Interaction between disclinations

The energy of deformation of an isolated disclination in a circular layer of radius R and of unit thickness 

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The disclination is supposed to have a core whose energy is not known. To allow for this, we postulate a cut-off radius around the disclination and integrate for distances greater than / , to obtain 

The energy of a single defect being proportional to according to the planar model that we have just considered, defects of strength. s should be unstable and should dissociate into 5 = defects. However, as is evident from fig. 1.1.6(A), stable defects of strength i = I (with four brushes) occur very frequently. The reason for this will be discussed in 3.5.3. 

The interaction between disclinations may be calculated by superposing solutions of the form (3.5.3), 

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The radial force of interaction between disclination will, of course, be modified because of anisotropy. In addition there will now be an angular component of the force. The physical basis for the angular force can be understood by referring to fig. 3.5.23, which shows the director patterns for two pairs of unlike defects, (+, — ) and ( +1, — 1), each in two different situations. It is seen that there are significant differences in the... 

The force of interaction between them dW/dp is proportional to l/pn- Here we see an analogy with the force of interaction between two infinite parallel wires carrying electric currents. For disclinations of opposite sign 51 2 < 0 the interaction energy is positive and decreases with decreasing distance. Therefore such disclinations attract each other. 

In this section we consider the interaction between a particle and a wall. We first present the results from the theory. Figure 1.2.2 shows the director profile and scalar order parameter maps for a particle of radius R = 0.36 located at the surface of a wall. Comparing these maps with those for a particle further away, (Fig. 1.2.2 ), we observe that the disclination line is no longer at the equatorial plane, but is shifted towards the wall. Similar deviations of the position of the ring are observed in the Monte Carlo simulations. 

The disclinations interact with each other through forces similar to the magnetic interactions between currents. Specifically, the interaction between two parallel threads p and q at and r, is given by ... 

Let us consider the role of the polymer in the polymer-stabilized blue phase (PSBP) from the viewpoint of the thermodynamic interaction between the hquid crystal and the polymer. A feature of the blue phase is that it must coexist with disclination lines, and the very existence of the disclinations is the reason why the temperature range of the blue phase is small. Looking at it from the opposite direction, the blue phase could be stabilized by the disclinations. 

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