The Rouse-Bueche Model for Unentangled Polymers

If one attempts to model the dynamics of a single long polymer molecule in a very dilute solution starting from an atomically detailed picture of the molecule, the task rapidly becomes impossibly complicated because of the number of bonds that must be taken into account and the limitations on the motion of the backbone bonds with respect to each other. Fortunately, it is possible to achieve a drastic simplification of the problem if we are not interested in the very short-range motions that are responsible for the initial, very fast stages of relaxation after 

The freely-jointed chain picture is used in the next section as the basis for a model for the viscoelastic behavior of a dilute solution. It is important to keep in mind, however, that we cannot expect the model to describe the very short-time behavior involving short-range interactions between segments of the molecule that are within a submolecule, as these are not accounted for in this coarse-grained picture. 

2 The Rouse Model for the Viscoelasticity of a Dilute Polymer Solution 

Relaxation after deformation results from the restoring entropic-spring force acting against the viscous resistance of the solvent, and the characteristic time of the relaxation is thus proportional to and inversely proportional to kT. Because of the many degrees of freedom in the chain, the relaxation process is actually governed by a series of relaxation times. In the Rouse model, there is one relaxation mode, with relaxation time Tp for each value of the index p, up to N the number of submolecules in the chain, as shown by Eq. 6.1  

We recall that the number of submolecules N is arbitrary within limits, so if terms for which p approaches N made a significant contribution to the sum, the Rouse model would not be valid, as this would imply that phenomena occurring within a submolecule (which are not accounted for in the model) are affecting the stress. It is thus required that the series converge for p somewhat less than N. If the series converges for p NI5, the relaxation time Tp can be accurately approximated by  

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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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