Rotational diffusion model

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. 

Eagles T. E., McClung R. E. D. Reorientational correlation functions and memory functions in the. /-diffusion limit of the extended rotational diffusion model, Chem. Phys. Lett. 22, 414-18 (1973). 

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 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. 

A small step rotational diffusion model has been used to describe molecular rotations (MR) of rigid molecules in the presence of a potential of mean torque.118 120,151 t0 calculate the orientation correlation functions, the rotational diffusion equation must be solved to give the conditional probability for the molecule in a certain orientation at time t given that it has a different orientation at t = 0, and the equilibrium probability for finding the molecule with a certain orientation. These orientation correlation functions were found as a sum of decaying exponentials.120 In the notation of Tarroni and Zannoni,123 the spectral denisities (m = 0, 1, 2) for a deuteron fixed on a reorienting symmetric top molecule are ... 

Another rotational diffusion model known as the anisotropic viscosity model156,157 is very similar to the above model, and its main feature is to diagonalize the rotational diffusion tensor in the L frame defined by the director. A similar (but not the same) expression as Eq. (71) is J R(r)co)... 

The rotational diffusion model was generalized by Gordon (1966) to include... 

This electro-optical effect, commonly observed as transient changes in optical birefringence of a solution following application, removal, or reversal of a biasing electric field E(t), has been used extensively as a probe of dynamics of blopolymer solutions, notably by O Konski, and is a valuable tool because it gives information different in form, but related to, results from conventional dielectric relaxation measurements. The state of the subject to 1975 has been comprehensively presented in two review volumes edited by O Konski (25). The discussion here is confined to an outline of a response theory treatment, to be published in more detail elsewhere, of the quadratic effect. The results are more general than earlier ones obtained from rotational diffusion models and should be a useful basis for further theoretical and experimental developments. 

Chapter 8 by W. T. Coffey, Y. P. Kalmykov, and S. V. Titov, entitled Fractional Rotational Diffusion and Anomalous Dielectric Relaxation in Dipole Systems, provides an introduction to the theory of fractional rotational Brownian motion and microscopic models for dielectric relaxation in disordered systems. The authors indicate how anomalous relaxation has its origins in anomalous diffusion and that a physical explanation of anomalous diffusion may be given via the continuous time random walk model. It is demonstrated how this model may be used to justify the fractional diffusion equation. In particular, the Debye theory of dielectric relaxation of an assembly of polar molecules is reformulated using a fractional noninertial Fokker-Planck equation for the purpose of extending that theory to explain anomalous dielectric relaxation. Thus, the authors show how the Debye rotational diffusion model of dielectric relaxation of polar molecules (which may be described in microscopic fashion as the diffusion limit of a discrete time random walk on the surface of the unit sphere) may be extended via the continuous-time random walk to yield the empirical Cole-Cole, Cole-Davidson, and Havriliak-Negami equations of anomalous dielectric relaxation from a microscopic model based on a... 

It has been recognized for a long time that the orientation dependence of a vector fixed to a polymer chain could not be represented by a simple isotropic rotational diffusion model. In such a model the orientation is assumed to follow a vector joining the center of a sphere to a point performing a random brownien diffusion on the surface of that sphere. According to this model which describes well the orientation of spherical objects or infinitely thin rigid rods, the OACF is an exponential function... 

Generalized rotational diffusion models were built up later to account for the orientational relaxation of anisotropic rigid bodies or for the orientational... 

With QENS, both the rotational and translational motions of CH4 can be observed. It was found that the rotational motion of CH4 in ZSM-5 can be described by an isotropic rotational diffusion model with a rotational diffusion time constant, Dr (72). The values of Dr for CH4 adsorbed at 250 K in ZSM-5 are of the order of 5 x 10 s . In MD simulations at 400 K, Dr was found to be of the order of 10 s (73). This difference is due to the fact that a radius of gyration of 0.15 nm was used in the computer fits of the QENS profiles. This radius is intermediate between a simple rotation model, with 0.11 nm for the distance between the protons and the center of mass of the methane molecule, and the radius of the channel in which the molecule performs oscillations. 

Above 100 K, motional effects on spectrum become pronounced with increasing temperature and, above 230 K, the spectra consist of essentially an isotropic and equally spaced hyperfine triplet, but with different relative intensities. The line shape simulations were carried out by adopting a Brownian rotational diffusion model in order to evaluate the associated (average) rotational correlation time, and its degree of anisotropy, = zpy, /... 

Figure 2. Theoretical spectra of NO2 anisotropically rotating about its x axis (b), y axis (c), and z axis (d). The spectra were calculated for Brownian rotational diffusion model by using R// = 5.27 X 10 sec- Rj = 5.27 x 10 sec and T2 =3.0 G. For isotropic rotation (a), R// = Rj = 1.67 X 10 sec l and T2 l=3.0 G were used. The rotational diffusion values used through (a) to (d) correspond to a constant value of r =l x 10 sec v, 9 167 GHz...

C. Anomalous Dielectric Relaxation in the Context of the Debye Noninertial Rotational Diffusion Model... 

The relaxation times x]nt and t ) were obtained in the context of the normal rotational diffusion model in Refs. 8, 65, and 67 and are given by (in our notation)... 

We remark that evaluation of the Kerr effect response in the context of the fractional noninertial rotational diffusion model has been carried out by Dejardin and Jadzyn [104],... 

A similar study has been reported on specifically deuterated pyridines in a series of aqueous solutions. (172) A rotational diffusion model is used to interpret the relaxation data. The diffusion constant for reorientation around the C2 axis of pyridine is found to increase as the viscosity of the solution increases whereas the constant for reorientation about the axis perpendicular fo the molecular plane decreases as the amount of water present in the solution increases. These observations are attributed to short range ordering due to hydrogen bond formation. 

The correlation time, in Eq. (4) is generally used in the rotational diffusion model of a liquid, which is concerned with the reorientational motion of a molecule as being impelled by a viscosity-related frictional force (Stokes-Einstein-Debye model). Gierer and Wirtz have introduced the idea of a micro viscosity, The reorientational... 

It has recently become more widely appreciated that the presence of rotational diffusional anisotropy in proteins and other macromolecules can have a significant affect on the interpretation of NMR relaxation data in terms of molecular motion. Andrec et al. used a Bayesian statistical method for the detection and quantification of rotational diffusion anisotropy from NMR relaxation data. Sturz and Dolle examined the reorientational motion of toluene in neat liquid by using relaxation measurements. The relaxation rates were analyzed by rotational diffusion models. Chen et al measured self-diffusion coefficients for fluid hydrogen and fluid deuterium at pressures up to 200 MPa and in the temperature range 171-372 K by the spin echo method. The diffusion coefficients D were described by the rough sphere (RHS) model invoking the rotation translational coupling parameter A = 1. 

For many cases of molecular reorientation in liquids, the isotropic rotational diffusion model is applicable and the dipole-dipole interaction provides the dominant relaxation mechanism. Then, the spectral densities are given by [FARl]... 

The basic concepts of linear response theory are best illustrated by considering the rotational diffusion model of an assembly of electric dipoles constrained to rotate in two dimensions due to Debye [14] which is governed by the Smoluchowski equation... 

We have illustrated the concept of linear response using the simple rotational diffusion model of Debye. We will now illustrate how the concept applies for a rotator in an external potential. For convenience we will describe the procedure for the itinerant oscillator model given by Coffey [72],... 

In the following sections we evaluate the orientational correlation functions [Eq. (7.2.6)] for the rotational diffusion model. 

Now we turn to a detailed discussion of rotational motion in water. As already mentioned, the Debye rotational diffusive model was initially widely employed to describe water reorientation. As explained above, it describes the reorientation as an... 

NO2 is a stable paramagnetic gaseous molecule at normal temperatures. The ESR parameters of NO2 trapped in a solid matrix have been well established from single-crystal ESR measurements and have been related to the electronic structure by molecular orbital studies [39]. Thus, the NO2 molecule has potential as a spin probe for the study of molecular dynamics at the gas-solid interface by ESR. More than two decade ago temperature-dependent ESR spectra of NO2 adsorbed on porous Vycor quartz glass were observed [40] Vycor is the registered trademark of Coming, Inc. and more information is available at their website. The ESR spectral line-shapes were simulated using the slow-motional ESR theory for various rotational diffusion models developed by Freed and his collaborators [41]. The results show that the NO2 adsorbed on Vycor displays predominantly an axial symmetrical rotation about the axis parallel to the O—O inter-nuclear axis below 77 K, but above this temperature the motion becomes close to an isotropic rotation probably due to a translational diffusion mechanism. 

The graphs of the experimentally derived correlation times of liquid and plastic crystalline PH3 [5, 11] and PD3 [5] (which are identical for both isotopomers) were compared with the curves calculated for a spherical top molecule in the two limiting cases of the classical extended rotational diffusion model. The experimental graphs were found to lie between the curves for the J-diffusion limit (where magnitude and orientation of the angular momentum of the molecule is randomized by collisions) and the M-diffusion limit (where only the orientation... 

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