A state of matter with an orientational order of building units—individual molecules or their aggregates—and complete or partial absence of the long-range positional order. Liquid crystals, discovered more than 100 years ago, are one of the best-studied classes of soft matter, along with colloids, polymer solutions and melts, gels, and foams.
The liquid-crystalline state, also known as the mesophase, is in between that of a regular solid having long-range positional order of atoms (or molecules which are also orientationally ordered in crystals) and an isotropic fluid in which the molecules show neither positional nor orientational order. Liquid crystals can flow, adopt the shape of a container, and form drops as regular isotropic fluids do; however, the molecules in the liquid-crystal sample are ordered. The direction of average orientation in liquid crystals is called the director n.
Molecular interactions responsible for the orientation order in liquid crystals are relatively weak, as most liquid crystals melt into the isotropic phase below 100–150°C (212–300°F). As a result, the structural organization of liquid crystals, most importantly the spatial configuration of the director and thus the optical properties, are very sensitive to the external factors such as electromagnetic field, shear deformations, and type of molecular orientation on bounding walls. This sensitivity allows for numerous applications of liquid crystals, including liquid-crystal displays (LCDs). In these devices, weak electric voltage pulses reorient the director and change the optical appearance of the liquid-crystal cell. See also: Optical materials
Depending on the way the liquid-crystalline state is produced, one distinguishes thermotropic and lyotropic liquid crystals.
Thermotropic (solvent-free) liquid crystals form either by heating a solid crystal or by cooling an isotropic fluid. They exist in a certain temperature range for the materials made of strongly anisometric (elongated or disklike) molecules. An individual substance forming a thermotropic liquid crystal does not require any solvent to exhibit the mesophase. In practical applications, a number of liquid-crystal materials are often mixed together to improve their functional properties (such as their useful temperature range) or doped with additives that are not necessarily mesomorphic (such as dyes, chiral molecules, and polymer inclusions).
Lyotropic liquid crystals form only in the presence of a solvent, such as water or oil. Most commonly, lyotropic mesophases are formed by solutions of amphiphilic molecules (such as soaps, phospholipids, and surfactants). Amphiphilic molecules have two distinct parts: a hydrophilic (polar) head and a hydrophobic (nonpolar) tail, which is generally an aliphatic chain. As a result, amphiphilic molecules in solvents give rise to “self-organization,” as manifested by the formation of micelles and bilayers. Mesomorphic states (between liquid and solid) may also form in solutions of certain polymers; polymers may also form thermotropic liquid crystals. See also: Micelle
There are four basic types of liquid-crystalline phases, classified according to the dimensionality of the positional order of building units: nematic (only orientational order and no positional order), smectic (orientational and 1D positional order, caused by a periodic variation of the molecular densities), columnar (orientational and 2D positional order), and various 3D-correlated structures (such as cubic phases and blue phases).
Upon heating, many thermotropic substances, such as octylcyanobiphenyl (8CB) [Fig. 1], yield the following phase sequence: solid crystal > smectic (or columnar phase) > nematic > isotropic fluid. On cooling, the sequence is generally reversed, although it often includes phases that are absent during heating.
There are three different variations of the nematic phase: uniaxial nematic (UN), biaxial nematic (BN), and cholesteric (Ch). The UN is formed by the molecules of a rodlike or disklike shape that can easily rotate around one of the axes. The average orientation of these axes of rotation coincides with the director n (Fig. 2a). Even when the molecules are polar, as the 8CB molecules are, their head-to-head overlapping and flip-flops establish centrosymmetric arrangement in the nematic bulk. The UN is an optically anisotropic medium; the optic axis coincides with the director (Fig. 2a). The speed of light propagation in the nematic medium depends on how the polarization of light is directed with respect to the optic axis, giving rise to the phenomenon of birefringence. See also: Birefringence
In BN, the shape of building units significantly deviates from axially symmetric and can be more appropriately described as boxlike. These materials exhibit the orientational order of all three axes and have three directors (Fig. 2b).
When the building unit (molecule or aggregate) is not equal to its mirror image (chiral), the director n might show a twisted structure. Most often, the twist is unidirectional and the director forms a helicoid by rotating around a single axis in space, which is the case of the Ch phase (Fig. 2c). The UN, BN, Ch, and blue phases are liquid phases (no correlations in molecular positions over large distances). However, Ch shows a periodic structure associated with the spatial twist of molecular orientation (Fig. 2c). The pitch (period) of this twist is much larger than the molecular size, as molecular interactions responsible for the twist are weak as compared to the interactions responsible for the mesomorphic state itself. With a typical molecular size of a liquid crystal being 1–10 nanometers, the pitch is often in the range of 0.1–1 micrometer, which overlaps with the range of the visible part of optical spectrum.
Smectics are layered phases with a quasi-long-range 1D translational order of centers of molecules in a direction normal to the layers. This positional order is not exactly the long-range order as in a regular 3D crystal. As shown by L. D. Landau and R. E. Peierls, the fluctuative displacements of layers in 1D lattice diverge with the linear size of the sample. However, for typical smectic materials with a period of the order of 1 nm, the effect is noticeable only on scales of 1 mm and larger; it is thus not noticeable in samples that are normally 1–100 μm thick. In smectic A (SmA), the molecules within the layers show fluidlike arrangement, with no long-range in-plane positional order; it is a uniaxial medium with the optic axis and the director perpendicular to the layers (Fig. 2d).
In the lyotropic version of SmA, the lamellar Lα phase, the amphiphilic molecules arrange into bilayers. If the solvent is water, the exterior surfaces of the bilayer are formed by polar heads; the hydrophobic tails are hidden in the middle of the bilayer. (The membranes of many biological cells are organized in the similar way.) The periodic structure of alternating surfactant and water layers gives rise to the Lα phase (Fig. 2e). The lamellar structure may retain its smectic order, even when strongly diluted, as it is stabilized by thermal fluctuations of bilayers.
Other types of smectics show in-plane order, caused, for example, by a collective tilt of the rodlike molecules with respect to the normals to the layers (the so-called SmC) [Fig. 2f]. In chiral materials, the tilt of the molecules might lead to the helicoidal structure. The resulting chiral SmC phase is of considerable interest for applications in fast-switching optical devices (Fig. 2g).
Columnar phases are most frequently formed by hexagonal packing of cylindrical aggregates, as in the case of thermotropic materials formed by disklike molecules. The positional order is 2D only, as the intermolecular distances along the axes of the aggregates are not regular (Fig. 2h).
These demonstrate a periodic structure along all three coordinates, but they are still different from the 3D crystals, as the periodicity is associated with repetition of the local director patterns rather than with a regular positional order of the molecular centers of mass. The molecules are free to move around, adjusting to the local director orientation. For example, in cubic lyotropic phases the 3D network is formed by periodically curved layers of amphiphilic molecules. Another example is the blue phases that occur in the narrow temperature range between the isotropic fluid phase and Ch phase, in which the director forms double-twisted structures. The double twist cannot expand through the entire volume of the material and must be accompanied by a 3D periodic lattice of defects.
The director specifies only the direction of average orientation, not the degree of orientational order. For the latter, one can chose the temperature-dependent quantity s(T) = ½〈3 cos2 θ − 1&x3009 x232A;, where θ is the angle between the axis of an individual molecule and the director, and 〈...&x3009 x232A; means an average over molecular orientations. The term s is often called the scalar order parameter, where s = 1 corresponds to an ideal order with all the molecules rigidly aligned along one direction and s = 0 corresponds to absence of orientational order (in which case one deals with regular isotropic fluids).
The order parameter of the liquid crystal can be related to the anisotropy of macroscopic properties such as diamagnetic or dielectric susceptibility. Measuring these anisotropies allows one to determine the degree of orientational order. The magnetic measurements are especially convenient compared with their electric counterparts, as in this case the local field acting on the molecules differs very little from the external field. In uniaxial nematics (UNs), the values of the magnetic susceptibility measured in the direction parallel to the director and perpendicular to it are different. This difference χa = χ∥ — χ ⊥ is called the anisotropy of the magnetic susceptibility. In most thermotropic UNs, χ∥ < 0 and χ ⊥ < 0 (diamagnetism), and χa < 0, so that the director orients along the applied magnetic field. In the isotropic phase, χa = 0. For practical applications of UNs, of prime importance is the anisotropy of two other properties: the anisotropy of dielectric permittivity εa = ε∥ — ε⊥ and optical birefringence—that is, the difference in the refractive indices for light polarized parallel to the director (the extraordinary index of refraction ne = n∥) and for light polarized perpendicularly to the director (the ordinary index of refraction no = n⊥, i.e., Δ n = ne — no . When εa > 0, the director reorients along the applied electric field; and when εa < 0, the director realigns perpendicularly to the field. Because the director is simultaneously the optical axis, and because of the nonzero birefringence, Δn≠ 0, these field-induced reorientations change the optical properties of the sample, such as the effective value of the refractive index. The latter effect is at the core of numerous devices that use the dielectric response of the UN to display information or to control optical properties of the liquid crystal cell. See also: Electric susceptibility; Magnetic susceptibility; Refraction of waves
Elasticity and the free energy of the nematic phase
The molecular orientation in a liquid-crystal sample might change from point to point because of the external fields, boundary conditions, presence of foreign particles, and so on. The order parameter becomes spatially nonuniform. In most problems of practical interest, the typical scale of distortions is much larger than the molecular scale. The deformations are thus weak in the sense that the scalar part of the order parameter, s(T), remains constant despite the spatial gradients of the director n(r).
There are three basic types of director distortions in uniaxial nematics, called splay, twist, and bend (Fig. 3). The free-energy density associated with these three types of (small) deformation is written in terms of the spatial director gradients as Eq. (1)
and is known as the Frank-Oseen energy density, with Frank elastic constants of splay (K1), twist (K2), and bend (K3); all three are necessarily positive-definite and the dimensionality is that of a force. The elastic constants can be estimated as the typical energy of molecular interactions responsible for the orientational order divided by the characteristic length,
which is the molecular size which is a very good estimate for many thermotropic uniaxial nematic materials, as the experimental values are between 1 and 10 piconewtons (pN). The energy density [Eq. (1)] is often supplemented with the so-called divergence terms that can influence the equilibrium director through boundary conditions at the surface. In the presence of an external field, the free-energy density acquires additional terms; for example, a diamagnetic term ffield = −1/2μ0−1 χa B · n)2 in the magnetic field B; here μ0 = 4π × 10−7 henry/m is the magnetic permeability of free space.
Another important contribution to the free energy is related to the phenomenon of director anchoring at the bounding surfaces. Any actual liquid-crystal cell is confined, say, by a pair of parallel glass plates. The molecular interactions between the liquid crystal and the boundary substrates are anisotropic. This anisotropy establishes one (sometimes more) preferred orientation of n at the boundary, the “easy axis.” The phenomenon is called surface anchoring. Orienting action of the substrates usually keeps the director uniform if the external field is absent. To change this orientation, one needs to perform some work.
The equilibrium of a uniaxial nematic sample of a given shape in an external field is determined by the minimum of the functrional, Eq. (2),
where fanch is the surface-anchoring energy density, often represented by the simple potential fanch = 1/2 Waα2, with Wa being the surface-anchoring coefficient and α the angle between the easy axis and the actual director orientation at the surface.
Elasticity of the smectic A phase
For the smectic A (SmA) phase, the elastic free-energy density, Eq. (3),
is different from Eq. (1) because of certain restrictions that the layered structure imposes onto the director distortions and the elastic cost of changes in the thickness of the layers. Here B is the Young modulus (layers compressibility modulus) and γ = (d−d0)/d0, the relative difference between the equilibrium period d0 and the actual layer thickness measured along the director n. The ratio of K1 to B defines an important length scale, Eq. (4),
called the penetration length γ, and is of the order of the layer separation but diverges when the system approaches the SmA-nematic transition. One expects that a SmA would have K1 of the same order as in a nematic phase stable at higher temperatures. With λ ≈ d0 ≈ (1−3) nm, one finds B∼ 106 ÷ 107 N/m2, a value that is 103 to 104 times smaller than the compressibility modulus in a solid.
Liquid crystals are fluids. They can flow while preserving the orientational order. Flow imposes an orientational torque on n. Most often, n tends to realign along the direction of flow. There is also a reverse effect by which director distortions can cause the flow. This “backflow” effect is of importance in liquid-crystal displays. In the approximation of a constant scalar order parameter, the hydrodynamics of liquid crystals is described in terms of seven variables: (1) mass density, (2) three components of the velocity field, (3) energy density, and (4) two components of the director field n(r, t). In contrast to an isotropic fluid, the stress tensor depends not only on the gradients of the velocity but also on the director components. The uniaxial nematic phase should be characterized by five different viscosity constants. The number of viscosities reduces to three when the director distortions are small. These three can be chosen as the effective viscosities for three idealized geometries of flow, also known as Miezowicz geometries, in which one assumes that the director is fixed (for example, by a strong magnetic field): (a) When n = (1,0,0) is perpendicular to both the flow direction and the velocity gradient, the UN behaves as an isotropic fluid with a viscosity ηa; however, director fluctuations coupled with the certain values of the viscosity coefficients might destabilize the initial orientation. (b) When n is parallel to the flow or (c) when n is parallel to the velocity gradient (Fig. 4), the corresponding viscosities ηb and ηc are generally different from ηa and from each other; ηb < ηa < ηc for a typical uniaxial nematic material composed of rodlike molecules; the result ηb < ηc can be explained by assuming that the friction correlates with the cross section of the molecules seen by the flow.
When a thick (100-μm) nematic sample with no special aligning layers is viewed with a microscope, one usually observes a number of mobile flexible lines, called disclinations. The disclinations are seen as thin and thick threads (Fig. 5). Thin threads strongly scatter light and show up as sharp lines. These are truly topologically stable defect lines, along which the nematic symmetry of rotation is broken. The disclinations are topologically stable in the sense that no continuous deformation can transform them into a state with the uniform director field, n(r) = const. Thin disclinations are singular in the sense that the director is not defined along the line. Thick threads are line defects only in appearance; they are not singular disclinations. The director is smoothly curved and well defined everywhere, except for a number of point defects in the nematic bulk, called hedgehogs.
Applications of liquid crystals
The possibility to orient the director by an applied electric field leads to numerous practical applications. In a flat cell, the orienting action of the homogeneously treated substrate usually keeps the director uniform if the external field is absent. However, if the liquid crystal is dielectrically anisotropic, a sufficiently high electric field can overcome both the anchoring effect and the elasticity of the nematic bulk and reorient the director, either parallel (εa > 0) to the field or perpendicular to it (εa < 0). This is the Frederiks effect, first discovered for the magnetic case. When the field is removed, the surface anchoring and bulk elasticity restore the original director structure. Thus, one can use the electric field and surface anchoring to switch the elastic liquid crystal back and forth.
Various implementations of the Frederiks effect in uniaxial nematic (UN) and cholesteric (Ch) are widely used in electrooptic devices, including displays. The liquid crystal is usually sandwiched between two transparent electroconductive plates (for example, glass covered with indium tin oxide) coated with a suitable alignment layer, most often chosen from the polyimide family. The voltage across the cell controls the director configuration and thus the optical properties of the cell. In one of the most recent and commercially successful embodiments, the initial field-free state is the one in which n is perpendicular to the bounding plates with electrodes (Fig. 6). The cell is sandwiched between two crossed light-polarizing films. The light beam becomes linearly polarized after passing the first polarizer. This state of polarization is not changed by the liquid crystal, and the light beam is thus blocked by the second polarizer. The UN material is of a negative dielectric anisotropy, εa < 0. When the electric field is applied across the cell, the director deviates from the vertical axis. In this “field-on” state, because of the finite birefringence, the cell allows the light beam to pass through the second polarizer. The direction of director tilt in the plane of the UN cell can be controlled by using specially patterned electrodes; thus the mode is known as the patterned vertical alignment (PVA) mode. The field can also be applied in the plane of the sample, comprising a layer of a UN with εa > 0, thus forming the in-plane switching (IPS) mode. Finally, the polarization of light can be controlled by using the so-called twisted nematic (TN) cell, in which the electric field switched the twisted (chiral) nematic structure into a vertically aligned director configuration, thus altering the appearance of the UN cell between two polarizers, either crossed or parallel. To create a full-color display using the PVA, IPS, or TN techniques, one adds a pixelized color filter with red, blue, and green pixels to the glass plate facing the viewer. The panel that controls the voltage across the liquid crystal also is pixelized, so that the intensity of light transmitted by each pixel and the corresponding color filter is controlled independently of other pixels. The most advanced scheme of electric-addressing the liquid crystal display is the active matrix scheme, based on the array of thin-film transistors (TFTs). The operating voltage needed in liquid-crystal displays is relatively low, usually between 1 and 10 V. See also: Electronic display; Electrooptics; Flat-panel display device
The flexibility of molecular architectures of liquid crystals allows many other types of applications, including the information displays based on the effects of light scattering rather than the changes in polarization state. Historically, one of the first liquid-crystal devices was the cholesteric thermometer that changed colors as the function of temperature. In Ch, the helicoidal pitch is often in the range of 0.1–1 μm (the molecular interactions responsible for the chiral structure are relatively weak, thus the pitch is much larger than the size of an individual molecule). The well-aligned periodic director structure is thus capable of selective reflection in the visible part of the spectrum, which brings a “colored” appearance to the sample viewed in the white light. The spectral range of this selective reflection is related to the value of the cholesteric pitch and can be changed by changing the pitch. Ch materials with electrically controlled orientation of the helicoids axis are used in bistable light-reflective liquid-crystals displays (Fig. 7). The Ch is switched by the electric voltage pulses between a well-aligned (planar) light-reflecting state and a disoriented light-scattering state with numerous defects of the director configuration. By adding a black absorbing layer to the bottom of the Ch film, one creates a bistable Ch display with a bright colored appearance in the planar state and black appearance in the distorted state. The field is needed only to switch the Ch sample from one state to another, which greatly reduces energy consumption. Although only the planar state is truly stable, the distorted state is relatively stable, as its relaxation into the planar state is hindered by a large energy barriers associated with the helix distortions. As the display works in the light-reflective mode, it does not need a special illuminating panel beneath the liquid-crystal layer, as most other liquid crystals displays do. The Ch reflective displays are used in “electronic books” and in development of flexible displays in which the traditional glass panels are replaced with plastic films.
Some applications use composite materials in which the liquid crystal is a component. Polymer-dispersed liquid crystals are one example. By dispersing the UN droplets in a polymer matrix (through a phase separation process), one can create an effective light-scattering medium, provided the effective index of the liquid-crystal droplet is different from the refractive index of the polymer. By applying the electric field, the droplets and the director inside them reorient along the field. If the materials are selected in such a way that the ordinary index of refraction of the UN matches the refractive index of the polymer, the composite film becomes transparent. The composite film thus represents an electrically switchable light-scattering panel and can be used in large-area (square meters) “privacy windows.”