Disclination lines

Inelastic deformation of any solid material is heterogeneous. That is, it always involves the propagation of localized (inhomogeneous) shear. The elements of this localized shear do not occur at random places but are correlated in a solid. This means that the shears are associated with lines rather than points. The lines may delineate linear shear (dislocation lines), or they may delineate rotational shear (disclination lines). The existence of correlation means that when shear occurs between a pair of atoms, the probability is high that an additional shear event will occur adjacent to the initial pair because stress concentrations will lie adjacent to it. This is not the case in a liquid where the two shear events are likely to be uncorrelated. 

Note 1 Diselinations are responsible for some optical textures seen with a polarizing microscope, such as the schlieren texture formed by disclination lines in nearly vertical orientations, whose projections are seen as dark points with two or four emerging dark stripes or brushes (see Definition 4.9.2). 

Note A nucleus can indicate either the end of a disclination line terminating at the surface of a sample or an isolated defect. 

Thick, thread-like disclination line anchored along its length to the upper or the lower of the surfaces between which a sample is placed. 

Figure 10. Pairing of two disclination lines of opposite signs (lamellar details are not featured) (top) a less probable model for the core of a dislocation (middle) and focal line appearing on the dislocation in order to release locally deformation energy (bottom)...

Figure 10.21 Classes of disclinations in a nematic, with the disclination line perpendicular to the plane of the page. [From Larson 1988 (adapted from Chandrasekhar 1992), with permission from Cambridge University Press.]...

Disclination lines are energetically disfavored because they produce gradients in the director profile and Frank stresses. So if the sample is left alone, the disclination lines and loops spontaneously shrink in length and annihilate one another (see Fig. 10-22) until no disclinations are left—except for any that are pinned by any impurities in the fluid or by wall irregularities, and those trapped because of incompatibilities in anchoring conditions at surfaces (Chuang et al. 1991 Nagaya et al. 1992). 

Figure 10.24 Areal density of disclination lines in steady-state shearing of 5CB as a function of shear rate for a gap h of 250 pm. Closed symbols are measurements during increasing shear rate open symbols are for decreasing shear rate. For increasing shear rate, at a critical shear rate of about 10.5 sec i, the density of disclinations jumps from zero to a high value. The critical shear rate corresponds to an Ericksen number Er = (yiA y/k s) of 4100. The inset shows the critical shear rate as a function of the increment in angular...

Parabolic focal conics are a special case of generic focal conic defects, which are composed of layers curved to form toroidal surfaces called Dupin cyclides (see Fig. 10-31). Each such structure contains a pair of disclination lines—one an ellipse and the other... 

Figure 10.31 Smectic A layers forming Dupin cyclides. In (a), the disclination lines are an ellipse and a hyperbola the ellipse passes through the focus of the hyperbola and the hyperbola passes through the focus of the ellipse. In the degenerate case (b), the ellipse becomes a circle and the hyperbola becomes a line. The cone is an isolated confocal domain. (From Lavrentovich, reprinted with permission from Sov. Phys. JETP 64 984, Copyright 1986, American Institute of Physics.)...

These different contrast mechanisms can all be used to reveal the scale of liquid crystalline polymer microstructures. In specimens that exhibit a mosaic texture, and in those that contain predominantly planar defects, domain size is easily defined in terms of areas that uniformly show extinction between crossed polars. However, if the defects are predominantly linear, as in specimens that exhibit schlieren textures, such simple characterization of microstructural scale is no longer possible. Here it is more convenient to look at the length of disclination line per unit volume, which is equivalent to the number of lines intersecting unit area, and analogous to the dislocation density as defined for crystalline solids. Good contrast is essential in order to obtain an accurate count. Technologically, microstructural scale is of growing interest because of its relevance to processability, mechanical properties and optical transparency. 

Fig. 5 Structures of Blue Phases I and II. The rods in (a) and (c) represent double-twist cylinder. The black lines in (b) and (d) represent disclination lines...

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... 

If the disclination line is a wedged one and is along the z axis, then the solution of Equation (1.30c) is... 

The configuration of Equation (1.32) is associated with a twist disclination. Both twist and wedge disclinations are shown in Figure 1.24, where the disclination line is normal to this sheet of paper. 

The disclination line of m = 1/2 and disclination point of m = 1 are frequently observed in liquid crystals. In some cases high strength discli-nations may be observed, such as in a thin layer of small molecular mass liquid crystals, polymer liquid crystals, lyotropic liquid crystals, binary thermotropic liquid crystals, etc. 

For a three-dimensional nematic liquid crystal for example, the r = 0 case corresponds for example to a defect with d = 2, which means a discli-nation wall for r = 1, d = 1 corresponds to a disclination line for r = 2, d = 0 corresponds to a disclination point. It is known that the order vector space of three dimensional nematic liquid crystals is the projection plane P2 Its homotopy group of the zero rank (r = 0) is... 

This is a cycle group of two elements, one of two elements is in fact the disclination line of m = 1/2, the other element corresponds to the m = 1 disclination line, which is not topologically stable. 

The threaded texture is also often observed for nematics especially when a thicker film is examined. In this texture disclination lines show up as dark lines swimming in the sample with thermal motion. The lines are observable without polars (Figure 4.17) but the thread contrast will be higher when crossed polars are used. Because of the stopless molecular reorientation with thermal motion in the sample the old dark lines may disappear and new ones may form. In cases when the two ends are anchored on the surfaces the ends will stay where they have been but the thread may swing about. 

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