Reptation of a branched chain

The distance CyC, measured in units of the mesh size, is S. The displacement Co C reduces the entropy of the side group by an amount S In z, where z is the number of gates surrounding one unit cell in the network. As soon as p exceeds a few units, this entropy defect is large, and there is a strong elastic force that tends to bring C back to Co- 

This leads us to compute the probability that E, retraces its steps exactly. The total number of paths with N, steps starting from C is where z is the number of nearest neighbors to one site on the lattice in Fig. VIII.7. We want to count the fraction P of these paths for which 1) the end point is at the origin, and 2) the path is topologically equivalent to zero—i.e., it has the tree structure sliown in Fig. VIII.8. 

We can assume that the tube mobility fitube for the backbone is essentially reduced by a factor/, or exp(-aN,), (omitting all weak prefactors). Thus, as soon as N, is large, the reptation time for a chain with one (or more) side groups becomes exponentially long 

This theoretical conclusion has not been confirmed by direct reptation experiments, but it has some implications. Mechanical measurements on strongly entangled, high molecular weight chains may be completely dominated by the presence of a few branch points. If exponential laws such as eq. (VIII.23) are involved, we need only a small fraction of branch points, and such fractions cannot be detected by standard physicochemical methods. We conclude that mechanical measurements in long chain systems can be extremely sensitive to certain chemical defects. Unfortunately, we do not have reptation data on controlled branched polymers. We do have data on mechanical properties of branched melts,but the melt problem is much more complex than the reptation problem, as shown in next section. 

A similar problem occurs when we have long dangling ends in the gel such as shown in Fig. Vin.9. The above discussion on renewal of conformations applies also to this type of chain which is attached at one end. Again we expect chain relaxation to be severely quenched whenever the dangling end is much longer that the mesh size. 

---