Rotational diffusion motion

The non-collective motions include the rotational and translational self-diffusion of molecules as in normal liquids. Molecular reorientations under the influence of a potential of mean torque set up by the neighbours have been described by the small step rotational diffusion model.118 124 The roto-translational diffusion of molecules in uniaxial smectic phases has also been theoretically treated.125,126 This theory has only been tested by a spin relaxation study of a solute in a smectic phase.127 Translational self-diffusion (TD)29 is an intermolecular relaxation mechanism, and is important when proton is used to probe spin relaxation in LC. TD also enters indirectly in the treatment of spin relaxation by DF. Theories for TD in isotropic liquids and cubic solids128 130 have been extended to LC in the nematic (N),131 smectic A (SmA),132 and smectic B (SmB)133 phases. In addition to the overall motion of the molecule, internal bond rotations within the flexible chain(s) of a meso-genic molecule can also cause spin relaxation. The conformational transitions in the side chain are usually much faster than the rotational diffusive motion of the molecular core. 

In conclusion, a typical time of 300 fs has been found for the excited-state intramolecular double proton transfer in TAB and DAC. The proton transfer dynamics is not influenced by aggregation. In addition, a vibronic cooling time of 20 ps has been measured for the probe molecules in the molecular and stacked configurations. Finally, aggregation is found to almost completely hamper the rotational diffusion motions of the molecules during the fluorescence state lifetime of 4 ns. 

Diffusion is the random movement of a particle because of an exchange of thermal energy with its environment. Membrane lipids and proteins participate in highly anisotropic translational and rotational diffusion motion. Translational diffusion in the plane of the membrane is described by the mean square lateral displacement after a time At (r ) = TD At. Lipid lateral diffusion coefficients in fluid phase bilayers are typically in the range Dj 10 to 10 cm /s (3). 

Counterions. 1. Sodium-23 Alkali metal MIR is a sensitive >robe of the immediate chemical environment and mobility of alkali metal ions in aqueous and nonaqueous solvents (7, 8). The chemical shifts of alkali metal nuclei will respond to" electronic changes only in the immediate environment of the cation since alkali metals rarely participate in covalent bonding (7). All alkali metal nuclei have spins greater than 1/2 and hence have quadrupole moments. The interaction of these moments with electric field gradients, produced by asymmetries in the electronic environment, is modulated by translation and rotational diffusive motions in the liquid. It is via this relaxation mechanism that the resonance line width is a sensitive probe of ionic mobility. 

Nuclei with spin I greater than 1/2 possess quadrupole moments (produced by asymmetries in the electronic environment) that interact with electric held gradients at the nucleus. Fluctuations in this interaction arising from translational or rotational diffusive motions in the liquid can provide an efficient mechanism of nuclear spin relaxation. Equation 1 governs this relaxation for 23Na (18,19). 

We used a simple simulation program that was developed in our laboratory. The program simulates an FCS experiment, i.e., it emulates the behavior of an ensemble of particles diffusing in a volume irradiated by a laser beam with a defined spatial intensity profile. The particles can be either point-like or finite-sized and they perform both translational and rotational diffusive motion. The probability of displacement in time A/ is drawn from the known probability distribution ... 

Broadly speaking, the aim of an incoherent neutron scattering study is to characterize the individual translational and rotational diffusive motions of the scatterer (in this case, the proton held by an ion or molecule) at small energy transfers in the so-called quasielastic neutron scattering (QNS) region and the quantized vibrations at higher energy transfers in... 

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.)...

Since this is a stationary process, the evolution operator Fq acting on random variables is independent of time and spin variables. When used to describe dynamically averaged EPR spectra, Fq is the operator of rotational diffusion motion, and random variables correspond to Euler angles (Q = a,p,Y). 

We call the correlation time it is equal to 1/6 Dj, where Dj is the rotational diffusion coefficient. The correlation time increases with increasing molecular size and with increasing solvent viscosity, equation Bl.13.11 and equation B 1.13.12 describe the rotational Brownian motion of a rigid sphere in a continuous and isotropic medium. With the Lorentzian spectral densities of equation B 1.13.12. it is simple to calculate the relevant transition probabilities. In this way, we can use e.g. equation B 1.13.5 to obtain for a carbon-13... 

Rotational diffusion coefficient, Dg, internal motion rate parameter, angle between the internal rotation axis and the internuclear axis... 

Models for description of liquids should provide us with an understanding of the dynamic behavior of the molecules, and thus of the routes of chemical reactions in the liquids. While it is often relatively easy to describe the molecular structure and dynamics of the gaseous or the solid state, this is not true for the liquid state. Molecules in liquids can perform vibrations, rotations, and translations. A successful model often used for the description of molecular rotational processes in liquids is the rotational diffusion model, in which it is assumed that the molecules rotate by small angular steps about the molecular rotation axes. One quantity to describe the rotational speed of molecules is the reorientational correlation time T, which is a measure for the average time elapsed when a molecule has rotated through an angle of the order of 1 radian, or approximately 60°. It is indirectly proportional to the velocity of rotational motion. 

Fig. 11a and b. Decay of the alignment echo height as a function of the mixing time x2 for different motional mechanisms, a Tetrahedral jumps as a model for conformational changes b Diffusive motion, the solid lines correspond to unrestricted rotational diffusion, the dashed lines to diffusion restricted to an angular region of 8°. Note the strong dependence of the decay curves on the evolution time t, in case of diffusive motion... 

Fig. 12a and b. Calculated 2H spin alignment spectra for diffusive motion, a unrestricted rotational diffusion for different mixing times x2 b diffusion restricted to angular regions as indicated for long mixing times t2... 

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 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. 

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 nitroxide moiety of 16-SASL and 16-PC exhibits such a great deal of motion that the rotational correlation time can be calculated (Berliner 1978). The rotational correlation time (assuming isotropic rotational diffusion of the nitroxide fragment) can be calculated from the linear term of the line width parameter ... 

We would like to point out that an order parameter indicates the static property of the lipid bilayer, whereas the rotational motion, the oxygen transport parameter (Section 4.1), and the chain bending (Section 4.4) characterize membrane dynamics (membrane fluidity) that report on rotational diffusion of alkyl chains, translational diffusion of oxygen molecules, and frequency of alkyl chain bending, respectively. The EPR spin-labeling approach also makes it possible to monitor another bulk property of lipid bilayer membranes, namely local membrane hydrophobicity. 

Figure 15 The model molecule used to demonstrate the possibilities of HOESY experiments in terms of carbon-proton distances and reorientational anisotropy. To a first approximation, the molecule is devoid of internal motions and its symmetry determines the principal axis of the rotation-diffusion tensor. Note that H, H,., H,-, H,/ are non-equivalent. The arrows indicate remote correlations.

Another example comes from the work of Johnson, et a/.18 These workers studied spin labels dissolved in lipid bilayer dispersions of dipalmitoylphos-phatidylcholine and cholesterol (9 1 by weight) in the hope that anisotropic rotational diffusion of the spin label would mimic the motion of the bilayer components. In addition to 5-DS, which is sensitive to rotational motion about the NO bond, they used the steroidal nitroxide 8, which tends to rotate about an axis perpendicular to the N-O bond. ESR measurements were carried out at both 9 and 35 GHz and at temperatures ranging from 30 to 30 °C. Rather different results were obtained with the two spin labels, largely as a result of the different axes of rotation. Because the rotation rates were very slow, ESR spectra appeared as powder patterns rather than isotropic spectra and special methods were needed to extract the motional data. 

---