Motional ellipsoid

Fig. 2. ORTEP drawing of the (CH3)2NPP2 molecule with 50% thermal motion ellipsoids for the nonhydrogen atoms. Bond lengths (in Angstroms) and angles (in degrees) are given with standard deviations expressed in units of the last significant figure 212).

Fig. 18. DHDK (5) 0.4 CHCI3 stereodiagram of. the crystal structure. The thermal motion ellipsoids represent 40 % probability distributions (taken from Ref.

The assumption that the motional ellipsoid is cixlally symmetric causes m = m" and simplifies equation (12) to... 

Here 0 is the angle between the unique principal axis of the spin interaction emd the symmet axis of the motional ellipsoid. 

Woessner D E 1962 Nuclear spin relaxation in ellipsoids undergoing rotational Brownian motion J. Chem. Rhys. 37 647-54... 

In the last section we noted that Simha and others have derived theoretical expressions for q pl(p for rigid ellipsoids of revolution. Solving the equation of motion for this case is even more involved than for spherical particles, so we simply present the final result. Several comments are necessary to appreciate these results ... 

The viscosity of a suspension of ellipsoids depends on the orientation of the particle with respect to the flow streamlines. The ellipsoidal particle causes more disruption of the flow when it is perpendicular to the streamlines than when it is aligned with them the viscosity in the former case is greater than in the latter. For small particles the randomizing effect of Brownian motion is assumed to override any tendency to assume a preferred orientation in the flow. 

Unit cell of /3-cristobalite. Left snapshot the numbers indicate the height of the atoms in the direction of view as multiples of. Right with ellipsoids of thermal motion at 300 °C... 

Figure 4. A perspective drawing of C.jH NjSi.B, with nonhydrogen atoms represented by thermal vibration ellipsoids drawn to encompass 50% of their electron density hydrogen atoms are represented by arbitrarily small spheres which are in no way representative of their true thermal motion.

A modified version of the TAB model, called dynamic drop breakup (DDB) model, has been used by Ibrahim et aU556l to study droplet distortion and breakup. The DDB model is based on the dynamics of the motion of the center of a half-drop mass. In the DDB model, a liquid droplet is assumed to be deformed by extensional flow from an initial spherical shape to an oblate spheroid of an ellipsoidal cross section. Mass conservation constraints are enforced as the droplet distorts. The model predictions agree well with the experimental results of Krzeczkowski. 311 ... 

Returning to Staudinger s derivation, it must be revised on two grounds. First the kinematics of motion is three rather than two-dimensional and the hydrodynamic volume spherical rather than cylindrical, i.e. 3. The detailed calculation for thin ellipsoidal particles (13) shows an approximate proportionality of the intrinsic viscosity with M1 7, a considerable difference from eq. (1) for large M. 

Molecular motion in solids has been the object of many studies in the field of physical chemistry of polymers , but dynamic processes in molecular crystals of organic and inorganic compounds are less well investigated. In fact, the average chemist is not aware of the fact that processes like internal rotation or ring inversion proceed in solids quite often with barriers which are not very different from those found for these types of internal motion in the liquid state. Thus, for the equatorial axial ring inversion of fluorocyclohexane values of 42.4 and 43.9 kJ mol have been measured in the liquid and the solid, respectively. The familiar thermal ellipsoids of individual atoms obtained from X-ray studies are qualitative indicators of molecular motion in the crystal, but a more quantitative study of such processes is only possible after appropriate solid state NMR techniques are applied. 

Moreover, ellipsoidal bubbles and drops commonly undergo periodic dilations or random wobbling motions which make characterization of shape particularly difficult. Chapter 7 is devoted to this regime. 

The generalized graphical correlation presented in Fig. 2.5 gives one method of estimating terminal velocities of drops and bubbles in infinite liquid media. For more accurate predictions, it is useful to have terminal velocities correlated explicitly in terms of system variables. To obtain such a correlation is especially difficult for the ellipsoidal regime where surface-active contaminants are important and where secondary motion can be marked. 

Surface-active contaminants play an important role in damping out internal circulation in deformed bubbles and drops, as in spherical fluid particles (see Chapters 3 and 5). No systematic visualization of internal motion in ellipsoidal bubbles and drops has been reported. However, there are indications that deformations tend to decrease internal circulation velocities significantly (MI2), while shape oscillations tend to disrupt the internal circulation pattern of droplets and promote rapid mixing (R3). No secondary vortex of opposite sense to the prime internal vortex has been observed, even when the external boundary layer was found to separate (Sll). 

We noted above that either solvation or ellipticity could cause the intrinsic viscosity to exceed the Einstein value. Simha and others have derived extensions of the Einstein equation for the case of ellipsoids of revolution. As we saw in Section 1.5a, such particles are characterized by their axial ratio. If the particles are too large, they will adopt a preferred orientation in the flowing liquid. However, if they are small enough to be swept through all orientations by Brownian motion, then they will increase [17] more than a spherical particle of the same mass would. Again, this is very reminiscent of the situation shown in Figure 2.4. 

Figure 1. Stereoscopic drawings of the molecular configuration of Mo2 r C H )2 CO)4t 2-H) g2-P CH )2) showing (a) thermal ellipsoids of nuclear motion for all atoms scaled to enclosed 50% probability (b) the atom labeling. The entire molecule possesses a pseudotwofold axis passing through the bridging hydrogen and phosphorus atoms.

Figure 8. The molecular structure of the [Cr2(CO)io(M2-tf)] anion for the bis(triphenylphosphine)-iminium salt showing (a) a view normal to the Cr-Cr axis (b) a view looking down the Cr-Cr axis. The Cr-Cr intemuclear separation is 3.349(13) A. The thermal ellipsoids of nuclear motion for all atoms are scaled to enclosed 50% probability.

Figure 5. Architecture of the [Ni 12(00)21-Hl2]2 dianion in Compound 3 with 20% isotropic ellipsoids of nuclear motion as determined from the neutron diffraction refinements. The two hydrogen atoms were found to occupy the two octahedral interstices.

Figure 7. The two analogous Ni H fragments (a and b) in [P/i4P]2+-[Ni 2(CO)2 H2]2 (3), and the Ni H fragment (c) in [Ph4As]3+-[Nii2(CO)2itf]3-.Me2CO (4), showing hydrogen atoms in the octahedral interstices. The mean Ni-H distances and 20% isotropic thermal ellipsoids of nuclear motion were obtained from the neutron diffraction experiments.

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