Free rotational diffusion

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. 

a rigid subunit of 5CB is chosen to define the molecular frame (Xm, 1m, Zm)- This subunit should be chosen so that, to a good approximation, the reorientation of this reference axis system relative to the laboratory frame is independent of the internal motions. The rotational diffusion tensor of the whole molecule is supposed to be diagonal in this molecular frame. As a result, the small step rotational diffusion model (Section 7.2.2) may be used to account for the reorientation of the whole molecule. The internal rotation axis Zj) linking the th fragment (CjH2) and j — l)th fragment (Cj iH2) is used to define the Z axis of the jth subunit, while its Y axis is taken to be perpendicular to the Zj and Zj i axes (see Fig. 8.2). The correlation functions for the deuterons on the may be calculated 

FIGURE 8.3. Euler angles used in coordinate transformation for internal rotation. 

A similar expression for J mLoS) was obtained [8.8] using gi t) given by Eq. (8.14). Thus, for 5CB-di5, there are six internal correlation times plus D and D that need to be determined. 

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Fig. 6.13. The theoretical width of the broad line (in units of 2a) as a function of k x y. The solid curved line corresponds to pure bound translational diffusion. The dashed line corresponds to a combination of bound translational and free rotational diffusion. The straight line corresponds to a linewidth of 2k D, obtained for free diffusion. The insert shows the theoretical width of the narrow line in units of F (the natural linewidth) as a function of 2k D/r, for spheres of radius of 20 nm participating in bound translational and free rotational diffusive motions. (Ofer et ai, 1984.)...

If the diffusing bound particle is composed of many molecules, as, for example, in the case of a rigid macroscopic particle, rotational diffusion may influence the shape of the Mossbauer or neutron scattering spectra. Ofer et al. (1984) treat the case in which the particle participates in both bound translational and free rotational diffusion. The Mossbauer spectrum in this case can again be represented by a sum of Lorentzian lines. In the... 

In the general case of a sphere performing bound translational and free rotational diffusion, the spectrum consists of a narrow line, the n=0 line of Equation 6.18 broadened by rotation, and a broad line corresponding to... 

Finally, restricted 7-motion within an apex angle 0o [Eq. (7.80)] has been employed to interpret [7.11] spectral densities of aromatic deuterons in the nematic and smectic A phases of 50.7-d4. Assuming that a-, and 7-motion are completely uncorrelated, and neglecting the -motion [i.e., set / (0) = /3 t)], the spectral densities were evaluated by describing the a-motion by a simplified model of uniaxial free rotational diffusion about the director and the 7-motion by the reduced correlated functions given in Eq. (7.80). It remains to be examined whether the anisotropic viscosity model in conjunction with restricted 7-motion [i.e., Eq. (7.82)] would be better in interpreting spectral densities of motion in various smectic phases. Also, there is still no convincing NMR evidence for biased 7-motion in nematic or smectic A phases. 

As the density of a gas increases, free rotation of the molecules is gradually transformed into rotational diffusion of the molecular orientation. After unfreezing , rotational motion in molecular crystals also transforms into rotational diffusion. Although a phenomenological description of rotational diffusion with the Debye theory [1] is universal, the gas-like and solid-like mechanisms are different in essence. In a dense gas the change of molecular orientation results from a sequence of short free rotations interrupted by collisions [2], In contrast, reorientation in solids results from jumps between various directions defined by a crystal structure, and in these orientational sites libration occurs during intervals between jumps. We consider these mechanisms to be competing models of molecular rotation in liquids. The only way to discriminate between them is to compare the theory with experiment, which is mainly spectroscopic. 

The perturbation theory presented in Chapter 2 implies that orientational relaxation is slower than rotational relaxation and considers the angular displacement during a free rotation to be a small parameter. Considering J(t) as a random time-dependent perturbation, it describes the orientational relaxation as a molecular response to it. Frequent and small chaotic turns constitute the rotational diffusion which is shown to be an equivalent representation of the process. The turns may proceed via free paths or via sudden jumps from one orientation to another. The phenomenological picture of rotational diffusion is compatible with both... 

Although long-time Debye relaxation proceeds exponentially, short-time deviations are detectable which represent inertial effects (free rotation between collisions) as well as interparticle interaction during collisions. In Debye s limit the spectra have already collapsed and their Lorentzian centre has a width proportional to the rotational diffusion coefficient. In fact this result is model-independent. Only shape analysis of the far wings can discriminate between different models of molecular reorientation and explain the high-frequency pecularities of IR and FIR spectra (like Poley absorption). In the conclusion of Chapter 2 we attract the readers attention to the solution of the inverse problem which is the extraction of the angular momentum correlation function from optical spectra of liquids. 

In this case one determines the spectral intensity solely in the centre, not over the whole frequency range. Therefore the analysis often refers not to the spectrum as a whole, but to relaxation times Tg,i or and their dependence on rotational relaxation time tj [85]. This dependence contains much information and can be easier to interpret. It enables one to determine when free rotation turns into rotational diffusion. 

Its experimental confirmation provides information about the free rotation time tj. However, this is very difficult to do in the Debye case. From one side the density must be high enough to reach the perturbation theory (rotational diffusion) region where i rotational relaxation which is valid at k < 1. The two conditions are mutually contradictory. The validity condition of perturbation theory... 

The Hubbard straight line corresponds to rotational diffusion, and quasistatic straight lines (6.65) to quasi-free rotation. One type of motion substitutes for the other in the vicinity of the minimum point of curve xe,2 (tj) and is accompanied by collapse of the anisotropic scattering spectrum. 

In the absence of a correlation between the local dynamics and the overall rotational diffusion of the protein, as assumed in the model-free approach, the total correlation function that determines the 15N spin-relaxation properties (Eqs. (1-5)) can be deconvolved (Tfast, Tslow < Tc) ... 

Here we describe the model selection algorithm that is used to derive microdynamic (model-free) parameters for each NH group from 15N relaxation data. It is implemented in our program DYNAMICS [9]. Given the overall rotational diffusion tensor parameters (isotropic or anisotropic) derived as described above, this analysis is performed independently for each NH-group in order to characterize its local mobility. 

Fig. 3. Schematic representation of the topological space of hydration water in silica fine-particle cluster (45). The processes responsible for the water spin-lattice relaxation behavior are restricted rotational diffusion about an axis normal to the local surface (y process), reorientations mediated by translational displacements on the length scale of a monomer (P process), reorientations mediated by translational displacements in the length scale of the clusters (a process), and exchange with free water as a cutoff limit.

Existence of a high degree of orientational freedom is the most characteristic feature of the plastic crystalline state. We can visualize three types of rotational motions in crystals free rotation, rotational diffusion and jump reorientation. Free rotation is possible when interactions are weak, and this situation would not be applicable to plastic crystals. In classical rotational diffusion (proposed by Debye to explain dielectric relaxation in liquids), orientational motion of molecules is expected to follow a diffusion equation described by an Einstein-type relation. This type of diffusion is not known to be applicable to plastic crystals. What would be more appropriate to consider in the case of plastic crystals is collision-interrupted molecular rotation. 

Accordance with the experiment (Fig. 27) is only achieved when there is diffusive and free rotation of benzenes. The smectic polyester LC-phase is apparently characterized by... 

The fluid-mosaic model for biological membranes as envisioned by Singer and Nicolson. Integral membrane proteins are embedded in the lipid bilayer peripheral proteins are attached more loosely to protruding regions of the integral proteins. The proteins are free to diffuse laterally or to rotate about an axis perpendicular to the plane of the membrane. For further information, see S. J. Singer and G. L. Nicolson, The fluid mosaic model of the structure of cell membranes, Science 175 720, 1972. 

The adsorbate is still relatively free to diffuse on the surface and to rotate. 

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