Jump reorientation models

Schematically, theories of rotational motion in liquids may be divided into two groups, which may be called classical reorientation and jump reorientation models. For the case that the rotation of a molecule in a liquid is regarded as a solid body moving in a fluid continuum the Debye-Stokes-Einstein relation [66] should apply. Thus for the reorientation of a spherical molecule...

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... 

The Hubbard relation is indifferent not only to the model of collision but to molecular reorientation mechanism as well. In particular, it holds for a jump mechanism of reorientation as shown in Fig. 1.22, provided that rotation over the barrier proceeds within a finite time t°. To be convinced of this, let us take the rate of jump reorientation as it was given in [11], namely... 

Another model of rotational reorientation is the jump-diffusion model first described by Ivanov (1964). In this model the molecule reorients by a series of discontinuous jumps (with an arbitrary distribution of jump angles). This should be contrasted with the Debye model, which involves infinitesimal jumps, and the Gordon model, which involves continuous free rotations between collisions. This model is probably applicable to the situation where the molecular orientation is frozen until a volume fluctuation occurs, at which time the molecular orientation jumps to a new frozen value. We present our own version of the jump model here. It is assumed that (a) the jump takes place instantaneously, (b) successive jumps are uncorrelated in time with an average time tv between jumps, and (c) the dihedral angle between the two planes defined by the orientation vector u in two successive jumps is randomized. 

In-depth numerical analysis of the ESR lineshape was first carried out by modeling the jump reorientation of TEMPOL in terms of the jump angle 6 and the mean... 

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 Debye phenomenology is consistent with both gas-like and solidlike model representations of the reorientation mechanism. Reorientation may result either from free rotation paths or from jumps over libration barriers [86]. Primary importance is attached to the resulting angle of reorientation, which should be small in an elementary step. If it is... 

It is well known [11] that the reorientation rate in the jump model... 

The description of the chain dynamics in terms of the Rouse model is not only limited by local stiffness effects but also by local dissipative relaxation processes like jumps over the barrier in the rotational potential. Thus, in order to extend the range of description, a combination of the modified Rouse model with a simple description of the rotational jump processes is asked for. Allegra et al. [213,214] introduced an internal viscosity as a force which arises due to a transient departure from configurational equilibrium, that relaxes by reorientational jumps. Thereby, the rotational relaxation processes are described by one single relaxation rate Tj. From an expression for the difference in free energy due to small excursions from equilibrium an explicit expression for the internal viscosity force in terms of a memory function is derived. The internal viscosity force acting on the k-th backbone atom becomes ... 

No single model can exactly describe molecular reorientation in plastic crystals. Models which include features of the different models described above have been considered. For example, diffusion motion interrupted by orientation jumps has been considered to be responsible for molecular reorientation. This model has been somewhat successful in the case of cyclohexane and neopentane (Lechner, 1972 De Graaf Sciesinski, 1970). What is not completely clear is whether the reorientational motion is cooperative. There appears to be some evidence for coupling between the reorientational motion and the motions of neighbouring molecules. Comparative experimental studies employing complementary techniques which are sensitive to autocorrelation and monomolecular correlation would be of interest. 

Let us discuss a system that consists of a number of particles where their relaxation is provided by the reorientations (a jump or another type of transition) of particles between two local equilibrium states. In the spirit of the Arrhenius model (26), the first requirement for the relaxation is that the particles have enough energy to overcome the potential barrier Ea between the states of local equilibrium for the elementary constituents of the system under consideration. Thus,... 

Fig. 14. Effects of small-amplitude reorientation on 2H NMR experiments, as calculated by means of RW simulations. In the model, C-2H bonds (<5 = 2n 125 kHz, rj = 0) perform rotational random jumps on the surface of a cone with a full opening angle % = 6°. (a) 2H NMR spectra for various solid-echo delays tp (tj = t = 30 pis), and (b) 2H NMR correlation functions Fcos(tm) for various evolution times tp (tj = t = 10ms). (Adapted from Ref. 76.)...

We shall now demonstrate how the CTRW in the diffusion limit may be used to justify the fractional diffusion equation. We consider an assembly of permanent dipoles constrained to rotate about a fixed axis (the dipole is specified by the angular coordinate unit circle with fixed angular spacing A. We note that A may not necessarily be fixed for example, if we have a Gaussian distribution of jumps, the standard deviation of A serves as a fixed quantity. A typical dipole may remain in a fixed orientation at a given site for an arbitrary long waiting time. It may then reorient to another discrete orientation site. This is the discrete orientation model. 

The nature of the orientational disorder in phases III and II was investigated first by powder experiments to give an overview of the evolution of the anisotropy and asymmetry parameter of the averaged EFG interaction. In all phases, the EFG asymmetry parameter is different from zero, and continuously decreases as temperature increases, with no observed anomaly at the transition temperature. That the asymmetry is definitively nonzero over the whole temperature range shows that the planar reorientational motion consists of jumps over six asymmetric potentials wells, which retain the centrosymmetry on average, and that the population of the different sites continuously evolves with temperature. The authors argued that a precise estimation of the two unknown occupation probabilities cannot be drawn from powder spectra without simplifying assumptions. The reasons come from the model of sixfold jump around the axis normal to the benzene... 

Four different models of molecular motion were in agreement with the jump angle determined by NMR. However, of these possible motions only one was in agreement with the dielectric relaxation results of Miyamoto et al. [69], This motion is defined by a dipole-moment transition and a conformational change (tg" tg <->g tg" t) yielding an effective dipole-moment reversal only along the chain axis and a reorientation angle of 113° for the C—bond directions. 

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