Reorientation angle distribution

Reorientation Angle Distribution and Time Correlation Functions... 

Fig. 12. Bond reorientation angle distribution W(0,t) obtained [9] for system of x = SOO at 300 K. The tune interval t allowed for reorientation is indicated for eadi curve...

F%. 14. Cam risoii of observed bond reorientation angle distribution W(8,t) (solid curves) with theoretical curves Wo(0,t) (dotted curved calculated on the asannption of a rotational diffusion with a single diffiiwn coefficient. The rotational diffusion coefficaents used for the calculation are 0.02 and OifN ps ibr tiie ase of t equal to 2 and 20 ps, respectivdy. These D valu are chosen to make the calculated peak angles match the observed ones... 

Any attempt to explain our result of bond reorientation dynamics on the basis of superposition of rotational diffusion processes encounters a contradiction. If such an explanation was to be valid, the same spectrum of D had to be able to explain the shape of the observed Mi(t) and MaCO functions and the broad nature of the reorientation angle distribution W(6,t) at the same time. The spectrum g(x) of correlation time t or the equivalent spectrum g(D) of the rotational diffusion coefficient D can be evaluated from the correlation functions by means of a numerical procedure such as CONTIN [45]. When the correlation function can be represented by an analytical function, the spectrum can be obtained more conveniently by means of inverse Laplace transformation. In the case of a KWW function, with t and p characterizing the function as given in Eq. (12), g(D) can be calculated by [46]... 

Fig. 17. Comparison of observed bond reorientation angle distribution W(6,t) (solid curves) with curves (dotted curves) calculated on the assumption that the distributions arise from superposition of rotational diffusions. The spectrum g(D) of diffusion coefficient obtained from analysis of Mzft) and shown as curve 2 in Fig. 16 is used...

Examination of the reorientation angle distribution W(6,t) obtained at Tg exhibits a secondary peak at around 6 = W that does not change its angular position with time. This is interpreted to indicate the presence of a second reorientation mechanism by which the bond undergoes a large angle jump. Such a revelation of the presence of the second mechanism would not have been possible if only the time-correlation functions, and not the full distributions, had been evaluated. This provides an example in which the molecular modeling offers information that cannot easily be obtained from experiment. 

T = 275 K. The powder spectra olThe deutcrons at different chemical shifts overlap. The material is completely amorphous. The molecular segments undergo reorientation by isotropic lotational diffusion, (e) Distribution of reorientation angles for atactic PP established within the mixing time f ,.. Adapted from Blu4 with permission from Wiley-VC H. 

Fig. 6.2.2. Left Simulated NMR lineshapes that are averaged by various characteristic segmental motions. In the case of fast rotation, y represents the angle between the rotation axis and the C—bond. For a two-site jump, j3 denotes the angle between the C—bond in the two configurations, and the effective asymmetry parameter becomes 17 7 0. Right Calculated 2D exchange spectra for a two-site jump with /3 = 120° (top), and for continuous diffusion (bottom). The distribution functions P(/3) of the reorientation angle are shown, together with the contour maps of the corresponding spectra. All data are displayed on the reduced frequency scale in units of Cq, and mixing times are set equal to the motional correlation time r,..

Among the various types of local motions that take place in polymer liquids, the change in the orientation of individual bonds with time is one that can be readily evaluated from the molecular dynamics simulations. The bond reorientation motions are also amenaUe to experimental measurement by means of a number of techniques. We investigate the bond reorientation dynamics by evaluating three types of quantities, the distribution W(0,t) of reorientation angle 6 that a bond has undergone during a time interval t, and the time correlation functions... 

Typical examples of the reorientation ang distribution function W(6,t) are given in Fig. 12, where results obtained [9] with the system of x = SX) at MO K are plotted at several values of t. W(6,t) is deflned so that the probability of finding the reorientation angle between 6 and 6 + dO after time interval t is given by W(0,t)dO, and it approaches W(0, c ) = (l/2)sin0 at long t. The time correlation functions Mi(t) and M2(t) obtained from the same simulation run are given... 

For a cylindrically symmetric rigid body (a symmetric top) undergoing rotational diffusion, the distribution of reorientation angle 6 is given by [38-41]... 

Fig. 15. Effects of small-amplitude reorientation on 2H NMR stimulated-echo experiments, as calculated by means of RW simulations. The C-2H bonds perform rotational random jumps on the surface of a cone with a full opening angle % = 6°, which are governed by a broad logarithmic Gaussian distribution of correlation times G(lgr) (a = 2.3). (a) Correlation functions m tp — 30 is) for the indicated mean logarithmic time constants lgr 1. The calculated data are damped by an exponential decay, exp[—(tm/rso)] with rSD = 1 s, so as to mimic effects due to spin diffusion. The dotted lines are fits with Fcos(tm tp) = (1—C) expHtm/t/l + Qexp[—Om/rso)]- (b) Amplitude of the decays, 1-C,p, for various t resulting from these fits. The dotted line is the value of the integral in Eq. (12) as a function of rm. (Adapted from Ref. 76).

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