Contour Length Fluctuations of the Primitive Chain

According to the Doi-Edwards theory, after time t = Teq following a step deformation at t = 0, the stress relaxation is described by Eqs. (8.52)-(8.56). In obtaining these equations, it is assumed that the primitive-chain contour length is fixed at its equilibrium value at all times. And the curvilinear diffusion of the primitive chain relaxes momentarily the orientational anisotropy (as expressed in terms of the unit vector u(s,t) = 5R(s,t)/9s), or the stress anisotropy, on the portion of the tube that is reached by either of the two chain ends. The theory based on these assumptions, namely, the Doi-Edwards theory, is called the pure reptational chain model. In reality, the primitive-chain contour length should not be fixed, but rather fluctuates (stretches and shrinks) because of thermal (Brownian) motions of the segments. 

Doi and Edwards treated the polymer chain as a Gaussian chain in the time region t T q. They assumed that the equilibration process of segmental redistribution took place before T q. Even though they did not deal with the dynamic behavior of the equilibration process, they used the Gaussian chain model to obtain the stress at the end of the equilibration process (Eq. (8.32)). To be consistent, the dynamic aspect of the Gaussian chain picture needs to be included for t Teq. 

While considering the thermal motions of the segments, the stretch-and-shrink motion of the primitive-chain contour length will help relax the tube stress at both ends of the tube. This effect occurs because when a chain moves out of the tube due to a stretching of the contour length following a 

We first regard the fixed primitive-chain contom length L in Eqs. (8.3) and (8.51) as the time average of the fluctuating length L(t). In other words. 

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 tb, the tube length that still possesses tube stress can be defined by 

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Appendix 9.A — Contour Length Fluctuations of the Primitive Chain... 

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