Rouse scaling

Fig. 13 shows this autocorrelation function where the time is scaled by mean square displacement of the center of mass of the chains normalized to Ree)- All these curves follow one common function. It also shows that for these melts (note that the chains are very short ) the interpretation of a chain dynamics within the Rouse model is perfectly suitable, since the time is just given within the Rouse scaling and then normalized by the typical extension of the chains [47]. 

The presence of entanglement constraints is expected to show itself (1) by a reduction in the time decay of S(Q,t) compared to the Rouse dynamic structure factor and (2) by the systematic Q-dependent deviation from the Rouse scaling properties. 

Fig. 23. Rouse-scaling representation of the spectra obtained from polyisoprene at a neutron wavelength of X = 11.8 A and T = 473 K (A Q = 0.074 A"1 Q = 0.093 A-1 Q = 0.121 A-1). The solid lines are the result of a fit with the Ronca model [50]. The arrows indicate Q2/2 J Wxe for each curve. (Reprinted with permission from [39]. Copyright 1992 American Chemical Society, Washington)...

The second relation arises from the unentangled (Rouse) scaling of with N/ in terms of a fundamental monomer timescale... 

Fig. 3.6 Single chain structure factor from PEE melts as a function of the Rouse scaling variable. The dashed line displays the Rouse prediction for infinite chains, the solid lines incorporate the effect of translational diffusion. The different symbols relate to the spectra displayed in Fig. 3.5. (Reprinted with permission from [40]. Copyright 2003 Springer, Berlin)...

We now use these data, in order to investigate the scaling prediction inherent in Eq. 3.24. Figure 3.6 presents a plot of the data of Fig. 3.5 as a function of the Rouse scaling variable (Eq. 3.25). 

In Fig. 3.16 dynamic structure factor data from a A =36 kg/mol PE melt are displayed as a function of the Rouse variable VWt (Eq. 3.25) [4]. In Fig. 3.6 the scaled data followed a common master curve but here they spht into different branches which come close together only at small values of the scahng variable. This splitting is a consequence of the existing dynamic length scale, which invalidates the Rouse scaling properties. We note that this length is of purely dynamic character and cannot be observed in static equilibrium experiments. 

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)...

Fig. 13 Single molecule tracking data of dye-labeled PMOx lipopolymers as a function of area per molecule. The plots of the lateral diffusion coefficient, D, vs area per molecule for DiCisPMOx3o and DiCisPMOxso show two different diffusion regions (labeled I and II). Unlike in Region II, D follows Rouse scaling in Region I [31] (reproduced with permission from the American Chemical Society)...

Hg. 3.3 Log-linear data collapsing plot of the decay of the dynamic structure factor S(k, t) for Nek = 60, in Rouse scaling form, t) vs. using u = 0.59, and restricting the data... 

Paul et al extended the above discussion to a variety of different relaxation times Tx. Here we confine our discussion to the time they call td, which is the time when g t)/t w const. This time is determined by the crossover from the slowed-down displacement at shorter times to the asymptotic free diffusion g3(f) oc t. As seen above starts out as gi t) oc i . Compared to the almost perfect Rouse scaling of the mode spectrum, this is a significant difference. The mode spectrum however only contains fluctuations inside the chains but not the overall motion. This effect on gT, t) probably reflects the difference between the local hopping rates and the mobility, however now on a more global scale than for the monomer motion. Following the Rouse and reptation concept rp oc independent whether N < Ne or N> Ne. 

Figure 43 Symbols indicate NSE data from hPEO/dPEO/dPMMA at 400K for different Q values displayed as function of the Rouse scaling variable. The dashed region indicates the elastic contribution from the frozen dPMMA contrast. The solid lines are the predictions of the simple Rouse theory. Reproduced with permission from Niedzwiedz, K. Wischnewski, A. Monkenbusch, M. etal. Phys. Rev. Lett. 2007, 98, 168301. ...

---