Relaxation time 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... 

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]...

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 ... 

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 ... 

The longest unshifted Rouse relaxation time is only proportional to M2. The extra power of M in the relaxation time is consistent with the circulation contribution to t]0 which was described in the preceeding section. This modification together with Eq.(6.57) suggests a simple Rouse spectrum shifted to longer times. 

In this case the theory, apart from the characteristic Rouse relaxation time r, contains three more parameters, namely the relaxation time r of the medium, the measure B of the increase in the resistance of the particle when it moves among the chains, and the measure of internal viscosity E associated with resistance to the deformation of the coil due to the present of ambient macromolecules. 

Here, are the relaxation times of the macromolecule in a monomer viscous fluid - Rouse relaxation times... 

These are exactly the known results (Doi and Edwards 1986, p. 196). The time behaviour of the equilibrium correlation function is described by a formula which is identical to formula for a chain in viscous liquid (equation (4.34)), while the Rouse relaxation times are replaced by the reptation relaxation times. In fact, the chain in the Doi-Edwards theory is considered as a flexible rod, so that the distribution of relaxation times naturally can differ from that given by equation (4.36) the relaxation times can be close to the only disentanglement relaxation time r[ep. 

One can see that the relaxation times at a > (ip/x) 2 are the Rouse relaxation times of the part of the macromolecule that correspond approximately to the length of the macromolecule between adjacent entanglements Me. There is an interval between slow and fast relaxation times, which is the bigger the longer the macromolecules. 

At the end of the first relaxation process, the chain is still inside the old tube that existed right after the deformation, in the sense that the orientations of its parts are those produced by the deformation. The chain renews the tube by reptation, thus relaxing those orientations. This process requires a time that is much greater than the Rouse relaxation time Xr. The two processes merge into a single one for unentangled chains. In this case the chains relax according to the Rouse time Xr. 

Zimm relaxation time corresponding to the correlation blob [Eq. (8.75)]. The longest time scale is the Rouse relaxation time of the chain of correlation blobs [Eq. (8.78)]. 

The Rouse relaxation time tr of a branched polymer of N monomers with size / is a generalization of Eq. (8.138) ... 

The reptation time of the P-mer is Tep(P) and the constraint release time Tube given in Eq. (9.85). The faster of the two types of motion controls the diffusion of the P-mer. For constraint release to significantly affect terminal dynamics, the Rouse relaxation time of the confining tube Ttube must be shorter than the reptation time of the P-mer Tep( ) ... 

The second critical shear rate is much higher (i.e., 100 times Chauveteau, 1981) than the first one. The first critical shear rate is equal to the inverse of the longest rotational relaxation time k in the solution. Dilatancy starts as soon as the product of Rouse relaxation time and the maximum stretch rate, e, is greater than 4 (Chauveteau, 1981). The Rouse relaxation time demarcates the onset of entanglement effects (Roland et al., 2004). Chauveteau reported that the ratio of shear rate y to the maximum stretch rate e at the contraction was about 2.5 by laser anemometry for similar polymer solutions and flow geometries. Therefore, the second stretch rate (elongation rate) corresponds to the product of shear rate and Rouse relaxation time equal to 10. 

Roland, C.M., Archer, L.A., Mott, P.H., Sanchez-Reyes, J., 2004. Determining Rouse relaxation times from the dynamic modulus of entangled polymers. Journal of Rheology (J. Rheol.) 48 (2), 395 03. 

Chauveteau and co-workers 24, 48) examined the flow of PEO and HPAA through the extensional flow produced in severe constrictions. They concluded that a coil-stretch transition was responsible for the dilatant behavior observed, and that the critical shear rate required was of the order of 10 times the reciprocal of the Rouse relaxation time. Perhaps the most extensive studies have been those of Haas and co-workers (25, 26, 49). They have explored the critical dilatant behavior on flow through porous media and pursued the hypothesis that the phenomenon is primarily due to a coil-stretch transition beyond a critical deformation rate. They attempted a semiquantitative description based upon the dependence of the lowest order relaxation time of the random coil upon polymer type, molecular weight, solvent quality, and ionic environment. 

Estimate the longest Rouse relaxation time, x, for a polyethylene chain with 154 carbon atoms, C154 at 448 K, making use of the following parameters ... 

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