Orientational order cylindrically symmetric

In biaxial orientation a cylindrically symmetric orientation distribution no longer exists about a reference axis. Therefore, one needs to tilt the sample in order to obtain the orientation function with respect to a second reference axis such as the normal to the plane of a film or the normal to the transverse direction. Knowing two such functions for two orthogonal axes, one can calculate the third function for the third orthogonal axis by use of the direction cosines [Eq. (4)]. 

Some of the generalized orientation factors defined by equations (15) and (16) have been independently reported by several authors. One of the second order orientation factors F200 has been proposed by Hermanns and Platzek to relate the birefringence of fibrous materials to the uniaxial orientation of cylindrical-symmetric structural units, and F q has been defined by Stein and Norris to characterize the uniaxial orientation of polyethylene crystals. F220 F 2 Iso the... 

Note 3 Molecules which constitute nematogens are not strictly cylindrically symmetric and have their orientational order given by the Saupe ordering matrix which has elements 5 aP = (3< a P> - 5ap)/2, where la and ip are the direction cosines between the director and the molecular axes a and P, 5ap is the Kronecker delta and a, p denote the molecular axes X, Y, Z. 

Molecular order is descnhedhy the orientational distribution function P(0) [Mcbl]. This is the probability density of finding a preferential direction n in the sample under an angle 0 in a molecule-fixed coordinate frame (Fig. 3.2.2(a)). For simplicity, macroscopically uniaxial samples with cylindrically symmetric molecules are considered. Then, one angle is sufficient to characterize the orientational distribution function. In practice, not the angle 0 itself but its cosine is used as the variable and for weak order the distribution function is expanded into Legendre polynomials P/(cos 0),... 

The Maier-Saupe theory of nematic liquid crystals is founded on a mean field treatment of long-range contributions to the intermolecular potential and ignores the short-range forces [88, 89]. With the assumption of a cylindrically symmetrical distribution function for the description of orientation of the molecules and a nonpolar preferred axis of orientation, an appropriate order parameter for a system of cylindrically symmetrical molecules is... 

So far we have considered the order parameter for a cylindrically symmetric liquid crystal phase formed by cylindrically symmetric molecules. If either the phase or the molecules are not cylindrically symmetric it is necessary to specify the orientational ordering using tensors. A second-rank tensor describes, to lowest order, the orientation of phases with an inversion plane of symmetry (N, SmA,...). Second-rank tensors were defined in Section... 

The first attempt to develop a theory for non-rigid mesogens was made by Marcelja who extended the Maier-Saupe theory for nematics composed of cylindrically symmetric particles to include molecular flexibility. The advent of studies of the variation of the orientational order... 

The molecules in a nematic liquid crystal tend to be parallel to a unique direction known as the director which is identical with the optic axis of the phase. The molecular orientation fluctuates with respect to the director and the extent of these fluctuations is reflected by the orientational order parameters they are defined in Section 6. As we shall see the order parameters for cylindrically symmetric particles are chosen to be unity in a perfectly ordered phase or crystal while for the disordered phase or liquid the order parameters vanish. In a nematic phase the order parameters are intermediate between these two extremes. The magnitude of the orientational fluctuations are controlled by the energy of the molecule as it changes its orientation with respect to the director. For a cylindrically symmetric molecule this energy is determined entirely by the angle between the director and the molecular symmetry axis. 

To locate the nematic-isotropic transition it is necessary to determine the temperature dependence of the free energy but this is not possible without making further approximations concerning the temperature variation of the segmental interaction parameters X. and This variation with temperature results primarily through their dependence on the orientational order of the system. A rigorous derivation of this dependence is extremely difficult and so we adopt a semi-intuitive approach. As we have seen, the orientational order in a mesophase is characterized by an infinite set of order parameters but the most important of these are the second-rank order parameters, at least close to the nematic-isotropic transition. Indeed for cylindrically symmetric particles both theory and experiment agree that the potential of mean torque is proportional to When the mesophase... 

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