Small angle libration

It has been shown in chapters 4 and 6 that the individual mechanisms describing polymer chain motion in the bulk are often complex. An individual segment may simultaneously execute small-angle libration or rotational diffusion (10 < Tc < 10 s) and rotational jump motion with a broad distribution of correlation times (10 < < 10 s). An amorphous material may... 

An alternative interpretation of the previous solid-state NMR studies of the backbone motion of collagen has been reported. Based on the analysis of the solid-state NMR chemical shift anisotropy and quadrupolar line shapes for five different isotope labelled collagens, it has been shown that motional averaging of the NMR interactions occurs primarily via small-angle librations about internal bond directions. 

The Tc values evaluated from NMR relaxation times are always shorter than the values estimated using the SED model. The NMR relaxation takes place for one motion of a single dipole-dipole flip and can describe the time interval for the motion but not the amplitude. As proposed in our previous pap>ers,i-4 we assume that the Tc process is the small-angle librational motion of the cation. Assuming the T2 value calculated from t] correspond to a 360 rotation, the angles for the librational motions were calculated and plotted versus temperature in Fig. 19. The flip angle for EMlm is particularly larger than those for the other anions. The molecular flip amplitude becomes smaller at lower temperature. 

In addition to the large amplitude motion, the libration is expected to exist at higher temperatures as well as at lower temperatures, so that the libration was taken into account as small angle three-site jumps. As a result, the C-2H bond undergoes motions superimposed with two independent three-site jumps. In the calculations, a 0 value of 14° at room temperature was used as the polar angle of the libration, and its distribution was not... 

We first focus our attention on the losses (or the absorption) in the frequency range where the relaxational type of spectrum and the resonant type tend to coexist, because from there the information about molecular mobiUty preparing a- and -relaxations may be extracted. We mean the small-angle torsional oscillations (librations) leading to Poley-type absorption at 10-130 cm. Rektionships between the spectral parameters of this type absorption and molecular characteristics of polymers were discussed above. 

The unique information about molecular mobility preparing a- and b-relaxations in the frequency range, where relaxation type of spectrum and the resonant tend to co-exists, may be extracted from analysis of far-infra-red spectra. This enabled one to assign a low-temperature d-relaxation (d-loss peak observed in dynamic mechanical measurements in the temperature range 20-70K) to the small-angle torsional vibration (libration) of some molecular unit close in size to a repeat unit of macromolecules (156). 

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... 

Using the so-called planar libration-regular precession (PL-RP) approximation, it is possible to reduce the double integral for the spectral function to a simple integral. The interval of integration is divided in the latter by two intervals, and in each one the integrands are substantially simplified. This simplification is shown to hold, if a qualitative absorption frequency dependence should be obtained. Useful simple formulas are derived for a few statistical parameters of the model expressed in terms of the cone angle (5 and of the lifetime x. A small (3 approximation is also considered, which presents a basis for the hybrid model. The latter is employed in Sections IV and VIII, as well as in other publications (VIG). 

A simplified version of this model, termed the hybrid model (VIG, p. 305) [32-34, 39] (see also Section IV.E) was proposed for the case of a small cone angle p. In this model the rotators move freely over the barrier U0 as if they do not notice the conical surface the librators move in the diametric sections of a cone—that is, they librate. The hybrid model was widely used for investigation of dielectric relaxation in a number of nonassociated and associated liquids, including aqueous electrolyte solutions (VIG, p. 553) [53, 54]. The hat model was recently applied to a nonassociated liquid [3] and to water [7, 12c]. 

P represents the instantaneous angle between the molecule HA and the inertial axis. The brackets indicate librational motion averaging. For small p angles, x can be approximated as... 

Estimates show that the potential minimum is low only for small l values (such tha l2 < 0.1). Therefore, the particles with small axial momenta yield the main contribution to the spectrum stipulated by reorienting dipoles. At such small /-values the angle 0 usually falls into the range 15-18 degrees that is, 0 is turned out to be rather close to the libration amplitude ft equal to 23°. 

Just as the equilibrium conformational properties of macromolecules, the theory of which has been developed in well-known classical works by Kuhn, Flory, Volken-stein and others " the kinetic properties of polymer chains can be determined by two main mechanisms of intramolecular mobility. First, it is the discrete rotational isomeric (rotameric) mechanism of mobility caused by the jump of small-chain segments (kinetic units) from certain energically stable allowed conformers into others ° 1S4-16S) gg ond it is the continuous mechanism of motion determined by the accumulation of vibrational displacements of units owing to the twisting or bending vibrations (and librations) or valence angles and angles of internal rotation 45 i - ... 

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