Tube and Primitive Chain

1 Tube and Primitive Chain In the preceding subsection, we paid attention to the short-time motion of monomers within a blob. The motion does not involve translation of the polymer chain as a whole. Here we look at the overall motion of the chain over a distance longer than the blob size. 

Polymer melts and semidilute and concentrated solutions of polymer are highly viscous. Even at a concentration of 1 wt %, solutions of polymer with a molecular weight greater than several million g/mol can flow only slowly. Their behaviors are even elastic like rubber at accessible time and frequency ranges. These exquisite properties had eluded researchers for decades until the tube model and the reptation theory elegantly solved the mystery. The tube model and the reptation theory were introduced by de Gennes. They were refined and applied to the viscoelasticity of semidilute solutions of polymers and polymer melts in the late 1970s by Doi and Edwards. Until then, there had been no molecular theory to explain these phenomena. We will leam the tube model and the reptation theory here. 

Later we will find that the test chain that follows the above assumptions makes a diffusional motion in the solution at sufficiently long times, although it is slow. The freedom allowed for the ends of the test chain makes the diffusion possible. 

We learned in Section 4.2.2 that the polymer chain in the sanidilute solution takes a conformation of an ideal chain. We therefore can use a random-walk model to construct the test chain. Let the random walk consist of N independent steps of step length b. Then, the contour length of the test chain is Nb, and the mean square end-to-end distance is Nb.  

It is convenient to define a primitive chain. It is the centerline of the tube. The test chain wiggles around the primitive chain (Fig. 4.31). The motion of the primitive chain is nothing more than a coarse-grained view for the motion of the test chain it represents. Wiggling motion in the short time scale is averaged to form the primitive chain. 

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