Rouse-modes

The summation extends over tire - 1 Rouse modes witli relaxation time i . ) = 1,. . AT> - 1, aiid c is tire... 

The normal modes (Rouse modes) that characterize the internal dynamics of the polymer can be computed exactly for a Gaussian chain and are given by... 

Rouse modes, multiparticle collision dynamics, polymers, 123-124... 

Fig. 17. Time-dependent correlators for different Rouse modes in the Hess model. The calculations where performed for PE of Mw = 6500 g/mol. The mode numbers increase from the top commencing with p = 1. (Reprinted with permission from [36]. Copyright 1994 American Chemical Society, Washington)...

A different estimate, which identifies xe with the slowest Rouse mode of a chain with the end-to-end distance, R2 = d2 leads to a quite similar result... 

Local reptation regime For times t > xe we have to consider curvilinear Rouse motion along the spatially fixed tube. The segment displacement described by Eq. (18) (n = m) must now take the curvilinear coordinates s along the tube into consideration. We have to distinguish two different time regimes. For (t < xR), the second part of Eq. (19) dominates - when the Rouse modes... 

In contrast to the Rouse modes, which have the dispersion xp 1 p2, the pure Zimm modes (B 0 and Bx/2N/p 1) lead to... 

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... 

Figure 13 Temperature dependence of the time scales for the first five Rouse modes in the bead-spring model in the vicinity of the MCT Tc. 

The movement from the tube is restricted to odd-numbered Rouse modes, and this is determined by the nature of the motion required for a defect ... 

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. 

A second appealing feature of tube model theories is that they provide a natural hierarchy of effects which one can incorporate or ignore at will in a calculation, depending on the accuracy desired. We will see how, in the case of linear polymers, bare reptation in a fixed tube provides a first-order calculation more accurate levels of the theory may incorporate the co-operative effects of constraint release and further refinements such as path-length fluctuation via the Rouse modes of the chains. 

Since l is proportional to and q is proportional to 1/L, i is proportional to. Substitution of Eq. (67) into Eq.(62) gives the Langevin equation for the Rouse modes of the chain within the approximations of preaveraging for hydrodynamic interactions and mode-mode decoupling for intersegment potential interactions. Equation (62) yields the following results for relaxation times and various dynamical correlation functions. 

Using the Rouse mode variable q and employing the preaveraging approximation, we obtain... 

According to the Rouse model the mode correlators (Eq. 3.14) should decay in a single exponential fashion. A direct evaluation from the atomic trajectories shows that the three major contributing Rouse modes decay with stretched exponentials displaying stretching exponents jSof (1 13=0.96 and 2,3 jS=0.86). We note, however, that there is no evidence for the extreme stretching of in-... 

In a real chain segment-segment correlations extend beyond nearest neighbour distances. The standard model to treat the local statistics of a chain, which includes the local stiffness, would be the rotational isomeric state (RIS) [211] formalism. For a mode description as required for an evaluation of the chain motion it is more appropriate to consider the so-called all-rotational state (ARS) model [212], which describes the chain statistics in terms of orthogonal Rouse modes. It can be shown that both approaches are formally equivalent and only differ in the choice of the orthonormal basis for the representation of statistical weights. In the ARS approach the characteristic ratio of the RIS-model becomes mode dependent. 

This is a large area of research, and significant progress in this direction has already been made by Schweizer [199] and by Freed and coworkers [200], However, these two schemes are entirely different. While the approach of Freed pays special attention to the coupling between different Rouse modes, the formulation of Schweizer is more in line with the mode coupling theory of liquids with monomer density as the slow collective mode. The main... 

The situation of a freely-draining macromolecule without excluded-volume effects and internal viscosity, when zv = 2, and the above eigenvalues reduce to (1.17), is especially simple. In this case, equation (2.29) describes Rouse modes, and it is convenient to use the largest orientation relaxation time... 

These equations describe the reptation normal relaxation modes, which can be compared with the Rouse modes of the chain in a viscous liquid, described by equation (2.29). In contrast to equation (2.29) the stochastic forces (3.47) depend on the co-ordinates of particles, equation (3.48) describes anisotropic motion of beads along the contour of a macromolecule. 

One can see that the approximation of the theory, based on the linear dynamics of a macromolecule, is not adequate for strongly entangled systems. One has to introduce local anisotropy in the model of the modified Cerf-Rouse modes or use the model of reptating macromolecule (Doi and Edwards 1986) to get the necessary corrections (as we do in Chapters 4 and 5, considering relaxation and diffusion of macromolecules in entangled systems). The more consequent theory can be formulated on the base of non-linear dynamic equations (3.31), (3.34) and (3.35). 

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