Tube model in concentrate solutions and melts

In the previous chapter, we discussed the dynamics of a polymer in a fixed network. We shall now discuss the polymer dynamics in concentrated solutions and melts. In these systems, though aU polymers are moving simultaneously it can be argu that the reptation picture will also hold. Consider the motion of a certain test polymer arbitrarily chosen in melts. If the test polymer moves perpendicularly to its own contour, it drags many other chains surroun hng it and will feel a large resistance. On the other hand the movement of the test polymer along its contour will be much easier. It will be thus plausible to assume that the polymer is confined in a tube-like region, and the major mode of the dynamics is reptation. 

Qearly it is a crude simplification to characterize the effect of the environment by a single parameter a. A more appropriate description would be to assign a viscoelastic character to the environment. However, we proceed here using the simplest possible model. As we shall show later various experimental results indicate that this model is adequate for linear polymers with narrow distribution of molecular weight. Limitations of the model will be discussed later. 

Within this framework, one can draw a rather simplified picture about the dynamics of the polymers in the entangled state. If the characteristic length scale of a motion is smaller than a, the entanglement effect is not important, and the dynamics is well described by the Rouse model (or the Zimm model if the hydrodynamic interaction is not screened). On the other hand, if the length-scale of the motion becomes larger than a, the dynamics is governed by reptation. 

That the short time-s e (or small length-scale) motion of the polymer 

On the othmr hand there is much experimental evidence which indicates that leptation is the dominant motion governing the dynamics in the highly entangled state. 

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