Segments Rouse

K(R),Kg Entropic spring constant, (7.41), spring constant for a Rouse segment, (7.55). 

Doi and Edwards (1978, 1979, 1986). They started with the Rouse-segmented chain model for a polymer molecule. Because of the presence of neighboring molecules, there are many places along the chain where lateral motion is restricted, as shown in Fig. 21. To simplify the representation of these restrictions, Doi and Edwards assume that they are equivalent to placing the molecule of interest in the tube as shown in Fig. 22. This tube has a diameter d and length L. The mean field is represented by a three-dimensional cage. The primitive chain can move randomly forward or backward only along itself. For a monodisperse polymer, the linear viscoelasticity is characterized by... 

Fig. 3.1 Chain dynamics at different length scales microstructure, Rouse segment, entanglement strand, and whole chain and some of the usual techniques for probing them.

In Chapter 3, we used the Rouse model for a polymer chain to study the diffusion motion and the time-correlation function of the end-to-end vector. The Rouse model was first developed to describe polymer viscoelastic behavior in a dilute solution. In spite of its original intention, the theory successfully interprets the viscoelastic behavior of the entanglement-free poljuner melt or blend-solution system. The Rouse theory, developed on the Gaussian chain model, effectively simplifies the complexity associated with the large number of intra-molecular degrees of freedom and describes the slow dynamic viscoelastic behavior — slower than the motion of a single Rouse segment. 

The elastic dumbbell model studied iu Chapter 6 is both structurally and djmamicaUy too simple for a poljmier. However, the derivation of its constitutive equation illustrates the main theoretical steps involved. In this chapter we shall apply these theoretical results to a Gaussian chain (or Rouse chain) containing many bead-spring segments (Rouse segments). First we obtain the Smoluchowski equation for the bond vectors. After transforming to the normal coordinates, the Smoluchowski equation for each normal mode is equivalent in form to the equation for the elastic dumbbell. 

From Chapter 1, we know that the probability for the end-to-end vector to be R in a Gaussian chain with N Gaussian (or Rouse) segments is given by... 

The force constant, 3kT/b, used in Eq. (9.A.2) originates from the entropy related to the number of microstructural configurations in a Rouse segment. The microstructural configurations have to be considered in three-dimensional space. Involving three-dimensional space is related to the idea that the size of the Rouse segment, b, is much smaller than the tube... 

Here, the internal viscosity is defined as the contribution of the glassy-relaxation process to the zero-shear viscosity. This definition is different from the common understanding of the term used in literature/ although both have similar notions as to the existence of an effect of fast sub-Rouse-segmental motions on polymer viscoelasticity. In the literature the term internal viscosity generally refers to the effect that would lead to a plateau value of the intrinsic viscosity at high frequencies. 

No shift along the modulus axis is involved in obtaining the very close superposition of the measured and calculated spectra as shown in Fig. 11.9. This indicates that the force constant on the Rouse segment, which gives... 

Consider a single chain having N entanglement strands with the slip-link positions at Rq, Ri,. .., Rat. Let the number of Rouse segments on the th entanglement strand be denoted by AT. Then the total number of Rouse segments of the chain is... 

Let Sn be the point on the primitive-chain contour corresponding to the nth Rouse segment. Denote the positions of Sn in three-dimensional space before and after a step deformation E is applied as R°(S ) and R(S (t)), respectively. Then, before the application of E, the length vector /()v° along the primitive chain corresponding to the nth Rouse segment is given by... 

And at time t following the initial application of E, the length vector corresponding to the nth Rouse segment is given by... 

If the molecular weight of a Rouse segment, m, is known, the upper modulus bound G (w —> oo) = pRT/m (or in terms of Young s modulus. 

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