Coherent dynamic structure factor

Coherent dynamic structure factor Characteristic frequency Adjustable parameter... 

The long-time behavior (Q(Q)t) > 1 of the coherent dynamic structure factors for both relaxations shows the same time dependence as the corresponding incoherent ones... 

For the case that both Rouse and Zimm relaxation are present, the coherent dynamic structure factor is also available [40]. For completeness, its analytical representation is included in Table 1. 

More general access to the coherent dynamic structure factor was provided by Akcasu and coworkers [93-95], starting from the assumption that the temporal evolution of the densities in Fourier space p(Q,t)... 

Similar to the case of good solvent conditions, the complete coherent dynamic structure factor S(Q,t) is not available in the transition range of the regimes I and T. However, the crossover behavior becomes accessible via the initial slope as a function of Q and t [105]. Typical results of this treatment are shown in Fig. 39, where... 

In contrast to -conditions a large number of NSE results have been published for polymers in dilute good solvents [16,110,115-120]. For this case the theoretical coherent dynamic structure factor of the Zimm model is not available. However, the experimental spectra are quite well described by that derived for -conditions. For example, see Fig. 42a and 42b, where the spectra S(Q, t)/S(Q,0) for the system PS/d-toluene at 373 K are shown as a function of time t and of the scaling variable (Oz(Q)t)2/3. As in Fig. 40a, the solid lines in Fig. 42a result from a common fit with a single adjustable parameter. No contribution of Rouse dynamics, leading to a dynamic structure factor of combined Rouse-Zimm relaxation (see Table 1), can be detected in the spectra. Obviously, the line shape of the spectra is not influenced by the quality of the solvent. As before, the characteristic frequencies 2(Q) follow the Q3-power law, which is... 

The crossover from 0- to good solvent conditions in the internal relaxation of dilute solutions was investigated by NSE on PS/d-cyclohexane (0 = 311 K) [115] and on PDMS/d-bromobenzene(0 = 357K) [110]. In Fig. 45 the characteristic frequencies Qred(Q,x) (113) are shown as a function of t = (T — 0)/0. The QZ(Q, t) were determined by fitting the theoretical dynamic structure factor S(Q, t)/S(Q,0) of the Zimm model (see Table 1) to the experimental data. This procedure is justified since the line shape of the calculated coherent dynamic structure factor provides a good description of the measured NSE-spectra under 0- as well as under good solvent conditions. 

Simulation Study of the a-Relaxation in a 1,4-Polybutadiene Melt as Probed by the Coherent Dynamic Structure Factor. 

A more realistic model for the secondary relaxation needs to consider motions of a molecular group (considered as a rigid object) between two levels. The group may contain N atoms with the scattering length h, at positions r (i=lj ). The associated motion may consist of a rotation aroimd an arbitrary axis, e.g. through the centre of mass depicted by a rotational matrix Q and a displacement by a translational vector . In order to evaluate the coherent dynamic structure factor, scattering amphtudes of the initial (1) and final (2) states have to be calculated ... 

The correlation function B(k, t) cannot be evaluated in closed form in the present case. However, for t comprised between T ,j and numerical calculations [54] show that B(0, t) cc whereas, for t x q), B(0, t) oc t due to globule diffusion. Consequently, the exponent of the relationship hiiQ = const has values 3 and 2 in the two respective regimes, as with the unperturbed chain [see Eqs. (3.2.15 ) and (3.2.16)]. We do not obtain any plateau of B 0,t) in this case, unlike the Rouse limit. Figure 7 shows the coherent dynamic structure factor S Q, t) as a function of t for two different Q values both for the unperturbed and for the collapsed chain. The two Q s correspond to observation distances /Q [ref. 15, note 6] just below and... 

Figure 7. Coherent dynamic structure factor S(Q,t)/S(Q,0) vs. t/to (to = U / bT) for atactic poiystyrene in collapsed state (continuous lines) and in unperturbed state (dashed lines). The Q values correspond to observation distances just below (Q = 0,0lA ) and above (Q = 0.003 A ) radius of gyration of collapsed chain (2 is 2.5 and 0.75, respectively). [Parameters JV = 5 x 10 chain atoms, J = 5 x 10, = 1450A, a, = 0.17,...

Figure 12. Half-peak time ti,2 of coherent dynamic structure factor for atactic polystyrene plotted vs. Q. Left, CgD solution right, CS2 solution continuous lines, calculated results at T = 30 °C dashed line, at F = 70 °C. Experimental points from ref. 14. [Model assumptions and parameters same as in Figures 10 and 11, tB = 0.012.] (Reprinted with permission from ref. 14, Copyright 1984, American Chemical Society.)...

Our results for the collective behavior for protein hydration water on length scales of angstroms length scales and timescales up to a few picoseconds are remarkable for two reasons. First, the coherent dynamic structure factors we have computed for... 

FIGURE 16.3 (See color insert following page 172.) Contour plots of the longitudinal current spectra, Ci Q,E) = (E /Q )S Q,E), computed from coherent dynamic structure factors for protein hydration water in the RNase crystal at 300K [60], The black lines trace the maxima of the two Brillouin side peaks in the spectra, one of which is dispersive (i.e., the excitation energy... 

Obviously, if found experimentally, the exponent -2 is only a sign of reptation motion but not a sufficient condition for it. Thus, in order to prove that reptation is a really dominant mode of chain motion in polymer concentrates, we have to test it not only with self-diffusion but also with other physical properties which reflect the local motion of polymer chains. One such property is the (coherent) dynamic structure factor 5(fc, r) (see Section 3.2 of Chapter 4 for its definition). In fact, it was predicted theoretically [45-47] that the k dependence of its decay with r in the range of k defined by... 

Fig. 1 were slightly raised to improve the agreement with the experimental data. Other experimental observables such as dielectric relaxation [12] and the coherent dynamic structure factor for neutron spin echo [13] have also been calculated from simulations and used to judge the realism of the local relaxation processes in the simulations. 

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