Weak anchoring effects

As an elementary example of a weak Preedericksz transition consider the same situation first encountered in Section 3.4.1 for a nematic but now with the director weakly anchored to both boundary plates, so that n satisfies 

In this particular example the total surface energy per unit area actually reduces to the sum of the energy integrand evaluated at the two points z = 0 and z = d, that is, 

The surface energy contribution introduced here only differs by a constant from that introduced by Rapini and Papoular [227], taking into account the slightly different notation. 

The bulk equilibrium equation for the director is exactly the same as (3.111), and if symmetric solutions satisfying 

This equation must be supplemented by the relevant boimdary equilibrium condition for weak anchoring mentioned in equation (2.241), giving 

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Fig. 11.29 Conversed flexoelectric effect in cells with homeotropic (a) and homogeneous (b) director alignment and electric field applied along the cell normal. Weak anchoring energy at the bottom plate allows the flexoelectric deflection of the director 3 at the surface propagating up in the vertical direction (e = 0)...

In a recent article Nehring, Kmetz and Scheffer described effects of weak anchoring on equilibrium configurations of twist cells.1 In order to obtain tractable equations for analytic solution, the three bulk elastic constants were made equal, the field-and-strain-free orientation at the surface was assumed to be parallel to the surface, and only nematic liquids were discussed. That treatment gives useful insights into the nature of the problem but leaves a number of interesting questions unanswered. 

We have used the full Leslie-Ericksen hydrodynamic equatlons2>3 for cholesterics in laminar flow between parallel surfaces, except for the justifiable omission of inertial terms. Use of cholesterics removes the difficulty mentioned in Ref. 1 of obtaining a 90 twist with very weak anchoring. The effects of using models for surface energy other than the symmetric sin (n,np) form used in previous work are investigated. 

To Illustrate the effects of viscous weak anchoring, we have computed the dynamic and optical behavior of cells with rigid anchoring and with weak anchoring. The material chosen for an example Is E-7, a widely used mixture of high dielectric anisotropy made by Anchoring based on surface energies... 

Thermal fluctuations probably would have the same effect on real cells. As with the rigid boundary cell, we are using three volts but the weak anchoring reduces the critical voltage considerably. Notice that turn-on time is faster but turn-off is much slower than with the rigid-boundary cell because of the... 

In conclusion, we have shown that the use of a HAN cell is convenient to get nonlinear change in the dielectric constant for the following reasons (a) an incidence angle = 0 can be used even with very low optical fields (b) the nonlinear effect can be greatly enhanced inducing a weak anchoring on the boundaries using a suitable surfactant on the walls. 

There is a complementary approach to studying the effects of asymmetry on the domain scaling, which is to choose a fixed xM and vary /a. This approach revealed new behavior very near the MST [6]. As the asymmetry increases, the domain size initially decreases but, near the MST, this decrease reverses and the domain size increases. The explanation for this is that the short minority blocks are only weakly anchored in their spherical domains, and can escape into the majority domain. In this realm, the spheres continue to shrink, but the overall domains increase in size. 

For weak anchoring there is a competition between the torque in the bulk due to one plate, and the torque resulting from the other surface this is usually confined to a boundary region as indicated in the figure. External electric and magnetic fields will also affect the equilibrium director distribution, and this is the basis of many liquid crystal applications these effects will be considered later. 

U.D. Kini, Crossed fields induced periodic deformations in nematics effect of weak anchoring, Liq. Cryst, 21, 713-726 (1996). 

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