The Dynamics of a Primitive Chain

Let us now consider the dynamics of a primitive chain. Within the concept of reptation, in the short timescale the motion of the polymer can be regarded as wriggling around the primitive path. On a longer timescale, the conformation of the primitive path changes as the polymer moves, creating and destroying the ends of the primitive path. In the absence of an external potential, the time evolution (i.e., the dynamics) of the primitive 

The fraction of all steps initially located at s that are still occupied after time t can be obtained by integrating Eq. (4.105), yielding 

Now the fraction of segments in the primitive chain at time t, which is still in the tube defined at time t = 0 (original tube), can be found by integrating Eq. (4.107), yielding 

Ihe firsl relaxalion process can be neglecled and thus the relaxation for f is only due to the disengagement. This can be described as follows. The timescale of the first relaxation process is essentially the Rouse relaxation times, thus from Eq. (4.53), by replacing N with N, we have 

For a very short time, say at t x, the chain segment does not feel the constraints of the tube, so that the mean-square displacement of a primitive chain segment is the same as that calculated for the Rouse model in free space. At time t x, the whole polymer is confined in a deformed tube. As time passes (i.e., at r t ), the Rouse behavior is stopped, because the chain feels the constraints imposed by the tube, and therefore the reptation behavior starts that is, there exists a time x at which the chain begins to feel the onset of the effect of tube constraints. For t 3 x, part of the polymer near the ends has disengaged from the deformed tube, while the part in the middle is still confined in the tube. Since only the segments in the deformed tube are oriented and contribute to the stress, the G t) in the terminal region is proportional to the fraction of the polymers still confined in the deformed tube 4 (1) (Doi and Edwards 1986), that is 

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