Liquid Crystals

Part of the Springer Handbooks book series (SPRINGERHAND)

Abstract

This chapter outlines the basic physics, chemical nature and properties of liquid crystals. These materials are important in the electronics industry as the electro-optic component of flat-panel liquid-crystal displays, which increasingly dominate the information display market.

Liquid crystals are intermediate states of matter which flow like liquids, but have anisotropic properties like solid crystals. The formation of a liquid-crystal phase and its properties are determined by the shape of the constituent molecules and the interactions between them. While many types of liquid-crystal phase have been identified, this Chapter focuses on those liquid crystals which are important for modern displays.

The electro-optical response of a liquid crystal display (LCD ) depends on the alignment of a liquid-crystal film, its material properties and the cell configuration. Fundamentals of the physics of liquid crystals are explained and a number of different displays are described.

In the context of materials, the relationship between the physical properties of liquid crystals and their chemical composition is of vital importance. Materials for displays are mixtures of many liquid-crystal compounds carefully tailored to optimise the operational behaviour of the display. Our current understanding of how chemical structure determines the physical properties is outlined, and data for typical liquid-crystal compounds are tabulated. Some key references are given, but reference is also made to more extensive reviews where additional data are available.

36.1 Introduction to Liquid Crystals

Liquid crystals have been known for almost 130 years but it is only in the last 40 years or so that their unique application in display devices has been recognised. Now they are seen as extremely important materials having made possible the development of thin screens for use with personal computers (PCs) and in televisions. In fact a wide range of different liquid-crystal (LC ) display devices has been developed. The common feature for each of these is that the optical characteristics of the display are changed on application of an electric field across a thin liquid-crystal film. The process causing this change is associated with a variation in the macroscopic organisation of the liquid crystal within the cell. The liquid crystal is, therefore, strictly behaving as a molecular material and not an electronic one. Nonetheless the display itself is closely integrated with electronic components. Since liquid crystals may be unfamiliar to those concerned with conventional electronic materials, this section begins with an introduction to liquid crystals and the compounds that form them. The following section describes the basic physics for liquid crystals which are needed to understand their use in display devices. The functioning of the most important displays is described in Sect. 36.3, which makes contact with the basic physics outlined in Sect. 36.2. The liquid-crystal materials used in display devices are discussed in the final section, where the necessary optimisation of a wide range of properties is addressed.

The majority of chemical compounds can exist in three states of matter, namely crystal, liquid or gas, each with its defining characteristics. There is a fourth state known as a liquid crystal and, as the name suggests, this state has characteristics of both crystals and liquids. Thus a liquid crystal flows when subject to a stress, like a liquid, but certain of its properties are anisotropic, like a crystal. This macroscopic behaviour, often used to identify the phase, implies that at the microscopic or molecular level the material has an element of long-range orientational order together with some translational disorder at long range. It is this combination of order and disorder that makes liquid crystals so fascinating and gives them their potential for applications, especially in the field of electro-optic displays.

A variety of different classes of materials are known to form liquid crystals at some point on their phase diagram [36.1]. These include organic materials where the liquid crystal is formed, on heating, between the crystal and isotropic liquid phases. Such materials are known as thermotropic liquid crystals and are the subject of this Chapter. Another class is formed by amphiphilic organic materials in which part of the constituent molecules favours one solvent, normally water, while the other part does not. When the amphiphile is dissolved in the water, the molecules form aggregates which then interact to give the liquid-crystal phase, the formation of which is largely controlled by the concentration of amphiphile. These are known as lyotropic liquid crystals; they underpin much of the surfactant industry, although they are not used in displays and so will not be considered further. Colloidal dispersions of inorganic materials such as clays can also form liquid-crystal phases depending on the concentration of the colloidal particles. Solutions of certain organic polymers also exhibit liquid-crystal phases and, like the colloidal systems, the solvent acts to increase the separation between polymer chains but does not significantly affect their state of aggregation. A prime example of such a system is the structural polymer Kevlar, which for the same weight is stronger than steel; it is formed by processing a nematic solution of the polymer.

The following sections return to thermotropic liquid crystals and describe the molecular organisation within the phases, mention some of their properties and briefly indicate the relationship between the phase and molecular structures.

36.1.1 Calamitic Liquid Crystals

In view of the anisotropic properties of liquid crystals, it seems reasonable that a key requirement for their formation is that the molecules are also anisotropic. This is certainly the case, with the majority of liquid crystals having rod-like molecules, such as those shown in Fig. 36.1. One of the simplest nematogenic rod-like molecules is p-quinquephenyl (Fig. 36.1a), which is essentially rigid. However, flexible alkyl chains can also be attached at one or both ends of the molecule (Fig. 36.1b,c) or indeed in the centre of the molecule (Fig. 36.1d), and rigid polar groups such as a cyano (Fig. 36.1d–f) may be attached at the end of the molecule. The rigid part is usually constructed from planar phenyl rings (Fig. 36.1a–e) but they can be replaced by alicyclic rings (Fig. 36.1f) which enhance the liquid crystallinity of the compound. The term calamitic, meaning rod-like, is applied to the phases that they form. There are, in fact, many different calamitic liquid-crystal phases but we shall only describe those which are of particular relevance to display applications.
Fig. 36.1

The molecular structures for a selection of compounds which form calamitic liquid crystals

At an organisational level the simplest liquid crystal is called the nematic and in this phase the rod-like molecules are orientationally ordered, but there is no long-range translational order. A picture showing this molecular organisation, obtained from a computer simulation of a Gay–Berne mesogen [36.2] is given in Fig. 36.2 b. The molecular shape is ellipsoidal and the symmetry axes of the ellipsoids tend to be parallel to each other and to a particular direction known as the director.
Fig. 36.2a–c

The molecular organisation in (a) the isotropic phase (b) the nematic phase and (c) the smectic A phase obtained from the simulation of a Gay–Berne calamitic mesogen

In contrast there is no ordering of the molecular centres of mass, except at short range. The essential difference between the nematic and isotropic phases (Fig. 36.2 a) is the orientational order, which is only short range in the isotropic liquid. At a macroscopic level the nematic phase is characterised by its high fluidity and by anisotropy in properties such as the refractive index. The anisotropic properties for a nematic have cylindrical symmetry about the director, which provides an operational definition of this unique axis and is the optic axis for the phase. The anisotropy in the refractive index combined with the random director distribution results in the turbidity of the phase, which contrasts with the transparency of the isotropic liquid. This on its own would not be sufficient to identify the liquid crystal as a nematic phase but identification is possible from the optical texture observed under a polarising microscope. These textures act as fingerprints for the different liquid-crystal phases. An example of such a texture for a nematic phase is shown in Fig. 36.3a; it is created by the anisotropy or birefringence in the refractive index combined with a characteristic distribution of the director in the sample.
Fig. 36.3a–c

Typical optical textures observed with a polarising microscope for (a) nematic, (b) smectic A and (c) columnar liquid-crystal phases

The next level of order within liquid crystal phases is found for the smectic A phase. Now, in addition to the long-range orientational order, there is translational order in one dimension, giving the layered structure shown in Fig. 36.2c [36.2]. The director associated with the orientational order is normal to the layers. Within a layer there is only short-range translational order as in a conventional liquid. In this structure the layer spacing is seen to be comparable but slightly less than the molecular length, as found experimentally for many smectic phases.

At a macroscopic level the layer structure means that the fluidity of a smectic A phase is considerably less than for a nematic phase. The properties are anisotropic and the birefringence is responsible for the turbidity of the phase, as found for a nematic. However, under a polarising microscope the optical texture is quite different to that of a nematic, as is apparent from the focal conic fan texture shown in Fig. 36.3b.

A variant on the smectic A is the smectic C phase. The essential difference to the smectic A phase is that the director in the smectic C phase is tilted with respect to the layer normal. The defining characteristic of the smectic C phase is then the tilt angle, which is taken as the angle between the director and the layer normal. This tilt in the structure reduces the symmetry of the phase to the point group C2 h in contrast to D∞h for nematic and smectic A phases. This lowering in symmetry naturally influences the symmetry of the properties. The fluidity of the smectic C phase is comparable to that of a smectic A phase. However, the optical texture can be quite different and it has elements similar to a nematic phase and to a smectic A; the focal conic fan structure is less well defined and is said to be broken. The nematic-like features result from the fact that the tilt direction of the director is not correlated between the smectic layers and so it adopts a distribution analogous to that of the director in a nematic phase.

The molecular factors which influence the ability of a compound to form a liquid-crystal phase have been well studied both experimentally [36.3] and theoretically [36.4]. Consider the simple nematic as formed, for example, by p-quinquephenyl (Fig. 36.1 a); this melts at 401C to form the nematic phase, which then undergoes a transition to the isotropic phase at 445C; the transition temperatures are denoted by TCrN and TNI, respectively. The very high value of TNI, which is a measure of the stability of the nematic phase, is attributed to the large length-to-breadth ratio of p-quinquephenyl. In contrast p-quaterphenyl, formed by the removal of just one of the five phenyl rings, does not exhibit a liquid-crystal phase at atmospheric pressure, even though its shape anisotropy is still relatively large. This occurs because, on cooling, the isotropic liquid freezes before the transition to the nematic phase can occur. Indeed many compounds with anisotropic molecules might be expected to form liquid-crystal phases, but do not because of their high melting points. As a consequence the molecular design of liquid crystals needs to focus not only on increasing the temperature at which the liquid crystal–isotropic transition occurs but also on lowering the melting point. One way by which this can be achieved is to attach flexible alkyl chains to the end of the rigid core (Fig. 36.1 ). In the crystal phase the chain adopts a single conformation but in a liquid phase there is considerable conformational disorder, and it is the release of conformational entropy on melting that lowers the melting point. The addition of the chain also affects the nematic–isotropic transition temperature, which alternates as the number of atoms in the chain passes from odd to even. This odd–even effect is especially dramatic when the flexible chain links two mesogenic groups (Fig. 36.1d) to give what is known as a liquid-crystal dimer [36.5]. The odd–even effect is particularly marked because the molecular shapes for the dimers with odd and even spacers differ significantly on average, being bent and linear, respectively.

The attachment of alkyl chains to the rigid core of a mesogenic molecule has another important consequence, as it tends to promote the formation of smectic phases (Fig. 36.1b,c). The reason that the chains lead to the formation of such layered structures is that, both energetically and entropically, the flexible chains prefer not to mix with the rigid core, and so by forming a layer structure they are able to keep apart. Indeed it is known that biphenyl (a rigid rod-like structure) is not very soluble in octane (a flexible chain). The lack of compatibility of the core and the chain increases with the chain length and so along a homologous series it is those members with long alkyl chains that form smectic phases. For example, 4-pentyl-4-cyanobiphenyl (Fig. 36.1e) forms only a nematic phase whereas the longer-chain homologue, 4-decyl-4-cyanobiphenyl only exhibits a smectic A phase. To obtain a tilted smectic phase such as a smectic C there clearly needs to be a molecular interaction which favours an arrangement for a pair of parallel molecules that is tilted with respect to the intermolecular vector. Such a tilted structure can be stabilised by electrostatic interactions; for example by off-axis electric dipoles (Fig. 36.1b,c) [36.6] or with a quadrupolar charge distribution.

The ability of a compound to form a liquid crystal is not restricted to just one phase. The delicate balance of the intermolecular interactions responsible for the various liquid-crystal phases means that transitions between them can result from modest variations in temperature. This is apparent for the 4,4-dialkyl-2,3-difluoroterphenyl shown in Fig. 36.1c, which on cooling the isotropic liquid forms nematic, smectic A and smectic C phases; such a compound is said to be polymorphic. Materials which form even more liquid-crystal phases are known [36.1]. The occurrence of several liquid-crystal phases in a single system can be of value in processing the material for display applications.

36.1.2 Chiral Liquid Crystals

The mesogenic molecules that have been considered so far are achiral in the sense that the molecule is superimposable on its mirror image. Molecules may also be chiral in that they are not superimposable on their mirror images; this chirality can result from the tetrahedral arrangement of four different groups around a single carbon atom. This is illustrated in Fig. 36.4 a which shows such an arrangement together with its mirror image; these are known as enantiomers. The presence of a chiral centre will certainly change the nature of the interactions between the molecules and it is relevant to see whether the chiral interactions might not influence the structure of the liquid-crystal phases exhibited by the material. From a formal point of view it might be expected that the molecular chirality of a mesogen should be expressed through the symmetry of the liquid-crystal structure. This proves to be the case, provided no other interactions oppose the chiral deformation of the liquid-crystal phase. In fact the first liquid crystal to be discovered [36.7] was chiral; this was cholesteryl benzoate where the cholesteryl moiety contains many chiral centres. The structure of the liquid crystal is nematic-like in that there is no long-range translational order but there is long-range orientational order. However, the difference between this phase and a nematic formed from achiral materials is that the director is twisted into a helix. The helix may twist in a left-handed or a right-handed sense and these structures, shown in Fig. 36.4b, are mirror images of each other. The phase structure is certainly chiral and so is known as a chiral nematic, although originally it was called a cholesteric phase. The symbol for the phase is N*, where the asterisk indicates that the phase has a chiral structure.
Fig. 36.4

(a) A chiral mesogenic molecule, (R) 2-[4-cyano-4-biphenyl]-hexane, together with its mirror image, (S) 2-[4-cyano-4-biphenyl]-hexane. (b) The left- and right-handed helical organisation of the director for a chiral nematic

The helical structure is characterised by the pitch of the helix p which is the distance along the helix axis needed for the director to rotate by 2π. Since the directions parallel and antiparallel to the director are equivalent the periodicity of the chiral nematic is p ∕ 2. For many chiral nematics the helical pitch is comparable to the wavelength of visible light. This, together with the periodic structure of the phase, means that Bragg reflection from a chiral nematic will be in the visible region of the spectrum and so this phase will appear coloured with the wavelength of the reflected light being related to the pitch of the helix. This pitch is sensitive to temperature, especially when the chiral nematic phase is followed by a smectic A. This sensitivity has been exploited in the thermochromic application of chiral nematics, where the reflected colour of the phase changes with temperature [36.1].

The chirality of cholesteryl benzoate clearly results from the chiral centres present in the mesogenic molecule. However, the chirality can also be introduced indirectly to a mesogen by simply adding a chiral dopant, which does not need to be mesogenic. The mixture will be chiral and this is sufficient to lead to a chiral nematic. The pitch of this mixture depends on the amount of the dopant and the inverse pitch, p−1, proves to be proportional to its concentration. The handedness of the helix induced by the dopant will depend on its stereochemical conformation and will be opposite for the two enantiomers. Accordingly, if both enantiomers are present in equal amounts, i. e., as a racemic mixture, then doping a nematic with this will not convert it to a chiral nematic.

Chiral smectic phases are also known in which the director adopts a helical structure as a result of introducing molecular chirality into the material either as a dopant or as an intrinsic part of the mesogenic molecule. The chiral smectic C phase, denoted SmC*, provides an appropriate example with which to illustrate the structure of such phases. In an achiral smectic C phase the tilt direction of the director changes randomly from layer to layer, analogous to the random director orientation in an achiral nematic. For the chiral smectic C phase, as might be anticipated, the tilt direction of the director rotates in a given sense, left-handed or right-handed, and by a small, fixed amount from layer to layer. Other structural features of the smectic C phase remain unchanged. Thus, the director of the chiral smectic C phase has a helical structure with the helix axis parallel to the layer normal. The pitch of the helix is somewhat smaller than that of the associated chiral nematic phase. The magnitude of the pitch is inversely related to the tilt angle of the smectic C, and since this grows with decreasing temperature so the pitch decreases. The reduced symmetry C2 of the SmC* phase leads to the introduction of a macroscopic electrical polarisation [36.8]. This is of potential importance for the creation of fast-switching displays.

The ability of the SmC* phase to adopt a helical structure results from the fact that the tilt direction for the director acts in an analogous manner to the director in a nematic, and importantly that the layer spacing is preserved in the helical structure. In marked contrast there are strong forces inhibiting the creation of a twisted structure for a smectic A composed of chiral molecules. The director is normal to the layers and so the creation of a twisted structure would require a variation in the layer thickness but this has a high energy penalty associated with it. Accordingly many of the smectic A phases formed from chiral molecules have the same structure as those composed of achiral molecules. There are, however, exceptions and these occur when the chiral interactions are especially strong and, presumably, the translational order of the layers is small. Under such conditions the smectic A structure is partially destroyed, creating small SmA-like blocks about 1000 Å wide, separated by screw dislocations [36.10, 36.9]. These defects in the organisation allow the directions for the blocks to rotate coherently to give a chiral helical structure; the pitch of the helix is found to be larger than that in an analogous chiral nematic phase. This chiral phase is just one example of a liquid-crystal structure stabilised by defects; it is known as a twist grain-boundary phase and denoted by TGBA*. The letter A indicates that the director is normal to the layers in the small smectic blocks; there is a comparable phase in which the director is tilted, denoted by TGBC*.

36.1.3 Discotic Liquid Crystals

The key requirement for the formation of a liquid crystal is an anisotropic molecule, as exemplified by the rod-like molecules described in the previous sections. However, there is no reason why disc-like molecules should not also exhibit liquid-crystal phases. Nonetheless, it was not until 1977 that the first example of a thermotropic liquid crystal formed from disc-like molecules was reported [36.11]. Since that time several liquid-crystal phases have been identified and these phases are known collectively as discotic liquid crystals. The range of compounds that exhibits these phases is now extensive and continues to grow [36.12], although it does not match the number that form calamitic liquid crystals. The molecular structures of three compounds which form discotic liquid crystals are shown in Fig. 36.5.
Fig. 36.5

A selection of molecular structures for compounds that form discotic liquid crystals

As for rod-like molecules, the simplest liquid-crystal phase formed by disc-like molecules is the nematic, usually denoted ND, where the D indicates the disc-like nature of the molecules. Within the nematic phase, shown in Fig. 36.6a, the molecular centres of mass are randomly distributed and the molecular symmetry axes are orientationally correlated. The nematic structure is the same as for that formed from rod-like molecules, the only difference being that the symmetry axes which are orientationally ordered are the short axes for the discs and the long axes for the rods. The point symmetry of the discotic and calamitic nematic is the same, namely D∞h. The discotic nematic is recognised in the same way as the calamitic nematic; that is it flows like a normal fluid and its anisotropy is revealed by a characteristic optical texture analogous to that shown in Fig. 36.3a. In fact, the refractive index of a discotic nematic along the director is smaller than that perpendicular to it, which is the opposite to that for a calamitic nematic. It has been suggested that this difference may be of value in display devices [36.13] but this concept has not as yet been commercialised.
Fig. 36.6a–c

The molecular organisation in discotic liquid-crystal phases, (a) nematic, (b) hexagonal columnar and (c) rectangular columnar

The other class of discotic liquid crystals possesses some element of long-range translational order and these are known as columnar phases, two examples of which are sketched in Fig. 36.6b,c. The disc-like molecules are stacked face-to-face into columns. A single column has a one-dimensional structure, and as such is not expected to exhibit long-range translational order, although this can result from interactions between neighbouring columns in the liquid-crystal phase. The column–column interactions will result in the columns being aligned parallel to each other; these interactions will also determine how the columns are packed. When the discs are orthogonal to the column axis the cross section is essentially circular and so the columns pack hexagonally, as shown in Fig. 36.6b. The symbol given to this phase is Colhd, where h denotes hexagonal packing of the columns and d indicates that the arrangement along the column is disordered. The point group symmetry of this phase is D6 h. The disc-like molecules may also be tilted with respect to the column axis, giving an elliptical cross section to the columns. As a result the columns are packed on a rectangular lattice; there are four possible arrangements and just one of these is indicated in Fig. 36.6 c. In general the mnemonic used to indicate a rectangular columnar phase is Colrd. The point group symmetry of the rectangular columnar phase is D2 h and the extent to which the structure deviates from that of the Colhd phase will depend on the magnitude of the tilt angle within the column. The columnar phases can be identified from their optical textures and an example of one is shown in Fig. 36.3c.

The columnar phases have potential electronic applications because of the inhomogeneity of the molecules that constitute them, i. e., the central part is aromatic while the outer part is aliphatic. As a result of the overlap between the π-orbitals on the centres of neighbouring discs it should be possible for electrical conduction to take place along the core of the column. This should occur without leakage into adjacent columns because of the insulation provided by the alkyl chains. It should also be possible to anneal these molecular wires because of their liquid-crystal properties [36.14]. This and the ability to avoid defects in the columns which can prevent electronic conduction in crystals mean that the columnar phase has many potential advantages over non-mesogenic materials. In addition, discotic systems are also used as compensating films to improve the optical characteristics for some liquid-crystal displays.

At a molecular level the factors that are responsible for the formation of the discotic liquid-crystal phases are similar to those for calamitic systems. Thus, the molecular design should aim to increase the liquid crystal–isotropic transition temperature while decreasing the melting point. The latter is certainly achieved by attaching flexible alkyl chains to the perimeter of the rigid disc. The creation of the columnar phases should be relatively straightforward provided the central core is both planar and large. Then, because of the strong attractive forces between the many atoms in the rigid core the molecules will wish to stack face-to-face in a column. The formation of the columns will also be facilitated by the flexible alkyl chains attached to the core. Clearly then it may prove to be difficult to create the nematic phase before the columnar phase is formed unless the disc–disc interactions can be weakened. One way in which this can be achieved is by destroying the planarity of the core, for example, by using phenyl rings attached to the molecular centre so that they can rotate out of the plane (Fig. 36.5 a). It is to be expected that the columnar phases should occur below the nematic phase, corresponding to an increase in order with decreasing temperature. This is usually observed, for example, for the hexasubstituted triphenylenes (Fig. 36.5b). However, the truxene derivatives, with long alkyl chains on the perimeter, (Fig. 36.5 c) exhibit quite unusual behaviour. For these compounds the crystal melts to form a discotic nematic and then at a higher temperature a columnar phase appears. This deviates from the expected sequence, and because the nematic phase appears at a lower temperature than the columnar phase it is usually referred to as a re-entrant nematic. The occurrence of a re-entrant phase is often attributed to a conformational change which strengthens the molecular interactions with increasing temperature thus making the more ordered phase appear at higher temperatures.

36.2 The Basic Physics of Liquid Crystals

36.2.1 Orientational Order

The defining characteristic of a liquid crystal is the long-range orientational order of its constituent molecules. That is, for rod-like molecules, the molecular long axes tend to align parallel to each other even when separated by large distances. The molecules tend to be aligned parallel to a particular direction known as the director and denoted by n. This is an apolar vector, that is \(\boldsymbol{n}=-\boldsymbol{n}\), because the nematic does not possess long-range ferroelectric order. The properties of the nematic phase are cylindrically symmetric about the director, which provides a macroscopic definition of this. The anisotropy of the properties results from the orientational order and the extent of this is commonly defined [36.15] by
$$S=\left\langle\frac{1}{2}\left(3\cos^{2}\beta-1\right)\right\rangle\;,$$
(36.1)
although other definitions are possible [36.16]. Here β is the angle made by a molecule with the director and the angular brackets indicate the ensemble average. In the limit of perfect order S is unity while in the isotropic phase S vanishes. The temperature dependence of S is shown in Fig. 36.7 for the nematogen, 4,4-dimethoxyazoxybenzene; this behaviour is typical of most nematic liquid crystals. At low temperatures S is about 0.6 and then decreases with increasing temperature, reaching about 0.3 before it vanishes discontinuously at the nematic–isotropic transition, in keeping with the first-order nature of this transition. It is also found, both experimentally and theoretically, that the orientational order of different nematic liquid crystals is approximately the same provided they are compared at corresponding temperatures, either the reduced, T ∕ TNI, or shifted, TNI − T, temperatures [36.16]. Since many properties of liquid crystals are related to the long-range orientational order these also vary with temperature especially in the vicinity of the transition to the isotropic phase.
Fig. 36.7

The temperature variation of the orientational order parameter S for 4,4-dimethoxyazoxybenzene; the different symbols indicate results determined with different techniques

36.2.2 Director Alignment

The director in a bulk liquid crystal is distributed randomly unless some constraint is applied to the system; a variety of constraints can be employed and two of these are of special significance for display applications. One of them is an electric field and because of the inherent anisotropy in the dielectric permittivity of the liquid crystal the director will be aligned. The electric energy density controlling the alignment is given by [36.17]
$$U_{{\text{elec}}}=-\frac{\varepsilon_{0}\Updelta{}\varepsilon}{2}(\boldsymbol{n}\cdot\boldsymbol{E})^{2}\;.$$
(36.2)
Here ε0 is the permittivity of a vacuum and the scalar product n ⋅ E is Ecosθ, where E is the magnitude of the field E and θ is the angle between the director and the field, Δε is the anisotropy in the dielectric tensor
$$\Updelta{}\varepsilon=\varepsilon_{||}-\varepsilon_{\bot}\;,$$
(36.3)
where the subscripts denote the values parallel (\({||}\)) and perpendicular (⊥) to the director. If the dielectric anisotropy is positive then the director will be aligned parallel to the electric field and, conversely, if Δε is negative, then the director is aligned orthogonal to the field. The molecular factors that control the sign of Δε will be discussed in Sect. 36.4 . Of course, intense electric fields are also able to align the molecules in an isotropic phase but what is remarkable about a nematic liquid crystal is the very low value of the field needed to achieve complete alignment of the director. Thus for a bulk nematic free of other constraints the electric field necessary to align the director is typically about 30 kV ∕ m, although the value does clearly depend on the magnitude of the dielectric anisotropy. This relatively small value results because of the long-range orientational correlations which mean that the field acts, in effect, on the entire ensemble of molecules and not just single molecules.

The other constraint, essential for display devices, is the interaction between the director and the surface of the container [36.18]. At the surface there are two extreme arrangements for the director. One is with the director orthogonal to the surface, the so-called homeotropic alignment. In the other the director is parallel to a particular direction in the surface; this is known as uniform planar alignment. The type of alignment depends on the way in which the surface has been treated. For example, for a glass surface coated with silanol groups a polar liquid crystal will be aligned homeotropically, while to achieve this alignment for a non-polar nematic the surface should be covered with long alkyl chains. In these examples the direct interaction of a mesogenic molecule with the surface produces an orthogonal alignment which is then propagated by the long-range order into the bulk. To achieve uniform planar alignment of the director the surface is coated with a polymer, such as a polyimide, which on its own would result in planar alignment. To force the director to be parallel to a particular direction in the surface the polymer is rubbed which aligns the director parallel to the direction of rubbing. There is still some uncertainty about the mechanism responsible for uniform planar alignment. It might result from alignment of the polymer combined with anisotropic intermolecular attractions with the mesogenic molecules, although it had been thought [36.19] to have its origins in surface groves and the elastic interactions which are described later.

The energy of interaction between the surface and the director clearly depends on the nature of the surface treatment and the particular nematic. Rapini and Papoular [36.20] have suggested the following simple form for the surface energy density
$$U_{{\text{S}}}=-\frac{A}{2}(\boldsymbol{n}\cdot\boldsymbol{e})^{2}\;,$$
(36.4)
where A is the anchoring energy and e is the easy axis or direction along which the director is aligned. Clearly it has an analogous form to that for the anisotropic interaction between the nematic and an electric field (36.2) but is essentially phenomenological. The anchoring energy is determined to be in the range \({\mathrm{10^{-7}}}{-}{\mathrm{10^{-5}}}\,{\mathrm{J/m^{2}}}\) [36.21]. The upper value corresponds to strong anchoring in that typical values of the electric field would not change the director orientation at the surface. In contrast the lower value is associated with weak anchoring and here the director orientation at the surface can be changed by the field.

36.2.3 Elasticity

As the name suggests, a liquid crystal has some properties typical of crystals and others of liquids. The elastic properties of crystals should, therefore, be reflected in the behaviour of liquid crystals. Here it is the director orientation which is the analogue of the atomic positions in a crystal. In the ground state of a nematic liquid crystal the director is uniformly aligned. However, the elastic torques, responsible for this uniform ground state, are weak and, at temperatures within the nematic range, the thermal energy is sufficient to perturb the director configuration in the bulk. This perturbed state can take various forms depending on a combination of factors but, whatever the form, it can be represented as a sum of just three fundamental distortion modes. These are illustrated in Fig. 36.8 and are the splay, twist and bend deformations [36.22]. At a more formal level these modes are also shown in terms of the small deviations of the director from its aligned state at the origin as the location from this is varied. For example, for the twist deformation away from the origin along the x-axis there is a change in the y-component of the director, n y , and the displacement along the y-axis causes a change in n x . The magnitude of the twist deformation is measured by the gradients \(\partial n_{y}/\partial x\) and \(\partial n_{x}/\partial y\).
Fig. 36.8a–c

The three fundamental deformations for the nematic director (a) splay, (b) twist and (c) bend

The energy needed to stabilise a given distortion of the director field is clearly related to the extent of the deformation via the gradients. This distortion energy density for a bulk nematic liquid crystal is given by continuum theory [36.22] as
$$\begin{aligned}\displaystyle f&\displaystyle=\frac{1}{2}\left[K_{1}(\nabla\cdot\boldsymbol{n})^{2}+K_{2}(\boldsymbol{n}\cdot\nabla\times\boldsymbol{n})^{2}\right.\\ \displaystyle&\displaystyle\left.\quad\,+K_{3}(\boldsymbol{n}\times\nabla\times\boldsymbol{n})^{2}\right]\;,\end{aligned}$$
(36.5)
where the terms ∇ ⋅ n, \(\boldsymbol{n}\cdot\nabla\times\boldsymbol{n}\) and \(\boldsymbol{n}\times\nabla\times\boldsymbol{n}\) correspond to the splay, twist and bend deformations, respectively. The contribution each makes to the free energy is determined by the proportionality constants K1, K2 and K3, which are usually known as the Frank elastic constants for splay, twist and bend, respectively. They are small, typically \({\mathrm{5\times 10^{-12}}}\,{\mathrm{N}}\), and their small magnitude explains why the thermal energy is able to distort the uniform director arrangement so readily. The elastic constants are not in fact constant but vary with temperature and nor are they equal as the results for 4-pentyl-4-cyanobiphenyl (5CB) shown in Fig. 36.9 demonstrate. The twist elastic constant is seen to be the smallest while the largest is the bend elastic constant. This means that it is easiest to induce a twist deformation in a nematic while a bend deformation is the most difficult to create. All three elastic constants decrease with increasing temperature in keeping with the decreasing order as the transition to the isotropic phase is approached. Like the orientational order the elastic constants vanish discontinuously at the first-order nematic–isotropic transition.
Fig. 36.9

The temperature dependence of the three elastic constants, K1, K2 and K3, for the nematogen 5CB

The continuum theory is especially valuable in predicting the behaviour of display devices and this is illustrated by considering one of the ingenious experiments devised by Fréedericksz to determine the elastic constants [36.23]. In these a thin slab of nematic is confined between two glass plates with a particular director configuration, either uniform planar or homeotropic produced by surface forces. These forces control the director orientation just at the two surfaces and the alignment across the slab is propagated by the elastic interactions. A field is then applied which will move the director away from its original orientation and the variation of the director orientation with the field strength provides the elastic constant. To employ the continuum theory in order to describe the experiment it is necessary to add the field energy density (36.2) to the elastic free-energy density in (36.5). The director configuration is then obtained by integrating the free-energy density over the volume of the sample, and minimising this, subject to the surface constraints. The geometry of the experiment is shown in Fig. 36.10a,b and provided the dielectric anisotropy Δε is positive the director will move from being orthogonal to the field to being parallel to it. However, the extent of this twist deformation will vary across the cell, being greatest at the centre and zero at the surfaces, in the limit of strong anchoring. The dependence of the director orientation, at the centre of the slab, with respect to the electric field is shown in Fig. 36.11 . As the field strength is increased from zero the director orientation remains unchanged until a threshold value Eth is reached when the angle between the director and the field starts to decrease continuously. At very high values of the field the director at the centre of the slab tends to be parallel to the field. This behaviour can be understood in the following simple terms. Below the threshold field the elastic energy exceeds the electrical energy and so the director retains its uniform planar alignment. Above the threshold field strength the elastic energy is less than the electrical energy and so the director begins to move to be parallel to the electric field. The threshold electric field is predicted to be
$$E_{{\text{th}}}=\frac{\uppi}{l}\sqrt{\frac{K_{2}}{\varepsilon_{0}\Updelta{}\varepsilon}}\;,$$
(36.6)
where l is the slab thickness. Once the threshold field has been measured and provided the dielectric anisotropy is known, the twist elastic constant would be available. Other Fréedericksz experiments with analogous expressions for the threshold field lead to the determination of the splay and bend elastic constants.
Fig. 36.10a,b

The geometry for the Fréedericksz experiment used to determine the twist elastic constant, K2; (a) in zero field and (b) above the threshold value

Fig. 36.11

The dependence of the director orientation θm at the centre of a nematic slab on the scaled field strength, \(E^{\ast}=E/E_{{\text{th}}}\)

For real display applications (Sect. 36.3) the director alignment at the surface deviates from either uniform planar or homeotropic alignment. This deviation is known as a surface pre-tilt and is illustrated in Fig. 36.12 a for near-uniform planar alignment with the tilt direction on the two surfaces differing by 180. The cell with this arrangement has what is known as antiparallel alignment, so named because of the difference in the tilt direction caused by the direction of rubbing on the surfaces being antiparallel. In zero field, therefore, the director is uniformly aligned across the cell but tilted with respect to the x-axis set in the surface. A continuum theory calculation analogous to the case when the pre-tilt angle θ0 is zero allows the dependence of the director orientation θm in the centre of the cell to be determined as a function of the strength of the field applied across the cell. The results of these calculations, for a nematic with positive Δε, are shown in Fig. 36.12b as a function of the scaled field strength E ∕ Eth, where Eth is the threshold field for zero pre-tilt. The theoretical dependence for a tilt angle of zero is analogous to that considered for the twist deformation. In other words, below the threshold field the director is parallel to the easy axis and then above this threshold the director moves to become increasingly parallel to the field. When there is a surface pre-tilt the behaviour is quite different and this is especially apparent for a pre-tilt angle of 10 (Fig. 36.12b). In zero field the angle θm made by the director with the x-axis is 10 and, as the field increases, so does θm. The rate of increase grows as the threshold field is approached and then is reduced as E ∕ Eth increases beyond unity. Comparison of this behaviour with the conventional Fréedericksz experiment (Fig. 36.12b) shows that a pre-tilt of 10 has a significant effect on the way in which the director orientation changes with the field strength. Indeed, even for a pre-tilt of just 2, there is a pronounced difference in behaviour in the vicinity of the threshold field. This clearly has important implications for the accurate determination of the elastic constants [36.20]. It also shows how unique the behaviour is when the surface pre-tilt angle is zero.
Fig. 36.12

(a) The director alignment in a cell with pre-tilt at the two surfaces assembled so that the rubbing directions are antiparallel. (b) The electric field dependence of the director orientation θm at the centre of the cell for values of the pre-tilt angle of 0 (solid line), 2 (dashed line) and 10 (dash-dotted line)

Chiral nematics are often employed in liquid-crystal display devices. Locally, their structure is analogous to a nematic but the director is twisted into a helical structure. The continuum theory for the chiral nematic must, therefore, be consistent with the helical ground state structure of the phase. To achieve this, a constant is added to the twist term in the elastic free-energy density. Thus (36.5) for a nematic becomes
$$\begin{aligned}\displaystyle f&\displaystyle=\frac{1}{2}\left[K_{1}(\nabla\cdot\boldsymbol{n})^{2}+K_{2}\left(\boldsymbol{n}\cdot\nabla\times\boldsymbol{n}-\frac{2\uppi}{p}\right)^{2}\right.\\ \displaystyle&\displaystyle\qquad\left.+K_{3}(\boldsymbol{n}\times\nabla\times\boldsymbol{n})^{2}\right]\end{aligned}$$
(36.7)
for a chiral nematic, where p is the pitch of the helix.

36.2.4 Flexoelectricity

Another property of solids which is mimicked by liquid crystals is piezoelectricity. For solids this is the generation of a macroscopic electrical polarisation as a result of the deformation of certain ionic materials. It is to be expected, therefore, that deformation of the director distribution for a liquid crystal will create a macroscopic polarisation; this proves to be the case [36.24] and the phenomenon is known as flexoelectricity. The origin of flexoelectricity can be understood in the following way. It is generally assumed that mesogenic molecules are cylindrically symmetric but examination of the molecular structure of real mesogens shows that this is not the case. The molecules are asymmetric and at one extreme can be thought of as bent, as shown in Fig. 36.13a,b; here z is the molecular long axis and x is the axis bisecting the bond angle. When the director is uniformly aligned the z-axis will be parallel to the director and the x-axis will be randomly arranged orthogonal to it (Fig. 36.13a,b ). If the director is now subject to a bend deformation the molecular long axis will still tend to be parallel to the local director, however, the x-axis will tend to align parallel to a direction in the plane formed by the bent director. This change in the distribution function for the x-axis will introduce polar order into the system and if there is an electrical dipole moment along the x-axis then the deformed nematic will exhibit a macroscopic polarisation P. A similar argument shows that, if the molecule is wedge-shaped, a splay deformation of the director distribution will also induce a polarisation in the nematic [36.24].
Fig. 36.13a,b

The orientational distribution of bent molecules in (a) a uniformly aligned nematic and (b) one subject to a bend deformation; the x-axes are denoted by arrows

The magnitude of P, the induced dipole per unit volume, clearly depends on the extent of the deformation in the director distribution. In the linear response regime the induced polarisation for mesogenic molecules of arbitrary asymmetry is given by [36.25]
$$\boldsymbol{P}=e_{1}\boldsymbol{n}\nabla\cdot\boldsymbol{n}+e_{3}\boldsymbol{n}\times\nabla\times\boldsymbol{n}\;,$$
(36.8)
where the vectors n∇ ⋅ n and \(\boldsymbol{n}\times\nabla\times\boldsymbol{n}\) represent the splay and bend deformations, respectively. Since the polarisation per unit volume is also a vector then the proportionality constants, e1 and e3, are scalars. These are known as the splay and bend flexoelectric coefficients, respectively, and their dimensions are C m−1. The determination of individual flexoelectric coefficients is challenging [36.26], however, it seems that (e1 + e3) is of the order of \({\mathrm{10^{-12}}}{-}{\mathrm{10^{-11}}}\,{\mathrm{C{\,}m^{-1}}}\), although from the previous discussion their magnitude should vary amongst the mesogens because of their dependence on the molecular asymmetry and the size and location of the dipole moment. The molecular model proposed to understand the flexoelectricity of nematics suggests that, for rod-like molecules, devoid of asymmetry, the flexoelectric coefficients should vanish, but that does not seem to be the case. Indeed, it has been proposed that polarisation can result even for rod-like molecules if they possess an electrostatic quadrupole moment [36.27]. This is not inconsistent with the polarisation predicted by (36.8) which follows from the reduced symmetry of a nematic with splay and bend deformations of the director [36.24]. Although the existence of flexoelectricity is of considerable fundamental interest the inverse effect in which the director is deformed from a uniform state by the application of an electric field is of relevance for liquid-crystal displays (Sect. 36.3). This deformation occurs because the polarisation induced by the director deformation can couple with the applied electric field and so stabilise the deformation [36.26]. Since the coupling is linear in the electric field then reversal of the field will simply reverse the deformation; a novel bistable device based solely on this reversal is described in Sect. 36.3.

36.2.5 Viscosity

For a nematogen the fluidity of the nematic phase is comparable to that of the isotropic phase appearing at a higher temperature. It is this fluidity, similar to that of a liquid, which is responsible, in part, for the display applications of nematics. An indication of the fluidity of conventional liquids is provided by a single viscosity coefficient η, measured from the flow of the liquid subject to an applied stress in a viscometer. The flow behaviour of a nematic is made more complex by its defining long-range orientational order and the resultant anisotropy of the phase.

This complexity can be appreciated at a practical level in terms of the Miesowicz experiments to determine the viscosity coefficients of a nematic [36.28]. In these experiments it is helpful to consider flow in a viscometer with a square cross section such that there is a velocity gradient orthogonal to the direction of flow. For a nematic, flow through the viscometer will now depend on the orientation of the director with respect to these two axes. A magnetic field is employed to align the director along a particular axis and it must be sufficiently strong that flow does not perturb the director alignment. There are three relatively simple flow geometries and these are shown in Fig. 36.14a-c: with the director parallel to the flow direction (Fig. 36.14a-ca), with the director parallel to the velocity gradient (Fig. 36.14a-cb), and with the director orthogonal to both the flow direction and velocity gradient (Fig. 36.14a-cc). The viscosity coefficients were denoted by Miesowicz as a) η1, b) η2, and c) η3, although other notation has been proposed in which η1 and η2 are interchanged [36.29]. The three viscosity coefficients are clearly expected to differ given the anisotropy of the nematic phase and these differences are found to be large. They are illustrated for the room-temperature nematogen 4-methoxybenzylidene-4-butylaniline (MBBA) [36.30] in Fig. 36.15 where the viscosities are plotted against temperature.
Fig. 36.14a–c

The principal flow geometries of the Miesowicz experiments with the director pinned (a) parallel to the flow direction, (b) parallel to the velocity gradient and (c) orthogonal to both the flow direction and velocity gradient

Fig. 36.15

The temperature dependence of the three Miesowicz viscosity coefficients, η1, η2, and η3, for the nematic phase of MBBA; the viscosity coefficient for the isotropic phase is also shown

It is immediately apparent that flow is easiest when the director is parallel to the direction of flow, as might have been anticipated. Conversely flow of the nematic is most difficult when the director is orthogonal to the flow direction but parallel to the velocity gradient. The intermediate viscosity η3 occurs when the director is orthogonal to both the flow direction and the velocity gradient. Here it is seen that this third viscosity coefficient for the nematic is very similar to the viscosity coefficient for the isotropic phase extrapolated to lower temperatures.

The viscosity coefficient η ( θ , φ )  for an arbitrary orientation (θ , φ) of the director with respect to the flow direction and velocity gradient can be related to the director orientation. It might be expected that this orientation dependence would involve just the three Miesowicz viscosity coefficients, rather like the transformation of a second-rank tensor from its principal axis system. In fact this is not quite the case and a fourth viscosity coefficient η12 needs to be introduced. The orientation dependence is then [36.28]
$$\begin{aligned}\displaystyle\eta(\theta,\phi)&\displaystyle=\eta_{1}\cos^{2}\theta+\left(\eta_{2}+\eta_{12}\cos^{2}\theta\right)\\ \displaystyle&\displaystyle\quad\times\sin^{2}\theta\cos^{2}\phi+\eta_{3}\sin^{2}\theta\sin^{2}\phi\;.\end{aligned}$$
(36.9)
We see that for the principal orientations (0, π ∕ 2), (π ∕ 2, 0) and (π ∕ 2, π ∕ 2) the expression gives η1, η2, and η3, as required. The optimum director orientation with which to determine η12 is with the director at 45 to both the flow direction and the velocity gradient; the viscosity coefficient for this is
$$\eta\left(\frac{\uppi}{4},0\right)=\frac{\eta_{12}}{4}+\frac{\eta_{1}+\eta_{2}}{2}\;.$$
(36.10)
The value of η12 determined for MBBA proves to be significantly smaller than the other three viscosity coefficients.
The four viscosity coefficients have been defined at a practical level in terms of flow in which the director orientation is held fixed. The converse of these experiments, in which the director orientation is changed in the absence of flow, allows the definition of the fifth and final viscosity coefficient. This is known as the rotational viscosity coefficient; it is denoted by the symbol γ1 and plays a major role in determining the response times of display devices (Sect. 36.4 ). To appreciate the significance of γ1, it is helpful, as for the other four viscosity coefficients, to consider an experiment with which to measure it. In this an electric field is suddenly applied an angle θ to a uniformly aligned director, then providing the dielectric anisotropy is positive the director orientation will be changed and rotates towards the field direction. The electric torque responsible for the alignment is given by
$$\Gamma_{{\text{elec}}}=-\frac{\varepsilon_{0}\Updelta{}\varepsilon}{2}\sin 2\theta\;,$$
(36.11)
which is the derivative of the electric energy in (36.2 ). The rotation of the director is opposed by the viscous torque
$$\Gamma_{{\text{visc}}}=\gamma_{1}\frac{\mathrm{d}\theta}{\mathrm{d}t}\;,$$
(36.12)
which is linear in the rate at which the director orientation changes, with the proportionality constant being γ1. Provided the only constraint on the director is the electric field and provided, \(\theta_{0}\leq{\mathrm{45}}^{\circ}\), the director will move as a monodomain so that the elastic terms vanish. The inertial term for a nematic is small and so the movement of the director is governed by the equation in which the two torques are balanced. That is, the electric torque causing rotation is balanced by the viscous torque opposing it; this gives
$$\gamma_{1}\frac{\mathrm{d}\theta}{\mathrm{d}t}=-\frac{\varepsilon_{0}\Updelta{}\varepsilon}{2}E^{2}\sin^{2}\theta\;.$$
(36.13)
The solution to this differential equation is
$$\tan\theta=\tan\theta_{0}\exp\left(-\frac{t}{\tau}\right),$$
(36.14)
where θ0 is the initial orientation of the director with respect to the electric field and τ is the relaxation time
$$\tau=\frac{\gamma_{1}}{\varepsilon_{0}\Updelta{}\varepsilon E^{2}}\;.$$
(36.15)
Measurement of the time-dependent director orientation allows τ to be determined and from this γ1, given values of Δε. For MBBA the rotational viscosity coefficient is found to be slightly less than η2 and to parallel its temperature dependence [36.31].

The five independent viscosity coefficients necessary to describe the viscous behaviour of a nematic have been introduced in a pragmatic manner by appealing to experiments employed to measure these coefficients. However, the viscosity coefficients can be introduced in a more formal way as has been shown by Ericksen and then by Leslie in their development of the theory for nematodynamics [36.32]. The Leslie–Ericksen theory in its original form contained six viscosity coefficients, but subsequently Parodi has shown, using the Onsager relations, that there is a further equation linking the viscosity coefficients, thus reducing the number of independent coefficients to five [36.33]. These five coefficients are linearly related to those introduced by reference to specific experiments.

36.3 Liquid-Crystal Devices

The idea to use liquid crystals as electro-optic devices goes back to the early days of liquid-crystal research. In 1918 Björnståhl, a Swedish physicist, demonstrated that the intensity of light transmitted by a liquid crystal could be varied by application of an electric field [36.34]. As optical devices of various types became established in the first decades of the 20th century, mainly in the entertainment industry, ways of controlling light intensity became important to the developing technologies. One device that soon found commercial application was the Kerr cell shutter, in which an electric field caused the contained fluid (usually nitrobenzene) to become birefringent. Placing such a cell between crossed polarisers enabled a beam of light to be switched on and off very rapidly. The first report of liquid crystals being of interest for electro-optic devices was in 1936, when the Marconi Company filed a patent [36.35] which exploited the high birefringence of nematic liquid crystals in an electro-optic shutter. However, it was another 35 years before commercial devices became available which used the electro-optic properties of liquid crystals. The long interruption to the development of liquid-crystal devices can be attributed to the lack of suitable materials. We shall see in this Section how the physical properties of liquid crystals determine the performance of devices.

The optical properties of liquid crystals are exploited in displays, although the operational characteristics of such devices also depend crucially on many other physical properties (Sect. 36.2). Since the devices to be described all depend on the application of an electric field, their operation will be influenced by electrical properties such as dielectric permittivity and electrical conductivity. There is a range of electro-optic effects that can be used in devices, and the precise manner in which the properties of the liquid-crystal materials affect the device behaviour depends on the effect and the configuration of the cell. Thus there is not a single set of ideal properties that can define the best liquid-crystal material, rather the material properties have to be optimised for a particular application. In this section, different devices will be described, and their dependence on different properties will be outlined. The way in which the material properties can be adjusted for any application will be discussed in the final section of this chapter.

36.3.1 A Model Liquid-Crystal Display: Electrically Controlled Birefringence (ECB) Mode

If an electric field is applied across a film of planar-aligned liquid crystal having a positive dielectric anisotropy, then the director of the liquid crystal will tend to align along the electric field. Thus the director will rotate into the field direction, and the optical retardation or birefringence of the film will change. If the film is placed between crossed polarisers, then the change in optical retardation will be observed as a change in the intensity of transmitted light. The configuration for an ECB-mode display is schematically illustrated in Fig. 36.16. This illustrates the principal components of a liquid-crystal display. The liquid crystal is contained between two glass plates that have been coated with a transparent conducting layer, usually an indium–tin oxide alloy. Such treatment allows the application of an electric field across the liquid-crystal film, which can then be viewed along the field direction. For most applications, the surfaces of the electrodes are treated so that a particular director orientation is defined at the surface. This can be achieved in a variety of ways, depending on the desired surface director orientation. For the ECB-mode display under consideration, the director alignment should be parallel to the glass substrates and along a defined direction, which is the same on both surfaces. The standard technique to produce this alignment is to coat the glass plates with a thin (0.5 nm) layer of polyimide, which is mechanically rubbed in a particular direction. The cell is then placed between crossed polarisers, the extinction directions of which make angles of \(\pm{\mathrm{45}}^{\circ}\) with the surface-defined director orientation.
Fig. 36.16

Electrically controlled birefringence (ECB) cell

The unperturbed state of the liquid-crystal film will be determined by the surfaces which contain it. If these have been treated in such a way that the liquid-crystal director is parallel to the surface along a particular direction, then the film will act as an optical retardation plate. Thus incident polarised light will, in general, emerge elliptically polarised, and there will be a phase retardation between components of the light wave parallel to the fast and slow axes of the retardation plate. For electromagnetic waves polarised at \(\pm{\mathrm{45}}^{\circ}\) to the director the phase retardation φ, in radians, will be determined by the intrinsic birefringence of the liquid crystal (\(\Updelta{}n=n_{{\text{e}}}-n_{{\text{o}}}\)), the film thickness ℓ and the wavelength of the light λ
$$\phi=\frac{2\uppi\ell}{\lambda}({n_{\text{e}}-n_{\text{o}}})\;;$$
(36.16)
ne and no are respectively the extraordinary (slow) and ordinary (fast) refractive indices of the liquid crystal (assuming that Δn is positive). If the emergent elliptically polarised light passes through a second polariser, crossed with respect to the incident polarisation direction, only a proportion of the incident intensity will be transmitted. The normalised intensity of light transmitted by a pair of crossed polarisers having a birefringent element between them, the axis of which is at \(\pm{\mathrm{45}}^{\circ}\) to the extinction directions of the polarisers is given by
$$T=\left({\frac{1}{2}}\right)\sin^{2}\frac{\phi}{2}$$
(36.17)
(the factor of one half appears for incident unpolarised light – if the light is polarised, as from a laser, then the factor is one).

Thus the initial appearance of the cell will be brightest if the cell thickness, birefringence and wavelength are chosen to give φ equal to π, 3π, 5π etc. It is normal to select the cell thickness to give φ = π, and under these conditions the display is known as normally white. It is possible, though less satisfactory, to configure the display so that it operates in a normally black state, corresponding to a phase retardation of a multiple of 2π.

Application of an electric field causes the director orientation to change such that the optical retardation of the cell decreases to zero, and hence the cell becomes non-transmitting, at least in the normally white configuration. Under these circumstances the optical retardation across the cell becomes
$$\phi=\frac{2\uppi}{\lambda}\int_{0}^{\ell}\left(n_{\text{e}}\left[\theta\left(z\right)\right]-n_{0}\right)\mathrm{d}z\;,$$
(36.18)
where the effective extraordinary refractive index ne [ θ ( z )  ]  depends on the angle \([{\mathrm{90}}^{\circ}-\theta(z)]\) between the director and the field and is a function of position z in the cell. This effective index is given by
$$\begin{aligned}\displaystyle\frac{1}{n_{\text{e}}^{2}[\theta(z)]}=\frac{\sin^{2}[\theta(z)]}{n_{\text{o}}^{2}}+\frac{\cos^{2}[\theta(z)]}{n_{\text{e}}^{2}}\;;\end{aligned}$$
(36.19)
when the director is along the field direction \(\theta={\mathrm{90}}^{\circ}\), so \(n_{{\text{e}}}({\mathrm{90}}^{\circ})=n_{{\text{o}}}\), and φ = 0.
The orientational distribution of the director in the cell in the presence of an applied electric field is determined by the strength of the field, the electric permittivity and elastic constants of the liquid crystal, and most importantly by the properties at the interface between the liquid crystal and the aligning surfaces. If the director at the surface satisfies the strong-anchoring condition, i. e., it is unaffected by the applied electric field, and the surface director is strictly perpendicular to the electric field, then the reorientation of the director exhibits a threshold response, known as a Fréedericksz transition. The change in transmitted intensity as a function of voltage can be calculated using continuum theory and simple optics [36.36], and a typical transmission curve for an ECB cell is illustrated schematically in Fig. 36.17.
Fig. 36.17

Variation of optical transmission with voltage for an ECB cell between crossed polarisers

The threshold voltage for director reorientation is independent of cell thickness, and is given by
$$V_{{\text{th}}}=\uppi\sqrt{\frac{K_{1}}{\varepsilon_{0}\Updelta{}\varepsilon}}\;,$$
(36.20)
where Δε is the anisotropy in the dielectric permittivity and K1 is the splay elastic constant. However, it is clear from (36.17) and (36.18 ) that the intensity of light transmitted by the ECB cell depends on the cell thickness and the wavelength of light. This undermines the usefulness of displays based on the ECB mode, since they require cells of uniform thickness and also they will tend to show colouration in white light. Another important operating characteristic of displays is the angle of view, i. e., how the image contrast changes as the angle of incidence moves away from 90 with respect to the plane of the cell. This is clear from Fig. 36.16, where the optical paths in the distorted state for observation to the left or right of the perpendicular to the electrodes are clearly different.
In order to construct a useful display from a simple on∕off shutter, it is necessary to consider how image data will be transferred to the display. This is known as addressing, and to a large extent it is determined by the circuitry that drives the display. However, we shall see that certain properties of liquid crystals also contribute to the effectiveness of different types of addressing. The simplest method of displaying images on a liquid-crystal display is to form an array of separate cells of the type illustrated by the ECB cell, each having a separate connection for the application of an electric field. Images can then be created by switching on, or off, those cells required to form the image. This technique is known as direct addressing, and can be illustrated by the seven-segment displays used in watches and numerical instrument displays (Fig. 36.18).
Fig. 36.18

Schematic of a directly addressed seven-segment liquid-crystal display

To create complex displays, a large number of separate cells, known as picture elements or pixels, must be fabricated, usually in the form of a matrix. Providing separate electrical connections for these pixel arrays (e. g., 640 × 480) for a standard visual graphics array (VGA ) computer screen, is impossible, and so other methods of addressing have had to be developed. Historically, the first was the technique known as passive matrix addressing, in which the array of cells are identified in rows and columns, and connections are only made to the rows and columns: for an array of n × m pixels, only n + m connections are made instead of n × m, as required for direct addressing. Each row is activated in turn (sequentially), and appropriate voltage pulses applied to the columns. Only those pixel elements for which the sum of column and row voltages exceeds a threshold are switched to an on-state. However, the problem with this method is that many unwanted pixels in the off-state still have a voltage applied, and may be partially activated in the display: so-called crosstalk. The time-sharing of activating signals is known as multiplexing, and it relies on the addressed pixels holding their signal while other pixels which make up the image are activated. Provided that the multiplexing is on a time scale of milliseconds, any fluctuation in the image goes unnoticed. However, if the multiplexing becomes too slow, the image starts to flicker. In fact there is a limit to the number of rows of the matrix which can be addressed (nmax), which is related to the ratio of the on-voltage to off-voltage by a result due to Alt and Pleshko [36.37]:
$$\frac{V_{{\text{on}}}}{V_{{\text{off}}}}=\sqrt{\frac{n_{\max}^{1/2}+1}{n_{\max}^{1/2}-1}}\;.$$
(36.21)
The cell characteristic which determines the values of Von and Voff is the optical transmittance curve, as illustrated in Fig. 36.17 . Depending on the desired contrast ratio Ion ∕ Ioff for a pixel, then the voltages Von ∕ Voff are determined. Thus, in the operation of a passively addressed matrix display, there is a tradeoff between the contrast ratio (Ion ∕ Ioff) and the number of rows, i. e., the complexity of the display. The shape of the transmittance curve is determined by the liquid-crystal material properties, and so these affect the resolution of the image displayed. For a desired contrast ratio of 4, the corresponding on∕off ratio for an ECB cell might be 1.82. This gives a maximum number of rows as four, which corresponds to a very-low-resolution display, which would be unusable except for a very basic device. This brief description of a simple electro-optic display operating in the ECB mode illustrates that its performance depends on the dielectric, optical and elastic properties of the liquid-crystal material used. Additionally, the properties at the liquid-crystal-substrate interface and the geometry of the cell will influence the electro-optic response of the cell.

One performance characteristic that is of very great importance is the speed with which the display information can be changed, since this determines the quality and resolution of moving images. The dynamics of fluids are related to their viscosity, and it has already been shown that the viscous properties of liquid crystals are complicated to describe, and are correspondingly difficult to measure. The complication arises because liquid crystals are elastic fluids, and so there is a coupling between the flow of the fluid and the orientation of the director within the fluid. We have seen that the motion of the director can be described in terms of a rotational viscosity, and the optical properties exploited in displays are related to changes in the director orientation. Thus it is the rotational viscosity that is of primary importance in determining the time response of displays. However, the fact that changes of director orientation cause fluid flow in liquid crystals complicates the process.

The time response of a liquid-crystal display pixel can be illustrated by reference to the ECB display, although other cell configurations modify the behaviour to some extent. In what follows, we shall assume that the reorientation of the director within a display pixel does not cause the nematic liquid crystal to flow. For a uniform parallel-aligned nematic-liquid-crystal film, the time for the director to respond depends on the magnitude of the electric field (or voltage) applied to the cell. If the voltage applied is only just greater than the threshold voltage, then the time is very long, while if a large voltage is applied, then the director responds quickly. It is found that the time response can usually be represented as an exponential behaviour, although effects of flow will change this. Neglecting these, a response time τon can be defined in terms of the change in transmitted-light intensity as
$$\frac{I(t)-I(0)}{I_{\text{on}}-I(0)}=1-\exp\left[-\left({\frac{t}{\tau_{{\text{on}}}}}\right)\right].$$
(36.22)
and the relaxation time for switching on the display is given approximately by [36.38, 36.39]
$$\tau_{{\text{on}}}=\frac{\gamma_{1}\ell^{2}}{\uppi^{2}K_{{\text{eff}}}}\left[{\left({\frac{V_{{\text{on}}}}{V_{{\text{th}}}}}\right)^{2}-1}\right]^{-1}\;.$$
(36.23)
On removing the applied voltage, the display element returns to its off-state, but with a different relaxation time which is independent of the applied voltage, such that
$$\tau_{{\text{off}}}=\frac{\gamma_{1}\ell^{2}}{\uppi^{2}K_{{\text{eff}}}}\;;$$
(36.24)
here γ1 is the rotational viscosity coefficient, and the effective elastic constant Keff that appears in these expressions is the splay elastic constant K1 for the ECB-mode display. These equations can be modified for other display configurations by changing Keff.

Although the ECB-mode display is the simplest that can be envisaged, based on the Fréedericksz effect, there are many disadvantages, in particular with respect to its viewing characteristics, and it has not been used commercially to any significant extent. However, the apparently simple modification of twisting the upper plate by 90 has resulted in the phenomenally successful twisted nematic display, which represents a large part of today’s multi-billion-dollar market.

36.3.2 High-Volume Commercial Displays: The Twisted Nematic (TN) and Super-Twisted Nematic (STN) Displays

The simple twisted-nematic display is essentially the same as the ECB display depicted in Fig. 36.16, except that the orientations of the surface director at the containing glass plates are rotated by 90. However, the TN cell, invented by Schadt and Helfrich in 1970 [36.40], represented a considerable improvement over earlier devices, and rapidly became the preferred configuration for commercial displays. A schematic representation of the TN cell is given in Fig. 36.19.
Fig. 36.19

Schematic of a twisted nematic display

This twisted configuration for a liquid crystal film was discovered by Mauguin, who found that instead of producing elliptically polarised light, such a twisted film could rotate the plane of polarisation by an angle equal to the twist angle between the surface directors of the glass plates [36.41]. In fact by working through the optics of twisted films, Mauguin showed that perfect rotation of the plane of polarisation only resulted if the film satisfied the following condition
$$2\Updelta{}n\ell\gg\lambda\;.$$
(36.25)
Thus for the cell illustrated in Fig. 36.19, the off-state would be perfectly transmitting. This is known as the normally white mode. In reality the cells used for TN displays do not meet the Mauguin condition, and the transmission for a 90 twisted cell between crossed polarisers and incident unpolarised light is given by [36.42]
$$T=\frac{1}{2}\left({1-\frac{\sin^{2}\left({\frac{\uppi}{2}\sqrt{1+u^{2}}}\right)}{1+u^{2}}}\right),$$
(36.26)
where \(u=\frac{2\Updelta{}n\ell}{\lambda}\). Equation (36.26) shows that, for sufficiently large u, the transmission T is indeed a maximum of 0.5, however it is also a maximum for \(u=\sqrt{3}\), \(\sqrt{15}\), \(\sqrt{35}\), etc. These points on the transmission curve correspond to the Gooch–Tarry minima; they are labelled as minima, since they were found for a TN cell operating in the normally black state. Most commercial cells operate under conditions of the first or second minima so that thin cells can be used, which give faster responses. It is, therefore, important that the birefringence of the liquid-crystal material can be adjusted to match the desired cell thickness, so that the display can have the best optical characteristics in the off-state.
Application of a sufficiently strong electric field across the twisted film of a nematic liquid crystal having a positive dielectric anisotropy causes the director to align along the field direction. Under these circumstances the film no longer rotates the plane of polarised light, and so appears dark. The transmission as a function of voltage for a twisted cell is similar to that shown in Fig. 36.17, except that the transmission varies more strongly with change in voltage above the threshold, and drops to zero much more rapidly. In contrast to the ECB cell discussed above, the threshold voltage for a TN cell depends on all three elastic constants
$$V_{{\text{th}}}^{{\text{TN}}}=\uppi\sqrt{\frac{{\left({K_{1}+\frac{\zeta}{2}(K_{3}-2K_{2})}\right)}}{\varepsilon_{0}\Updelta{}\varepsilon}}\;,$$
(36.27)
where ζ is the twist angle (usually π ∕ 2). The relative change of the transmission intensity with voltage of the TN cell is greater than for the ECB cell, and it can be shown that the steepness of the transmission curve increases as the property ratios K3 ∕ K1 and \(\Updelta{}\varepsilon/\varepsilon_{\bot}\) decrease. The on∕off voltage ratio for a TN cell is closer to unity, than for an ECB cell, and so for similar contrast ratios more lines can be addressed: up to about 20 for typical cells and materials. This is significantly larger than for the ECB cell, and so the TN cell allows more complex images to be displayed. There is still a wavelength dependence for the transmission, although this is less marked than for the ECB mode. However, even with the improved multiplexing capabilities of the TN display over the ECB cell, it is still not good enough to use for computer screens. One very successful approach to solve this problem was to modify the TN cell geometry so that instead of a 90 twist, the directors on opposite sides of the cell are rotated by about 270. This is known as a super-twisted nematic cell.

The concept of the 270 super-twisted nematic (STN) display seems at first sight to be irrational [36.43, 36.44]. The director is an apolar vector and so there should be no difference between a 270 and a 90 twisted cell. However, it is possible to maintain a director twist greater than 90 if the liquid crystal is chiral. The use of chiral additives in 90 TN cells was already established, since a very small quantity of chiral dopant added to a TN mixture would break the left/right twist degeneracy in the cell and so remove patches of reversed twist, giving a much improved appearance to the display. If the amount of chiral dopant was increased, then the chiral liquid-crystal mixture would develop a significant intrinsic pitch. By adjusting the concentration of the chiral dopant, the pitch of the mixture could be matched to the 270 twist across the cell thickness of ℓ, i. e., \(p\approx 4\ell/3\). An STN cell operates in the same way as a TN cell, so that an applied electric field causes the director to rotate towards the field direction, thereby changing the optical retardation through the film and the transmission between external crossed polarisers. However, the additional twist in the STN cell has a significant effect on the optical properties of the nematic film.

The optical characteristics of the STN off-state are usually outside the Mauguin condition, which means that polarised light passing through the cell is not guided, and emerges elliptically polarised. The degree of ellipticity is wavelength dependent, and so in white light the off-state appears coloured, as does the on-state. An ingenious solution to this problem is to have two identical STN cells, one behind the other, but where the second compensating cell has a twist of the opposite sense. In operation, only the first of the cells has a voltage applied to it. The compensation cell acts to subtract the residual birefringence of the liquid crystal layer, and the display now switches between white and black. Despite this additional complexity, the huge advantage of the STN display is the rapid change in optical transmission with increasing voltage, and a full optical analysis shows that under optimum conditions the rate of change of transmission with voltage can become infinite. In modern implementations of the STN display, the residual birefringence can be compensated by an optical film, avoiding the need for double cells. It is easy to operate an STN display with a Von ∕ Voff ratio of 1.1, which corresponds to an nmax of 100. The shapes of the transmission∕voltage curve of ECB, TN and STN cells are a direct consequence of the dielectric and elastic properties of the liquid-crystal material, but also depend strongly on the configuration of the cells and the surface alignment

In the description given of displays and their performance, some important aspects have been ignored. The surface orientation of the directors has been assumed to be pinned in the surface plane, which is the requirement for a threshold response. However, it has been found that the performance of displays can be greatly improved if this condition is relaxed, and a pre-tilt is introduced to the cell, such that the surface director may make an angle of up to 60 to the plane of the containing glass plates. This pre-tilt can be introduced by different surface treatments, and it depends on the interfacial properties of the liquid crystal. An important performance characteristic of displays is the angle dependence of the contrast ratio, or more simply how the appearance of the displayed image changes with angle of view. This is largely determined by the display design, and can be accurately calculated from the optical properties of the cell. The refractive indices of the liquid crystal will affect the contrast ratio and angle of view, but precise control of these performance parameters is difficult. Improvements to the appearance in terms of the angle of view or brightness of displays have been achieved by placing precisely manufactured birefringent polymer films behind or in front of liquid-crystal cells.

36.3.3 Complex LC Displays and Other Cell Configurations

The STN configuration described above used passive matrix addressing, and this opened up the possibility of relatively large-area, high-resolution displays, which could be used in laptop computers and other hand-held displays. The next step was to introduce colour by dividing each picture element into three sub-pixels with red, green and blue filters. However, the demand in some market sectors for larger displays with improved appearance having higher resolution (extended graphics array (XGA ) displays have 1024 × 768 pixels) overwhelmed the capability of passive matrix addressing, and the alternative method of active matrix addressing is now used for more complex displays. This technique requires each pixel to have its own switch, as in the simple seven-segment display already described. For high-resolution displays hundreds of thousands of switches are provided by thin-film transistors deposited onto the glass substrate, which forms the screen. These are known as thin-film-transistor twisted-nematic (TFT -TN) displays [36.45]. The sophistication of these displays relies on the capabilities of integrated circuit technology, but the properties of the liquid-crystal materials must still be optimised for the device configuration. A representation of a TFT-TN display is given in Fig. 36.20.
Fig. 36.20

Schematic of a complex colour TFT-TN display

Although being described as an active matrix, the TFT-TN display still uses sequential addressing of pixel rows, and so activated pixels must remain switched on while the rest of the display image is created. The TFTs provide a source of voltage to each liquid-crystal pixel, which must then hold its charge, as a capacitor, until the image is changed. So another property of liquid crystals becomes important, that of low electrical conductivity, since the charge on a pixel will be lost by conduction through the liquid crystal. This determines the choice of materials for TFT-TN displays. Generally, high dielectric anisotropy is a desirable property for liquid-crystal display mixtures, since it reduces the operating voltage. However, materials with a high dielectric constant tend to have a high electrical conductivity, since charges either from impurities or leached from surfaces will be stabilised in high-dielectric-constant fluids. Thus the selection of suitable materials requires a compromise between its dielectric and conductance properties, and, of course, the all important refractive indices.

Many different cell configurations, which exploit the optical properties of liquid crystals in different ways, have been tried, and some of these have been commercialised to meet particular market requirements. One rather successful approach to the problem of restricted viewing angle has been the development of the in-plane switching (IPS ) mode twisted-nematic display [36.46, 36.47]; this is illustrated in Fig. 36.21. The two optical states of the cells are (i) twisted, and (ii) planar (parallel aligned film), and the director is switched between these states by application of an electric field across electrodes on a single plate of the cell. The state that is stabilised by the electric field depends on the dielectric anisotropy of the liquid-crystal material. The preferred configuration uses materials having a negative dielectric anisotropy, so that the off-state is a planar-aligned liquid-crystal film. Application of an electric field to the in-plane electrodes will cause the director at the bottom surface to align perpendicularly to its initial direction, and so induce a twist through the cell (Fig. 36.21).
Fig. 36.21

Schematic of an in-plane switching mode display

From an optical point of view, the director is always in the plane of the cell, and this means there is less distortion of an image when viewed at angles other than 90. Another advantage of the IPS device is that the electric field is confined to the lower plate, and the lines of force do not extend across the cell to the grounded upper plate. This means that a very low electrical conductivity of the liquid-crystal material is less important than for conventional TFT-TN displays. The threshold voltage for the IPS-mode device is given by
$$V_{{\text{th}}}^{{\text{IPS}}}=\frac{\uppi d}{\ell}\sqrt{\frac{K_{2}}{\varepsilon_{0}\Updelta{}\varepsilon}}\;,$$
(36.28)
where ℓ is the thickness of the liquid-crystal film, and d is the separation of the in-plane electrodes. Not surprisingly the threshold depends only on the twist elastic constant, which is usually smaller by about a factor of two than the splay and bend elastic constants. While this helps to reduce the operating voltage, the smaller elastic energy associated with the pure twist deformation results in longer switching times. A further disadvantage of the IPS display is that the optical transmission of the cell is reduced by the requirement to have both electrodes deposited on one plate, thereby making smaller the active area available to display the image.
A configuration which shares some of the characteristics of the IPS cell and has been successfully commercialised is the twisted vertically aligned nematic (TVAN ) cell. The two optical states for this configuration are uniform vertical (homeotropic) alignment of the director for the off-state, and a twisted geometry for the on-state (Fig. 36.22 ). The liquid-crystal material used has a negative dielectric anisotropy, so application of an electric field between the plates causes the director to align perpendicularly with respect to the field direction. A small quantity of optically active material (chiral dopant) is added to the liquid-crystal mixture to ensure that the switched director adopts a twisted configuration through the cell. The advantages of this cell are good viewing-angle characteristics and high optical contrast. Improvements in display technology continue to be made, often simplifying earlier devices. For example, high-quality displays described as vertically aligned nematic (VAN ) devices are now available based on the TVAN configuration, but without the twist. The material requirement here is for a liquid crystal of negative dielectric anisotropy, that will align perpendicularly to an applied electric field.
Fig. 36.22

Schematic of a twisted vertically aligned display

In the devices described above, one state is defined by the surface conditions of the cell, while the other is defined by the action of the applied electric field. A bistable device is one in which two stable field-free states exist, both of which are accessible by switching with an external field. The first bistable liquid-crystal display to be developed was based on a ferroelectric effect observed in chiral tilted smectic C liquid crystals [36.48]. This ferroelectric smectic display has achieved some limited commercial success in specialist markets, but relies on a surface stabilisation of smectic layers, which is very sensitive to mechanical shock. Recently [36.49], bistable nematic displays have been developed in which two alignment states within a liquid-crystal cell, having different optical transmission, can be stabilised. If one of the substrates of a normal cell is replaced by a surface which has potentially two states of minimum energy corresponding to two surface alignments of the director, then it becomes possible to switch these states selectively using an electric field. A suitable bistable surface is provided by a grooved surface (grating) which has been treated with a surfactant to favour homeotropic alignment of the director at its surface [36.50]. Thus the two surface states correspond to (i) that determined by the grating, and (ii) that determined by the surfactant where the director is perpendicular to the substrate. Combining this intrinsically bistable substrate with a second substrate having a director alignment direction perpendicular to the grating direction gives a cell configuration capable of supporting two optically distinct stable states, which can be switched between using an applied voltage. Various director configurations are possible with this type of cell, and one example is illustrated in Fig. 36.23.
Fig. 36.23

Schematic of a zenithal bistable device

In the absence of any perturbation, the director orientation within the cell will be determined by the homeotropic alignment at one substrate and the alignment at the grating substrate. This hybrid (uniform planar and homeotropic) alignment causes a spatially varying director tilt through the sample. Application of an electric field to a positive-dielectric-anisotropy material will cause the director to align parallel to the applied field, and eventually a fully homeotropic configuration for the director is stabilised. The switch back from the homeotropic state to the hybrid state is thought to be due to a flexoelectric interaction. Other alignment configurations are also possible for the so-called zenithal bistable nematic (ZBD) cell. Displays based on these configurations share the good viewing characteristics of both the IPS and TVAN configurations, but they have the great advantage that the image is retained when the voltage is removed.

All the displays described so far rely on the coupling between an applied electric field and the dielectric properties of the liquid-crystal material, but, as we have shown, other material properties are just as important to the operating characteristics of the display. The appearance of a display depends on the optical properties of the liquid crystal and the cell configuration, but the operating voltage and switching times of a display are crucial in determining the types of application. Changing the nature of the interaction between the switching electric field and the liquid crystal gives rise to another range of possibilities for liquid-crystal devices. Under certain circumstances, a liquid crystal can be made to exhibit permanent ferroelectric (or spontaneous) polarisation, and this couples linearly with an external electric field, in contrast to dielectric properties which couple with the square of the electric field strength. Not surprisingly this makes a big difference to the switching behaviour of liquid crystals.

The final display to be considered is based on flexoelectric coupling between the electric field and the liquid crystal. Flexoelectricity occurs, in principle, with all liquid crystals, chiral or not, and shows itself as a bulk electric polarisation induced by an elastic strain. Conversely application of an electric field can cause an elastic strain. In general, flexoelectricity is rather small and difficult to detect, however it is thought to be responsible for an electro-optic effect observed in chiral nematic liquid crystals, which is being investigated for display applications. The effect, sometimes known as the deformed helix mode [36.51], is similar in some respects to the ferroelectric switching observed in chiral smectic C phases, but there is no longer a requirement for a layered structure. Chiral nematic liquid crystals spontaneously form helical structures in which the director rotates with a pitch determined by the molecular structure. If an electric field is applied perpendicularly to a chiral-nematic helix, then there is a tendency for the helix to unwind, depending on the sign of the dielectric anisotropy. Even if the dielectric anisotropy of the material is zero, there is an elastic strain which can generate a polarisation (flexoelectric polarisation), which will interact with an applied electric field. This may be exploited in a device configuration, where a thin film of a chiral nematic liquid crystal, having a small or zero dielectric anisotropy, is aligned such that its helix axis is parallel to the containing glass plates (Fig. 36.24).
Fig. 36.24

Schematic of a deformed helix mode flexoelectric display

Application of an electric field across the plates will cause a distortion of the helix through the splay and bend flexoelectric coefficients, which appears as a rotation of the optic axis in the plane of the film [36.25]. Reversal of the electric field direction will reverse the rotation of the optic axis, with an intrinsic switching time about one hundred times faster than conventional nematic displays. Optically, the effect observed is very similar to that exhibited by smectic ferroelectric displays.

There are many cell configurations that can be used with liquid crystals to produce optical switches, displays or light modulators, and some of the more important have been described. The precise operation and performance of these liquid-crystal devices depends on both the cell design and the material properties of the liquid crystal. To a large extent the configuration of the liquid crystal within the cell is determined by such factors as the surface treatment of the plates enclosing the liquid crystal and the interactions between the surfaces and the liquid crystal. Our understanding of these interactions is very limited at the present time, and much more research is necessary before a quantitative theory can be formulated. However, given the cell configuration, the performance of the liquid-crystal device depends critically on the physical properties of the liquid-crystal material. Thus the electrical switching characteristics will depend on the dielectric properties, while the optical appearance of the device will be determined by the refractive indices. Elastic properties contribute to both the electric field response and the optical appearance, since any deformation of the director will be determined by the elastic properties of the liquid crystal. Finally, the all-important dynamical behaviour will be controlled by the viscous properties of the liquid crystal. All these material properties will be discussed in the next section.

36.4 Materials for Displays

The most important requirement for a liquid-crystal display material is that it should be liquid crystalline over the temperature range of operation of the device. Despite this, some of the first experimental display devices incorporated heaters in order to maintain the material in the liquid-crystal phase: for example, the first liquid-crystal shutter patented by Marconi and early prototype displays developed by the Radio Corporation of America (RCA). It was not until the late 1960s that room-temperature nematic liquid crystals suitable for display applications were discovered. The first of these were based on Schiff’s bases, which although easy to prepare, were difficult to purify and were susceptible to chemical decomposition in a device. One material which attracted particular attention from experimental physicists was MBBA, which has a nematic range of 22–47C (Table 36.1). The basic two-ring core linked by an imine group was the structural unit of many compounds having different terminal groups which were prepared for display mixtures in the early 1970s. It was found that mixtures of Schiff’s bases often had lower crystal-to-nematic transition temperatures than any of the components, and furthermore would often remain liquid crystalline even below the thermodynamic crystallisation temperature. These two phenomena of eutectic behaviour and super-cooling have been exploited in the development of materials for devices. The early experiments on liquid-crystal displays were primarily focused on nematic or chiral compounds, but we have seen that other displays have been developed which use different liquid-crystal phases, most importantly the chiral smectic C phase.
Table 36.1

Some typical liquid-crystal materials, including selected physical properties ((ex) – extrapolated from measurements on a nematic solution)

Compound

Transition temperatures (C) [reference]

Δε

(T C)

Δn

(T C)

Rotational viscosity

(γ1 ∕ m Pa s)

Open image in new window

TCrN 22C TNI 47C [36.53]

≈ 1

0.2

γ1 = 109 [36.52]

(37C)

Open image in new window

TCrN 22C TNI 35C [36.53]

8.5

(29C)

0.18

(25C)

γ1 = 102 [36.52]

(25C)

Open image in new window

TCrI 34C [36.54]

3.2 (ex)

0.05 (ex)

 

Open image in new window

TCrN 31C TNI 55C [36.53]

9.9

(48C)

0.12

(40C)

γ1 = 128 [36.52]

Open image in new window

TCrN 62C TNI 85C [36.53]

3.5

(78C)

0.05

 

Open image in new window

TCrN 44C TNI 57C [36.53]

19.9

(40C)

0.15

 

Open image in new window

TCrN 67C TNI 82C [36.55]

≈ 0

0.15

 

Early studies established guiding principles for the development of display materials. First, the phase behaviour must be acceptable, i. e., the right phase stable over a suitable temperature range. Secondly, the material must have the correct electrical and optical properties for the particular display application envisaged, and above all must be of sufficient chemical purity to prevent any deterioration in performance over time. Again, guided by the early experiments, suitable display materials require the synthesis of compounds of appropriate chemical structure, and then the formulation of mixtures to optimise the properties. There have been a number of reviews of liquid-crystal materials for displays [36.53, 36.54, 36.56, 36.57, 36.58, 36.59] and these contain many tables of data on a wide range of compounds. In this Section, we will give a brief account of the basic chemical structures used for materials in modern liquid-crystal displays, and then show how mixtures are devised to give the best possible performance characteristics for different displays. It has to be recognised that many of the details of display materials are matters of commercial confidentiality, and so it is not possible to give precise accounts of materials currently used or under investigation. However, the generic chemical structures and principles used in developing suitable mixtures are generally applicable.

36.4.1 Chemical Structure and Liquid-Crystal Phase Behaviour

There is a huge literature on the relationship between the structure of mesogens and the nature and stability of the liquid-crystal phases they form [36.60]. The studies have embraced empirical correlations of chemical structure and phase behaviour, theoretical calculations for simple particles (hard rods, spherocylinders etc.) representing mesogens, and computer simulations of collections of particles of varying complexities. For the display applications considered in this Chapter, the desired phases are nematic, and occasionally chiral nematic or chiral smectic C. Such phases are formed by molecules having extended structures, which usually require the presence of terminal alkyl chains to reduce the crystallisation temperatures. Components in nematic display mixtures typically have two, three or four carbocyclic rings joined directly or through a variety of linking groups.

36.4.2 The Formulation of Liquid-Crystal Display Mixtures

The two requirements for a liquid crystal to be used in a display are a suitable temperature range of phase stability and appropriate physical properties. These requirements cannot be satisfied for complex displays by a single compound, and commercial display materials may contain up to twenty different components. The formulation of these mixtures is essentially an empirical process, but guided by the results of thermodynamics and experience. The principles behind the preparation of multicomponent mixtures can be illustrated initially by consideration of a binary mixture.

It is well-known that the melting point of a binary mixture of miscible compounds is depressed, sometimes below the melting points of both components. Furthermore, the melting point of the binary mixture may exhibit a minimum at a particular composition, known as the eutectic. This occurs with liquid-crystalline compounds, and provides a method of reducing the lower temperature limit for liquid-crystal phase stability in mixtures. The upper temperature limit of the liquid-crystal range is fixed by the transition to an isotropic liquid. The phase rule of Willard Gibbs predicts that in binary mixtures there will always be a region of two-phase coexistence in the vicinity of a phase transition; that is, the transition from liquid crystal to isotropic occurs over a range of temperatures for which both the isotropic liquid and liquid crystal are stable in the mixture. Because of the weak first-order nature of most liquid crystal to isotropic phase transitions, the two-phase region is small. The character of phase transitions is determined by the corresponding entropy change, and a weak first-order transition has a small \(\big(\approx{\mathrm{2}}\,{\mathrm{J{\,}K^{-1}{\,}mol^{-1}}}\big)\) associated entropy. If the latter were zero, then the transition would be second order, and there would no longer be a region of two-phase coexistence. The phase diagrams of multicomponent nematic mixtures can be calculated by thermodynamic methods [36.61, 36.62] and the transition temperatures of the mixtures can vary with composition in a variety of ways. For mixtures of two liquid-crystalline compounds of similar chemical constitution, the variation of the nematic to isotropic transition temperature is approximately linear with composition [36.63].

It is possible to calculate the variation of the melting point with composition using an equation attributed to Schroeder and van Laar. For each component i, the mixture composition (mole fraction x i  )  and the melting point of the mixture T are related by
$$\ln x_{i}=-\frac{\Updelta{}H_{i}}{R}\left({\frac{1}{T}-\frac{1}{T_{i}}}\right),$$
(36.29)
where ΔH i and T i are, respectively, the latent heat of fusion and melting point of the pure component i. For a binary mixture there are two such equations which can be solved to give the eutectic temperature and composition. In a multicomponent mixture, the set of equations (36.29) can be solved subject to the condition,
$$\sum_{i}{x_{i}=1}$$
(36.30)
to predict the eutectic of the mixture.

While there is a reasonable thermodynamic basis to the prediction of the phase diagrams of mixtures, the determination of the physical properties of mixtures from the properties of individual components is much more difficult. Given the absence of any better theories, it is common to assume that in mixtures, physical properties such as dielectric anisotropy, birefringence and even viscosity vary linearly with the amount of any component, at least for small concentrations. While this may give an indication of the effect of different components on the properties of a display mixture, it can also be very misleading. One theoretical problem is that, for a mixture at a particular temperature, the orientational order parameters of the different components are not equal. The more anisometric components (e. g., three-ring mesogens) are likely to have a larger orientational order parameter than smaller (two-ring) mesogens. Since the various physical properties of interest in displays depend on the order parameter in different ways, it is difficult to predict the contribution of different components to the overall mixture properties. Despite this, many tables of data for liquid-crystal compounds of interest for display mixtures are prepared [36.60] on the basis of extrapolated measurements on mixtures at low composition, normally < 20 wt%. There is always a problem concerning the temperatures at which to compare the physical properties of liquid crystals and their components. Many measurements are made at room temperature, so that this becomes the temperature for comparison. However, a more useful approach is to compare properties at equal reduced temperatures (or at the same shifted temperatures, TNI − T), since under these conditions the orientational order parameters are likely to be similar.

36.4.3 Relationships Between Physical Properties and Chemical Structures of Mesogens

Electrical and Optical Properties

These properties include the dielectric permittivity, electrical conductivity and refractive indices. The magnitude of the dielectric anisotropy determines the threshold voltage necessary to switch a display, and influences the transmission∕voltage characteristics of the cell. Depending on the particular display configuration, a positive or negative dielectric anisotropy may be required. Refractive indices strongly affect the appearance of a display. Usually the refractive indices or birefringence must be adjusted for a particular cell configuration to give the optimum on∕off contrast ratio. Coloration in displays can sometimes occur in materials of high refractive index, and so it is desirable to keep the birefringence as low as possible, compatible with an acceptable contrast ratio. For twisted structures, the magnitude of the birefringence also determines the efficiency of light guiding, and so close control of the values of the principal refractive indices of a display mixture is important. For non-conducting materials, the refractive indices are measures of the dielectric response of a material at very high, i. e., optical frequencies, and it is possible to formulate a single theory which relates the dielectric and optical properties of a liquid crystal to its molecular properties. Unfortunately this is not possible for the electrical conductivity. The latter is largely determined by the purity of the liquid crystal, but it is found that the higher the value of the permittivity, the larger the electrical conductivity. Materials of high electrical conductivity tend to leak charge, and so an image may deteriorate during a multiplexing cycle. In general it is desirable to minimise the conductivity of a display mixture, although this was not the case for the first liquid-crystal displays reported [36.65]. These utilised the strong light scattering which results when an electric field is applied to certain nematic materials. The scattering is due to electrohydrodynamic instabilities in liquid-crystal materials which have a significant electrical conductivity. Such materials are not suitable for use in modern, fast-multiplexed displays.

The dielectric anisotropy Δε and birefringence Δn of a nematic can be related to molecular properties of polarisability and dipole moment using a theory originally developed by Maier and Meier [36.66]. The birefringence is given by
$$\Updelta{}n\approx\frac{NS}{\varepsilon_{0}}\big(\alpha_{\mathrm{l}}-\alpha_{\text{t}}\big)\;,$$
(36.31)
where N is the density in molecules per m3 and \(\Updelta{}\alpha=(\alpha_{\mathrm{l}}-\alpha_{{\text{t}}})\) is the anisotropy of the molecular polarisability. S is the order parameter, defined in Sect. 36.2.1, and small corrections due to the local field anisotropy have been neglected. Such corrections cannot be ignored in the corresponding expression
$$\Updelta{}\varepsilon=\frac{NhFS}{\varepsilon_{0}}\left[\Updelta{}\alpha+\frac{\mu^{2}}{2k_{\text{B}}T}({3\cos^{2}\beta-1})\right]$$
(36.32)
for the dielectric anisotropy, especially for materials of high permittivity. In (36.32) h and F are local-field correction factors, while μ is the molecular dipole moment, and β is the angle between the dipole direction and the long axis of the molecule. For molecules containing a number of dipolar groups, μ is the root mean square of the vector sum of all contributing groups. Both the birefringence and the dielectric anisotropy increase with decreasing temperature, and the detailed variation with temperature is largely determined by the temperature dependence of the order parameter S.
The manipulation of birefringence is achieved by changing the chemical constitution of the mesogen. Thus extending the electronic conjugation along the axis of a mesogen will result in an increase in longitudinal polarisability, and hence an increase in birefringence. Saturated carbocyclic rings, such as cyclohexane, and aliphatic chains generally have small polarisabilities and mesogens containing a predominance of such moieties form low-birefringence liquid crystals. Mixtures for displays require a positive birefringence, which is associated with calamitic or rod-like mesogens. In order to improve the viewing characteristics of displays, optically retarding films are placed in front of the display, and depending on their function, these may be of negative or positive birefringence. The latter can be fabricated by encapsulation or polymerisation of suitable molecules of an extended structure. On the other hand, films of negative birefringence have been made using discotic materials, i. e., mesogenic molecules formed from disc-like structures which have a negative polarisability anisotropy (Sect. 36.1.3). Some examples of liquid crystals having different birefringence are shown in Table 36.2.
Table 36.2

Materials of high, low and negative birefringence

Compound

Transition temperatures

Δn[reference]

(T C)

Open image in new window

TNI 112C

0.31 [36.64]

(92C)

Open image in new window

TCrSmB 23C TSmBN 35C TNI 49C

0.052 [36.59]

Open image in new window

TNI 93.5C

0.074 [36.64]

(58.5C)

Open image in new window

TCrN 80C TNI 96C

−0.193 [36.64]

(61C)

The introduction of chirality into liquid crystals has important consequences for their optical properties. The selective reflection of coloured light from the helical structure of a chiral nematic has already been mentioned in Sect. 36.1.2. All chiral materials will rotate the plane of incident polarised light, and the particular optical properties associated with chiral thin films are exploited in many liquid-crystal device applications.

The dipole moment of a molecule is increased if strongly electronegative or electropositive groups are substituted into the structure, with the result that the dielectric permittivity increases. For mesogenic molecules, the locations of the electropositive or electronegative groups are important, since not only the magnitude but also the orientation of the molecular dipole strongly influences the dielectric anisotropy. From (36.32) it can be seen that the dipolar contribution to the dielectric anisotropy may be positive or negative depending on the value of the angle β, since for values of β greater than 54.7 the dipolar contribution to the permittivity anisotropy becomes negative. This is illustrated by the mesogens shown in Table 36.3, where different structures can be designed to give large positive, negative or zero dielectric anisotropy.
Table 36.3

Materials of high, low and negative dielectric anisotropy; the inset figure indicates the direction of the total dipole moment with respect to the core of the molecule \(({\mathrm{1}}\,{\mathrm{D}}={\mathrm{3.33564\times 10^{-30}}}\,{\mathrm{C{\,}m}})\)

Compound

Transition temperatures

Δε[reference]

(T C)

Total dipole moment

μ (D)

Open image in new window

TCrN 22.5C TNI 35C

11.5 [36.67] (25C)

4.8

Open image in new window

TCrN 143C TNI 150C

−10.0 [36.68] (145C)

6.4

Open image in new window

TCrN 13C TNI 64C

0.0 [36.69] (58C)

1.4

The most effective substituent for producing materials of high dielectric anisotropy is the cyano group, and mixtures containing cyano-mesogens were the basis for the rapid development of complex displays in the 1980s and early 1990s. However, these mixtures tended to have relatively high viscosities, which gave rise to slow switching times. Another disadvantage was the high electrical conductivity associated with the high dielectric anisotropies which caused charge leakage during multiplexing, and hence degradation of the image.

As the demands placed on liquid-crystal materials by more sophisticated display technologies have increased, new families of molecules have been synthesised and screened for their physical properties. However, it is no longer the properties of the pure mesogens that are of interest, rather how they behave in mixtures. For this reason, the physical properties of most components of display mixtures are measured in mixtures, and values for the pure mesogens are obtained by extrapolation. Data derived in this way are useful in designing display mixtures and for comparison purposes, but cannot be relied upon to give quantitative information about the relationship between molecular structure and physical properties.

The major display technologies using TN, STN or TFT-TN configurations require display mixtures having a positive dielectric anisotropy. Many materials have been developed to improve the electro-optical behaviour of these displays, particularly using fluorine-substituted mesogens to provide the required dielectric and optical properties (for examples see [36.54, 36.58, 36.70]). Some of these fluorinated mesogens are shown in Table 36.4. However, within the past seven years, new display configurations have emerged, such as the in-plane switching (IPS) and vertically aligned (VAN and TVAN) nematic modes, which require mixtures with negative dielectric anisotropy. Using the design strategy illustrated above for simple mesogens, it has been possible to prepare a large number of materials with the desired negative dielectric anisotropy. These are again mostly based on fluorine-substituted compounds, and as before their properties have mostly been determined by extrapolation of measurements on mixtures.
Table 36.4

Fluorinated mesogens of positive and negative dielectric anisotropy used in liquid-crystal mixtures for modern displays. All measurements listed have been obtained by extrapolation from measurements on nematic solutions

Compound

Transition temperatures (C) [reference]

Δε

Δn

Rotational viscosity

(γ1 ∕ m Pa s)

Open image in new window

−6.2

0.1

γ1 = 110

TCrN 49C TNI 13C [36.59]

   

Open image in new window

−5.9

0.156

γ1 = 233

TCrN 80C TNI 173C [36.59]

   

Open image in new window

11.3

0.134

γ1 = 191 [36.73]

TCrN 30C TNI 58C [36.54]

   

Open image in new window

9

0.14

γ1 = 180 [36.73]

TCrSmB 43C TSmBN 128C TNI 147C [36.54]

   

Open image in new window

−2.7

0.095

γ1 = 218

TCrN 67C TNI 145C [36.59]

   

Open image in new window

−4.3

0.2

γ1 = 210

TCrN 88C TNI 89C [36.59]

   

Open image in new window

11.1

0.067

γ1 = 175

TCrN 56C TNI 117C [36.59]

   

Elastic Properties

The property known as elasticity is characteristic of liquid crystals, and distinguishes them from isotropic liquids. It has been shown in Sect. 36.2.3 that the macroscopic orientational disorder of the director in liquid crystals can be represented in terms of three normal modes, designated as splay, twist and bend, and associated with each of these deformations is an elastic constant. Since the elastic properties of display materials contribute to the electro-optic response, their optimisation for particular display configurations is important to maximise the performance of commercial devices. However, despite their importance, the relationships between the magnitudes of elastic constants and the chemical structure of mesogens are poorly understood. There is a good reason for this; the optical and dielectric properties are to a first approximation single particle properties. That is they are roughly proportional to the molecular number density and are also linearly dependent on the order parameter. Because elastic properties are a measure of the change in energy due to displacements of the director, they are related to the orientation-dependent intermolecular forces. Thus, at a molecular level, elastic properties are two-particle properties, and are no longer linearly proportional to the number density. A further consequence is that the elastic properties depend to lowest order on the square of the order parameters. Molecular theories of elasticity in nematic liquid crystals have been developed [36.71] and the simplest results suggest that the different elastic constants can be related to molecular shape
$$K_{1}=K_{2}\propto\langle{x^{2}}\rangle\quad\text{and}\quad K_{3}\propto\langle{z^{2}}\rangle\;,$$
(36.33)
where ⟨ z2 ⟩  and ⟨ x2 ⟩  are the average intermolecular distances parallel and perpendicular to the molecular alignment direction, respectively. Thus theory predicts that for rod-like molecules the splay elastic constant should be smaller than the bend elastic constant, and increasing the molecular length should increase K3, while increasing the molecular width should increase K1. This is roughly in accord with experimental results, except that the prediction of equal splay and twist elastic constants is not confirmed (Fig. 36.9 ). In general, the twist elastic constant is about one half of the splay elastic constant. Hard particle theories [36.72] evaluated for spherocylinders provide further guidance on the relationship of elastic constants to molecular shape. These theoretical results can be presented in a simplified way as follows
$$\begin{aligned}\displaystyle&\displaystyle\frac{K_{1}-K}{K}=\Delta(1-3\sigma)\;;\\ \displaystyle&\displaystyle\frac{K_{2}-K}{K}=-\Delta(2+\sigma)\;;\\ \displaystyle&\displaystyle\frac{K_{3}-K}{K}=\Delta(1+4\sigma)\;,\end{aligned}$$
(36.34)
where \(K=\frac{1}{3}({K_{1}+K_{2}+K_{3}})\). The quantities Δ and σ are parameters of the theory, where Δ is approximately equal to the square of the length:width ratio of the spherocylinder, and σ depends on the degree of orientational order. Despite the fact that details of internal chemical structure are ignored, these theoretical results for nematics are in approximate agreement with experimental measurements on simple nematics. If the nematic material has an underlying smectic phase, or if there is a tendency for local smectic-like ordering, this can strongly affect the elastic constants. Both the twist and bend elastic constants are infinite in a smectic phase, and in a nematic phase they diverge as the transition to a smectic phase is approached.

The elastic constants contribute directly to the threshold voltage and the response times of displays. Threshold voltages increase with increasing elastic constants, and the elastic constants responsible depend on the configuration of the display. Thus for the planar-aligned electrically controlled birefringence display (ECB), the threshold voltage depends on K1, while the switching voltage for TN displays depends on a combination of all three elastic constants (36.27). The IPS display voltage depends only on K2, and for VAN and TVAN devices, the threshold voltage is determined by K3. Different combinations of elastic constants determine the transmission∕voltage curves, which are important in the multiplexing of complex displays. For example, decreasing the ratio K3 ∕ K1 increases the steepness of the curve for TN displays, and so increases the number of lines that may be addressed. On the other hand, for the STN display, if the ratio K3 ∕ K1 is decreased, the number of lines that may be addressed also decreases. Identification of the important elastic constants necessary to optimise the operation of these displays is relatively straightforward; however, manipulation of the components of displays mixtures to give the best results is much more difficult.

Ferroelectric and Flexoelectric Properties

The electro-optic properties considered so far result from interaction of an electric field with the anisotropic permittivity of a material. This might be termed a quadratic response since the dielectric term in the free energy (36.2 ) is quadratic in the electric field, and as a consequence the electro-optic response does not depend on the sign of the electric field. For materials having a centre of symmetry, such as achiral nematic and smectic liquid crystals, this response is the only one possible. However, if the centro-symmetry of the liquid crystal is broken in some way, then a linear electric polarisation becomes possible, which results in a linear response to an applied electric field. One example of this, in the context of displays, is the chiral smectic C phase, which in the surface-stabilised state exhibits ferroelectricity, i. e., a spontaneous electric polarisation. The origin of the symmetry breaking in this case is the chirality of the material, and the polarisation is directed along an axis perpendicular to the tilt plane of the smectic C. Another way in which the symmetry can be broken is through elastic strain. This effect was first described by Meyer [36.24], and it can be represented as a polarisation resulting from a splay or bend deformation (Sect. 36.2.4 ). Since at a molecular level, strain is related to molecular interactions, the flexoelectric response depends on a coupling of intermolecular forces and the molecular charge distribution. Two molecular mechanisms have been identified which contribute to the strain-induced polarisation. If the molecules have a net dipole moment, then the longitudinal component can couple with the molecular shape to give a splay polarisation along the director axis, while the transverse component couples with the shape to give a bend polarisation perpendicular to the director axis. Even in the absence of a net dipole, a quadrupolar charge distribution in a molecule can result in strain-induced polarisation [36.26]. Both contribute to the splay and bend flexoelectric coefficients, but only the dipolar part persists in the sum e1 + e3. Thus it is common to quote flexoelectric coefficients as a sum and difference rather than as separate coefficients.

The measurement of flexoelectric coefficients has been a challenge to experimentalists, and there is a wide range of values in the literature for standard materials (Sect. 36.2.4). It is, therefore, premature to draw any conclusions about structure∕property relationships for flexoelectricity from the limited experimental data available. There have been attempts [36.74, 36.75] to model flexoelectricity for collections of Gay–Berne particles simulating wedge-shaped molecules. Application of the surface interaction model to flexoelectric behaviour [36.76] has allowed the calculation of flexoelectric coefficients for a number of molecules; these calculations include the quadrupolar contribution. The importance of molecular shape is clearly demonstrated, and in particular changes of shape, either through conformational changes or cis–trans isomerisation, have large effects on the magnitude and sign of the flexoelectric coefficients.

Flexoelectric effects contribute to the electro-optic response of nematic displays, especially those with hybrid alignment, i. e., planar on one electrode and homeotropic on the other electrode, but they are not usually considered in the optimisation of mixture properties. However, flexoelectric properties are of direct importance to the operation of displays based on the switching of the direction of the optic axis in chiral nematics: the so-called deformed helix mode [36.25].

Viscous Properties

As explained in Sect. 36.2.5 , the flow properties of liquid crystals are complicated. Since the materials are anisotropic, the viscosities in different directions are different, furthermore because of the torsional elasticity, viscous stress can couple with the director orientation to produce complex flow patterns. Thus there are five viscosity coefficients necessary for nematics, in addition to the elastic constants, and as many as 20 viscosities for smectic C liquid crystals [36.77]. To relate all or indeed any of these to molecular structure is a formidable challenge. However, for most liquid-crystal displays, the only viscosity of interest is that which relates to the reorientation of the director: the so-called rotational viscosity. This depends on the temperature and order parameter, and on the forces experienced by the rotating director. The rotational viscosities for all liquid-crystalline materials can be represented by one or other of the following parameterised relations
$$\begin{aligned}\displaystyle&\displaystyle\gamma_{1}=aS^{x}\exp\left({\frac{A(T\kern 0.3pt)}{k_{\text{B}}T}}\right)\\ \displaystyle&\displaystyle\text{or}\\ \displaystyle&\displaystyle\gamma_{1}=bS^{y}\exp\left({\frac{B}{T-T_{0}}}\right),\end{aligned}$$
(36.35)
where a, b, A and B are material parameters, and S is the order parameter raised to a power of x or y between 0 and 2. The first of these expressions emphasises the diffusional nature of rotational relaxation in a liquid crystal, that is molecules rotating in an external potential. The second expression taken from polymer physics describes rotation in terms of free volume, where T0 is the temperature at which the free volume becomes zero, and the rotational viscosity infinite, i. e., a glass transition.
At the simplest level, the rotational viscosity depends on the molecular shape and size. As the length of the molecule increases, from two rings to three rings etc. γ1 increases, similarly it increases with the length of the alkyl chain. Varying the nature of the rings in the mesogenic core can have a dramatic effect on the rotational viscosity, which correlates well with free volume, as shown in Fig. 36.25, and the glass temperature of the material.
Fig. 36.25

Relationship between the rotational viscosity coefficient γ1 (\({\text{P}}={\mathrm{10^{-1}}}\,{\mathrm{Pa{\,}s}}\)) and the geometric free volume (Vfg) at 25C for bicyclic polar mesogens. Compound numbers represent alkyl series as follows: 1-cyanophenylalkylcyclohexanes; 2-alkylcyanobiphenyls; 3-cyanophenylalkylpyridines; 4-cyanophenylalkylbicyclooctanes; 5-alkoxycyanobiphenyls. (After [36.52])

There is a correlation between the increasing dielectric anisotropy and increasing rotational viscosity which can be attributed to local dipolar intermolecular forces which impede end-over-end rotation of molecules. Thus mesogens having cyano-groups in a terminal position tend to have relatively high viscosities. Other dipolar groups such as fluorine do not have such a deleterious effect on viscosity as the cyano-group, and so fluorine-containing mesogens are increasingly preferred in the formulation of display mixtures.

As with the determination of other properties, the rotational viscosities of mesogenic components are often determined by extrapolation from measurements on mixtures doped with the component under investigation. Such a method only provides approximate values to compare different components, but in the formulation of mixtures for display applications it is only the rotational viscosity of the final mixture that is important. Rotational viscosities of some mesogens of interest for display mixtures are given in the Tables 36.1 and 36.4 accompanying this section. In many instances, the viscosities, as with other properties, have been determined by extrapolation from measurements on mixtures. The measurement of rotational viscosities is experimentally difficult, and some authors prefer to quote the results of bulk-viscosity measurements in terms of a kinematic viscosity. In fact there is a good correlation between kinematic viscosity and rotational viscosity, and where possible both values have been included in the tables. From the various tables, the effect of increasing the molecular length on the viscosity is clearly seen, as is the effect of replacing an F atom with a CN group. Lateral substitution, which produces materials of negative dielectric anisotropy, tends to increase the rotational viscosity. Despite this the fastest nematic displays now use vertical alignment and materials of negative dielectric anisotropy. The operating characteristics of display mixtures depend on the physical properties of individual components in a very complex manner, and optimisation of mixture properties has to be carried out in a concerted way.

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© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.ChemistryUniversity of SouthamptonSouthamptonUK

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