Planar Rotation

By virtue of the assumptions of unidirectional electrostatic fields and planar mechanical stress, the electromechanically coupled constitutive relations have been modified significantly. In example, the formulation on the left-hand side of Eq. (4.10a) reduces to Eq. (4.28) or (4.30). A transformation of coordinates on the considered plane may be performed as a rotation around the axis normal to this plane. In the case of Eq. (4.30), the base vector C2 represents the axis of rotation and thus this planar rotation may be formulated as follows  

The transformation matrix T can be adapted from the spatial case of Eq. (4.14) by removing the unnecessary rows and columns. To account for the modified arrangement of components in Eq. (4.30), the respective rows and columns also need to be exchanged in the transformation matrix it takes the following form  

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Measurements of Pe in fixed-pH solutions but at various different stirring speeds need to be made. The double-reciprocal analysis, HPe versus 1/v , for Caco-2 permeability measurements in the Transwell (Corning Costar) system produced a linear plot for x- 0.8 [514]. The intercept yields the membrane permeability for the particular pH value in the study the slope determines the k constant. From the analysis of testosterone transport, for the stirring speed of 25 rpm (planar rotating shaker), the thickness of each UWL (assuming symmetric geometry) was calculated to be 465 pm at 150 rpm, haq= 110 pm [514], Karlsson and Artursson [512] found x = 1.0 to best represent their stirring-based analysis of the UWL permeability. 

Figure 8 Appearance kinetics of radiolabeled solutes that diffuse across Caco-2 cell monolayers via the paracellular pathway. The Transwell system consisted of a donor and receiver solution at pH 7.4. Stirring by planar rotation up to 100 rpm had no effect. The insert with filter, cell monolayer, and donor were transferred to a new receiver chamber at time intervals to maintain sink conditions.

Table 12 Effective Permeability Coefficients and Thicknesses of the Aqueous Boundary Layer of the Caco-2 Cell Monolayer/ Transwell System as a Function of Stirring by Planar Rotating Shaker3...

In both the syn and the anti isomers of benzaldoxime, the entire molecule was found to be planar. Rotating the aromatic ring ont of the plane of the other atoms is energetically unfavorable. The compnted energy difference between the isomers is again quite small, but the syn is favored, the syn anti equilibrium constant being 1.7 x 10 . ... 

Stretching vibration (sym.) Stretching vibration (asym.) Planar rotation (rocking)... 

The planar rotation of the Green s vector results in the additivity of the angles between all consecutive vectors g, therefore the sum of these angles is equal to ,... 

This corresponds to an increase in room temperature half-hfe for interconversion of 10 s to 4 x 10 s nevertheless, the principal path for interconversion is still a planar rotation. ... 

Another feature that is crucial in considering rearrangements in monosubstituted allyls is the effect on the chirahty and stereochemistry. In crotyl complexes, formation of a a-bond at the unsubstituted terminus provides a path for racemization for the stereogenic center at the substituted terminus (equation 21). Formation of the a-bond at the monosubstituted terminus, however, results in conversion to a different isomer (equation 22). The most stable isomer is the syn isomer (72) and, in the absence of a substituent on the central carbon, the anti isomer (74) will only occur to the extent of f 5Vo. Thus if one considers complexes hke (acac)Pd(allyl), some racemize, whereas others only isomerize because there is no path for racemization (equation 23). These concepts have been used effectively by Bosnich in the design of systems for asymmetric allylic alkylation. These concepts also allow the rationalization of why certain substrates give low enantiomeric yields. It should be noted here that the planar rotation found in some of the molybdenum complexes retains the chirahty in the allyl moiety. 

Under proper mass-scaling the three possible Jacobi vectors sets can be related to each other by a planar rotation by an angle which depends only on the masses of A,... 

For o=l, Eq. (116) yields the Green function for normal rotational diffusion of a planar rotator, namely,... 

So far we have considered the planar rotator model. However, the above equations can readily be generalized to rotation in space. Here, the space coordinate is the polar angle i) and the Fokker-Planck operator for normal rotational diffusion assumes the form [8] [cf. Eq. (80)]... 

We now recall that the classical planar rotator model may be used as a model of superfluid He4, 0 being the phase of the condensate wave function, S being related to the superfluid density ps as S = ps(hjm)2, m being the mass of a He4 atom. Thus one can have superfluid-normal fluid transition in d = 2 dimensions, despite the lack of conventional long range order This conclusion seems to be corroborated by experiments on He4 films (Bishop and Reppy, 1978). 

Fig. 24. Herringbone (a) and pinwiiccl (b) orientational ordering of uniaxial diatomic molecules on a triangular lattice. The heavy bars represent planar rotators and the circles denote vacancies, p is the degeneracy of the ordering. From Mourilsen (1985).

There exists many generalizations and variants of the isotropic planar rotator model, eq. (159). Here we only mention the anisotropic planar rotor mode (Mouritsen and Berlinsky, 1982 Harris et al., 1984)... 

The temperature dependence of the Pake pattern can be used to deduce that the bound dihydrogen ligand undergoes a torsional or hindered rotation motion around an axis perpendicular to the metal-dihydrogen axis. The bound hydrogen is characterized as a rigid planar rotator. In some cases, the potential surface for this rotation can be characterized by these measurements. 

In section 3.2 we have seen how rigid rotations, meanders, and drifts are closely tied to equivariance with respect to the Euclidean group SE 2) of planar rotations and translations. We therefore prefer to not describe our curvature flows in terms of the position vector Z t,s) e C, directly. Instead, we work with the curvature scalar... 

It is suggested that for 9 e [(-/ , Pf] the dependence U(9) is analogous. The smaller the/, the narrower the calculated absorption loss spectrum. Substituting Eq. (214) into the Hamiltonian for planar rotation, we have29... 

The EBC is thus defined by specifying N -1 rotation angles, 2, 3, , Sf, describing successive planar rotations in the space of qj and each of the other bath coordinates. Writing U (0) = exp[-Si, (0)], the overall transformation operator is... 

F 7. Dependence of calculated NMR powder spectra on type and timescale of various motions a) Two-site jumps, 6k = 60°, b) two-site jumps, 6k = 109°, c) tteee-site jumps, = 109°, d) planar rotational (fusion, 6k = 109°, e) tetrahedral jumps, I) isotropic spherical diffusion. 6r = angle between rotation axis and C-D bond direction... 

The sensitivity of typical NMR experiments to the type and time scale of various motions is illustrated in Fig. 7. For simplicity, we have chosen equal correlation times for all motions, namely = 1 x 10 s, = 1 x 10 s and Xg = 5 x 10 s. The powder spectra refer to quadrupole echo sequences [57], and characterize two-site jumps (a, b), three-site jumps (c), planar rotational diffusion (d), tetrahedral jumps (e) and isotropic spherical diffusion (f), respectively. The significant differences of the lineshapes arise from the different motional anisotropies. Evidently, quadrupole echo spectra [57] contain valuable information on the type of motion [10,48,49, 58]. 

Some of the earliest DNMR observed for organometallics involved the planar rotation of olefins. This process effectively involves a rotation about an axis from the metal to the midpoint of an -alkene and is sometimes called a propeller rotation. In unsymmetrical metal complexes, there is the possibility of several conformers based on orientations of the olefin. This can generally be attributed to differing donor properties of non-equivalent other ligands. The faces of many substituted olefins are enantiotopic, and binding to the metal produces... 

Each of these is, it turns out, easily integrated as a planar rotation (see Exercise 7). This yields a simple, efficient algorithm for the free rigid body which is symplectic in a generalized sense. 

Fig.6 shows an example for the fragmentation of an ABC molecule, in which a force located in the ABC-plane leads to the excitation of planar rotational motion in the AB product. Planar forces are expected for example from antibonding orbitals that are symmetric to the nuclear plane. 

Figure 6 Planar rotational excitation of a triatomic molecule ABC...

Another feature, that is typical for the case of planar rotational excitation, is the relative orientation of the velocity vector v to the rotational angular momentum vector N. In contrast to the case of torsional excitation, v is perpendicular to N (compare figure 8). 

The obvious difference from the case of planar fragmentation is the different direction of the relative velocity vector with respect to the rotational vector N. For the case of planar rotational excitation, V is perpendicular to N, whereas in the case of torsional motion v is parallel to N. With modern experimental techniques it is possible to distinguish between these two cases (compare section 2.3) ... 

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