Symmetry cholesteric phase

Some cholesteric materials show the blue phase as the temperature increases from that of the cholesteric phase and before it reaches that of the isotropic phase. The blue phase is a cubic phase. There have been three blue phases found so far BP I, BP II and BP III phases. It is now understood that the BP I phase is a body-centered cubic, the BP II phase is a primitive cubic and the BP III phase is a fog phase with no structural symmetry. Generally the temperature range of the blue phase is quite narrow, less than 1 degree... 

Many synthetic polymers form cholesteric phases, and even solids showing certain of the fundamental symmetries of cholesteric liquids. The purpose of this paper is to review the main examples of biological polymers assembling into cholesteric liquids or into more or less solid analogues. We will present them according to the main chemical classes of polymers to which they belong. We will also indicate the main forces involved in creating the cholesteric twist. 

Like in the nematic phase, the textures of SmC reveal blurred Schlieren patterns with linear singularities of strength s = 1. The singularities of 5 = 1/2 are not observed due to the reduced symmetry (C2h) of the SmC phase. Chiral smectics C are periodic structures and the helical pitch can be measured under a microscope either from the Grandjean lines or as a distance between the lines indicating periodicity, like in Fig. 8.22 for the cholesteric phase. On the other hand, like in... 

The limitations on multiplexing any rms-responding monostable liquid crystal effect have been mentioned in Section II.A. Active matrix addressing, described in Sectin IV.A, is one way of overcoming these limitations. Another is to consider alternative liquid crystal effects that are bistable, or at least non-rms responding. With such effects, the maximum number of rows that can be multiplexed is usually determined by the ratio of the frame time (the time period during which the whole picture must be refreshed or updated) to the line time (the time required to address one row of pixels). This is quite demanding of the line time a frame time of 40 msec (only 25-Hz frame rate) would require a line time of 40 /xsec for 1000 lines. Bistable behavior is associated with smectic and cholesteric phases, both of which in completely different ways have translational symmetries added to nematiclike orientational order. In this section, the ferroelectric tilted smectic devices are reviewed, while (untilted) smectic A and cholesteric devices are described in Section IV.C. 

Cholesteric liquid crystals consist of chiral molecules and therefore do not have reflection symmetry. The symmetry group of cholesteric hquid crystals is >2 [1,3]- A cholesteric liquid crystal is invariant for the two-fold (180°) rotation around n, which rules out the possibility of spontaneous polarization perpendicular to n. It is also invariant for the two-fold rotation around an axis that is perpendicular to the n - (the hehcal axis) plane, which mles out the possibility of spontaneous polarization parallel to n. Therefore there is no ferroelectricity in the cholesteric phase. 

As pointed out by Meyer [14], the reflection symmetry of smectic-C liquid crystals can be removed if the constiment molecules are chiral, and thus it becomes possible to have spontaneous polarization. This phase is called the chiral smectic-C or smectic-C, and its stmcture is shown in Figure 4.7. Within a layer, the structure is the same as in smectic-C. The liquid crystal director n is, however, no longer oriented unidirectionally in space but twists from layer to layer as in the cholesteric phase [15]. The symmetry group is C2. The two-fold rotational symmetry axis is perpendicular to both the layer normal a and the director n. Now it is possible to have spontaneous polarization along the two-fold rotational symmetry axis. 

A Landau theory for blue phase was proposed by Brazovskii, Dmitriev, Homreich, and Shtrik-man [7-10]. In this theory, the free energy of the blue phase is expressed in terms of a tensor order parameter which is expanded in Fourier components. The free energy is then minimized with respect to the order parameter with the wave vector in various cubic symmetries. In a narrow temperature region below the isotropic transition temperature, the stmctures with certain cubic symmetries have free energy lower than both the isotroic and cholesteric phases. 

The advantage of this, sometimes cumbersome description, is the fact that the transformation properties of the macroscopic and microscopic coordinates can be seen immediately. Because the anisotropy of most of the phases is sufficiently described by tensors of second rank for the microscopic as well as the macroscopic properties, only the orientational distribution coefficients gijki are mentioned here. As an example for the description of order by these orientational distribution coefficients, the order of a cholesteric phase will be briefly discussed. Four different order parameters are needed for a molecule of point symmetry group Di and local symmetry Z>2 for the cholesteric phase. The stars as a suffix indicate that (, 33 is given in its system of principal axes. In general, the convention... 

Chiral liquid crystals belong to a wide class of soft condensed phases. The director field in the ground state of chiral phases is nonuniform because molecular interactions lack inversion symmetry. Among the broad variety of spatially distorted structures the simplest one is the cholesteric phase in which the director n is twisted into a helix. The spatial scale of background deformations, e.g., the pitch p of the helix, is normally much larger than the molecular size ( > 0.1 pm) since the interactions that break the inversion symmetry are weak. 

The cholesteric phase appears in organic compounds which consist of elongated (nematogenic) molecules without mirror symmetry (chiral molecules) [l]-[3]. Typical representatives of these compounds are the derivatives of... 

Microtomy and freeze-etching techniques have revealed many similarities between liquid crystals and a large series of biological materials, which show the same symmetries as nematic, smectic and cholesteric phases, but without being fluid [27-34]. The biological materials are stabilized... 

A property that may be admitted by noncen-trosymmetry may very well be ruled out by one of the other symmetry operations of the medium. As an example we will finally consider whether some of the properties discussed so far would be allowed in the cholesteric liquid crystals, which lack a center of inversion. A cholesteric is simply a chiral version of a nematic, abbreviated N, characterized by the same local order but with a helical superstructure, which automatically appears if the molecules are chiral or if a chiral dopant is added (see Fig. 30). Could such a cholesteric phase be spontaneously polarized If there were a polarization P, it would have to be perpendicular to n, because of the condition of Eq. (6), and thus along the helical axis direction m. However, the helical N phase has an infinity of twofold rotation axes perpendicular to m and the symmetry operation represented by any of these would invert P. Hence P=0. A weaker requirement would be to ask for piezoelectricity. (Due to the helical configuration, the liquid has in fact some small... 

Figure 30. The symmetry of the cholesteric phase (a, b) and the effect of a shear (c).

The nonchiral nematic is optically a positive uniaxial medium. A cholesteric is a nematic with twist. The local structure of a cholesteric is believed to be the same as that of the nematic except that it lacks reflection symmetry. This means that the director and therefore the local extraordinary optic axis is rotating around the helix axis making the cholesteric a negative uniaxial medium with the optic axis coinciding with the twist axis. The question has been asked as to why the nematic with twist could not be biaxial, and attempts have been made to measure a slight biaxiality of the cholesteric phase. In other words, why could the twist not be realized in such a way that the long molecular axis is inclined to the twist axis Why does it have to be perpendicular ... 

The symmetry and structure of the smectic-A liquid crystals are reviewed the natural order parameters are identified. The relationship of the smectic-A phase to the nematic (or cholesteric) and isotropic phases in homologous series is also examined. The McMillan form of the single molecule potential function is then deduced starting from the Kabayashi form of the potentiaP and using the formal development presented earlier. The derivation of the statistical thermodynamics then follows, along with a presentation of McMillan s numerical results and a comparison with experiment. Improvements in the theory introduced by Lee et al are also considered. In the last section, the important question of whether the smectic-A to nematic (cholesteric) phase transition can ever be second order is examined. 

The structure of liquid crystals can broadly be classified as nematic, cholesteric and smectic, see Fig. 1. None of them have full three-dimensional (3-D) positional order, but some degree of orientational order. Most often the constituent molecules are elongated, as indicated in Fig. 1, but distinctly flat molecules make up the socalled discotic liquid crystals. The nematic phase has only orientational ordering of the molecules. The collection of molecules have one symmetry axis called the director n. The cholesteric phase has only orientational order, formed by the constituent chiral molecules. The director twists with a pitch comparable to the wavelength of light. 

When the mesogenic compounds are chiral (or when chiral molecules are added as dopants) chiral mesophases can be produced, characterized by helical ordering of the constituent molecules in the mesophase. The chiral nematic phase is also called cholesteric, taken from its first observation in a cholesteryl derivative more than one century ago. These chiral structures have reduced symmetry, which can lead to a variety of interesting physical properties such as thermocromism, ferroelectricity, and so on. 

Experimental evidence was reported for the existence of various additional phases a pre-cholesteric order in the form of a network of double-twisted cylinders, analogous to the thermotropic blue phases [27], a hexatic phase that replaces the hexagonal columnar in very long DNA fragments [31], and a structure with orthorhombic symmetry appearing in the transition to crystalline order [27]. 

A vast array of covalent molecules have been synthesised over the years in the search for LCs that show the useful cholesteric and ferroelectric smectic C phases, often on a trial and error basis ignoring the interactions between the molecules. The idea that one could think of the interactions between the molecules as a kind of molecular recognition came from the careful analysis of the conformations of molecules in the layers [77,78]. The arguments are based on the symmetry limitations of the angle formed by the alkyl chain and the phenyl benzoate moiety in the molecules that were the subject of this study. A molecular recognition site within the phase was used as the basis for these speculations , which have actually proved rather successful. The actual interactions between molecules are usually weak, but the formation of layers of aromatic and aliphatic units in these mesophases gives rise to their unique properties. 

In this liquid crystal phase, the molecules have non-symmetrical carbon atoms and thus lose mirror symmetry. Otherwise optically active molecules are doped into host nematogenic molecules to induce the chiral liquid crystals. The liquid crystals consisting of such molecules show a helical structure. The most important chiral liquid crystal is the cholesteric liquid crystals. As discussed in Section 1.2, the cholesteric liquid crystal was the first discovered liquid crystal and is an important member of the liquid crystal family. In some of the literature, it is denoted as the N phase, the chiral nematic liquid crystal. As a convention, the asterisk is used in the nomenclature of liquid crystals to mean the chiral phase. Cholesteric liquid crystals have beautiful and interesting optical properties, e.g., the selective reflection of circularly polarized light, significant optical rotation, circular dichroism, etc. 

When the nematic phase is composed of optically active materials (either a single component or a multicomponent mixture made up of chiral compounds or chiral compounds mixed with achiral materials), the phase itself becomes chiral and has reduced environmental space symmetry. The structure of the chiral nematic (or cholesteric) modification is one where the local molecular ordering is identical to that of the nematic phase, but in the direction normal to the director the molecules pack to form a helical macrostructure, see Fig. 5. As in the nematic phase the molecules have no long-range positional order, and no layering exists. The pitch of the helix can vary from about 0.1 x 10 m to almost infinity, and is dependent on optical purity and the degree of molecular... 

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