Rotational diffusion model small 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. 

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 ... 

The small step rotational diffusion model has been extensively applied to interpret ESR linewidth [7.4, 7.9], dielectric relaxation [7.2], fluorescence depolarization [7.19], infrared and Raman band shapes [7.24], as well as NMR relaxation in liquid crystals [7.14, 7.25]. When dealing with internal rotations in flexible mesogens, they are often assumed to be uncoupled from reorientation to give the so-called superimposed rotations model. Either the strong collision model or the small step rotational diffusion model may be used to describe [7.26, 7.27] molecular reorientation. 

In this section, two models related to the small step rotation diffusion model are briefly surveyed. First, a model that has a much simplified ordering... 

The small step rotational diffusion model has been employed to extract rotational diffusion constants Dy and D from the measured deuterium spectral densities in liquid crystals [7.25, 7.27, 7.46, 7.49 - 7.53]. Both the single exponential correlation functions [Eq. (7.54)] and the multiexponential correlation functions [Eq. (7.60)] have been used to interpret spectral densities of motion. However, most deuterons in liquid crystal molecules are located in positions where they are rather insensitive to motion about the short molecular axis. Thus, there is a large uncertainty in determining D or Tq (t o) because of Dy > D and the rather small geometric factor [doo( )] for most deuterons in liquid crystal molecules. For 5CB, it is necessary to fix [7.52] the value of D using the known activation... 

First, 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... 

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... 

Consider for a moment a 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, D - id, random walk along the surface of the unit sphere. Debye [16] in 1929 developed a model for the reorientation process based on the assumption that collisions are so fiiequent that a particle can rotate throu only a very small angle before having another reorienting collision (i.e., small step diffusion). Debye began with the diffusion equation... 

The complex rotational behavior of interacting molecules in the liquid state has been studied by a number of authors using MD methods. In particular we consider here the work of Lynden-Bell and co-workers [60-62] on the reorientational relaxation of tetrahedral molecules [60] and cylindrical top molecules [61]. In [60], both rotational and angular velocity correlation functions were computed for a system of 32 molecules of CX (i.e., tetrahedral objects resembling substituted methanes, like CBt4 or C(CH3)4) subjected to periodic boundary conditions and interacting via a simple Lennard-Jones potential, at different temperatures. They observe substantial departures of both Gj 2O) and Gj(() from predictions based on simple theoretical models, such as small-step diffusion or 7-diffusion [58]. Although we have not attempted to quantitatively reproduce their results with our mesoscopic models, we have found a close resemblance to our 2BK-SRLS calculations. Compare for instance our Fig. 13 with their Fig. 1 in [60]. 

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]. 

Debye (1929) developed a model for the reorientation processes based on the assumption that collisions are so frequent in a liquid that a molecule can only rotate through a very small angle before suffering a reorienting collision (small-step diffusion). We give here a heuristic treatment of the Debye model. 

To be historically fair, other people did observe the existence of jump motions in the rotation of water molecules in the liquid state but detailed analysis of the dynamics of an individual event was not carried out before. Given that perspective, the Laage-Hynes mechanism of water rotation by large-amplitude jumps is indeed a departure from conventional and prevailing wisdom that water rotation is Brownian that is, it occurs differently in water from in other liquids where motion by small steps dominates. Experimental verification of the jump diffusion model came from a beautiful study of the temperature-dependent rate of water rotation. However, both the experiments and the interpretation of results are quite involved. We shall discuss the results as simply as possible. 

There exist a number of models predicting the time evolution of r t) for small llu-orophores in the solution. They differ in accuracy and detail of physical description and in mathematical approximations. The simplest rotational model is based on the Debye hydrodynamic theory [9]. It assumes that the rotational diffusion proceeds in small steps between collisions of the fluorophore with surrounding molecules. An analytical expression for r t) as a sum of several exponentials was first derived by Favro [10] ... 

Neat liquids are, in a way, difficult objects for NMR relaxation studies. The simple modelling of reorientational motion as small-step rotational diffusion is based on hydrodynamics (large body immersed in continuum solvent) and becomes problematic if we deal with a liquid consisting of molecules of a single kind. Deviations from the models based on few discrete correlation times can therefore be expected. 

Jump reorientation models may involve activation over barriers to rotation or the migration of lattice defects or holes. Reorientation is in both cases discontinuous and changes in orientation occur-ing in one step are assumed to be large. Both types of jump reorientation models have been discussed by O Reilly [68], In his quasilattice random flight model, for example, O Reilly [69 70] assumes that the liquid structure up to the first coordination shell may be approximated by a lattice. Some of the properties of the solid state such as vacancies and translational diffusion by vacancy migration are considered present. In general difficulties arise when these jump reorientation models are compared with experimental data because several parameters are needed in the analysis. Furthermore, it appears that O Reilly [71] employs results obtained by Huntress [55] which apply only in the limit of small-step reorientation to treat the case of... 

In the limit of small-step diffusion where Xj J-diffusion reduces to Hubbard s relation (2.12). Thus in this limit the extended J-diffusion model agrees with the classical theory. This is not true for the M-diffusion model. In the other limit when Xj becomes large, as in a dilute gas, where many rotations occur between... 

The standard model for diffusive motion in polymers is Brownian diffusion, which occurs as a series of infinitesimal reorientational steps. This model is most appropriate for intermediate-to-large sized spin probes and spin-labeled macromolecules, where the macromolecule is much larger than any solvent molecules. Because of this broad applicability, the Brownian diffusion model is the most widely used. This type of rotational diffusion is completely analogous to the one-dimensional random walk used to describe translational diffusion in standard physical chanistry texts, with the difference that the steps are described in terms of a small rotational step 59 that can occur in either the positive or negative direction. In three dimensions, rotations about each of three principal axes of the nitroxide must be taken into account. A diffusion constant may be defined for each of these rotations motions, in a way that is completely analogous to the definition of translational diffusion constant for the one-dimensional random walk. 

These predictions of the simple phenomenological model are in accord with experimental dielectric data for amorphous solid polymers (4-7). The model does not specify detailed mechanisms for a and B processes, so, historically, the next stage was to develop such models. Many attempts were made and Table 1 summarizes a number of one-body models and their generalizations to include chain dynamics. Those for chain dynamics incorporate the basic models for one body motion e.g. the theory of Yamafuji and Ishida (22) is for coupled units each undergoing small-step rotational diffusion, while those of Jernigan (29) and Beevers and Williams (30) are for coupled units each undergoing motion in local (conformational) barrier systems. All the models in Table 1 exclude the short time effects associated with inertial factors and damped librations in a local potential. 

Note that the steady-state a.c. Kerr-effect is not simply related in the general case to the transient Kerr-effect in the time-domain (86-88), but exact relations can be given if the model for reorientation is specified. For example Benoit gave the necessary relations for the small-step diffusion model for axially-symmetric molecules (85). In this case Kjjj(t) = exp[-m(nH-l)Dj t] where Dr is the rotational diffusion coefficient (see also refs. 86,89). Thus for the case where the permanent dipole moment contribution to K greatly exceeds the "induced dipole moment contribution, the effective relaxation time for step-on response,... 

Small Step Rotational Diffusion and Strong Collision Models... 

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