Rapini energy

In order to solve equations (10.20) and (10.21) we must know the explicit angular dependence of functions F and El. Their simplest form is the so-called Rapini energy [16] ... 

When both angles 9 and cp are changed, two Rapini energies should be introduced Azimuthal (for fixed 9 ) ... 

When both angles 0 and (/ are allowed to be changed, two Rapini energies (for the polar and azimuthal angles) are introduced... 

An essential requirement for device applications is that the orientation of the molecules at the cell boundaries be controllable. At present there are many techniques used to control liquid crystal alignment which involve either chemical or mechanical means. However the relative importance of these two is uncertain and the molecular origin of liquid crystal anchoring remains unclear. Phenomenological models invoke a surface anchoring energy which depends on the so-called surface director , fij. In the case where there exists cylindrical symmetry about a preferred direction, hp the potential is usually expressed in the form of Rapini and Popoular [48]... 

There have been several theoretical investigations analysing the nematic distortions of the LC surrounding anisotropic particles for different anchoring conditions at the LC-particle interface. In modelling the surface-anchoring free-energy density fa it is usual to adopt the Rapini-Papular expressions, which can be written as... 

Within the mean-field theory there are no spontaneous elastic deformations since any deformation increases the free energy. However, when a nematic LC is subject to interactions with the confining walls the homogeneous order can be perturbed. On the microscopic level, the molecules of the walls and of the liquid crystal attract each other via a short-range van der Waals interaction. In the macroscopic description this is modeled with a contact quadruple-quadruple interaction, known as the Rapini—Papoular model [32,33], which to the lowest order reads... 

In the Rapini approximation, the zenithal and azimuthal anchoring energies are defined for the director n in terms of angles a and p ... 

To discuss a bistability, we should leave the small — cp approximation and go back to the equation (13.29) with the Rapini surface energy added. 

The simplest surface energy is of the form first proposed by Rapini and Papoular ([14] and [23, p.289])... 

We consider a nematic cell located between the two planes located at X3 = 0 and Xs= as illustrated schematically in FIGURE 1. The easy directions at the top and bottom substrate surfaces are denoted by the unit vectors and e , respectively. Generally the Rapini-Papoular energy has been written as a linear combination of a polar angle anchoring energy... 

Using the Rapini-Papoular energy, the intofiicial effect on the bulk orientation of die director has been investigated [1] and in this way an attempt was made to detomine the values of Ae and A4. 

For a twisted nematic sample the Rapini-Papoular energy for the director orientation must be extended to the more genial form [1]... 

The expression for the sur ce energy can also be predicted for other shapes of the sur ce energy which are different from the Rapini-Papoular energy type, for example the elliptic type, Legendre expansion [14], and so oa Thwe is presently a widespread intwest in the inclusion of the surface elastic moduli Kb and K24 in the theory to allow for novel contributions to the surfrice anchoring energy to be taken into account [23] (see Datareview 5.3). 

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. 

Notice that the constraint (6.4) is equivalent to the condition aij = Uj i in Cartesian component form. Other equivalent forms for the energy in terms of any two of the vectors a, b and c are available [172] and these allow comparisons with earlier results obtained by other workers, particularly the Orsay Group [213], Rapini [228], Dahl and Lager wall [62] and Nakagawa [209]. It is worth remarking here that three surface terms have been identified for the SmC phase, namely [172],... 

This formulation, which shares some of the features of that first introduced by Rapini [228], has two illustrative advantages firstly, it can be related directly to the Orsay Group formulation and, secondly, a physical interpretation of the basic elastic constants and their related deformations can be visualised easily. This version of the energy can be constructed by simple combinations of the basic deformations, as we shall now show with the aid of Fig. 6.3. 

The contribution of the confining surface to the free energy depends strongly on the characteristics of the substrate. In the case where the surface induces a preferred orientation v of the director at the boundary, the intrinsic interactions orient the whole sample and the sample is aligned. Any deviation of the director at the surface from the induced orientation increases the free energy, and the term added to (4.3) is the Rapini-Papoular surface anchoring term [10]... 

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