Superimposed rotations model

The deuteron spin-lattice relaxation rates R, where j labels the position of the deuteron in the alkyl chains, were found to decrease monotonically along the chains of mesogens [8.7, 8.15-8.17]. The spin-lattice relaxation rates in liquid crystals were also found to follow the same trend [8.18, 8.19]. These observations may be accounted for by a model that considers the contributions made to the rotations within the molecule and reorientation of the whole molecule. Beckmann et al. [8.7] found that a model of superimposed rotations is consistent with a monotonic decrease of relaxation rates along the pentyl chain of 5CB-di5 (Fig. 8.2). 5CB will be used as a model liquid crystal. 

FIGURE 8.2. Sketch of 5CB, the location of the molecular frame and internal rotation axes. Ym points out of the page. 

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The small step rotational diffusion model has been extensively applied to interpret ESR linewidth [7.4, 7.9], dielectric relaxation [7.2], fluorescence depolarization [7.19], infrared and Raman band shapes [7.24], as well as NMR relaxation in liquid crystals [7.14, 7.25]. When dealing with internal rotations in flexible mesogens, they are often assumed to be uncoupled from reorientation to give the so-called superimposed rotations model. Either the strong collision model or the small step rotational diffusion model may be used to describe [7.26, 7.27] molecular reorientation. 

The second term in the above expression represents a cross-term between the two types of motion, but is zero except when rriL = 0. Unless it is necessary to calculate Jo (a ), or the spin-spin relaxation time, the overall correlation functions will be approximated by linear combinations of the products of the correlation functions for each motion [i.e., retain only the first term in Eq. (8.10)]. To discuss the superimposed rotations model, it is assumed that internal rotations about different C-C bonds are independent and use additional coordinate frames to carry out successive transformations from the local a frame to the molecule-fixed frame. Free rotational diffusion will be used to describe each bond rotation in the following section. 

It has been clear for some years that internal motions are important in determining nuclear spin relaxation in liquid crystals. This mechanism has been neglected in the early theories [8.27-8.29] proposed to explain nuclear spin relaxation in ordered mesophases. Recently, large quantities of experimental data were collected [8.7, 8.11, 8.15, 8.30, 8.31] in liquid crystal phases to give spectral densities for the different nuclear sites in the molecules. This led to the development of the theoretical models described in previous sections, which aim at treating internal motions in ordered mesophases. Thus far, only several liquid crystals has been studied in detail to examine the relaxation effects of internal motions. Beckmann et al. [8.7] were first to apply the superimposed rotations model (Section 8.21) to account for the site dependence of the deuteron spin-lattice relaxation rates at 30.7 MHz in the pentyl chain of 5CB. The spectral densities were calculated in the fast motion limit to give equations like Eq. (8.15). Since only the spin-lattice relaxation rate see Eq. (5.40)]... 

The superimposed free rotations model has also been used to interpret the spectral densities of the chain deuterons and the linkage methine deuteron measured at 15.3 MHz in the nematic liquid crystal MBBA-dia [8.30]. The overall motion is discussed in terms of the small step rotational diflFusion model of Nordio. The correlation times mi,2... 

In order to obtain proper results with these formulas, it should be remembered that they are projections and must be treated differently from the models in testing for superimposability. Every plane is superimposable on its mirror image hence, with these formulas there must be added the restriction that they may not be taken out of the plane of the blackboard or paper. Also, they may not be rotated 90°, though 180° rotation is permissible ... 

This velocity profile is commonly called drag flow. It is used to model the flow of lubricant between sliding metal surfaces or the flow of polymer in extruders. A pressure-driven flow—typically in the opposite direction—is sometimes superimposed on the drag flow, but we will avoid this complication. Equation (8.51) also represents a limiting case of Couette flow (which is flow between coaxial cylinders, one of which is rotating) when the gap width is small. Equation (8.38) continues to govern convective diffusion in the flat-plate geometry, but the boundary conditions are different. The zero-flux condition applies at both walls, but there is no line of symmetry. Calculations must be made over the entire channel width and not just the half-width. 

This simple relaxation theory becomes invalid, however, if motional anisotropy, or internal motions, or both, are involved. Then, the rotational correlation-time in Eq. 30 is an effective correlation-time, containing contributions from reorientation about the principal axes of the rotational-diffusion tensor. In order to separate these contributions, a physical model to describe the manner by which a molecule tumbles is required. Complete expressions for intramolecular, dipolar relaxation-rates for the three classes of spherical, axially symmetric, and asymmetric top molecules have been evaluated by Werbelow and Grant, in order to incorporate into the relaxation theory the appropriate rotational-diffusion model developed by Woess-ner. Methyl internal motion has been treated in a few instances, by using the equations of Woessner and coworkers to describe internal rotation superimposed on the overall, molecular tumbling. Nevertheless, if motional anisotropy is present, it is wiser not to attempt a quantitative determination of interproton distances from measured, proton relaxation-rates, although semiquantitative conclusions are probably justified by neglecting motional anisotropy, as will be seen in the following Section. 

The literature offers numerous calculation models for mass transfer in single liquid particles. However, they provide only a rough approximation to reality in industrial columns, since the processes in droplet swarms are much more complicated, especially when pulsing and rotating motion are superimposed. For estimation, the following relationships are sufficient ... 

Figure 1. Theoretical line profiles of the final model with 8=0.5 superimposed on the observations. The profiles are rotationally broadend by 450 km/s... 

FIGURE 1.16 Isomers of cyclopropane-1,2-dicarboxylic acid, (a) is-l,2-Dicarboxycyclopropane (trans isomers) (b) Z-l,2-Dicarboxycyclopropane isomer, (meso isomer) (c) Molecular models of E- (trans-) 1,2-dimethylcyclo-propane shown in the tube representation. Rotation of the right hand structure about a vertical axis through the center of the cyclopropane will superimpose the two methyl groups. The methylene of the rotated structure will be in the back, rather than the front, and not superimposed, (d) A mirror plane through the methylene and the back carbon-carbon bond is a plane of symmetry. The two carboxyl groups appear not to reflect each other in the model shown but they can rotate freely and will reflect each other on an instantaneous basis. 

Fig. 32 Enantiospecific recognition between the nucleic acid base adenine and the amino acid phenylglycine on Cu(110) [102]. a STM image (15nmxl0nm) of coadsorbed S-phenylglycine and alanine. Only the CW-rotated adenine double rows, marked by arrows, are decorated with S-phenylglycine double rows, b Structure model superimposed on the STM image of adenine/S-phenylglycine. The carboxylate oxygens and the N atom have been placed atop Cu atoms. Reprinted with permission of the authors...

Barker and Grimson (1991) modeled the flow of deformable particles after a free-draining floe whose shape, orientation, and internal structure ranged between the extremes of an extended chain and a folded globule. They interpreted the unhindered motions of free-flowing, deformable droplets to result from an unbalanced force imposed by the flow field, resulting in rotations around the particles center of mass this rotation is superimposed on the steady translational motion. 

Build models of the compounds represented by these Fischer projections. Determine whether the models superimpose. (Note that these Fischer projections are related by a 90° rotation in the plane of the page.)... 

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