Saupe tensor

It proves convenient to represent the Saupe tensor, S, in terms of just its five independent elements. These five independent tensorial elements are written as a vector, s, and are related as follows to the elements of the relevant 3x3 Saupe tensor described in Eq. (5). 

The direction cosine functions, cosajj, used to describe the rigid molecular geometry can be rearranged into a tensor R1- in analogy to the Saupe tensor S... 

By increasing the chemical potential we will start to fill the box and at some point observe a gas-liquid transition either with or without isotropic-nematic transition accompanying it. The nematic order can be monitored with the help of the Mayer-Saupe tensor... 

Another method to determine the magnitude and rhombicity of the alignment tensor is based on the determination of the Saupe order matrix. The anisotropic parameter of motional averaging is represented by this order matrix, which is diagonalized by a transformation matrix that relates the principal frame, in which the order matrix is diagonal,... 

It is common to work with an expression for RDCs written in the PAS of alignment since this leads to simplification of Eq. (8), with the remaining nonzero elements (Y29( (f)> (0)) related to the diagonalized Saupe order tensor elements (Szz and (Sxx—Syy)) which describe the magnitudes of alignment. When written in its PAS, Eq. (8) becomes,... 

The residual anisotropy A describes the orientation assumed by heavy water molecules. It is expressed as a function of the Saupe order tensor and of the Euler angles describing the orientation of the D20 water molecule with respect to the local director (29) ... 

S. principal axes of the Saupe ordering tensor B - 1/2 (DOD angle)... 

The original nomenclature introduced by Saupe22 uses the Saupe-matrix S instead of the alignment tensor A, which are related by... 

If Eq. (11-3) is multiplied by uu and integrated over the unit sphere, one obtains an evolution equation for the second moment tensor S (Doi 1980 Doi and Edwards 1986). In this evolution equation, the fourth moment tensor (uuuu) appears, but no higher moments, if one uses the Maier-Saupe potential to describe the nematic interactions. Doi suggested using a closure approximation, in which (uuuu) is replaced by (uu) (uu), thereby yielding a closed-form equation for S, namely. 

In eq. (4.25), is the orientational order parameter (relative to the external magnetic field) of the vector mn, and < > indicates averaging over internal vibrations. The coefficients sfp in eq. (4.26) are Saupe orientational order tensor elements relative to the external magnetic field in a molecular-fixed frame. Consequently, it is necessary to know the tensor in order to be able to determine experimentally the direct dipolar contribution from... 

Optical anisotropies characteristic of nematogenic molecules, comprising a linear array of aromatic groups, appear to play an important role in thermotropic LCs. Anisotropies of the polarizability tensors of the interacting molecules enhance parallel molecular alignment, providing a basis for the Maier-Saupe [1959, 1960] theory of the nematic LC state. The internal energy per mole is defined as... 

S and D are the Saupe order parameters that describe the order of the molecules in the absorbing sample. s% (i = 1, 2, 3) are the diagonal elements of the transition moment tensor. They are proportional to the squares of the components of the transition moments (<(/< > i = 1,2,3)) given with respect to a molecule fixed-coordinate system (xj) joc(Circular dichroism of chiral anisotropic phases without suprastructural chirality). 

The latter is calculated as a tensor product of the spin-spin coupling tensor J l expressed in the molecular fixed coordinate frame and the Saupe orientation tensor S ... 

The chirality interaction tensor Wy is responsible for the interaction of the chiral guest, the chirality of which is described by the chirality tensor Cy, with the anisotropic host Wy = ikLkj- Ly covers the anisotropic host properties. W = Tr JT), is different from zero. The gyki are the orientational distribution coefficients of the guest in the molecular ensemble of the guest-host phase. From the two possible representations to avoid non-diagonal elements in (3.19), the representation in the system of principal axes of the order tensor is chosen instead of the principal axes of the chirality interaction tensor Wy. With Saupe s order parameters the HTP is also a sum of three terms... 

For asymmetric molecules the Hamiltonian may be written in an analogous form if the principal axes system of the g- and the hyperfine tensor coincide with the principal axes system of the Saupe order matrix S [65]. 

The Maier-Saupe theory assmnes high symmetry for molecules forming liquid crystals. In reahty, this is usually not the case and the theory has been extended [3.18] to lath-like molecules. The order parameter tensor S is given by Eq. (3.8) for a biaxial molecule in a uniaxial phase. In the principal axis x y z) system of 5, only two order parameters, Szz and D = Sxx — Syy, are needed, which are related to the Wigner matrices according to Eq. (2.43) ... 

For D D , the reciprocals of these correlation times are raised by an amount [(D /D ) — 1] as indicated in Eq. (7.55). As noted previously, K rriL, ttim) values at a particular temperature are computed from (P2) and (P4). The fourth-rank order parameter P4) cannot be directly measmed from a NMR spectrmn, but may be derived from measurements of the mean square value of a second-rank quantity [7.19-7.22]. In the Raman scattering technique [7.21], the second-rank molecular quantity is the differential polarizability tensor of a localized Raman mode. In fluorescence depolarization [7.19], the average of the product of the absorption and emission tensors is used to determine (P4). Since there is a lack of experimental determination of (P4) in liquid crystals, this may be calculated based on the Maier-Saupe potential... 

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