Disclinations axial

One solution to this equation, called an axial disclination, depends linearly on ( ) and does not depend on p that is, G( ) = [ti(t + 0(j, where m and 0 are constants. One restriction... 

The free energy per unit volume of an axial disclination can be found by writing F. in cylindrical coordinates. Using... 

It is not difficult to visualise the director configuration for some of these axial disclinations. If m, which is called the strength of the disclination, is positive, then the director rotates counter-clockwise in traversing a counter-clockwise path around the disclinatiom If m is negative, the director rotates clockwise in traversing this same... 

Figure 2.13. Three director configurations around an axial disclination.

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... 

Figure 2.14. Vertical cross-sectional view of an axial disclination (a) and an escaped axial disclination (b) in a capillary tube. The dot in (b) is a point defect.

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. 

In order to simplify the presentation the one-constant approximation will be assumed for the nematic energy density given by equation (2.67). To analyse axial disclinations we consider configurations in which the director referred to Cartesian axes takes the form... 

For axial disclinations, 6 is expected to be independent of r and so in this case equation (3.338) collapses to... 

We axe now in a position to picture the flux lines following the orientation of n around an axial disclination for various values of the FVank index and constant o-To find the flux lines in particular cases we solve the differential equation (3.342) after substituting for 0 in the solution (3.341) for fixed values of the arbitrary constant < o- In all cases, except the solution forn = 2 in equation (3.348), changing the constant o merely rotates the flux lines shown in Fig. 3.20 below, and so we set ( 0 = 0 except for two of the instances when n = 2. It suffices to demonstrate the technique for a few cases the others are in a similar style. For n = —2 and 00 = 0, equation (3.342) can be solved to find that the flux lines away from the disclination are given by... 

The energy W in equation (3.335) for an axial disclination can be evaluated most easily by transforming to cylindrical coordinates. For 6 given by (3.341) we have, upon using equation (3.337) and the usual chain rule for partial derivatives. 

The energy of a disclination at one of the cusp points must also be added to Fsurf [107]. This can be estimated by treating the disclination as an axial disclination of strength s = 1 (see Fig. 3.20 on p.ll4), which is limited to a half-space, since we are examining a disclination at a surface. The integration in 0 in the energy equation (3.352) therefore only takes place over the interval 0 < 0 < tt with Frank index n = 2. The relevant energy per unit area around the disclination is then... 

When a nematic-smectic A transition occurs in a capillary tube (Fig. 8 a), smectic layers nucleate at disclination points and a singular line forms along the axis, with cylindrical layers (as shown in Fig. 8 b), but the presence of beads along the axial defect shows that the situation is less schematic [42]. 

Figure 3.20 Examples of flux lines which are tangential to the orientation of the director n around an axial line disclination located perpendicular to the page and passing through the point indicated by a black dot. Various cases of Rrank index n are given together with the associated solution B provided by equation (3.341). The strength of such disclinations is often defined by s = f. The bold lines represent the singular radial lines obtained from equation (3.347). The constant 0o has been set to zero except for the examples of Prank index n = 2.

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