Liquid crystal director disclination

As in crystals, defects in liquid crystals can be classified as point, line or wall defects. Dislocations are a feature of liquid crystal phases where tliere is translational order, since tliese are line defects in tliis lattice order. Unlike crystals, tliere is a type of line defect unique to liquid crystals tenned disclination [39]. A disclination is a discontinuity of orientation of tire director field. 

Figure 1.20 The liquid crystal director configurations of the disclinations with variety of strengths.

For a disclination with the strength S, as one approaches the center of the disclination, the elastic energy diverges, as shown by Equation (1.124). In reality this will not occur. The liquid crystal will transform either into isotropic phase at the center of the disclination or a different deformation where there is no singularity. Here we only discuss the cases of cylindrical confinements (two-dimensional confinement) where it is possible to obtain analytical solutions. The mechanism of liquid crystal director escape in spherical confinement (three-dimensional confinement) is similar to that of two-dimensional. 

Figure 13.9 (a) The liquid crystal director configuration of the disclination formed between the doubletwist cylinders, (b) The structure of the disclinations in the simple cubic packing of the double-twist cylinders, (c) The structure of the disclinations in the body-centered cubic packing of the double-twist cylinders. 

Liquid crystals may have line defects called disclinations. The name comes from discontinuity and inclination. The director rotates about a line normal to the disclination. The strength of a disclination, S, is defined by... 

Small-angle light scattering has also been extensively applied to PLCs subject to flow [173]. As in the case of scattering dichroism, SALS patterns arise principally from fluctuations in orientation, and these arc strongest in the vicinity of disclinations, or defects in the director field. The experimental geometries used for SALS in liquid crystals normally use polarizers placed before and after the sample. The arrangements include VV scatter-... 

The dislocations have been extensively studied so we limit ourselves mainly to the disclination. Figure 1.21 sketches a sample of nematic liquid crystals, the short bars denoting the directors which all lie on the sheet of paper. Now cut the sample along the normal and via the line L, see... 

Figure 1.21(a), and rotate the two opposite slips of the cut A1 and A2 by 7T. 7r is a symmetry element. Insert a new sample piece of liquid crystals and arrange their director to fit the directors on the A1 and A2 slips, as shown Figure 1.21(b). The liquid crystals then relax themselves to the configuration shown in Figure 1.21(c). A disclination with the strength of —1/2... 

There is a simple process to produce a disclination rotate the directors on two slips respectively by uq and wo and make lo — luo = w. Thus the same disclination line is produced. The process is named the de Gennes-Friedel process. One can prove that the de Gennes-Friedel process is equivalent to the Volterra process for nematic liquid crystals. The operation Pv of the Volterra process can in fact be divided into the translation and rotation steps, i.e., first, translate the directors (T) and then rotate them around themselves (IV). The latter is actually the de Gennes-Friedel process. In other words... 

Fig. 1. Schematic diagrams of the director field distortions black lines) around particles in an aligned nematic liquid crystal. For a normal anchoring of the liquid crystal molecules at the surface of the particles, there are two possible configurations, a Dipole configuration with a companion point defect (indicated by an arrow) located in the immediate vicinity of the particle, b Quadrupolar Saturn-ring configuration with a disclination ring surrounding the particle at the equator...

The study of defects in liquid crystal systems is rooted in the understanding of defects in the solid state. For instance, crystals are rarely perfect and usually contain a variety of defects, e.g., point defects, line defects, or dislocations, and planar defects such as grain boundaries. In addition to these typical imperfections of the solid state, liquid crystals can also exhibit defects known as disclinations. These defects are not usually found in solids and result from the fact that mesophases have liquid-like structures that can give rise to continuous but sharp changes in the orientations of the molecules, i.e., sharp changes in orientation occur in the director field. 

Fig. 8.13 Two disclinations fixed by their end at the two glasses limiting a layer of a nematic liquid crystal. They interact with each other by the elastic force proportional to 1/Pi2 (a). The structure of the director field n(r) near the two disclinations of positive and negative strength and four dark brushes corresponding to the j = 1 disclinations (b)...

The problem is to find the distribution of the director around a disclination [14]. To solve it we can use the elasticity theory discussed in Section 8.3. Let a liquid crystal layer is situated in the y plane of drawing, and singularity L is parallel to the... 

Another example is formation of boodjooms at the cell surfaces. Now we are interested not in the linear disclinations responsible for the SchUeren texture but in their nuclei at the solid substrates limiting a liquid crystal cell. The linear discUna-tions of strength s = 1 may annihilate within the bulk due to some reconstruction of the director field induced, for instance, by temperature or a flow of the material. For example, a bulk discUnation of strength s = +1 shown by the solid vertical line in Fig. 8.18b disappears but its nuclei localized at the surfaces transform into new, surface defects. Fig. 8.18c illustrates the situation at one of the two surfaces. The escaped line leaves behind it a boodjoom. We meet such a situation in thick planar cells where the Schlieren textures with four brushes are observed. 

Fig. 8.21 Structure of the director field around different singular lines (disclinations) in a cholesteric liquid crystal x , X and x, Signs (—) and (+) correspond to different Volterra...

The curves represent the projection of the director field in the xy plane. For S I, a change in C merely causes a rotation of the figure by C/(l — S), while for 5 = 1 the pattern itself is changed. The disclinations are characterized by their strengths , which are defined as the number of multiples of 2n that the director rotates in a complete circuit around the diselination core. A value of+1 indieates that the director is rotated through 2n. The micrograph of a liquid crystal material, shown below, illustrates the various disclinations described above. 

Usually the Cano method [8] is used for chiral nematic liquid crystals. It can also be applied to SmC phases, but then the demands on the orientation of the liquid crystal are more extensive as a nearly perfect orientation of the layer normal k as well as of the c-director are required. The presented method utilizes the different thicknesses which occur in a sample if a lens is placed on top of it. A sketch of this is shown in Fig. 4.6a. Due to the anchoring conditions, only helical structures with integer multiples N of the pitch p are allowed and regions with different values of N are separated by disclination lines [14]. Thus, a picture similar to the one shown in Fig. 4.6b occurs [15]. If the radius of curvature Rc of the lens is known, the value of the pitch p can be calculated according to [9]... 

As mentioned earlier, most studies of field interactions with liquid crystals are done using thin films with a well-defined initial state, usually a monodomain or a thin film with a simple distortion induced by incommensurate surface anchoring. These conditions simplify observation and theoretical analysis. However, most liquid crystal materials that are not specially prepared contain topological defects that are very important to their response to external fields. One class of defect commonly observed in nematics is the disclinalion line. At a disclination line the director field is ill defined. The director field turns around the disclination line a multiple of half-integer times. Several disclination lines are shown in Fig. 8. 

What do these axial disclinations look like in a hqnid crystal The best way to observe them is with the sample between crossed polarisers. As will be discussed at length later, liquid crystals are birefringent, so only light that is polarised parallel or perpendicular to the director is transmitted unchanged. This means that regions of the liquid crystal where the director is parallel or perpendicular to one of the crossed polariser axes are dark. The rest of the liquid crystal is bright. This results in dark bands emanating from a disclination where the director points in either of two perpendicular directions. Thus there... 

There are other dischnations besides axial disclinations that form in nematic liquid crystals. In axial dischnations, the rotation axis of the director in traversing a loop aroimd the disclination is parallel to the disclination. In a twist dischnation, the rotation axis is perpendicular to the disclination. Figure 2.15 shows +1/2 and +1 strength twist dischnations in which the rotation axis for the director is along the y-axis and the dischnation points along the z-axis Due to the fact that the director twists, an entirely new class of dischnations form in chiral nematic liquid crystals. Likewise, the spatial periodicity of both chiral nematic and smectic hquid crystals ahows for defects in the perio(hc stmcture in addition to defects in the director configuration. These additional defects are quite different and resemble dislocations in solids. 

Disclinations are like dislocations in crystalline solids, where domains of differing orientations meet. The disclinations cause distortion of the director field of the polymer chains, giving rise to an excess free energy of the liquid crystal material (28). 

The point defect at a surface of an ordered medium can represent either the end of a line that is topologically stable in the bulk or a true surface point defect with no bulk singularity attached [61]. In cholesteric liquid crystals, all points with A = i 1 are the ends of bulk disclinations. Only when k = 2 An rotations of the director field), the point defect might be an isolated surface singularity. However, even in this case one should take care of the requirement of the layers equidistance. For example, the classical boojum configuration cannot be observed in a cholesteric vessel when 1. 

Carrying this idea over to the helical-isotropic transition, there are two differences. First, we must use disclinations topological line singularities in the director field of the liquid crystal—rather than crystal defects. The second difference is that the helical phase, which has no defects, melts to the blue phase, which is characterized by a stable defect lattice of line disclinations rather than by a random collection of defects. Indeed, there is more than one way to create such a lattice thus BPI and BPII. The helical phase therefore melts to BPI, BPI melts to BPn, and, with a final onset of randomly positioned defects, BPII melts to the isotropic phase. 

A. Saupe, Disclinations and properties of the director fleld in nematic and cholesteric liquid crystals. Mol. Cryst. Liq. Cryst. 21, 211-238 (1973). 

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