Primitive chain contour length

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. 

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... 

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... 

The average primitive path contour length (L) is shorter than the contour length of the chain bNhy the factor afb because each entanglement... 

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

Let Sn be the point on the primitive-chain contour corresponding to the nth Rouse segment. Denote the positions of Sn in three-dimensional space before and after a step deformation E is applied as R°(S ) and R(S (t)), respectively. Then, before the application of E, the length vector /()v° along the primitive chain corresponding to the nth Rouse segment is given by... 

The primitive path is therefore much smaller than the contour length along the backbone of the chain. 

The tube itself is a random walk, each step of which has length a. This random walk is called the primitive path of the chain. The contour length of the tube, or the primitive path, is therefore Lj — aMfMg. For polymers of high molecular weight, the tube s contour length is much less than the contour length of the chain (see Fig. 3-24). Thus, the chain meanders about the primitive path. Some values for the tube diameter a for typical polymer melts are presented in Table 3-3. 

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

The conformation of the primitive chain becomes Gaussian on a large length scale. This means that if the position of two points on the primitive chain are r(, t) and r(s, t), where 5 and s are the contour lengths measured from the chain end, then... 

The average contour length (L) of the primitive path (the centre of the confining tube, see Fig. 7.10) is the product of the entanglement strand length a and the average number of entanglement strands per chain N/N. ... 

At the same time, the contour length of the primitive chain, L", recovers its equilibrium value, L. Thus,... 

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... 

At t = Teg, the tube stress resides over the whole tube length — the average contour length of the primitive chain L (Eqs. (8.3) and (8.51)). From Eq. (9.2) we have... 

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

The time-correlation function 5L 0)5L t)) of Eq. (9.3) will be derived by considering the polymer chain as a Gaussian chain consisting of No segments each with the root mean square length b. Let 5 (t) be the contour position of the nth bead relative to a certain reference point on the primitive path. Then the contour length of the primitive chain at time t is given by... 

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... 

To denote a point on the primitive chain, we use the contour length s measured from the chain end and call this the primitive chain segment s. If R(s, t) is its position at time t, the vector... 

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... 

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