Diffusional rotation

Here, tr and tj are the correlation times for the diffusional rotation and the isotropic random motion, respectively. 0r is the angle between the C—H internuclear vector and the z axis. 

In contrast, Howarth [12] derived J,(w) for the 3t model corresponding to p = 3, where three independent motions are assumed to be superposed for the overall motion of the C—H vector as shown in Fig. 3.6. Namely, the C—H vector undergoes diffusional rotation about the Z axis in the Oi frame, whereas the zi axis librates within a cone whose axis is parallel to the Z2 axis in the O2 frame. Moreover, the Z2 axis undergoes the isotropic random reorientation in the laboratory frame. Although an empirical approximation was made in the previous calculation, we obtained the following equations by the exact mathematical derivation [8-10] ... 

Here, tr, tj and ti are the correlation times for the diffusional rotation, libration and isotropic motion, respectively. and are halves of the vertical angles of the cones associated with the corresponding motions. The librational motion is defined here as time-dependent reorientation in which the zi axis randomly changes in direction within the larger cone with the Z2 axis as shown in Fig. 3.6. 

Several types of dispersions show strain rate thinning, and a quantitative explanation is not easily given. We will briefly consider two cases. The first one concerns shear flow. As discussed (above, Factor 4), anisometric particles show rotational diffusion and thereby increase viscosity. This effect will be smaller for a higher shear rate when the shear-induced rotation is much faster than the diffusional rotation, the latter will have no effect anymore. The shear rate thinning effect is completely reversible. Something comparable happens in polymer solutions (Section 6.2.2). 

If the mode of the time-fluctuation of the C-H vector under consideration is designated, the spectral densities appearing in eqns (33) and (34) can be formulated. For example, if the C-H vector r undergoes random diffusional rotation while the internuclear length is held constant, i.e. if the vector r undergoes rotational diffusional motions around the x, y, and z-axes with a common rotational diffusion constant R, all of the correlation functions of the orientation functions of r can be described by a correlation time Tc as... 

These relations are together generally referred to as single correlation time theory and used to correlate the relaxation phenomena for monomeric substances in solution to their molecular motion. Nevertheless, in the case of macromolecules, the C-H vectors in the molecular structure are not thought to undergo such isotropic spherical diffusional rotation. In fact, the relaxation phenomena of macromolecules seldom follow these relations and particular modes of motion must be assumed for the internuclear vectors considering the detailed molecular structure. 

Many models for the motional mode of the internuclear vector (C-H vector) to describe the magnetic relaxation of macromolecules have been proposed. Examples are the ellipsoidal or spherical rotational models where an ellipsoidal or spherical molecule undergoes independent diffusional rotations around the long and short axes and the C-H vectors are embedded in the ellipsoidal or spherical molecule.Here consider one model including three independent motions as a pertinent model for long-chain molecules (referred to as the 3t model). In this model, schematically depicted in Fig. 6, the... 

The time-constant for diffusional rotation of many organic molecules in fluid solvents is also in the region of 10- to 10- ° s. 

The analysis of the observed concentration dependence of relaxation times Tjp(c) of the ion-pair relaxation region, reveals two superimposed modes, a diffusional rotation mode of the ion-pair dipoles and a kinetic mode due to the re-establishment of the undisturbed equilibrium = Cd2+, M +)... 

The rotational mode yielding the relaxation time may be compared to the relaxation time for diffusional rotation as given by the Stokes-Einstein-Debye relation, Eq. (21), where V is the volume of the prolate ellipsoid representing the ion-pair, q and f having their usual meaning. The kinetic mode results from a normal mode analysis35 5 ... 

If a CDj group of an organic molecule undergoes fast diffusional rotation about the C-CD bond with an angle a = 109.47° between the rotation axis Cj and the direction... 

Consider the example of motions of phenyl rings [53]. Two different motional models are possible. Reorientation about the chain ring C -Cv bond axis may occur either by 180° flips 0umps between two indistinguishable conformations) or by fast-diffusional rotational motion with small angular displacements. The principal axis system is shown in Fig. 8.12. 

The effects of fast-diffusional rotation have been calculated by transforming the tensor from the static system to a rotating system. For comparison, the motional model that considers rapid jumps between two equivalent sites is calculated by using rotation matrices to construct an average flipping-frame tensor from the static one. 

These experimental findings on the rotational dynamics of anthraquinone dyes in liquid hosts are also interesting as they show deviations from hydro-dynamic theories usually used for describing the rotational diffusion of dye molecules in liquid solvents [6-12]. These theories are commonly gathered in the Uterature under the name of Stokes-Einstein-Debye theory (SED). SED models treat the solvent as a macroscopic continuum, in which the diffusional rotation of the solute is only affected by the viscosity and temperature of the hosting solvent. However, as expected, the validity of this continuum description breaks down when the size of the solute molecules approaches that of the solvent molecules or becomes smaller. In this regime that includes the important case of pure materials, the specific intermolecular interactions between the solute and the solvent molecules start to play a fundamental role in their rotational dynamics [9,11-16]. 

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