Orientational distribution coefficients

The advantage of this, sometimes cumbersome description, is the fact that the transformation properties of the macroscopic and microscopic coordinates can be seen immediately. Because the anisotropy of most of the phases is sufficiently described by tensors of second rank for the microscopic as well as the macroscopic properties, only the orientational distribution coefficients gijki are mentioned here. As an example for the description of order by these orientational distribution coefficients, the order of a cholesteric phase will be briefly discussed. Four different order parameters are needed for a molecule of point symmetry group Di and local symmetry Z>2 for the cholesteric phase. The stars as a suffix indicate that (, 33 is given in its system of principal axes. In general, the convention... 

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... 

I Nn -> I Kk) = orientational distribution coefficients h = Planck constant I = identity matrix I = light intensity [M] = molar rotation M = molecular mass (g moH) n = refractive index, number density ... 

K. M. Beatty, K. A. Jackson. Orientation dependence of the distribution coefficient obtained from a spin-1 Ising model. J Cryst Growth 774 28,... 

We turn now to the orientational correlations which are of particular relevance for liquid crystals that is involving the orientations of the molecules with each other, with the vector joining them and with the director [17, 28]. In principal they can be characterised by a pair distribution function but in view of the large number of orientational coordinates the representation of the multi-dimensional distribution can be rather difficult. An alternative is to use distance dependent orientational correlation coefficients which are related to the coefficients in an expansion of the distribution function in an appropriate basis set [17, 28]. 

One of the primary features of the Gay-Berne potential is the presence of anisotropic attractive forces which should allow the observation of thermally driven phase transitions and this has proved to be the case. Thus using the parametrisation proposed by Gay and Berne, Adams et al. [9] showed that GB(3.0, 5.0, 2, 1) exhibits both nematic and isotropic phases on varying the temperature at constant density. This was chosen to be close to the transitional density for hard ellipsoids with the same ellipticity indeed it is generally the case that to observe a nematic-isotropic transition for Gay-Berne mesogens the density should be set in this way. The long range orientational order of the phase was established from the non-zero values of the orientational correlation coefficient, G2(r), at large separations and the translational disorder was apparent from the radial distribution function. 

Polarized Raman and fluorescence spectroscopies, NMR and X-ray diffraction allow the determination of at least (P2) and (P4) for uniaxial systems. This is a great advantage since the shape of the orientation distribution can then be estimated [7], even if not all the coefficients of the ODF s expansion are known. While P2 has fixed boundary limits, those of (P4) depend on the (P2) value such as... 

Each coefficient of the multipole expansion is computed by a numerical integration - after aligning and normalizing the found orientation distribution. 

Note 5 Even for molecules with cylindrical symmetry, does not provide a complete description of the orientational order. Such a description requires the singlet orientational distribution function which can be represented as an expansion in a basis of Legendre polynomials with L an even integer. The expansion coefficients are proportional to... 

Here, 7 is the magnitude of the strain rate tensor and C/ is a phenomenological coefficient which models the interactions between the fibers, usually referred to as the Folgar-Tucker interaction coefficient. The coefficient varies between 0, for a fiber without interaction with its neighbors, and 1, for a closely packed bed of fibers. For a fiber reinforced polyester resin mat with 20-50% volume fiber content, CV is usually between 0.03 and 0.06. When eqn. (8.153) is substituted into eqn. (8.152), the transient governing equation for fiber orientation distribution with fiber interaction built-in, becomes... 

Write a short explicit finite difference program to compute the fiber orientation function in a compression molding process with stretching only in the x-direction. Test the program with an example, where a 3 mm thick 50 x 50 cm plate is compression molded with an initial mold coverage of 50%. Assume an interaction coefficient C/ = 0.05. Assume initial fiber orientation distribution that is random. 

One approach to a solution of this problem was put forward by Hansen (14), who derived general equations which express the overall experimental film absorbance in terms of the external reflectance of the substrate. These relations contain within them expressions for the individual anisotropic extinction coefficients in each geometric orientation. Solution of these general equations for the anisotropic extinction coefficients allows for an unambiguous description of the dipole orientation distribution when combined with a defined orientation model. 

The optical apparatus used in this work was described in section 8.6 and has the capability of providing both Raman scattering and birefringence measurements simultaneously. The Fourier expansion of the overall Raman scattering signal is given by equation (8.51), and the coefficients are given by equations (8.52) to (8.54). In these expression, a simple, uniaxial form for the Raman tensor was assumed. From these coefficients, the anisotropies in the second and fourth moments of the orientation distribution can be solved as... 

For objects more complicated than spheres (e.g., rods or flexible chains), exclusion can be triggered not only by proximity to a section of pore wall (as above) but also by any inappropriate orientations and conformations that lead to overlap. This is illustrated by Figure 2.3. The distribution coefficient K for such complex bodies can no longer be considered as a simple volume ratio as expressed in Eq. 2.42. Instead, K becomes a ratio of volumes in multidimensional configuration space in which all possible positions, orientations, and conformations must be considered [27]. 

For the determination of the orientation distribution function it is necessary to record diffraction patterns successively by rotating the sample on a goniometer, as was shown in Section 12.1.4.2. The patterns must be measured in a large number of points (, y) scattered more or less uniformly on a hemisphere. It is difficult to evaluate beforehand how many such points are necessary for a reliable determination of the ODE. For a calcite sample previously used in a texture round robin Von Dreele recorded neutron time of flight diffraction patterns in about 50 points (T, y). All patterns were processed by GSAS simultaneously and six pole distributions calculated from the refined harmonic coefficients were further used as input in the WIMV inversion routine. An ODF similar to those obtained in the texture round robin resulted, but its dependence on the number of points in the space (T, y) was not examined. 

From the knowledge of the order parameter from the measurement of the variation of the optical absorption spectrum due to poling, one can estimate the x parameter (Eq. (43)) intervening in the above developments. For poled polymers one can use also another alternative description of linear and nonlinear optical susceptibilities by expanding the orientation distribution function in the series of Legendre polynomials, where the expansion coefficients are order parameters ... 

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