Splay geometry

When the tilt angle is small, the splay elastic energy dominates and the ceU geometry is called splay geometry. The electric energy is negative and is approximately given by 

The total free energy (per unit area) of the system is 

Using the Euler-Lagrange method to minimize the free energy, we obtain 

When 0 is small, we use the approximations sin0 = 0 and cos0=1. Neglecting the second-order terms. Equation (5.21) becomes 

Now let us look at the boundary condition. Under infinitely strong anchoring, the boundary conditions are 0(z = 0) = 6(z = h) = 0. Therefore B = 0 and 

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Now the free energy density has a form (11.45) wherein, due to a large the field E becomes dependent oti coordinates. In this case, one should operate with electric displacement D. For example, in the case of the Frederiks transition and the splay geometry of Fig. 11.15a the field strength is ... 

Consider the flexoelectric effect in the splay geometry as shown in Figure 4.5. The cell thickness A is 5 microns. The splay elastic constant Kn of the liquid crystal is 10 N. The flexoelectric coefficient is 2 x 10 V. Calculate the tilt angle 6 at the cell surface when the applied field is 1 V/pm. 

Figure 5.2 Schematic diagram of Freedericksz transition in the splay geometry.

Figure 5.3 The tilt angle at the middle plane vs. the applied field in splay geometry. i = 6.4 x 10" ...

Figure 5.4 The tilt angle as a function of position at various fields in splay geometry. fsTi i = 6.4 x 10 and Jir33 = 10 X 10 N are used.

Although the dynamics of Freedericksz transition in splay geometry, bend geometry, and twisted geometry is more complicated, the response time is still of the same order and has the same cell thickness dependence. The rotational viscosity coefficient is of the order O.IN - s/m. When the elastic constant is 10 "N and the cell thickness is 10pm, the response time is of the order 100 ms. Faster response times can be achieved by using thinner cell gaps. 

We consider the dynamics of the Freedericksz transition in the splay geometry upon the removal of the applied field [24-27]. Initially the liquid crystal director is aligned vertically by the applied field, as shown in Figure 5.17(a). When the applied field is removed, the liquid crystal relaxes back to the homogeneous state. The rotation of the molecules induces a macroscopic translational motion known as the backflow effect. The velocity of the flow is... 

Figure 5.17 Schematic diagram showing the relaxation of the liquid crystal in the splay geometry.

Freedericksz in splay geometry. Use the parameters in Figure 5.3 and Equation (5.30) to calculate and plot the tilt angle at the middle plane as a function of the normalized field / ,. 

In some cases, it is simpler to describe the liquid crystal director n in terms of the polar angle 0 and azimuthal angle The angles may vary in one or two or three dimensions. We first consider a simple case Fr edericksz transition in splay geometry. The liquid crystal director is represented by the tilt angle 6 n = cos 0 z)x + sin 0 z)z, where the z axis is in the cell normal direction. The electric field is applied in the cell normal direction. From Equation (4.17) and (7.8) we have the free energy density... 

There are two other geometries besides the twist geometry in which a Freedericksz transition takes place. Figure 10.9 shows the splay geometry, where the botmdary conditions again favottr a director oriented alortg the x-axis, but now the electric field is applied in the z direction. 

In these geometries, the sample cell is usually observed in the transmitted light with crossed polarizers inserted in 45° orientation to the director tilt plane. While the ordinary polarized light in the birefringent medium always experiences the ordinary index of refraction and is unaffected by the deformation, the optical path of the extraordinary wave is sensitive to the director tilt. Both waves are brought to interference at the analyser and the optical interference can be related in a direct way to the director deflection. Saupe first used this magneto-optical method with splay geometry to determine the splay and bend constants of p-azoxyanisol (PAA) [13]. 

In order to relate the cell capacitance or optical transmission curves in splay or bend geometry to the elastic ratios, exact anal3rt-ical equations for the director field in the cell are used. The one-dimensional deformation of the director field n=[cos 6(z), 0, sin0(z)] induced by electric or magnetic fields in splay geometry is found by minimizing the free energy... 

The introduction of is not as straightforward as that of K24, and causes serious mathematical problems. The free energy contribution corresponding to the 13 term is not bound from below, and the simple application of the variational principle with a nonzero Ki coefficient may lead to discontinuities in the director field at the boundaries. This problem is known as the Oldano-Bar-bero paradox [213, 325]. Consider the simple one-dimensional splay geometry having the director field n=[sin d(z), 0, cos 0(z)] in one constant approximation Kn=K =K. The free energy density is... 

The fluctuation modes with the corresponding displacements 5, and < 2 of the director in the ej and 2 directions are shown in Fig. 8. The modes are named according to the effective elastic coefficients. The main term for a viscous director rotation is, in all cases, the rotational viscosity 7i. The additional terms are caused by the backflow. The backflow term in the splay geometry is very small, and therefore its determination is difficult. 

A similar strategy may be followed in the bend and splay geometries, but each case depends on the two different Frank constants Ki and Kj instead of just one. We will highlight the differences for the bend geometry, and the splay geometry can then be found by swapping the roles of Ki and K. ... 

Optical detection. The distortion in thin films is most accurately detected optically [37,41-44,52]. Consider the splay geometry in FIGURE 5. For light polarised along the y axis the refi active index is no irrespective of the director pattern. For light polarised along the x axis the refractive index is ne at d/2 boundaries, but elsewhere is... 

The three classical Freedericksz transitions will be considered in what are commonly called the splay, twist and bend geometries. Full details will be given for the splay geometry, with the corresponding results for the twist and bend geometries being stated since their analysis is analogously similar. The Section ends with a qualitative discussion for the one-constant approximation. 

The equilibrium equation for this twist geometry can be derived in a similar fashion to that above for the splay geometry and is found to be... 

Section 5.9 introduces the dynamics of the Preedericksz transition in the classical geometries described in Chapter 3 in Section 3.4.1. The switch-on and switch-off times will be defined when flow is considered to be negligible in the usual twist geometry, as detailed in Section 5.9.1. In some instances, however, flow turns out to be quite influential and leads to the phenomena of backflow and kickback y as to be discussed in detail in the case of the splay geometry in Section 5.9.2. Backflow in the bend geometry is discussed in Section 5.9.3. 

Backflow in the Planar to Homeotropic Transition Splay Geometry... 

For the physical parameters for MBBA in Table D.3, the ratio 7j /7i a is approximately 0.18 and therefore the effective viscosity is greatly reduced at high field strengths. This obviously enhances the switch-on time and can considerably reduce the value of Ton- The effect of backflow upon the dynamics of the Freeder-icksz transition in the homeotropic to planar bend geometry is therefore significant compared to that for the planar to homeotropic splay geometry. 

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