General Rouse model

Some years ago, on the basis of the excluded-volume interaction of chains, Hess [49] presented a generalized Rouse model in order to treat consistently the dynamics of entangled polymeric liquids. The theory treats a generalized Langevin equation where the entanglement friction function appears as a kernel... 

Fig. 26. NSE spectra in polyethyleneo melts at 509 K in a Rouse scaling plot (o Q = 0.078 A-1 Q = 0.116 A-1 A Q = 0.155 A-1. Above Spectra in comparison to a fit of the generalized Rouse model [49]. Below comparison of the data with the predictions of the local reptation model [53] omitting the measurement points which correspond to the initial Rouse relaxation. The arrows indicate Q2/2V/Wxe. (Reprinted with permission from [39]. Copyright 1992 American Chemical Society, Washington)...

In generalized Rouse models, the effect of topological hindrance is described by a memory function. In the border line case of long chains the dynamic structure factor can be explicitly calculated in the time domain of the NSE experiment. A simple analytic expression for the case of local confinement evolves from a treatment of Ronca [63]. In the transition regime from unrestricted Rouse motion to confinement effects he finds ... 

Hess derived a similar expression from his microscopic model by explicitly considering the effective entanglement as a dynamic effect. Hess included the important many chain cooperative effects of constraint release and tube renewal, which are necessary in order to get quantitative predictions for the stress relaxation functions. Ultimately this does not affect the N dependence of the relaxation time. He found that after an initial fast Rouselike decay up to time r, Tp Hess = / irp Rep- Both models describe essentially the same physical picture. For the generalized Rouse model, Kavassalis and Noolandi found that Tp rm N /p. MD simulation results of Kremer and Grest could not distinguish between the standard reptation and Hess models but could rule out the generalized Rouse model. 

The validity of Eqs.(4.10)(4.12) probably extends well beyond the Rouse model itself [characterized by the specific set of rt values in Eq. (4.5)1 and it seems likely that they will apply, at least for small disturbances, whenever the elements supporting the stress are joined by sufficiently flexible connectors and configurational relaxation is driven by simple Brownian diffusion. One might speculate further that these same forms would apply even in concentrated systems, with Eq.(4.10) expressed in a somewhat more general form because of intermolecular interactions ... 

For Q<0, this distribution function is peaked around a maximum cluster size (2Q/(2Q-1))< >, where < > is the mean cluster size. 2Q=a+df1 is a parameter describing details of the aggregation mechanism, where a1 is an exponent considering the dependency of the diffusion constant A of the clusters on its particle number, i.e., A NAa. This exponent is in general not very well known. In a simple approach, the particles in the cluster can assumed to diffusion independent from each other, as, e.g., in the Rouse model of linear polymer chains. Then, the diffusion constant varies inversely with the number of particles in the cluster (A Na-1), implying 2Q=-0.44 for CCA-clusters with characteristic fractal dimension d =l.8. 

Calculate the stress relaxation modulus G(t), valid for all times longer than the relaxation time of a monomer, for a monodisperse three-dimensional melt of unentangled flexible fractal polymers that have fractal dimension V <1. Assume complete hydrodynamic screening. Hint Keep the fractal dimension general and make sure your result coincides with the Rouse model for V — 2. 

This latter model was employed by Rouse (27) and by Bueche (28) in the calculation of viscoelasticity and is sometimes called the Rouse model. It was used later by Zimm (29) in a more general calculation which may be regarded as an application of the Kirkwood theory. As illustrated in Fig. 2.1, the Rouse model is composed of N + 1 frictional elements represented by beads connected in a linear array with N elastic elements or springs, hence the bead-spring model designation. The frictional element is assumed to represent the translational friction... 

There are two forms of phenomenological equations for describing Brownian motion the Smoluchowski equation and the Langevin equation. These two equations, essentially the same, look very different in form. The Smoluchowski equation is derived from the generalization of the diffusion equation and has a clear relation to the thermodynamics of irreversible processes. In Chapters 6 and 7, its application to the elastic dumbbell model and the Rouse model to obtain the rheological constitutive equations will be discussed. In contrast, the Langevin equation, while having no direct relation to thermodynamics, can be applied to wider classes of stochastic processes. In this chapter, it will be used to obtain the time-correlation function of the end-to-end vector of a Rouse chain. 

The illustrative applications in Sects. 14-16 to the momenum, mass, and energy fluxes are given for the Hookean dumbbell and Rouse models. These models are known to be very inadequate because of their mfinite extensibility. They have been used solely to show how to apply the general formulas of Tables 1 and 2. The next phase of study should emphasize calculations using beadspring chain models with finitely extensible springs, using molecular or Brownian dynamics [23, 33d, 33e, 33f]. 

It should be emphasized that the essence of the Rouse model is in the universal nature of the modelling of the dynamics of a connected object. The central assumption in the Rouse model is that the dynamics is governed by the interactions localized along the diain. In fact, if one assumes a linear Langevin equation for R with localized interaction, one ends up with the Rouse model in the long time-scale behaviour. To see this, consider the general form of the linearized Langevin equation... 

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