Relaxation Rouse

There are three basic time scales in the reptation model [49]. The first time Te Ml, describes the Rouse relaxation time between entanglements of molecular weight Me and is a local characteristic of the wriggling motion. The second time Tro M, describes the propagation of wriggle motions along the contour of the chain and is related to the Rouse relaxation time of the whole chain. The important... 

Fig, 14. Disentanglement mechanism (A) Tightened slack between entanglements. (B) Retraction and disentanglement by Rouse relaxation. (C) Critically connected state. 

The main predictions of the scaling theory [40], concerning the dynamics behavior of polymer chains in tubes, deal with a number of characteristic times the smallest time rtube measures the interval of essentially Rouse relaxation before the monomers feel the tube constraints significantly, 1 < Wt < Wrtube = and diffusion of an inner monomer is... 

As in the case of the Rouse relaxation time, it is possible to express x2 in term of the intrinsic viscosity either by using Eq. (24) or the Kirkwood-Riseman theory [24, 47] ... 

Tlim is the limiting value for a fully stretched chain and should correspond to the Rouse relaxation time when there is no deformation, t must equal the Zimrn... 

Fig. 5.13. Relaxation time r3 plotted vs. temperature for the coarse-grained model of PE with N = 20, using the random hopping algorithm (upper set of data) or the slithering snake algorithm (lower set of data), respectively. The time r3 is of the same order as the Rouse relaxation time of the chains, and is defined in terms of a crossing criterion for the mean-square displacements [41], g3(t = r3) = g2(t = r3) [See Eqs. (5.2) and 5.3)]. From [32]...

Fig. 7. Characteristic relaxation rate for the Rouse relaxation in polyisoprene as a function of momentum transfer. The insert shows the scaling behavior of the dynamic structure factor as a function of the Rouse variable. The different symbols correspond to different Q-values. (Reprinted with permission from [39]. Copyright 1992 American Chemical Society, Washington)...

Figure 14 shows the results for Wp, the mode-dependent relaxation rate, for the different molecular masses as a function of the mode index p. For the smallest molecular mass Mw = 2000 g/mol relaxation rates Wp are obtained which are independent of p. This chain obviously follows the Rouse law. The modes relax at a rate proportional to p2 [Eq. (17)]. If the molecular weight is increased, the relaxation rates are successively reduced for the low-index modes in comparison to the Rouse relaxation, whereas the higher modes remain uninfluenced within experimental error. 

From Fig. 35, where the normalized coherent scattering laws S(Q, t)/S(Q,0) are plotted as a function of 2 (Q)t for Zimm as well as for Rouse relaxation, one sees that hydrodynamic interaction results in a much faster decay of the dynamic structure factor. 

To guarantee the transformational invariance of DG = kBT/(NQ (see Table 4) in the case of Rouse relaxation, the replacement of N by N/A, requires the simultaneous replacement of the segmental friction coefficient by X which is natural, since friction is proportional to the number of segments involved. 

Fig. 59. Incomplete screening of hydrodynamic interactions in semi-dilute polymer solutions. Presentation of different regimes which are passed with increasing concentration. A,C Unscreened and screened Zimm relaxation, respectively, B enhanced Rouse relaxation. (Reprinted with permission from [12]. Copyright 1987 Vieweg and Sohn Verlagsgemeinschaft, Wiesbaden)...

If a residual hydrodynamic interaction over large distances does not exist (l/r H(c) = 0), the regime of screened Zimm relaxation vanishes, and only the crossover from unscreened Zimm to enhanced Rouse relaxation remains. 

Figure 64 presents the results of the line-shape analysis for c = 0.18 and c = 0.45. In the first case the polymer relaxation is still determined by the Zimm modes at larger Q-values, while at smaller Q the Rouse modes become dominant. Qualitatively, this behavior is expected for the crossover from unscreened Zimm to enhanced Rouse relaxation. At c = 0.45 the Q-dependence P is... 

With respect to the screening of hydrodynamic interactions, one is confronted with the occurrence of a multiple-transition behavior. Instead of the expected crossover from ordinary (unscreened) Zimm to enhanced Rouse relaxation, one observes, at increasing concentrations, additional transitions from enhanced Rouse to screened Zimm and from screened Zimm to enhanced Rouse relaxation. This sequence of crossover effects are highly indicative of an incomplete screening of hydrodynamic interactions. 

This relationship between the relaxation modes and the low shear viscosity is an important one. It indicates that the longest Rouse relaxation time, i.e. the p = 1 mode ... 

We can define the Rouse relaxation time by Equation (5.92) multiplied by an additional factor for the shortened chains ... 

Figure 6.21 Comparison of Graessley and Doi-Edwards models for normalised viscosity versus normalised shear rate. Also shown is an estimate of the role of short time Rouse relaxation mechanisms within the tube...

It is fairly clear that as re approaches rd the role of Rouse relaxation is significant enough to remove the dip altogether in the shear stress-shear rate curve. As the relaxation process broadens, this process is likely to disappear, particularly for polymers with polydisperse molecular weight distributions. The success of the DE model is that it correctly represents trends such as stress overshoot. The result of such a calculation is shown in Figure 6.23. 

Different equilibrium, hydrodynamic, and dynamic properties are subsequently obtained. Thus, the time-correlation function of the stress tensor (corresponding to any crossed-coordinates component of the stress tensor) is obtained as a sum over all the exponential decays of the Rouse modes. Similarly, M[rj] is shown to be proportional to the sum of all the Rouse relaxation times. In the ZK formulation [83], the connectivity matrix A is built to describe a uniform star chain. An (f-l)-fold degeneration is found in this case for the f-inde-pendent odd modes. Viscosity results from the ZK method have been described already in the present text. 

These are the Rouse relaxation times. For Gaussian chains (fi = ), they become... 

We will take a somewhat different but equivalent criterion in order to describe the crossover. As the crossover time r, we take the Rouse relaxation time of a polymer section, spanning the tube diameter ... 

Fig. 3.13 Schematic sketch of the different time regimes of reptation a unrestricted Rouse motion for b local reptation, i.e. Rouse relaxation along the confining tube, and...

---