Boundary conditions homeotropic

To solve these equations we need suitable boundary conditions. In the following we will assume that the boundaries have no orienting effect on the director (the homeotropic alignment of the director is only due to the layering and the coupling between the layer normal p and the director h). Any variation of the layer displacement must vanish at the boundaries ... 

Figure 10.11 (a) Apparent viscosity for Poiseuille flow of p-azoxyanisole (PAA) as a function of the effective shear rate AQjnR y where Q is the volumetric flow rate and R is the tube radius, for homeotropic boundary conditions. (b) The data of (a) are replotted against AQjnR, which is proportional... 

Figure 10.29 Response of an aligned smectic to layer dilation, (a) Initial equilibrium sample, (b) For a very small dilation Sh < Ink, the layer spacing simply increases, (c) A uniform rotation of the layers decreases the spacing toward that of equilibrium, but doesn t satisfy the boundary conditions, (d) Hence, the sample undergoes an mdulational instability, which also narrows the layer spacing while satisfying homeotropic boundary conditions, (e) For a large enough dilation, the undulation instability leads to formation of parabolic focal conic defects. (From Rosenblatt et al. 1977, with permission from EDP Sciences.)...

Experiments and theoretical considerations have shown that the key parameters are the symmetry of the system (planar or homeotropic boundary conditions), the dielectric and the conductivity anisotropies. It is therefore convenient to categorize the various combinations of parameters as listed in Table 1. In the last column the structures predicted and/or observed are summarized and will be discussed below systematically. 

Eq. (8) is similar to Eq. (7) however, the effective quantities and in Eq. (8) are defined differently. In fact, they can be transformed into each other by interchanging the subscripts <->--L and the material parameters K K, rji <-> r]2 and 0 2 -0 3. These transformations are natural consequences of switching the boundary conditions between planar and homeotropic cases. 

The essential difference between cases A and B lies in the symmetry of the system. In case A the planar geometry is anisotropic, and the wavevector direction is selected hy the boundary conditions. In contrast, the homeotropic alignment in case B provides isotropic conditions in the plane of the patterns, and the direction of the wavevector of the striped patterns is chosen accidentally at threshold, which corresponds to a spontaneous breaking of the rotational symmetry. 

If a smectic A sample is homeotropically aligned between two glass plates having a separation L, the boundary conditions require that = mnIL, where m is an integer. We shall coniine the discussion to m = 1. When = 0 we have from (5.3.6)... 

Flexoelectric patterns also exist for nematic layers with asjrmmetric boundary conditions, i.e. with homeotropic anchoring on one surface and planar anchoring on the other one hybrid-aligned nematics).The critical voltage and the critical wave number obtained with the one-elastic-constant approximation are in a good agreement with experimental re-sults. ... 

Fig. 7.8. Even homeotropic boundary conditions allow for a standing helix structure. Thus, homeotropic anchoring conditions will not ensure a long-term stable lying helix structure for the flexoelectro-optic effect. ...

To avoid the influence of surface interactions on the measurement of the helical pitch length, two alterations of the experiment are possible. The first one is to enlarge the thickness of the sample significantly. The second is to measure the helical pitch length under homeotropic boundary conditions which are known to not afifect its value [33]. Both methods, i.e. the direct method cf. Sect. 4.5.1) with samples of 250 pm thickness and the Cano method cf. Sect. 4.5.2), were applied to determine the influence of the solvent concentration on the helical pitch length. 

Santamato et ai assumed that with homeotropic boundary conditions the normal component of the optical torque cannot be balanced by an elastic torque, hence the molecules are set into rotation. is balanced by a viscous torque arising from the precession of the director. They established a simple relation between the angular velocity of the director rotation and the total change of the Stokes parameter, S3. This relation was experimentally verified. 

We have considered several examples which illustrate the diversity of the interesting effects associated with lightwave-liquid-crystal interaction in cells with a nonuniform initial director distribution. Many such cells can be constructed. We have already seen that their qualitative properties depend not only on the boundary conditions but also on the specific physical properties of the LC material. This fact should be stressed, since in cells with a uniform director orientation the differences in the Franck constants, say, from one LC to another lead merely to quantitative differences. In many cases, even the description of the equilibrium structure for nonuniform cells encounters serious mathematical difficulties. Conservation laws may provide a powerful technique for solving problems of this type. For instance, a theorem of E. Noether was used in Ref 23 to derive analytic expressions for the equilibrium structure of complex configurations such as homeotropic-planar oriented cholesterics and cholesterics in magnetic fields with a homeotropic orientation at the walls. 

The appropriate boundary conditions for Eq. (S) are 6 z = d/2)=Q and y,(z = d/2)=0, assuming strong anchoring of the homeotropically aligned molecules at the windows. If the optical-fleld-induced molecular reorientation is much weaker than the dc magnetic-fleld-induced reorientation, 6 and Vx in Eq. (5) can be written approximately in the form... 

The electrooptical behavior of quasi-homeotropic oppositely pretilted nematic layers has been investigated [11]. In these layers, due to the boundary conditions... 

Theoretical investigations of the electrooptical phenomena in nonuniform fields were performed [157], for different boundary conditions (homogeneous, homeotropic, and twisted) and different types of spatially nonuniform field. It was shown that the sensitivity and spatial resolution of a liquid... 

The role of the boundary conditions was discussed theoretically in [37, 38]. When the helical axis is parallel to the limiting walls and the field E h, the threshold for the helix untwisting decreases when following from the planar to the homeotropic anchoring. For cell thickness a < y/2Pofn,... 

In a continuous description, anchoring terms introduce boundary conditions at interfaces. On isotropic substrates like oxidized wafers, water, or glycerol and for flat films (without thickness gradient), the only condition is on the preferred orientation with respect to the surface normal, that is, the polar angle 0. In the cases considered here, the preferred orientation at the free interface is along the normal (9 = 0, homeotropic anchoring). The preferred... 

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