Primitive chain

Figure 5.26. Running along the centre of the tube is a primitive chain. This is the shortest path down the tube. The deviations of the polymer chain from this path can be considered as defects. The motion of these kinks or defects in the chain away from the primitive path allows the chain to move within the tube. The polymer creeps through the tube, losing its original constraints and gradually creating a new portion of tube. This reptilian-like motion of the chain was named by de Gennes from the Latin reptare, to creep, hence reptation.

Figure 6 The tube concept The real chain is trapped between entanglements and is wriggling aroud the "primitive chain" (full line).

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

The primitive chain reptates along itself with a diffusion constant that can be identified as the diffusion coefficient of the Rouse model. Under the action of a force /, the velocity of the polymer in the tube is v =f /, where is the overall friction coefficient of the chain. It is expected that C is related to the friction coefficient of the individual segments, Q, by the expression... 

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 three parameters necessary for the characterization of the primitive chain, L, D, and a can be expressed in terms of the Rouse model parameters N, b, and Thus D is given by Eq. (11.14), while La is equal to Nb, the mean square end-to-end distance of the Rouse chain. As a result, the length of the primitive chain can be written as... 

The theory predicts (Ref. 3, p. 214) that x Z Xr and x Zx, where Z = Lja = Nb /( is the number of steps of the primitive chain, often referred to as the number of entanglements per chain. These relationships have been confirmed by computer simulations (12). 

Doi and Edwards (1978, 1979, 1986). They started with the Rouse-segmented chain model for a polymer molecule. Because of the presence of neighboring molecules, there are many places along the chain where lateral motion is restricted, as shown in Fig. 21. To simplify the representation of these restrictions, Doi and Edwards assume that they are equivalent to placing the molecule of interest in the tube as shown in Fig. 22. This tube has a diameter d and length L. The mean field is represented by a three-dimensional cage. The primitive chain can move randomly forward or backward only along itself. For a monodisperse polymer, the linear viscoelasticity is characterized by... 

Doi and Edwards considered the primitive chain as a freely jointed chain with step length a. The positions of the joints (or links) can be labeled as... 

Then R — Rn-i = o,. Assume that in a time interval At, the primitive chain jumps forward or backward with equal probability one step of length a. Then the curvilinear diffusion constant can be defined by... 

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

Equation (8.36) becomes Eq. (8.2) for a polymer melt and as mentioned above, G r is smaller by a factor of 4/5 than the result based on the theory of rubber elasticity (Eq. (8.1)), where each entanglement is treated as a permanent cross-link. The reduction factor 4/5 is entirely due to the the segmental redistribution along the primitive chain. Prom Eq. (8.31), it can be shown that N" = N m the linear region in other words, the reduction of the number of entanglement strands occurs only in the nonlinear region. 

The segmental distribution along the primitive chain has reached the equilibrium state at t = Teq. In this state, by using Eq. (8.27), Eq. (8.32)... 

Assume in a short time step At the primitive chain moves a curvilinear distance As. Then... 

As discussed in the last chapter, the Doi-Edwards theory describes how the stress initiated by a step deformation relaxes by the reptational process after the equilibration time Teq of the segmental redistribution along the primitive chain. As will be shown below, the reptational process plays the most important role in the terminal region of the relaxation modulus. 

Doi has proposed a theory, which will be discussed in detail in Chapter 12, describing A) as the relaxation of tension on the primitive chain. The theory predicts that the t, A) process is not observable in the linear region, which has been found to be in agreement with experiment. However, corresponding to the dynamics of A), there is a process... 

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... 

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 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

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... 

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... 

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