Smectic A Liquid Crystals: Overview Smectic-A is the simplest form of the smectic liquid crystalline phase (Chandrasekhar 1992, deGennes and Prost 1993). In this phase, the molecules form ‘‘layers’’ that are perpendicular to the director, as illustrated in Fig. 1. The term ‘‘layer’’ is an overstatement of the ordering. In most smectic-A materials the layers are not well defined, and are more accurately described as a modulation of the density whose wave vector is along the director. In fact, the smectic-A order is commonly represented (de Gennes 1972) by the order parameter ψ in the following expression for the density of the liquid: ρ(r) l ρ [1jRe ψ(r) exp(iq z)] ! ! where d l 2π\q is the wavelength of the density wave or the spacing ! of the smectic ‘‘layers.’’ The order parameter ψ is a complex number with two degrees of freedom: ψ(r) l Q ψ Q exp(iq u) ! The magnitude Q ψ Q is the amplitude of the density modulation, and the phase represents a displacement u of the ‘‘layers’’ in the wave-vector direction, which is usually taken to be along z. Smectic-A order is really a one-dimensional density wave in a three-dimensional liquid, and is the simplest kind of freezing or melting to occur in nature. The smectic-A phase remains translationally invariant (liquid-like) in directions perpendicular to the density wave vector.
1971). The analogy with superconductivity may be seen through the following arguments. Suppose, while moving through smectic-A material from point A to point B, one computes the line integral 1 d
& n:dr B
where n is the liquid crystal director. This integral calculates the number of ‘‘layers’’ that have been crossed. If the smectic-A is free from defects, and considering the layers to be incompressible, the result must be independent of the path followed, which means that the integral carried out around a closed path must yield 0. Applying Stokes’ theorem, we find that ]in l 0 in the smectic-A phase, and so both bend and twist director distortions are excluded from the smectic-A phase (see article Nematic Liquid Crystals: Elastic Properties). This is analogous to the exclusion of magnetic field lines from a superconductor (the Meissner effect). In fact, the phenomenological theories of the smectic-A to nematic phase transition and the superconducting to normal-state transition are mathematically similar. The liquid crystal director, n, is analogous to the magnetic vector potential, A, of the superconductor. As the smectic-A phase is approached from the nematic phase, the presence of smectic short-range order in the nematic phase close to the smectic-A—nematic transition causes a divergence of the bend and twist elastic constants. This effect is the analogue of fluctuation diamagnetism in superconductors. A phase, termed twist grain boundary phase, which is analogous to the Abrikosov flux lattice in superconductors was recently predicted (Renn and Lubensky 1988) and also independently discovered (Goodby et al. 1989).
1. Analogy to Superconductivity The two-component order parameter of the smectic-A phase is similar to the order parameter used in the Ginzburg–Landau phenomenological model (Ginzburg and Landau 1950) of superconductivity (de Gennes 1972; see also Kobayashi 1970, McMillan
Figure 1 Schematic diagram of the layer structure and density modulation in the smectic-A liquid crystalline phase.
2. The Absence of True Long-Range Order As the number of spatial dimensions is reduced, ordered phases lose their order. No long-range order is believed possible in one dimension, because the entropy of the disordered state is overwhelmingly high. Solids, for example, cannot exist in two dimensions, because the fluctuations of the atomic positions in the crystal lattice due to thermally excited lattice vibrations become infinite in two dimensions and destroy the order (Landau and Lifshitz 1980). In the language of phase transitions, we say that the lower marginal dimensionality, do, for the existence of solid crystals is two. As stated earlier, the smectic-A liquid crystal is an example of a one-dimensional density wave in a threedimensional liquid. Extending the Landau–Peierls argument (Landau 1965, Peierls 1934) to liquid crystals, it has been predicted that do for such a material is three (Caille! 1972). Precisely at do the effects of the 1
Smectic A Liquid Crystals: Overview Longitudinal wave vector q11/q0 –0.001 0 +0.001 0.005 100 80CB Longitudinal profile T = Tc – 0.315 °C
10 –2 80CB SmA Simulated 80CB (001) Direct beam 10 –3 –0.005
0 0.005 0.010 0.015 Analyzer Bragg angle misset (h–hB) (deg)
Figure 2 Longitudinal line profile for the smectic-A phase of 8OCB, direct beam and simulated profile of smectic-A if it had truly long range order (after Als-Nielson et al. 1980).
fluctuations are subtle. Whether or not long-range order exists can be discussed quantitatively by defining a correlation function: G(r) l feiG.[u(0)−u(r)]g where G is a reciprocal-lattice vector of the crystal, and u(r) is the displacement of an atom at r from its equilibrium position. In the case of a smectic-A phase, G would be the smectic density wave vector, q , and u the displacement ! of a smectic layer from its equilibrium position. This function is useful because its Fourier transform gives the x-ray Bragg scattering from the crystal lattice or smectic density wave. If there is long-range order, G(r) will have a finite value as r _. However, if there is only short-range order, G(r) decays exponentially to zero as e−r/ξ, where ξ is the correlation length. At do, G(r) goes to zero at large distances, but does so algebraically as r−η, where η " 0.1. The signatures of the different types of order in the x-ray diffraction peaks are very different. When there is long-range order, the Bragg scattering gives a resolution-limited peak, while short-range order gives a broad peak whose width is inversely proportional to ξ. The algebraic decay at the do is expected to give a scattering that has a cusp-like peak. 2
To experimentally distinguish this cusp from the resolution-limited peak of true long-range order is non-trivial. To probe this subtle difference, one has to use a high-resolution spectrometer with a sharply decaying resolution function (Als-Nielsen et al. 1980), obtained from multiple Bragg reflections of perfect collimating crystals. The longitudinal line profiles obtained in the smectic-A phase using such a setup show the cusp profile clearly (Fig. 2) and are in agreement with the theoretical predictions. Most of the properties of smectic-A (and smectic-C) liquid crystals are not significantly different from what they would be if the order were truly long range. However, they have been of fundamental interest because of a certain uniqueness—they enable us to demonstrate the existence of do and its associated behavior in three dimensions. Such an effect is observed only in two dimensions in most other materials. 3. Frustration of Two Length Scales of the Layer Structure: Polymorphism of Smectic-A and Reentrant Behavior The smectic-A phase discussed above has been assumed to have a layer spacing, d, that is approximately equal to the length, l, of a molecule. However, if the molecules have a strong longitudinal dipole moment (such a smectic-A is generally referred to as strongly polar smectic-A), like a cyano group shown in Fig. 3, there will be near-neighbor antiparallel correlations. These correlations in strongly polar smectic-A can lead to different polymorphic forms which have the same basic smectic-A symmetry and can be characterized by the ratio d\l (Sigaud et al. 1978, Hardouin et al. 1983). Four forms of smectic-A are now recognized: smectic-A with d " l, smectic-A with d " 2l, smectic# M . The smectic-AM Ad with l" d 2l, and smectic-A phase has the same local symmetry as smectic-A but # with a transverse dipolar modulation of the structure. All the phases have been identified by the x-ray diffraction features of aligned samples (see Fig. 4). The existence of these polymorphic smectic-A phases has been explained by a phenomenological model (Prost 1980, Prost and Barois 1983). This theory defines the free energy of strongly polar smectic-A in terms of two coupled-order parameters associated with a mass-density wave and a dipolar-density wave. The competition between these two types of order leads to different types of smectic-A phases. The same competition also leads to the re-entrant behavior. In the definition involving the two order parameters Ψ (the dipolar order parameter) and Ψ (the density" # the smecticorder parameter), Ψ l 0 and Ψ 0 for " # A . In the case of both smectic-Ad and smectic-A , Ψ as" well as Ψ has a non-zero value, but Ψ Ψ# in" # the case of #smectic-Ad, while Ψ $ Ψ in " the case " # of smectic-A . This theory also predicts an incom# mensurate smectic-A phase with two incommensurate collinear density modulations. The existence of
Smectic A Liquid Crystals: Overview
Figure 3 Chemical structure of 4-n-octyloxy-4h-cyanobiphenyl (8OCB).
Figure 5 Phase diagram of a binary liquid crystal system showing the smectic-Ad to smectic-A critical point # (after Shashidhar et al. 1987). Figure 4 Schematic of the layer structures of the different smectic-A phases along with their characteristic x-ray diffraction spots. The solid circle represents condensed reflection, while the shaded ellipse represents a diffuse reflection corresponding to fluctuations.
such an incommensurate phase is yet to be clearly established experimentally (Kumar 1991). Phase transitions between polymorphic forms of smectic-A are of considerable interest, since they are transitions between two phases with quasi-long-range translational order (Shashidhar and Ratna 1989). As mentioned earlier, the different forms of smectic-A have the same macroscopic symmetry, and differ from each other only by the wavelength of their periodic modulations (Fig. 4). Hence, it should be possible to transform from smectic-A to smectic-Ad or from " varying only the layer smectic-Ad to smectic-A by # spacing. There can be a first-order transition between these phases at which the wavelength characterizing the phase shows a discontinuous jump. A line of such first-order transitions can terminate at a critical point when the difference between the wave vectors in the smectic-A phase goes to zero (Shashidhar et al. 1987). This critical point (see Fig. 5) provides a continuous path between the smectic-Ad and smectic-A phases. # More recently, an experimental phase diagram has
demonstrated a continuous path between smectic-A , " smectic-Ad, and smectic-A phase (Pfeiffer et al. 1992) # (Fig. 6). The situation regarding the smectic-A to smectic" or second A transition is different. It can be either first # order, this being due to the exact doubling of the layer spacing, i.e., the layer periodicity doubles as a result of the continuous vanishing of a spatial subharmonic. These predictions have been verified experimentally (Chan et al. 1985). Although the mean field Landau theory predicts that the smectic-Ad–smectic-A critical # (Ising point is similar to the gas–liquid critical point symmetry), a more extensive theoretical treatment (Park et al. 1988), which includes fluctuations using the ε expansion, appears to indicate that it should belong to a new universality class, with dc, the upper critical dimension, of six. The theory also makes important predictions concerning anisotropic scaling. Some of these have indeed been verified experimentally (Wen et al. 1992). The situation concerning the terminus of the smectic-A to smectic-Ad phase boundary is quite " Landau theory predicts a bicritical point different. The where the nematic–smectic-A (second-order), nematic–smectic-Ad (second-order) "and smectic-A to smectic-Ad (first-order) phase boundaries meet. "However, when the effect of fluctuations is considered, the existence of the bicritical point becomes questionable. 3
Smectic A Liquid Crystals: Overview
Figure 6 Phase diagram of a binary liquid crystal system showing the continuous progression from the smectic-A1 to smectic-Ad to smectic-A phases # (after Pfeiffer et al. 1992).
In fact, it has been shown experimentally that the bicritical point actually splits into a tricritical point and a critical end point (Raja et al. 1988). This situation is analogous to the one predicted in magnetic systems (Rohrer et al. 1980, Galam and Aharony 1980). It should also be pointed out that the re-entrant nematic behavior (Cladis 1975, Cladis et al. 1977) observed before the discovery of the polymorphic smectic-A, as well as the exotic re-entrant smectic behavior seen in a few cases (Prost 1980), all essentially arise due to the competing smectic-A and smectic-Ad " interactions.
4. Smectic-A of Chiral Molecules When the smectic-A liquid crystal is composed of chiral molecules, an electric field applied in the layer plane leads to several symmetry-breaking effects, due to the coupling of the applied field to the transverse dipole moment of the molecule. The most well-studied of these symmetry-breaking effects is the ‘‘electroclinic effect.’’ Garoff and Meyer (1978) demonstrated this effect, wherein the applied field induces a tilt of the molecules in a plane perpendicular to the electric field. The electroclinic effect was demonstrated as a pretransition effect in the smectic-A phase arising from the increased tilt susceptibility as the smectic-A to smectic-C* (smectic C phase of chiral molecules) transition is approached. It is also referred to as the soft mode effect, in analogy to the softening of a vibration mode near the paraelectric to ferroelectric transition in solid ferroelectrics. 4
The amount of tilt per unit applied field is defined as the electroclinic coefficient. A feature of the electroclinic effect is that the amount of the induced tilt is proportional to the applied field and is continuously variable. Hence, if placed between suitably oriented crossed polarizers, the smectic-A composed of chiral molecules can exhibit continuously varying gray scale (also referred to as analog gray scale) capability. In addition, the switching times associated with the electro-optic switching of the molecules between the field on and off states can be very fast (tens of microseconds). This combination of analog gray scale and fast switching times makes the smectic-A phase very attractive for many applications (Shashidhar et al. 2000). Although the electroclinic effect has been known for a long time, smectic-A materials suitable for applications did not exist until recently. This is due to the fact that whenever there is a large field-induced tilt angle, there is always an accompanying layer contraction leading to a buckling of the layers. The resulting optical stripe texture (which can be observed in a polarizing microscope) strongly reduces the contrast ratio, making it unacceptable for applications. This problem has been overcome by the development of a new family of smectic-A liquid crystals, in which one or more siloxane units have been attached to the hydrocarbon chains at one end of the molecule. These siloxane-based materials (Shashidhar et al. 2000) are unique—they exhibit large induced optical tilt angles ( 30m) with practically no layer contraction. Also, without an applied field, hardly any change in layer spacing is observed (Spector et al. 2000) in these siloxane materials across the phase transition between the smectic-A and smectic-C phases. In fact, such a phase was predicted by de Vries some time ago (1977). The electroclinic effect is one of the manifestations of the effect due to the field-induced breaking of the rotational symmetry in chiral smectic-A liquid crystals. Several other such symmetry-breaking effects, like field-induced dielectric biaxiality (Kimura et al. 1993), field-induced optical biaxiality (Bartoli et al. 1997), and optical second harmonic generation (Kobayashi et al. 1996) have all been experimentally demonstrated.
5. Layer Distortions and Defects of Smectic-A The layered structure of smectic-A imposes restrictions on the type of deformations that can exist. A compression of the layers requires considerable energy, and hence only those deformations that preserve the interlayer separation are preferred. When a homeotropically aligned slab of smectic A (molecules perpendicular to the bounding surfaces) is subjected to mechanical tension, the layers undergo a periodic undulation. This effect, known as the Helfrich–
Smectic A Liquid Crystals: Overview Hurault instability (Helfrich 1970, Hurault 1973), has been observed by monitoring the light diffracted by the undulations (Delaye et al. 1973, Clark and Meyer 1973). The fundamental reason for this effect is that when the system is expanded (due to tension), the layers tend to conserve their layer thickness and fill the available space by undulations. Static as well as thermal contributions to the fluctuations of the layer displacement have been studied by dynamic light scattering. However, in the case of x-ray scattering technique, the thermal contribution usually swamps the static part. Recently it has been possible to probe the static undulations using off-specular diffuse x-ray scattering on samples of smectics in which the thermal contributions were quenched by molecular phase separation (Geer et al. 1993, Geer et al. 1995). The flexibility of the smectic-A layers and the tendency to preserve the interlayer spacing is the underlying reason for the occurrence of the defects in smectic-A. When observed between crossed polarizers in an optical microscope (with no special treatments to obtain a monodomain sample), the inherent inhomogeneity at the surface, or undissolved impurities, or even dust particles, can distort the smectic-A structure, resulting in optically identifiable textures like focal conics, ellipses, or hyperbolas and batonnets. Such textures have been extensively studied and well understood. Edge dislocations, screw dislocations, and disclinations have also been observed in smectic-A (Chandrasekhar 1992). Topologically the edge disclinations of smectic-A are very similar to those in crystals, albeit important differences do exist. The screw dislocations, on the other hand, are exactly analogous to the screw dislocations in crystals. See also: Smectic A Liquid Crystals: Continuum Theory; Nematic Liquid Crystals: Defects
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R. Shashidhar and B. R. Ratna
Copyright ' 2001 Elsevier Science Ltd. All rights reserved. No part of this publication may be reproduced, stored in any retrieval system or transmitted in any form or by any means : electronic, electrostatic, magnetic tape, mechanical, photocopying, recording or otherwise, without permission in writing from the publishers. Encyclopedia of Materials : Science and Technology ISBN: 0-08-0431526 pp. 8664–8670 6