Twisted boundary conditions

The use of twisted boundary conditions is commonplace for the solution of the band structure problem for a periodic solid, particularly for metals. In order to calculate properties of an infinite periodic solid, properties must be averaged by integrating over the first Brillouin zone. 

Note the presence of the bend-twist coupling stiffnesses in the boundary conditions as well as in the differential et uation. As with the specially orthotropic laminated plate, the simply supported edge boundary condition cannot be further distinguished by the character of the in-plane boundary conditions on u and v because the latter do not appear in any plate problem for a symmetric laminate. 

The buckling load will be determined for plates with various laminations specially orthotropic, symmetric angle-ply, antisymmetric cross-ply, and antisymmetric angle-ply. The results for the different lamination types will be compared to find the influence of bend-twist coupling and bending-extension coupling. As with the deflection problems in Section 5.3, different simply supported edge boundary conditions will be used in the several problems addressed for convenience of illustration. 

For a degenerate Fermi liquid, finite-size shell effects are much reduced if the twist angle is averaged over twist averaged boundary conditions (TABC) [30]. For any given property A the TABC is defined as... 

C. Lin, F. H. Zong, and D. M. Ceperley (2001) Twist-averaged boundary conditions in continuum quantum Monte Carlo algorithms. Phys. Rev. E 64, 016702(1-12]... 

There is a new twist in process design though. Chemical engineering processes should be optimized not only for economic but also for environmental performance. This places new constraints and boundary conditions on process modeling and computer simulation. Chemical engineers are required to have in-depth knowledge not only of process efficiency but also of the chemical pathways involving both common and uncommon raw materials. 

Textures correspond to various arrangements of defects. When the isotropic liquid is cooled, the nematic phase may appear at the deisotropization point in the form of separate small, round objects called droplets (Fig. 12). These can show extinction crosses, spiral structures, bipolar arrangements, or some other topology depending on boundary conditions. Theoretical studies based on a simple model confirm the stability of radial or bipolar orientation (Fig. 5) [22]. Considerations based on improved theoretical models yield stable twisted... 

It is significant that the fpL,max calculated above occurred for pinning by a twist boundary, where the grain boundary and H are parallel to the [111] direction of one grain and the [100] direction of the other. But these specimens contain tilt boundaries with the [111] tilt axis. The condition for the peak in these... 

According to Eq. (7.1) P is zero for the two cases of uniform director fields and pure twist. Hence both cases can serve as a zero state as far as flexoelectric excitations are concerned. It is important to note that a twist is not associated with a polarization (i.e. C2 is identically zero, cf. Fig. 7.2). An imstrained nematic has a centre of symmetry (centre of inversion). On the other hand, none of the elementary deformations - splay, twist or bend have a centre of symmetry. According to Curie s principle they could then be associated with the separation of charges analogous to the piezoeffect in solids. This is true for splay and bend but not for twist because of an additional symmetry in that case if we twist the adjacent directors in a nematic on either side of a reference point, there is always a two-fold symmetry axis along the director of the reference point. In fact, any axis perpendicular to the twist axis is such an axis. Due to this symmetry no vectorial property can exist perpendicular to the director. In other words, a twist does not lead to the separation of charges. This is the reason why twist states appear naturally in liquid crystals and are extremely common. It also means that an electric field cannot induce a twist just by itself in the bulk of a nematic. If anything it reduces the twist. A twist can only be induced in a situation where a field turns the director out of a direction that has previously been fixed by boundary conditions (which, for instance, happens in the pixels of an IPS display). 

Here, we have splay, twist and bend terms corresponding to a particular bulk distortion. In experiment, they can be realized in different way using variable cell geometry and boundary conditions. For example, such distortions may be created mechanically as shown in Fig. 8.6. The important condition is to anchor the director firmly at the boundaries. 

Fig. 10.17 Twisted structure with a rigid boundary condition at z = 0 and soft boundary condition at z = d. The geometry of the director distortion (a) and illustration of the extrapolation length b and linear dependence of the director angle 4)(z) (b)...

As the baseline for our calculations described in the Appendix, we take the typical LCD cell configuration of thickness d=8 ym filled with E7 liquid crystal mixture with a cholesteric additive giving a natural cholesteric pitch P=40 ym. The boundary conditions imposed on the liquid crystal are total twist 0q=9O , pretilt ao=3 and strong surface anchoring. The polarizers are crossed and parallel to the nematic director at the neighboring liquid crystal surfaces. In calculations where the total twist... 

Using helical boundary conditions within an objective molecular-dynamics scheme it is possible to describe chiral nanotubes with reasonable computational effort. With this approach it could be shown that chiral M0S2 and T1S2 nanotube families are intrinsically twisted. However, this effect is only small. Additionally, it was shown that also the mechanical and electronic properties depend on the chirality, if the tubes have diameters smaller than 7 nm. On the other hand, tubes with larger diameters are chirality-independent. 

As pointed out in the previous section, there are voids between the double-twist cylindas when they are packed in 3-D space. Liquid crystal must fill the void. Because of the boundary condition imposed by the cylinders, the hquid crystal director is not uniform in this space and forms a defect... 

The effects of bistability and hysteresis in supertwisted nematic layers were first investigated in [122]. To obtain twist angles larger than 90 , nematics were doped with a small amount of an optically active material. Thus a cholesteric (or chiral nematic) with a large pitch P was created, so that the pitch value had to adjust the boundary conditions for the directors on the substrates. The corresponding texture was first discovered by Grandjean and is discussed in Chapter 6. In 1984 the display based on the Supertwist Birefringent Effect (SBE) was proposed [123]. 

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... 

---