The tube model

The movement of a chain in a dense polymeric system is highly constrained. Due to entanglements  with other chains lateral motions of the chain at many points are highly improbable.

  

Figure 8.1: An entangled chain.

Qualitatively we may imagine that the chain has available a tube  in which it may move with some freedom. In order to move over large distances the chain has to leave the tube by means of longitudinal motions.

The concept of tube introduced above, clearly has only a statistical meaning. The tube can change by two mechanisms. First by means of the motion of the central chain itself, by which the chain leaves parts of its original tube, and generates new parts. Secondly the tube will fluctuate  because of the motions of the chains which built up the tube.

Situations where the second cause of tube fluctuations are reduced to a minimum, are those of a long chain in a melt of chains which are even much longer, and a chain in a gel. Also in the case of a chain in its own melt, tube fluctuations due to the motions of the environmental chains may be assumed to be negligible. This clearly amounts to a mean field treatment, which will not be able to describe certain collective motions of the system.

Let us now look at the mechanism which allows the chain to move along the tube axis, which is also called the primitive chain .

  

Figure 8.2: Reptation.

The chain fluctuates around the primitive chain. By some fluctuation it may store some excess mass in part of the chain. This mass may diffuse along the primitive chain and finally leave the tube. The chain thus creates a new piece of tube and at the same time destroys part of the tube on the other side. This kind of motion is called reptation .

It is clear from the above picture that the reptative motion will determine the long time motion of the chain. The main concept of the model is the primitive chain. The details of the polymer itself are to a high extent irrelevant. We may therefore choose a convenient polymer as we wish. In this chapter our polymer will be a Gaussian chain . Its motion will be governed by the Langevin equations  at the Smoluchowski time scale . Our basic chain therefore is a Rouse chain .