Primitive chain with fluctuating contour

Fig. 9.3 The Brownian motion of a primitive chain (a) with fixed contour length, and (b) with fluctuating contour length. The oblique lines denote the region that has not been reached by either end of the primitive chain. The length of the region

To sort out such a complicated dynamic situation, we first assume that the primitive chain is nailed down at some central point of the chain, i.e. the reptational motion is frozen only the contour length fluctuation is allowed. This is equivalent to setting rg —> oo while allowing the contour length fluctuation 5L(t) to occur with a finite characteristic relaxation time Tb- In this hypothetical situation, the portion of the tube that still possesses tube stress tt fa tb is reduced to a shorter length Lq, because of the fluctuation SL(t). Then, tt tube length that still possesses tube stress can be defined by... 

Statistical distribution of the contour length In the previous sections we regarded the primitive chain as an inexten-sible string of contour length L. In reality, the contour length of the primitive chain fluctuates with time, and the fluctuation sometimes plays an important role in various dynamical processes. 

For linear polymers, primitive path fluctuations (PPF or CLF for contour length fluctuations ) occur simultaneously with reptation. At short times (or high frequencies) the ends of the chain relax rapidly by primitive path fluctuation. But primitive path fluctuations are too slow to relax portions of the chain near the center, and these portions therefore relax only by reptation. However, the relaxation of the center by reptation is speeded up by primitive path fluctuations, because the tube remaining to be vacated by reptation is shortened, since its ends have already been vacated by primitive path fluctuations. As a result, the longest reptation time Tj (i.e., the terminal relaxation time) and zero-shear viscosity, are lower than in the absence of the fluctuations and can be approximated by the following equation [ 1 ] ... 

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