Rouse model dynamic structure factor

For different momentum transfers the dynamic structure factors are predicted to collapse to one master curve, if they are represented as a function of the Rouse variable. This property is a consequence of the fact that the Rouse model does not contain any particular length scale. In addition, it should be mentioned that Z2/ or the equivalent quantity W/4 is the only adjustable parameter when Rouse dynamics are studied by NSE. 

Figure 6 shows the measured dynamic structure factors for different momentum transfers. The solid lines display a fit with the dynamic structure factor of the Rouse model, where the time regime of the fit was restricted to the initial part. At short times the data are well represented by the solid lines, while at longer times deviations towards slower relaxations are obvious. As it will be pointed out later, this retardation results from the presence of entanglement constraints. Here, we focus on the initial decay of S(Q,t). The quality of the Rouse description of the initial decay is demonstrated in Fig. 7 where the Q-dependence of the characteristic decay rate R is displayed in a double logarithmic plot. The solid line displays the R Q4 law as given by Eq. (29). 

Fig. 6. Dynamic structure factor as observed from PI for different momentum transfers at 468 K. ( Q = 0.038 A"1 Q = 0.051 A-1 A Q = 0.064 A-1 O Q = 0.077 A"1 Q= 0.102 A-1 O Q = 0.128 A 1 Q = 0,153 A "" 11. The solid lines display fits with the Rouse model to the initial decay. (Reprinted with permission from [39]. Copyright 1992 American Chemical Society, Washington)...

Like the dynamic structure factor for local reptation it develops a plateau region, the height of which depends on Qd. Figure 20 displays S(Q,t) as a function of the Rouse variable Q2/ 2X/Wt for different values of Qd. Clear deviations from the dynamic structure factor of the Rouse model can be seen even for Qd = 7. This aspect agrees well with computer simulations by Kremer et al. [54, 55] who found such deviations in the Q-regime 2.9 V Qd < 6.7. 

Fig. 20. Single-chain dynamic structure factor of the Ronca model as a function of the Rouse variable for different values of Qdt (dt tube diameter dt = d). (Reprinted with permission from [50]. Copyright 1983 American Institute of Physics, Woodbury N.Y.)...

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

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... 

Since the Rouse model does not contain an explicit length scale, for different momentum transfers the dynamic structure factors are predicted to collapse... 

In order to learn more about the Rouse model and its limits a detailed quantitative comparison was recently performed of molecular dynamics (MD) computer simulations on a 100 C-atom PE chain with NSE experiments on PE chains of similar molecular weight [52]. Both the experiment and the simulation were carried out at T=509 K. Simulations were imdertaken,both for an explicit (EA) as well as for an united (l/A) atom model. In the latter the H-atoms are not explicitly taken into account but reinserted when calculating the dynamic structure factor. The potential parameters for the MD-simulation were either based on quantum chemical calculations or taken from literature. No adjusting... 

In generalized Rouse models, the effect of topological hindrance is described by a memory function. In the border line case of long chains the dynamic structure factor can be explicitly calculated in the time domain of the NSE experiment. A simple analytic expression for the case of local confinement evolves from a treatment of Ronca [63]. In the transition regime from unrestricted Rouse motion to confinement effects he finds ... 

Rubber-like models take entanglements as local stress points acting as temporary cross finks. De Cloizeaux [66] has proposed such a model, where he considers infinite chains with spatially fixed entanglement points at intermediate times. Under the condition of fixed entanglements, which are distributed according to a Poisson distribution, the chains perform Rouse motion. This rubber-like model is closest to the idea of a temporary network. The resulting dynamic structure factor has the form ... 

Fig. 3.15 Dynamic structure factors from PE melts at 509 K a M =2,000 [69, 70] and b 12,400 [71]. The solid lines display the predictions of the Rouse model. The Q-values are noted adjacent to the respective lines. Note that the time frame in b is extended by an order of magnitude compared to a. (a Reprinted with permission from [69]. Copyright 1993 The American Physical Society)...

Fig. 3.16 Scaling presentation of the dynamic structure factor from a M =36,000 PE melt at 509 K as a function of the Rouse scaling variable. The solid lines are a fit with the reptation model (Eq. 3.39). The Q-values are from above Q=0.05,0.077,0.115,0.145 A The horizontal dashed lines display the prediction of the Debye-Waller factor estimate for the confinement size (see text)...

Recently a very detailed study on the single chain dynamic structure factor of short chain PIB (M =3870) melts was undertaken with the aim to identify the leading effects limiting the applicability of the Rouse model toward short length scales [217]. This study was later followed by experiments on PDMS (M =6460), a polymer that has very low rotational barriers [219]. Finally, in order to access directly the intrachain relaxation mechanism experiments comparing PDMS and PIB in solution were also carried out [186]. The structural parameters for both chains were virtually identical, Rg=19.2 (21.3 A). Also their characteristic ratios C =6.73 (6.19) are very similar, i.e. the polymers have nearly equal contour length L and identical persistence lengths, thus their conformation are the same. The rotational barriers on the other hand are 3-3.5 kcal/mol for PIB and about 0.1 kcal/mol for PDMS. We first describe in some detail the study on the PIB melt compared with the PDMS melt and then discuss the results. 

Figure 5.3 presents NSE results obtained on PIB at 470 K together with a fit with the Rouse dynamic structure factor Eq. 3.19. The Rouse model provides a good description of the spectra for Q<0.15 A In this range, the elementary... 

Fig. 5.3 Single chain dynamic structure factor from PIB in the melt at 470 K and Q=0.04 A" (empty circle), 0.06 A (filled triangle), 0.08 A (empty diamond), 0.10 A" (filled circle), 0.15 A (empty triangle), 0.20 A (filled diamond), 0.30 A (empty square), and 0.40 A (plus). The solid lines show the fit of the Rouse model to the data. (Reprinted with permission from [217]. Copyright 1999 American Institute of Physics)...

Fig. 5.6 Single chain dynamic structure factor measured for PDMS chains at 373 K in the melts compared to the standard Rouse model (lines) at the Q-values (A Q indicated. (Reprinted with permission from [186]. Copyright 2001 American Chemical Society)...

Fig. 6.5 Comparison between NSE spectra at Q=0.1 A and T=473 K of both binary blends (filled circle and filled triangle), and those of the isotopic PDMS blends filled square). The solid lines result from a fit t with the dynamic structure factor for the Rouse model filled square) and of a spatially limited Rouse dynamics, as derived in [256] filled square and filled triangle). (Reprinted with permission from [255]. Copyright 2003)...

For the Rouse model the mean-square displacement is proportional to the square root of time [Eq. (8.58)]. The logarithm of the dynamic structure factor of the Rouse model then also scales as the square root of time for... 

Calculate the Rouse model prediction for the dynamic structure factor for gels in the gelation regime. 

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