Thermotropic liquid crystals

Thermotropic phases are those that occur in a certain temperature range. If the temperature rise is 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 temperature, most LC materials will form a conventional crystal.^[19][20] Many thermotropic LCs exhibit a variety of phases as temperature is changed. For instance, a particular type of LC molecule (called a mesogen) may exhibit various smectic phases followed by the nematic phase and finally the isotropic phase as temperature is increased. An example of a compound displaying thermotropic LC behavior is para-azoxyanisole.^[24]

Nematic phase

See also: Biaxial nematic and Twisted nematic field effect

Alignment in a nematic phase

Phase transition between a nematic (left) and smectic A (right) phases observed between crossed polarizers. The black color corresponds to isotropic medium.

The simplest liquid crystal phase is the nematic. In a nematic phase, calamitic organic molecules lack a crystalline positional order, but do self-align with their long axes roughly parallel. The molecules are free to flow and their center of mass positions are randomly distributed as in a liquid, but their orientation is constrained to form a long-range directional order.^[25]

The word nematic comes from the Greek νήμα (Greek: nema), which means "thread". This term originates from the disclinations: thread-like topological defects observed in nematic phases.

Nematics also exhibit so-called "hedgehog" topological defects. In two dimensions, there are topological defects with topological charges +1/2 and -1/2. Due to hydrodynamics, the +1/2 defect moves considerably faster than the -1/2 defect. When placed close to each other, the defects attract; upon collision, they annihilate.^[26][27]

Most nematic phases are uniaxial: they have one axis (called a directrix) that is longer and preferred, with the other two being equivalent (can be approximated as cylinders or rods). However, some liquid crystals are biaxial nematic, meaning that in addition to orienting their long axis, they also orient along a secondary axis.^[28] Nematic crystals have fluidity similar to that of ordinary (isotropic) liquids but they can be easily aligned by an external magnetic or electric field. Aligned nematics have the optical properties of uniaxial crystals and this makes them extremely useful in liquid-crystal displays (LCD).^[7]

Nematic phases are also known in non-molecular systems: at high magnetic fields, electrons flow in bundles or stripes to create an "electronic nematic" form of matter.^[29][30]

Smectic phases

Schematic of alignment in the smectic phases. The smectic A phase (left) has molecules organized into layers. In the smectic C phase (right), the molecules are tilted inside the layers.

The smectic phases, which are found at lower temperatures than the nematic, form well-defined layers that can slide over one another in a manner similar to that of soap. The word "smectic" originates from the Latin word "smecticus", meaning cleaning, or having soap-like properties.^[31] 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 it. These phases are liquid-like within the layers. There are many different smectic phases, all characterized by different types and degrees of positional and orientational order.^[19][20] Beyond organic molecules, Smectic ordering has also been reported to occur within colloidal suspensions of 2-D materials or nanosheets.^[32][33] One example of smectic LCs is p,p'-dinonylazobenzene.^[34]

Chiral phases or twisted nematics

Schematic of ordering in chiral liquid crystal phases. The chiral nematic phase (left), also called the cholesteric phase, and the smectic C* phase (right).

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 can give rise to such a phase. This phase exhibits a twisting of the molecules perpendicular to the director, with the molecular axis parallel 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 (an asterisk denotes a chiral phase), the molecules have positional ordering in a layered structure (as in the other smectic phases), with the molecules tilted by a finite angle with respect to the layer normal. The chirality induces a finite azimuthal twist from one layer to the next, producing a spiral twisting of the molecular axis along the layer normal, hence they are also called twisted nematics.^[20][22][23]

Chiral nematic phase. The numerator p refers to the chiral pitch (see text).

The chiral pitch, p, refers to the distance over which the LC molecules undergo a full 360° twist (but note that the structure of the chiral nematic phase repeats itself every half-pitch, since in this phase directors at 0° and ±180° are equivalent). The pitch, p, typically changes when the temperature is altered or when other molecules are added to the LC host (an achiral LC host material will form a chiral phase if doped with a chiral material), allowing the pitch of a given material to be tuned accordingly. In some liquid crystal systems, the pitch is of the same order as the wavelength of visible light. This causes these systems to exhibit unique optical properties, such as Bragg reflection and low-threshold laser emission,^[35] and these properties are exploited in a number of optical applications.^[3][22] For the case of Bragg reflection only the lowest-order reflection is allowed if the light is incident along the helical axis, whereas for oblique incidence higher-order reflections become permitted. Cholesteric liquid crystals also exhibit the unique property that they reflect circularly polarized light when it is incident along the helical axis and elliptically polarized if it comes in obliquely.^[36]

A planar cell, filled with achiral LC host doped with an optically active Tröger base analog, placed between a pair of parallel (A) and crossed (B) linear polarizers. This doped mesogenic phase forms self-organized helical superstructures, that allow specific wavelengths of light to pass through the crossed polarizers, and selectively reflects a particular wavelength of light.^[37]

Lyotropic liquid crystals

Structure of lyotropic liquid crystal. The red heads of surfactant molecules are in contact with water, whereas the blue tails of surfactant molecules are immersed in oil:bilayer (left) and micelle (right).

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.^[46] 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 that 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 thermotropic 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 concentrations, 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 cubic phase)
• Hexagonal phase (hexagonal columnar phase) (middle 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 soap-like molecules leads to a series of new phases that show a variety of liquid crystalline behavior both as a function of the inorganic-organic composition ratio and of temperature. This class of materials has been named metallotropic.^[47]

Laboratory analysis of mesophases

Thermotropic mesophases are detected and characterized by two major methods, the original method was use of thermal optical microscopy,^[48][49] in which a small sample of the material was placed between two crossed polarizers; the sample was then heated and cooled. As the isotropic phase would not significantly affect the polarization of the light, it would appear very dark, whereas the crystal and liquid crystal phases will both polarize the light in a uniform way, leading to brightness and color gradients. This method allows for the characterization of the particular phase, as the different phases are defined by their particular order, which must be observed. The second method, differential scanning calorimetry (DSC),^[48] allows for more precise determination of phase transitions and transition enthalpies. In DSC, a small sample is heated in a way that generates a very precise change in temperature with respect to time. During phase transitions, the heat flow required to maintain this heating or cooling rate will change. These changes can be observed and attributed to various phase transitions, such as key liquid crystal transitions.

Lyotropic mesophases are analyzed in a similar fashion, though these experiments are somewhat more complex, as the concentration of mesogen is a key factor. These experiments are run at various concentrations of mesogen in order to analyze that impact.

Director

As we already saw above, the nematic liquid crystals are composed of rod-like molecules with the long axes of neighboring molecules aligned approximately to one another. To describe this anisotropic structure, a dimensionless unit vector n called the director, is introduced to represent the direction of preferred orientation of molecules in the neighborhood of any point. Because there is no physical polarity along the director axis, n and -n are fully equivalent.^[20]

Order parameter

The local nematic director, which is also the local optical axis, is given by the spatial and temporal average of the long molecular axes.

The description of liquid crystals involves an analysis of order. A second rank symmetric traceless tensor order parameter ${\displaystyle Q}$ is used to describe the orientational order of the most general biaxial nematic liquid crystal. However, to describe the more common case of uniaxial nematic liquid crystals, a scalar order parameter is sufficient.^[61] To make this quantitative, an orientational order parameter is usually defined based on the average of the second Legendre polynomial:

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

where ${\displaystyle \theta }$ is the angle between the liquid-crystal molecular axis and the local director (which is the 'preferred direction' in a volume element of a liquid crystal sample, also representing its local optical axis). The brackets denote both a temporal and spatial average. 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 the system undergoes a phase transition from an LC phase into the isotropic phase.^[62] The order parameter can be measured experimentally in a number of ways; for instance, diamagnetism, birefringence, Raman scattering, NMR and EPR can be used to determine S.^[23]

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]

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 Frank free energy density. 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 molecular axis. All three of these types of distortions incur an energy penalty. They are distortions that are induced by the boundary conditions at domain walls or 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. Elastic continuum theory is an effective tool for modeling liquid crystal devices and lipid bilayers.^[70][71]

Electric and magnetic field effects

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. 5CB and E7 liquid crystal mixture are two ${\displaystyle \Delta \epsilon >0}$ liquid crystals commonly used. MBBA is a common ${\displaystyle \Delta \epsilon <0}$ liquid crystal.

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 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.