Isotropic series, defined

A series of lead silicate glasses and their crystalline equivalents has revealed a linear relationship between the number of Pb atoms in the second coordination sphere of the structural Si tetrahedra (p) and the isotropic Si chemical shift (Bessada et al. 1994), defined by ... 

Figure 8.31. A. Relationship between the " i nuclear quadrupole coupling constants of ATiOs-type titanates and their structural shear strain i(( defined in equation (8.3). B. Relationship between the Ti isotropic chemical shifts (ppm) and the mean Ti-O bond lengths (A) for a series of ATi03 compounds with the perovskite structure. Note that compounds with the ilmenite stmcture do not fit this relationship and are not included here. From Padro et al. (2000), by permission of the copyright owner.

Listed are the molecular model, the type of phases in coexistence, the plane in which the integration was conducted, the known coexistence datum used to initiate the series, and the literature citation. Abbreviations for the various phases in the Coexistence Phenomenon column are as follows vapor (V), liquid (L), solid (S), fluid (F), isotropic (I), nematic (N), rotator (R) and smectic (Sm). Abbreviations for the Integration Plane column are as defined in the text. 

Starting from the isotropic phase, where the molecules have all three degrees of freedom, cooling will increase the density and rotation about the long axis becomes restricted. Series of models have been developed that consider the density of liquid in terms of the restriction of the order.These theories identify a critical density at which the isotropic to nematic transition would be predicted. Constraint of the molecule in terms of its rotation about the long axis defines the nematic phase. If now the translational freedom is restricted and layered alignment is imposed on the molecules, then smectic order is created. The smectic phase can still retain disorder in rotational freedom about the short axis. Loss of this final degree of freedom will lead to the creation of a erystalline ordered structure. This simple approach provides a description for the isotropic nematic smectic crystalline transitions. 

Consider the polymer to be comprised of N identical, transversely isotropic but randomly oriented cubes. For the case of uniform stress (Figure 7.14(a)) the cubes are arranged end-to-end, forming a series model in which the direction of elastic S3mmetry is defined by the angle d that the cube axis makes with the direction of external stress a. The strain in a single cube e is given by the compliance formula... 

Let us now discuss more specifically experiments on extended nematic films, either spin-coated on silicon wafers or deposited on water or glycerol. Usually, a solution of the liquid cr5 stal in hexane is deposited, allowing a fast spreading on the substrate, followed by a rapid evaporation of hexane. The first experiments with 5CB on wafers reported before have been completed by a systematic investigation of the series of n-cyanobiphenyls on water and glycerol in the nematic range. The aim of the study is to define a common scheme of behavior for the nCB series on the three isotropic substrates, as a base for the theoretical analysis. Specific deviations from the common scheme must also be understood. 

The n term in the series represents an n order interaction with the light field. The polarization, here defined as Tr[/6/t], is the measurable quantity in an optical experiment. Each term in the series has a specific meaning. The first term, usually referred to as the susceptibility, gives rise to absorption. The second term, which averages to zero in an isotropic medium, gives rise to second-order processes the third term is the relevant term for virtually all nonlinear optical experiments, also called experiments for the name of the susceptibility matrix. Higher order terms are rarely considered in view of the complicated experimental configuration needed, and the equally involved mathematical analysis needed to separate it from lower order processes. Some experiments such as Stark fluorescence, reported for a few tautomeric compounds, can be classified as measurements. 

In order to transform the interaction vector in the laboratory coordinate frame defined by the stationary magnetic field to the final coordinate frame F, defined by the jump axis, which is diagonal with the DD vector or aligned with the principal axis system of the CSA tensor, it may be useful to express the overall transformation as a series of transformations through intermediate coordinate frames N. This will enable the amplitude and frequency of each motion to be explicitly included in the correlation function. In the case of overall isotropic motion with diffusion constant D, the dipolar correlation function can be expressed as (Keepers and James, 1982)... 

---