Nematics biaxial

The applicability of homotopic theory becomes much less obvious for liquid crystal phases with more complicated order parameters such as biaxial nematics and cholesterics, which are both locally defined by three directors forming a tripod. This gives rise to a description of the line singularities in terms of the quaternion group, Q. This is particularly interesting because the quaternion group Q is non-Abelian, a property that 

II Defects and Textures in Nematic Main-Chain Liquid Crystalline Polymers 

The coefficients in the Landau-de Gennes expansion are in first-line formal parameters, and the theory does not require assumptions in the microscopic interpretation of these parameters. The main impact of the Landau-de Gennes expansion on elastic theory is the well-founded explanation of the leading term in the bulk elastic constants, and the prediction of a linear term in 13. 

A number of publications deal explicitly with the theory of elastic constants at phase transitions. The behaviour of nematic elastic constants in the vicinity of the nematic-smectic transition has been analysed by means of the Ginzburg-Landau Hamiltonian [295]. At the transition to the smectic A phase, a critical divergence of K22 and A 33 is predicted. 

Simple mean field theory yields a divergence of the form in the 

In biaxial nematic mesophases, the rotational symmetry of the phase around its director n is broken. It is commonly understood that the term biaxial nematic refers to an orthorhombic nematic phase. With respect 

In achiral nematic phases the first three terms vanish (ki=k =k -0). The (i-[l, m, n ) terms ctm be transformed into surface integrals and, therefore, do not contribute to the equilibrium bulk free energy. The remaining 12 terms correspond to the 12 basic types of bulk elastic deformation of an orthorhombic biaxial non-chiral nematic. 

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Disk-shaped molecules based on a metal atom possess discotic Hquid crystal phases. An example is octasubstituted metaHophthalocyanine. FiaaHy, metallomesogens which combine both rod-like and disk-like features iato a single molecule adopt the biaxial nematic phase. In addition to there being a preferred direction for orientation of the longest molecular axis as is tme for the nematic phase, perpendicular to this direction is another preferred direction for orientation of the shortest molecular axis (12). NonmetaHomesogens which combine both rod- and disk-like features iato a single molecule also adopt a biaxial nematic phase, but at least ia one case the amount of biaxiaHty is very small (15). 

Note 1 See Fig. 16 for an illustration of the molecular arrangement in a Nb mesophase. Note 2 From a crystallographic point of view, the biaxial nematic structure is characterised by the symbol D2h in the Schoenflies notation (2 m, m in International System). 

Note 3 In lyotropic systems, biaxial nematic mesophases have been identified from the biaxial symmetry of their tensorial properties. 

Note 5 A biaxial nematic has the same structure as a disordered sanidic mesophase (see Definition 3.4, Note 2) it is recommended that the latter name be discontinued and the name biaxial nematic be used. 

Note 2 Short board-like shaped molecules usually form biaxial nematic mesophases. It is recommended that the use of the term disordered sanidic mesophases for such mesophases be discontinued (see Definition 3.3.1, Note 5). 

Fig. 3 Hekates with flexible spacers. Compounds 1, 2 Thermotropic properties as a function of spacer length. Compounds 3, 4 LC properties with unchanging spacer length and variable peripheral chains. Compounds 5, 6 Nematogens claimed to form biaxial nematic phases. Cr crystal, g glass, N nematic, N cholesteric, SmX unidentified smectic phase, Colh columnar hexagonal phase. All temperature are given in °C...

An interesting similarity of what we discussed here appears if one deals with mixtures of rodlike and disklike micelles. These systems could behave very similarly to a truly biaxial nematic, but show interesting differences to them. Whereas for the usual orthorhombic biaxial nematics both directors are perpendicular to each other by construction, in mixtures there is no need to impose this restriction. Pleiner and Brand [70] investigated how mixtures are influenced by an external field (magnetic field or shear flow) and found that the angle between the two directors exhibits a flow aligning behavior similar to the one studied in [42,43],... 

The bent core molecules do not only exhibit spontaneous resolution in smectic phases. One achiral derivative resolves in a nematic phase in this fluid state [ 145], while a substituted oxadiazole which forms a biaxial nematic phase also segregates [ 146]. The bent core clearly has a special stereochemical influence as a result of the effects it induces beyond the molecule, at least for liquid crystals. 

NMR experiments carried out on one of these co-dendrimers have proved that the symmetry of the nematic phase is imiaxial, in contrast to the biaxial nematic phase observed for some side-on-polymers [219]. 

In the simplest liquid-crystalline phase, namely the uniaxial nematic, there is at rest a special direction designated by a unit vector n called the director (see Fig. 10-2). In the plane transverse to the director, the fluid is isotropic. The most common nematics are composed of oblong molecules that tend to point in a common direction, which defines the director orientation. Oblate, or disc-like, molecules can also form uniaxial nematics for these discotic nematics, the director is defined by the average orientation of the short axis of the molecule. Lath-like molecules or micelles (shaped like rectangular slabs), in which all three dimensions of the molecule are significantly different from each other, can form biaxial nematics (Praefcke et al. 1991 Chandrasekhar 1992 Fialtkowski 1997). A biaxial... 

The fluctuation relations for the viscosities have also been generalised to biaxial nematic liquid crystals. They have been evaluated numerically for a biaxial version of the Gay-Beme fluid consisting of a linear string of oblate Gay-Beme ellipsoids, the axes of which point in the same direction. The flow... 

S. Sarman, Molecular Dynamics of Biaxial Nematic Liquid Crystals, J. Chem. Phys. 104 (1996) 342. 

However, in 1986, Chandrasekhar [70] published a derivative (Figure 35) which he claimed to show a biaxial nematic phase. This report was interesting because the biaxial nematic phase (Nb), demonstrated in lyotropic systems [71], had been long sought after in thermotropic materials. Further, the molecules were described as bridging the gap between rod- and disc-like materials (a reference perhaps better reserved for polycatenar liquid crystals—vide infra)... 

The existence of the biaxial SmA phase (SmAj,), also known as the McMillan phase, has been demonstrated in some charge-transfer complexes (CT complexes) formed with the metallomesogens 126 and TNF by textural observations and X-ray investigations. (Like the biaxial nematic phase mentioned earlier, the SmAt phase has additional long-range order in a direction perpendicular to the principal director, n (Figure 7)). Two novel... 

Conoscopic studies suggest that this compound in fact exhibits a biaxial nematic phase although these experiments indicated only a small opticd biaxiality. Indeed, more recently, NMR spectroscopy using selectively deute-rated materials has shown that for other materials the assignment of a biaxial nematic phase on the basis of conoscopy alone is not necessarily reliable [118], This is discussed further in Sect, 3,4,... 

The existence of the thermotropic biaxial nematic phase was theoretically predicted almost thirty years ago [119] but such a phase has yet to be unambiguously identified. Indeed, as we saw in the previous section, a biaxial nematic phase has been claimed for a class of discotic dimers [111], Theory predicts that mixtures of rod-like and disc-like molecules should exhibit the biaxial nematic phase [120] but experimentally such systems phase separate... 

This experimental difficulty was overcome by simulating the behaviour of a model system and a biaxial nematic phase was indeed obtained [122], One way by which the problem of phase separation could be solved experimentally was suggested by Fletcher and Luckhurst and involved covalently linking a rod-like and disc-like unit via a flexible spacer yielding a non-symmetric discotic-calamitic liquid crystal dimer [123] ... 

Molecules which combine the features of the rod and the disc may be expected to form new types of mesophases. An example is the biaxial nematic phase reported in thermotropic systems (see 6.6). Malthete et a/. have prepared an interesting series of mesogens shaped like stick insects called phasmids (fig. 6.1.5(n)). Some of them form columnar mesophases the structure proposed for the hexagonal phase is shown schematically in fig. 6.1.5( >). 

Fig. 6.6.1. Two examples of molecules which exhibit the biaxial nematic phase (a) (bis-l-(p-n-decylbiphenyl)-3-(p-ethoxyphenyl) propane-1,3-dionatocopper (II))(8i.82) (hereafter referred to as complex A) (b) I,12- >w(pentakis((4-pentyl-phenyl)ethy nyl)pheny loxy)dodecane. ...

The biaxial nematic (NJ phase was first identified by Yu and Saupe in a ternary amphiphilic system composed of potassium laurate, 1-decanol and D2O. In such systems the constituent units are molecular aggregates, called micelles, whose size and shape are sensitive to the temperature and concentration the phase was found to occur over a range of temperature/concentration. There are obvious advantages in having a single-component, low-molar-mass thermotropic phase. The sugges-... 

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