Two-dimensional nematics

When the polar groups are ordered on the micellar face in two-dimensional nematic order, x and y correspond to the directions on the micellar face corresponding to the minimum and maximum values of R respectively (3). 

Cholesteric liquid crystals form in thin layers, each one molecule thick and within each layer the molecules are arranged with their long axes in the plane of the layer and parallel to each other, as a two-dimensional nematic structure. 

I. Musevic, M. Skarabot, U. Tkalec, M. Ravnik, S. Zumer, Two-dimensional nematic colloidal crystals self-assembled by topological defects. Science 313, 954—958 (2006)... 

We begin with a not particularly realistic example, a two-dimensional nematic. Here the classification of defects is quite illustrative, and in the one-constant approximation (12.20) a calculation of structures with point defects is simple. The equilibrium condition for a point defect, located at the center of the coordinate system, reads... 

External field distortions in SmC and chiral SmC phases have been investigated [38], but the large number of elastic terms in the free-energy, and the coupling between the permanent polarization and electric fields for chiral phases considerably complicates the description. In the chiral smectic C phase a simple helix unwinding Fr6ede-ricksz transition can be detected for the c director. This is similar to the chiral nematic-nematic transition described by Eq. (83), and the result is identical for the SmC phase. Indeed it appears that at least in interactions with magnetic fields in the plane of the layers, SmC and SmC phases behave as two dimensional nematics [39]. 

As discussed previously the extra hydrodynamic mode due to the molecular tilt in the SmC phase can be described by the variable 2. As demonstrated by Meyer et al. [14], free-standing films offer a unique geometry to single out this variable. For wavelengths that are large compared to the film thickness (approx. 2.5 nm/layer) and at a temperature far away from other transitions, without the complications of the surface-enhanced order, SmC free-standing films behave essentially like two-dimensional nematics. 

McMillan s model [71] for transitions to and from tlie SmA phase (section C2.2.3.2) has been extended to columnar liquid crystal phases fonned by discotic molecules [36, 103]. An order parameter tliat couples translational order to orientational order is again added into a modified Maier-Saupe tlieory, tliat provides tlie orientational order parameter. The coupling order parameter allows for tlie two-dimensional symmetry of tlie columnar phase. This tlieory is able to account for stable isotropic, discotic nematic and hexagonal columnar phases. 

These structures were firstly observed for terminally polar mesogens [11, 12]. However, recent experiments give clear evidence of the presence of smectic A layering [37, 38], re-entrant nematic behaviour [39], two-dimensional lattices [40, 41] and smectic layering with incommensurate periodicities [42] for non-polar sterically asymmetric LCs. 

Fig. 6. Phase diagram for the binary mixture of terminally polar mesogens showii the reentrant nematic (N), monolayer (Ai), partially bilayer (Ad) and two-dimensional (C) phases (Lobko et al. [47])...

Figure 6.2 Discotic molecules in a (a) nematic state ND, (b) twisted nematic discotic state Np (P/2 is half of cholesteric pitch), (c) columnar state, ordered D0 and disordered Dd, (d) hexagonal ordered columnar state Dho two-dimensional packing arrays for columnar structures in (e) hexagonal, Colh rectangular, Colr oblique, Col0b.

With further cooling, the SmA LC, which is more ordered than the nematic, becomes the thermodynamic minimum. In the SmA, there is a spontaneous formation of layers, with long-range positional order normal to the layer planes. Thus, the SmA can be considered a stack of two-dimensional fluid layers with crystalline (long-range positional) order in the third dimension, but no... 

Discotic LC are formed by disk-like molecules with aromatic cores and side chains that are either hydrophobic (i.e., thermotropic) or hydrophilic (i.e., lyotropic). The discotic nematic (No) phase behaves like a normal nematic phase formed by rod-like molecules, and the disk-like molecules are oriented with their short molecular axes parallel to the director but show no positional order. More ordered columnar phases are commonly formed by thermotropic discotics. The two-dimensional structure can pack the columns into a hexagonal or rectangular columnar phase, while within the columns, disks can be... 

Since many planar metal complexes have nearly axially symmetric g and AM tensors, two-dimensional powder ENDOR spectra can easily be obtained from such compounds oriented in nematic glasses84. As mentioned, interpretation of this type of spectra will be discussed in Sect. 4.3. 

Fig. 18 a, b. DOUBLE ENDOR spectrum of Cu(TPP) in a frozen nematic glass (Merck Phase 5). a) Two-dimensional nitrogen ENDOR spectrum with B0 in the complex plane. ENDOR observer frequency (v = 19 MHz) used in b) is marked by an arrow, b) DOUBLE ENDOR spectrum the corresponding ENDOR frequencies cN( 1/2) = AJ72 3/2 Qj vN obtained from single crystal work1 are marked by arrows. (Ref. 84)... 

Discotic liquid crystals arise from disk-shaped molecules as nematic or cholesteric mesophases. Their structural characteristics are similar to the respective ealamitie mesophases, that is, the normals of the disks are oriented parallel. Instead of the smectic mesophases, diseotie columnar liquid crystals arise from eonnecting the disks to each other. The columns of the discotic columnar mesophase form a two-dimensional lattice whieh is in a hexagonal or rectangular modification. In addition, the columns may be tilted (Fig. 2f,g). 

In the last few years disc-like molecules have been shown to form liquid crystals (Chandrasekhar, 1994). Typical of them are hexasubstituted esters of benzene (I) and certain porphyrin esters (II) (see below). In the liquid crystalline state, the disc-like molecules are stacked aperiodically in columns (liquid-like), the different columns packing in a two-dimensional array (crystal-like). The phases have translational periodicity in two dimensions but liquid-like disorder in the third. In addition to the columnar phase(D), the disc-like molecules also exhibit a nematic phase (Nj,). A transition between D and phases has been reported. 

Prominent exceptions are studies on the liquid crystal phase formation and self-assembly of two-dimensional disc- or sheet-like nanomaterials such as the organization of nanodiscs or nanoplatelets into nematic, smectic, or columnar morphologies [263-270] (see Fig. 2 for an example of the self-assembly of nanoclay in aqueous suspensions) or the synthesis of CuCl nanoplatelets from ionic liquid crystal precursors as described by Taubert and co-workers [271-273]. 

Beneficial electro-optic effects have also been reported for semiconductor quantum dots doped into nematic liquid crystals. Khoo and Mallouck et al. published one of the earlier reports on suspensions of quantum dots in nematic liquid crystals [331], This work, however, focused on CdSe nanorods and will be discussed in a later section on two-dimensional nanomaterials in liquid crystals. 

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