Zimm model dynamic structure factor

The dynamic structure factor is S(q, t) = (nq(r) q(0)), where nq(t) = Sam e q r is the Fourier transform of the total density of the polymer beads. The Zimm model predicts that this function should scale as S(q, t) = S(q, 0)J-(qat), where IF is a scaling function. The data in Fig. 12b confirm that this scaling form is satisfied. These results show that hydrodynamic effects for polymeric systems can be investigated using MPC dynamics. 

Fig. 35. Dynamic structure factors S(Q,t)/S(Q,0) as dependent on Qt for the Rouse and the Zimm model...

Fig. 40a, b. NSE spectra of a dilute solution under 0-conditions (PDMS/ d-bromobenzene, T = = 357 K). a S(Q,t)/S(Q,0) vs time t b S(Q,t)/S(Q,0) as a function of the Zimm scaling variable ( t(Q)t)2/3. The solid lines result from fitting the dynamic structure factor of the Zimm model (s. Tablet) simultaneously to all experimental data using T/r s as adjustable parameter. 

Obviously, in the case of PS these discrepancies are more and more reduced if the probed dimensions, characterized by 2ti/Q, are enlarged from microscopic to macroscopic scales. Using extremely high molecular masses the internal modes can also be studied by photon correlation spectroscopy [111,112], Corresponding measurements show that - at two orders of magnitude smaller Q-values than those tested with NSE - the line shape of the spectra is also well described by the dynamic structure factor of the Zimm model (see Table 1). The characteristic frequencies QZ(Q) also vary with Q3. Flowever, their absolute values are only 10-15% below the prediction. 

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. 

Figure 61 presents the Q(Q)/Q2 relaxation rates, obtained from a fit with the dynamic structure factor of the Zimm model, as a function of Q. For both dilute solutions (c = 0.02 and c = 0.05) Q(Q) Q3 is found in the whole Q-range of the experiment. With increasing concentrations a transition from Q3 to... 

Fig. 70. NSE spectra for 2% linear h-PI in deuterated n-decane at Q/A 1 values of 0.064 ( ), 0.089 ( ) and 0.115 ( ). The solid lines represent a common fit with the dynamic structure factor of the Zimm model (see Table 1) neglecting possible effects of translational diffusion. (Reprinted with permission from [174]. Copyright 1993 The American Physical Society, Maryland)...

Fig. 5.10 Chain dynamic structure factor of PDMS (a) and PIB (b) in toluene solution at 327 K at the Q-values 0.04 A" (empty circle), 0.06 A (filled circle), 0.08 A" (empty square), 0.10 A (filled square), 0,15 A (empty diamond), 0,20 A (filled diamond), 0,25 A (empty triangle), 0.30 A" (filled triangle), 0.40 A" (plus). Solid lines correspond to fitting curves Rouse-Zimm model for PDMS and Rouse-Zimm with intrachain viscosity for PIB (see the text). (Reprinted with permission from [186]. Copyright 2001 American Chemical Society)...

For the Zimm model the mean-square displacement of monomers is faster [Eq. (8.70)] leading to the logarithm of the Zimm dynamic structure factor scaling as the 2/3 power of time for tq < r < zz-... 

Figure 3.54. Dynamic structure factor 5 j(k, t) is plotted as a function of tjtj for various values of AJfg. Calculation was done by using the Zimm model for the theta solvent.

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