Rouse rate

Over the entire Q-range within experimental error the data points fall on the line and thus exhibit the predicted Q4 dependence. The insert in Fig. 7 demonstrates the scaling behavior of the experimental spectra which, according to the Rouse model, are required to collapse to one master curve if they are plotted in terms of the Rouse variable u = QV2 /wt. The solid line displays the result of a joint fit to the Rouse structure factor with the only parameter fit being the Rouse rate W 4. Excellent agreement with the theoretical prediction is observed. The resulting value is W/4 = 2.0 + 0.1 x 1013 A4s 1. 

Fig. 22. Scaling representation of the spectra obtained from a Polyisoprene melts in terms of the Rouse variable. The solid line displays the Rouse structure factor using the Rouse rate determined at short times, (o Q = 0.153 A"1 V Q = 0.128 A-1 Q = 0.102 A"1. Q = 0.077 A"1 A Q = 0.064 A-1) (Reprinted with permission from [39]. Copyright 1992 American Chemical Society, Washington)...

With respect to the intensity resolution relationship of NSE, PEB-2 [essentially PE with one ethyl branch every 50 main chain bonds the sample is obtained by saturating 1-4 polybutadiene, the residual 1-2 groups (7%) cause the ethyl branches Mw = 73200 g/mol Mw/Mn = 1.02] has two advantages compared to PEP (1) the Rouse rate W/4 of PEB-2 is more than two times faster than that of PEP at a given temperature [W/pEP (500 K) = 3.3 x 1013 A4s 1 W/pEB (509 K) = 7 x 1013A4s-1] (2) at the same time, the topological constraints are stronger. 

Here is the Rouse time - the longest time in the relaxation spectrum - and W is the elementary Rouse rate. The correlation function x(p,t) x p,0)) of the normal coordinates is finally obtained by ... 

Table 3.2 Rouse rates mean-square monomer length monomeric...

Moreover, from Fig. 3.18 it is apparent that the model of des Cloizeaux also suffers from an incorrect Q-dependence of S(Q,f) in the plateau region, which is most apparent at the highest Q measured. It is important to note that the fits with the reptation model were done with only one free parameter, the entanglement distance d. The Rouse rate was determined earlier through NSE data taken for Kr. With this one free parameter, quantitative agreement over the whole range of Q and t using the reptation model with d=46.0 1.0 A was found. 

Inserting the Rouse rate W(, 3Q9 K)=(7 0.7)xl0 AVns (Table 3.2) obtained from single chain structure factor measurement into Eq. 3.18 the solid line is obtained. It quantitatively corroborates the correctness of the Rouse description at short times. The data also reveal clearly a transition to a law, though Eq. 3.36 would predict the dotted line. The discrepancy explains itself in considering the non-Gaussian character of the curve-linear Rouse motion (Eq. 3.38). Fixing and d to the values obtained from the single chain struc-... 

Fig. 3. 27 NSE data from PE melts with a molecular weight of M =36,000 g/mol (left) and 190,000 g/mol (right) (M /M <1.02) for Q=0.03 (crosses), 0.05 (squares), 0,077 (circles), 0,096 (diamonds), 0.115 (triangles) and 0.145 A" (stars). For the common Q-values the data are identical. The lines represent a fit with the reptation model (using W =7xl0 AVns for the Rouse rate). (Reprinted with permission from [74]. Copyright 2000 EDP Sciences)...

Rouse rate at 470 K is 8100 AVns. The corresponding mode-dependent characteristic times are represented by the dashed-dotted line in Fig. 5.1. Taking into account the relationship between Dr and the corresponding diffusion coefficient Dr becomes Dr=1.22 0.17x10" mVs. 

While the NSE results show that, within the experimental accuracy, in the range Q<0.15 A the Rouse model gives a good account for the internal modes as well as for the diffusion of the chain centre of mass, it is also clear that for higher Q-values the experimental structure factors decay significantly more slowly than the Rouse model would require. These deviations are quantified in fitting the Rouse model to the different spectra separately. This procedure results in a strong dispersion of the elementary Rouse rate. The values determined for at Q>0.15 A" follow a Q-dependence, which can be described by the power law ... 

Q-dependent Rouse rates were obtained by fitting each spectrum separately. For a comparison with the BS data (see below) the obtained values for Wf (Q) were transformed into average relaxation times for the Rouse self-correlation function by (r (Q))=18TiQ V[ Wf (Q)] (Eq. 3.18). 

Fig. 6.8 Q dependence of the two eigenvalues Ai(Q) solid line) and A2(Q) dotted line) predicted by a two-component dynamic RPA approach for the case of an hA-dB labelled diblock copolymer melt. Calculations were performed with/=0.5, Rg =Rg =40 A, Na=Ny=200, Ku=0, Ai(Q) describes the collective mode of the diblock copolymer chains. The Rouse rates were taken from PE and PEE at 473 K (see Table 6.2). (Reprinted with permission from [44]. Copyright 1999 American Institute of Physics)...

Table 6.2 Experimental Rouse rates (Wi ) for the pure homopolymers and for the hlock-copolymer melts obtained from the high Q behaviour of JXQ) compared to the predictions of the RPA theory. Values of are given in 10s ...

Figure 6.3 presents the displacement pattern associated with the lowest order Rouse-mode with m = 1 (the solution for m = 0 describes just a free translation). The relaxation rate of this mode, shortly called Rouse-rate follows from Eq. (6.38) together with Eq. (6.28), as... 

Fig. 6.25 Inverse aetivation plot for the Rouse rate, W, and the chain self-dillasion coefficient, D. The normalization values for T = oo for both quantities were measured in independent simulations. VF laws are straight lines extrapolating to a finite value on the abscissa, whereas an Arrhenius law extrapolates to 7 = 0.

Fitting yields a surface tension )> = 4.5mNm" that compares to the prediction using the expression of Helfand and Sapse, which yields )> = 3.3mNm". The local viscosity as derived from the Rouse theory implies the involvement of 13 PI units. The Rouse rate also controlling the 2D segmental diffusion corresponds to the homopolymer value. 

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