the emergence of vortices^[74]).

The difference ${\displaystyle \Delta \epsilon =\epsilon _{\parallel }-\epsilon _{\bot }}$ is called dielectrical anisotropy and is an important parameter in liquid crystal applications. There are both ${\displaystyle \Delta \epsilon >0}$ and ${\displaystyle \Delta \epsilon <0}$ commercial liquid crystals.

${\displaystyle \Delta \epsilon >0}$

${\displaystyle \Delta \epsilon <0}$

The effects of magnetic fields on liquid crystal molecules are analogous to electric fields. Because magnetic fields are generated by moving electric charges, permanent magnetic dipoles are produced by electrons moving about atoms. When a magnetic field is applied, the molecules will tend to align with or against the field. Electromagnetic radiation, e.g. UV-Visible light, can influence light-responsive liquid crystals which mainly carry at least a photo-switchable unit.^[75]

Surface preparations

In the absence of an external field, the director of a liquid crystal is free to point in any direction. It is possible, however, to force the director to point in a specific direction by introducing an outside agent to the system. For example, when a thin polymer coating (usually a polyimide) is spread on a glass substrate and rubbed in a single direction with a cloth, it is observed that liquid crystal molecules in contact with that surface align with the rubbing direction. The currently accepted mechanism for this is believed to be an epitaxial growth of the liquid crystal layers on the partially aligned polymer chains in the near surface layers of the polyimide.

Several liquid crystal chemicals also align to a 'command surface' which is in turn aligned by electric field of polarized light. This process is called photoalignment.

Fréedericksz transition

The competition between orientation produced by surface anchoring and by electric field effects is often exploited in liquid crystal devices. Consider the case in which liquid crystal molecules are aligned parallel to the surface and an electric field is applied perpendicular to the cell. At first, as the electric field increases in magnitude, no change in alignment occurs. However at a threshold magnitude of electric field, deformation occurs. Deformation occurs where the director changes its orientation from one molecule to the next. The occurrence of such a change from an aligned to a deformed state is called a Fréedericksz transition and can also be produced by the application of a magnetic field of sufficient strength.

The Fréedericksz transition is fundamental to the operation of many liquid crystal displays because the director orientation (and thus the properties) can be controlled easily by the application of a field.

Effect of chirality

As already described,

Chiral phases usually have a helical twisting of the molecules. If the pitch of this twist is on the order of the wavelength of visible light, then interesting optical interference effects can be observed. The chiral twisting that occurs in chiral LC phases also makes the system respond differently from right- and left-handed circularly polarized light. These materials can thus be used as polarization filters.^[76]

It is possible for chiral LC molecules to produce essentially achiral mesophases. For instance, in certain ranges of concentration and

Liquid crystals find wide use in liquid crystal displays, which rely on the optical properties of certain liquid crystalline substances in the presence or absence of an electric field. In a typical device, a liquid crystal layer (typically 4 μm thick) sits between two polarizers that are crossed (oriented at 90° to one another). The liquid crystal alignment is chosen so that its relaxed phase is a twisted one (see Twisted nematic field effect).^[7] This twisted phase reorients light that has passed through the first polarizer, allowing its transmission through the second polarizer (and reflected back to the observer if a reflector is provided). The device thus appears transparent. When an electric field is applied to the LC layer, the long molecular axes tend to align parallel to the electric field thus gradually untwisting in the center of the liquid crystal layer. In this state, the LC molecules do not reorient light, so the light polarized at the first polarizer is absorbed at the second polarizer, and the device loses transparency with increasing voltage. In this way, the electric field can be used to make a pixel switch between transparent or opaque on command. Color LCD systems use the same technique, with color filters used to generate red, green, and blue pixels.^[7] Chiral smectic liquid crystals are used in ferroelectric LCDs which are fast-switching binary light modulators. Similar principles can be used to make other liquid crystal based optical devices.^[80]

Thermotropic chiral LCs whose pitch varies strongly with temperature can be used as crude liquid crystal 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 as well as on thermometers for infants or baths.^[83] Other liquid crystal materials change color 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.^[84]

Liquid crystal lenses converge or diverge the incident light by adjusting the refractive index of liquid crystal layer with applied voltage or temperature. Generally, the liquid crystal lenses generate a parabolic refractive index distribution by arranging molecular orientations. Therefore, a plane wave is reshaped into a parabolic wavefront by a liquid crystal lens. The

Many common fluids, such as soapy water, are in fact liquid crystals. Soap forms a variety of LC phases depending on its concentration in water.^[89]

Liquid crystal films have revolutionized the world of technology. Currently they are used in the most diverse devices, such as digital clocks, mobile phones, calculating machines and televisions. The use of liquid crystal films in optical memory devices, with a process similar to the recording and reading of CDs and DVDs may be possible.^[90][91]

Liquid crystals are also used as basic technology to imitate quantum computers, using electric fields to manipulate the orientation of the liquid crystal molecules, to store data and to encode a different value for every different degree of misalignment with other molecules.^[92][93]

See also

References

External links

Wikimedia Commons has media related to Liquid crystal.

• "History and Properties of Liquid Crystals". Nobelprize.org. Retrieved June 6, 2009.
• Definitions of basic terms relating to low-molar-mass and polymer liquid crystals (IUPAC Recommendations 2001)
• An intelligible introduction to liquid crystals from Case Western Reserve University
• Liquid Crystal Physics tutorial Archived August 5, 2007, at the Wayback Machine from the Liquid Crystals Group, University of Colorado
• Liquid Crystals & Photonics Group – Ghent University (Belgium) Archived June 13, 2007, at the Wayback Machine, good tutorial
• Simulation of light propagation in liquid crystals^{[permanent dead link]}, free program
• Liquid Crystals Interactive Online Archived March 18, 2021, at the Wayback Machine
• Liquid Crystal Institute Archived September 20, 2017, at the Wayback Machine Kent State University
• Liquid Crystals a journal by Taylor&Francis
• Molecular Crystals and Liquid Crystals a journal by Taylor & Francis
• Hot-spot detection techniques for ICs
• What are liquid crystals? from Chalmers University of Technology, Sweden
• Progress in liquid crystal chemistry Thematic series in the Open Access Beilstein Journal of Organic Chemistry
• DoITPoMS Teaching and Learning Package- "Liquid Crystals"
• Bowlic liquid crystal from San Jose State University
• Phase calibration of a Spatial Light Modulator