Bueche-Rouse model

In this article and its precursor (4) we have presented the mathematical and physical consequences of a model of polymer dynamics which consider the reorientation of monomer level damped torsional oscillators (DTO model). This mechanism was compared and contrasted with the Rouse-Bueche (RB) model which is concerned with motions of large scale segments of the macromolecular chain. The discussion accounts for certain viscoelastic and dielectric properties of polymers. 

In an earlier section, we have shown that the viscoelastic behavior of homogeneous block copolymers can be treated by the modified Rouse-Bueche-Zimm model. In addition, the Time-Temperature Superposition Principle has also been found to be valid for these systems. However, if the block copolymer shows microphase separation, these conclusions no longer apply. The basic tenet of the Time-Temperature Superposition Principle is valid only if all of the relaxation mechanisms are affected by temperature in the same manner. Materials obeying this Principle are said to be thermorheologically simple. In other words, relaxation times at one temperature are related to the corresponding relaxation times at a reference temperature by a constant ratio (the shift factor). For... 

Calculations with molecular structure models, as performed by Bueche, Rouse and others, in general based on bead spring models, predict that for monodisperse polymers the largest relaxation time is equal to... 

We assume that the basic relaxation strengths in the polystyrene relaxation spectrum as described by Ferry and others [5], based on a Rouse-Bueche [6] model, are maintained in the ionomer. A normal wedge-box distribution is assumed. The ions are in multiplets below 6 mole % ionic groups [2]. If a segment... 

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... 

Chapter 6 treats mean-field theories of melt behavior. We begin with the Rouse model for molecules in dilute solution and its modification by Bueche to deal with unentangled melts. The longest Rouse relaxation time emerges from this treatment and plays an important role in all molecular models. The tube model is introduced, in which the basic relaxation... 

While the Rouse model was originally intended to describe dilute polymer solutions, Bueche [6] noted that the freely-jointed chain model should be able to describe the behavior of an unentangled melt. It has been found experimentally that the static interactions between a polymer molecule and its surroundings are normally the same in the melt as in a solution in its theta state, although Krishnamoorti et al. [7] have noted a few cases where chain dimensions are different in the melt and at the theta state. They attribute this to the ability of some theta solvents to induce a conformer population different from what is favored in the melt state . 

We refer to this model as the bead-spring model and to its theoretical development as the Rouse theory, although Rouse, Bueche, and Zimm have all been associated with its development. 

The Rouse-Bueche model (97,98) replaces the real molecule of n main chain atoms by a mechanical chain of N +1 beads joined in sequence by N linear springs. The frictional interactions with the medium, which are distributed uniformly along the length of the real molecule to give a molecular frictional coefficient n(0, ate concentrated at regular intervals in the beads. The frictional... 

Thus the relaxation spectrum resulting from the average coordinates equation11 of our model has the same form as that of Rouse, of Kargin and Slonimiskii, or of Bueche. In order to relate the parameters of the model to those of the Rouse theory, the time scale factor a must somehow be connected to the frictional coefficient for a single subchain of a Rouse molecule. To achieve this comparison, we may23 study the translational diffusion coefficients as computed for the two models. 

The DTO model predicts that rmax is relatively insensitive to polymer concentration, in distinction to the results of Rouse and Bueche. The concentration dependence of the dielectric should therefore be a most interesting point of study and should distinguish between the two mechanisms. 

Rouse-Bueche model, the relaxation rate constants of a segment containing n ions is K/n2. Such an assumption is also justified by the relaxation behavior of ionomers of various degrees of neutralization [7]. The model detailed below was applied to the data of Navratil in order to quantify it [4,8],... 

Compare the Rouse-Bueche theory with the de Gennes theory. How do they model molecular motion ... 

Figure 5.12 Sketch of steady-state compliance versus molecular weight for samples of a monodisperse, linear polymer. Below the linear increase is in accord with the Rouse-Bueche model (Eq. 5.9), while above this critical molecular weight, a further increase is suppressed by entanglements (Eq. 5.10).

As the arm length increases above M q, we expect the onset of entanglement to cause marked deviations from this relationship. However, for star polymers it is observed that continues to increase linearly with M, in accord with the Rouse-Bueche model for Hnear, unentangled polymers. This is in contrast to the behavior of entangled, linear, monodisperse melts, for which 7s° is independent of M at large M as shown by Eq. 5.10. Figure 5.21 shows data of Graessley and Roovers for four and six arm polystyrenes [90]. The horizontal line is based on... 

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