Modelling defects, nematics

We note that earlier research focused on the similarities of defect interaction and their motion in block copolymers and thermotropic nematics or smectics [181, 182], Thermotropic liquid crystals, however, are one-component homogeneous systems and are characterized by a non-conserved orientational order parameter. In contrast, in block copolymers the local concentration difference between two components is essentially conserved. In this respect, the microphase-separated structures in block copolymers are anticipated to have close similarities to lyotropic systems, which are composed of a polar medium (water) and a non-polar medium (surfactant structure). The phases of the lyotropic systems (such as lamella, cylinder, or micellar phases) are determined by the surfactant concentration. Similarly to lyotropic phases, the morphology in block copolymers is ascertained by the volume fraction of the components and their interaction. Therefore, in lyotropic systems and in block copolymers, the dynamics and annihilation of structural defects require a change in the local concentration difference between components as well as a change in the orientational order. Consequently, if single defect transformations could be monitored in real time and space, block copolymers could be considered as suitable model systems for studying transport mechanisms and phase transitions in 2D fluid materials such as membranes [183], lyotropic liquid crystals [184], and microemulsions [185],... 

The presence, and to some degree the discovery, of TGB phases in liquid-crystalline systems stems from theoretical studies by de Gennes [18], Through modeling of the N-A transition, de Gennes predicted that, for a second order nematic to smectic A phase transition, a defect stabilized phase could occur... 

Textures correspond to various arrangements of defects. When the isotropic liquid is cooled, the nematic phase may appear at the deisotropization point in the form of separate small, round objects called droplets (Fig. 12). These can show extinction crosses, spiral structures, bipolar arrangements, or some other topology depending on boundary conditions. Theoretical studies based on a simple model confirm the stability of radial or bipolar orientation (Fig. 5) [22]. Considerations based on improved theoretical models yield stable twisted... 

We can imagine a cholesteric as a smck of nematic quasi-layers of molecular thickness a with the director slightly turned by ( ) from one layer to the next one. In fact it is Oseen model [18]. Such a structure is, to some extent, similar to lamellar phase. Indeed, the quasi-nematic layers behave like smectic layers in formation of defects, in flow experiments, etc. Then, according to the Landau-Peierls theorem, the fluctuations of molecular positions in the direction of the helical axis blur the one-dimensional, long-range, positional (smectic A phase like) helical order but in reality the corresponding scale for this effect is astronomic. 

The concept of defects came about from crystallography. Defects are dismptions of ideal crystal lattice such as vacancies (point defects) or dislocations (linear defects). In numerous liquid crystalline phases, there is variety of defects and many of them are not observed in the solid crystals. A study of defects in liquid crystals is very important from both the academic and practical points of view [7,8]. Defects in liquid crystals are very useful for (i) identification of different phases by microscopic observation of the characteristic defects (ii) study of the elastic properties by observation of defect interactions (iii) understanding of the three-dimensional periodic structures (e.g., the blue phase in cholesterics) using a new concept of lattices of defects (iv) modelling of fundamental physical phenomena such as magnetic monopoles, interaction of quarks, etc. In the optical technology, defects usually play the detrimental role examples are defect walls in the twist nematic cells, shock instability in ferroelectric smectics, Grandjean disclinations in cholesteric cells used in dye microlasers, etc. However, more recently, defect structures find their applications in three-dimensional photonic crystals (e.g. blue phases), the bistable displays and smart memory cards. 

The distribution of defects in mesophases is often regular, owing to their fluidity, and this introduces pattern repeats. For instance, square polygonal fields are frequent in smectics and cholesteric liquids. Such repeats occur on different scales - at the level of structural units or even at the molecular level. Several types of amphiphilic mesophase can be considered as made of defects . In many examples the defect enters the architecture of a unit cell in a three-dimensional array and the mesophase forms a crystal of defects [119]. Such a situation is found in certain cubic phases in water-lipid systems [120] and in blue phases [121] (see Chap. XII of Vol. 2 of this Handbook). Several blue phases have been modeled as being cubic centred lattices of disclinations in a cholesteric matrix . Mobius disclinations are assumed to join in groups of 4x4 or 8x8, but in nematics or in large-pitch cholesterics such junctions between thin threads are unstable and correspond to brief steps in recombinations. An isotropic droplet or a Ginsburg decrease to zero of the order parameter probably stabilizes these junctions in blue phases. 

The main defect in this kind of theory is the assumption that the orientational order is nematic like in the interfacial region, which is generally not true (see Section 10.3.1). Another problem is linked to the fact that the surface anchoring energy is unknown. This surface anchoring energy is difficult to measure directly and the result of any indirect measurement depends on how the response of the system to a disorientation is modeled. One way out is to use the orientational distribution of the molecules in the surface layer as a boundary condition of the nematic order [47] (see also Section 10.3.2]. 

Very large GB systems (over 80,000 molecules) have also been recoitly studied to investigate some of the most distinctive features of liquid crystals topological defects [27,28], until now simulated only with lattice models [29]. In particular, the twist grain boundary phase in smectics [27] and the formation of a variety of defects in nematics by rapid quenching [28] have been examined. 

It is instructive to examine boundary disclination lines at a nematic surface or interface by introducing some additional modelling and approximations that incorporate surface tension and the effect due to gravity. Many of these aspects introduced here in this Section are common throughout the literature on liquid crystals and will also form a basis for the discussion on point defects at a free surface of nematic discussed in the next Section. The results presented below are based on those derived by de Gennes [107] and have been further elucidated in physical terms by de Gennes and Prost [110]. 

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