Rouse-time

When M M, disentanglement is nearly instantaneous but approaches tro when M 8Me, which is the strain hardened (A. 4) upper bound for chain pullout without bond mpture. For welding, the relaxation times Trq refer to the minor chains of length /(/) such that the retraction time is approximated by Tro /(/). When M > 8Me, the chains cannot disentangle completely at the Rouse time and... 

The other important physical assumption is that the friction is local (hydro-dynamic interactions are screened in the melt [2]) so that D -(N /N)D with the diffusion constant in the melt of an unentangled chain of segments. Now the characteristic relaxation (Rouse) time of an entanglement segment % is just a /D. so that... 

For times less than the Rouse time of an entanglement segment, Tg and short distances, the chain behaves as if it were free since no section has moved far enough to be strongly affected by the tube constraint. The characteristic decay-rate of the scattering function at wavevector k is dominated by the Rouse-time of chain segments whose size is the order of k % k. A detailed calculation gives for t % [2]... 

Henceforth we take the primitive path co-ordinate s=L-z from the free end inwards to the branch point so that t(s) is an increasing function of s. The prefactor Tq is an inverse attempt frequency for explorations of the potential by the free end, and may be expected to scale as the Rouse time for the star arm (in fact this is not quite true - the actual scaling is as [25,26]). The relaxation mod-... 

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 ... 

Characteristic Rouse times for chains with bending elasticity Terminal time for reptation Crossover time Rouse, local reptation Characteristic Rouse times for the all-rotational model Rouse time... 

The short chains with MRouse chains with relaxation times on the order of the experimental time-scale. For example, the Rouse times of PSD3 and of PSDIO are approximately 9s and 19s seconds, respectively at 115°C [35,36]. Consequently, the residual orientation at long times for these chains can be attributed to orientational coupling interactions with the long chains of the polymer matrix. Similar orientational correlations have been observed on various systems by NMR spectroscopy studies of stretched elastomers where even dissolved solvent molecules and free chains were shown to possess a very short-length scale local orientation [37]. 

If the Rouse time is determined from the experimental value of the zero-shear viscosity, one finds X.i 615s for sample SI at 123°C, which is not too far from the experimental value of the weight-average relaxation time (X. 380s). Clearly, the determination of and Xi from the experimental value of qo implies the assumption of monomeric friction enhancement by entanglements [20], since for sample SI M is of the order of 3xMc. 

Rouse time of an N-chain with an elementary time x directly connected to the reptation time Xc of each passing chain. 

For weakly entangled monodisperse and polydisperse polymer melts, J. des Cloizeavuc [26] proposed a theory based on time-dependent diffusion and double reptation. He combines reptation and Rouse modes in an expression of the relaxation modulus where a fraction of the relaxation spectrum is transferred from the Rouse to the reptation modes. Furthermore, he introduces an intermediate time Xj, proportional to M2, which can be considered as the Rouse time of an entangled polymer movii in its tube. But, in the cross-over region, the best fit of the experimental data is obtained by replaced Xj by an empirical combination of... 

The Carreau model not only described well the flow data of LB gum solutions, but the magnitudes the time constant (Ac) were in good agreement with those of Rouse time (tr) constant derived from solution viscosity data while the Cross (oc) time constants were lower in magnitudes both the Carreau and the Cross time constants followed well power relationships with respect to the concentration (c) of the solutions (Lopes da Silva et al., 1992) ... 

Relaxation time, time constant, s Retardation time, time constant, s Rouse time constant, s Dimensionless time, t/< tV/L Cone angle, radians... 

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. 

The polymer diffuses a distance of the order of its size during a characteristic time, called the Rouse time, Tr ... 

The Rouse time has special significance. On time scales shorter than the the chain exhibits viscoelastic modes that shall be described in... 

Section 8.4. However, on time scales longer than the Rouse time, the motion of the chain is simply diffusive. 

The reciprocal of the fractal dimension of the polymer (see Section 1.4) is p. For an ideal linear chain p= j2 and the fractal dimension is l/i = 2. The Rouse time of such a fractal chain can be written as the product of... 

For an ideal linear chain, and the Rouse time is proportional to the... 

The Zimm time is proportional to the pervaded volume of the chain. Note that the Zimm time xz has a weaker dependence on chain length than the Rouse time tr [Eq. (8.16)]. 

Comparison of Eqs (8.16) and (8.25) reveals that the Zimm time is shorter than the Rouse time in dilute solution. In principle, a chain in dilute solution could move a distance of order of its size by Rouse motion, by Zimm motion, or some combination of the two. The chain could simply move its monomers by Rouse motion through the solvent without dragging any of the solvent molecules with it, or it could drag all of the solvent in its pervaded volume with it, thereby moving by Zimm motion. In dilute solution, Zimm motion has less frictional resistance than Rouse motion, and therefore, the faster process is Zimm motion. The chain effectively moves as though it were a solid particle with volume of order of its per-... 

The relaxation time of a monomer, tq [Eq. (8.15)] is the shortest relaxation time of the Rouse model, with mode index p = N, making xjyi = TQ. The mode with index p = 1 is the longest relaxation mode of the chain with relaxation time equal to the Rouse time ti tr, and corresponds... 

The Rouse time tr is the longest stress relaxation time [Eq. (8.18)]. 

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