Primitive chain contour length fluctuation

The effect of the primitive-chain contour length fluctuation was first considered by Doi. In an approach which was meant to be an approximation, Doi obtained the zero-shear viscosity and the steady-state compliance as... 

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. 

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... 

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

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

While the [t, E) relaxation is going on, the relatively slow relaxation of (v v ) by the reptational motion also gets under way. At t when the primitive chain has recovered its equilibrium contour length, the effect of the contour length fluctuation on the terminal relaxation should basically be the same as that in the linear region. In other words, the relaxation of (v (t)vn(t)) should be described by Eqs. (9.11) and (9.12). Thus from Eq. (12.24), we write the stress relaxation after t as... 

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 ] ... 

The primitive chain has a constant contour length L, so fluctuations of the contour length are neglected. 

The first assumption corresponds to neglecting the fluctuations of the contour length. The second states that the motion of the primitive chain is reptation. The third guarantees that the conformation of the primitive... 

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. 

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