Rotational diffusion step

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. 

Small-step rotational diffusion is the model universally used for characterizing the overall molecular reorientation. If the molecule is of spherical symmetry (or approximately this is generally the case for molecules of important size), a single rotational diffusion coefficient is needed and the molecular tumbling is said isotropic. According to this model, correlation functions obey a diffusion type equation and we can write... 

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 ... 

In the most general case of a completely anisotropic diffusion tensor, six parameters have to be determined for the rotational diffusion tensor three principal values and three Euler angles. This determination requires an optimization search in a six-dimensional space, which could be a significantly more CPU-demanding procedure than that for an axially symmetric tensor. Possible efficient approaches to this problem suggested recently include a simulated annealing procedure [54] and a two-step procedure [55]. 

In the isotropic model, the overall rotational diffusion is characterized by a single parameter, the overall correlation time zc. The following steps could be used to determine zc. 

That steps involving atomic or molecular motion can be rate determining, even in fluids, is well known through diffusion limited reaction rates and the solvent cage effect. In solids, motion more subtle than translational diffusion can be influential, and cases of rotational diffusion control are familiar [7],... 

Since these isotope effects have been interpreted in terms of a physical transition state, it is instructive to contrast this phenomenon with studies in which the absence of an isotope effect was used to demonstrate a physical rate determining step [97]. The most relevant example is rotational-diffusion control of radical disproportionation in the solid-state photolysis of azobisis-obutyronitrile (AIBN). Since there is normally a primary isotope effect on the disproportionation of cyanoisopropyl radicals to methacrylonitrile and isobutyronitrile, the absence of such an effect in the solid-state photolysis of... 

The next step in the calculation of the rotational diffusivity is to evaluate the radius, ac. The subscript c on this parameter is meant to remind us that it is a function of concentration. Clearly, as the concentration is increased, this radius must decrease. In addition, one can anticipate that ac will depend on the orientation of the rods. As the rods become increasingly aligned parallel to one another, the hindcrance to rotation will drop off. To obtain ac, one must determine the number of rods, A (ac), which can cut through the imaginary cylinder constraining the test rod. To determine the probability of such an intersection, consider the diagram below. 

Rotational Diffusion in a Shear Field Consider for a moment rod-shaped particle of unit length. The orientation of the rod, u, can be specified by a unit vector u directed along its axis with spherical polar coordinates, Q = (0, used to describe its orientation. Because the solvent molecules are expected to frequently collide with the rod, it should exhibit a random walk on the surface of the unit sphere (i.e., r = 1.0). Debye [15] in 1929 developed a model for the reorientation process based on the assumption that collisions are so frequent that a particle can rotate through only a very small angle before having another reorienting collision (i.e., small step diffusion). 

Generation of the Brownian trajectories for rodlike molecules requires simulation of the anisotropic translational diffusion and rotational diffusion. The rotational and translational diffusion are coupled in this case, however, taking a sufficiently small time step enables the computation of the different components... 

Fluorescence correlation spectroscopy analyses the temporal fluctuations of the fluorescence intensity by means of an autocorrelation function from which translational and rotational diffusion coefficients, flow rates and rate constants of chemical processes of single molecules can be determined. For example, the dynamics of complex formation between /3-cyclodextrin as a host for guest molecules was investigated with singlemolecule sensitivity, which revealed that the formation of an encounter complex is followed by a unimolecular inclusion reaction as the rate-limiting step.263... 

Measurement of diffusion coefficient by rotation speed.step... 

It is evident in Fig. 25b that the ratio has a value close to 3 at high temperatures (T > 1.0) and declines steadily below T 1.0 until it reaches a value nearly equal to unity at low temperatures. While the Debye model of rotational diffusion, which invokes small steps in orientational motion, predicts the ratio ti/t2 to be equal to 3, a value for this ratio close to 1 is taken to suggest the involvement of long angular jumps [146, 147]. The ratio was observed to deviate from the Debye limit at lower temperatures in a recent molecular dynamics simulation study as well [148]. The onset temperature was thus found to mark the breakdown of the Debye model of rotational diffusion [145]. Recently, the Debye model of rotational diffusion was also demonstrated to break down for calamitic liquid crystals near the I-N phase boundary due to the growth of the orientational correlation [149]. 

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