Primitive chain segments

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 function tjtis, t) will appear frequently in the subsequent discussions. This function has been defined as the probability that the original tube segment s remains at time t. As will be shown in Section 6.3.3, ip s, t) also represents the probability that the primitive chain segment s is in e original tube at time t. (Note the distinction between the tube segment and the primitive chain segment the former is fixed in space, while the latter moves with the primitive chain.)... 

Fig. 6.5. The probability i >(s, t) that the tube segment s is remaining at time t. This is also equal to the probability that the primitive chain segment s remains in...

Stochastic equation for reptation dynamics Although the above probabilistic description is quite useful in understanding the essence of reptation dynamics, it becomes progressively more difficult to proceed with the calculation for other types of time correlation function. For example, it is not easy to calculate the mean square displacement of a primitive chain segment (R(s, t)-R(s, 0)) ) by this method. In this section we shall describe a convenient method" for calculating general time correlation functions. 

For a very short time, say at t < x, the chain segment does not feel the constraints of the tube, so that the mean-square displacement of a primitive chain segment is the same as that calculated for the Rouse model in free space. At time t x, the whole polymer is confined in a deformed tube. As time passes (i.e., at r > t ), the Rouse behavior is stopped, because the chain feels the constraints imposed by the tube, and therefore the reptation behavior starts that is, there exists a time x at which the chain begins to feel the onset of the effect of tube constraints. For t 3> x, part of the polymer near the ends has disengaged from the deformed tube, while the part in the middle is still confined in the tube. Since only the segments in the deformed tube are oriented and contribute to the stress, the G t) in the terminal region is proportional to the fraction of the polymers still confined in the deformed tube 4 (1) (Doi and Edwards 1986), that is... 

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... 

Doi and Edwan argue for a particular relationship betwera stress and orientation They assume that the primitive path stepi deform affinely (Fig. 4) and that the chain segments contained in those steps respond initially like indepemlent Gau an strands. 

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... 

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)... 

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. 

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 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... 

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... 

---