Primitive chain relaxation time

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. 

Fig. 10 - A given chain embedded in a network of chemically identical chains. Different relaxation stages of the primitive chain after a step uniaxial stretching (a) initial iso] ropic state (b) step-strained primitive chain at time 0 (c) primitive chain at the end of the self-retraction process (d) primitive chain during reptation (e) primitive chain after complete disengagement. Slip-links are represented ty small circles. For easier comparison, in each figure the state of the primitive chain at the end of the previous stage is represented by a dotted line. Newly created parts of the tube are drawn in heavy line.

If both ends of the molecule are free to move, and so the chain can reptate, segments in the interior of the chain will relax faster by reptation than by primitive-path fluctuations, and so reptation will control the longest relaxation time of the chain. However, because primitive-path fluctuations are so much faster for the chain ends than for the chain center, the chain ends will still relax by primitive-path fluctuations. Only for very high molecular weights (MfMg 100) are the contributions of fluctuations confined to small enough portions of the chain ends that these effects can be neglected. 

In the above, X is the chain stretch, which is greater than unity when the flow is fast enough (i.e., y T, > 1) that the retraction process is not complete, and the chain s primitive path therefore becomes stretched. This magnifies the stress, as shown by the multiplier X in the equation for the stress tensor a, Eq. (3-78d). The tensor Q is defined as Q/5, where Q is defined by Eq. (3-70). Convective constraint release is responsible for the last terms in Eqns. (A3-29a) and (A3-29c) these cause the orientation relaxation time r to be shorter than the reptation time Zti and reduce the chain stretch X. Derive the predicted dependence of the dimensionless shear stress On/G and the first normal stress difference M/G on the dimensionless shear rate y for rd/r, = 50 and compare your results with those plotted in Fig. 3-35. 

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. 

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... 

The longest relaxation time of (P(l) P(0)) is given by tj. This is called the reptation or disengagement time, since it is the time needed for the primitive chain to disengage from the tube it was confined to at r = 0. 

Ilg. 7J2. Explanation of the stress relaxation after large step strain, (a) Before deformation the conformatian of the fnimitive chain is in equilibrium (r = —0). (b) Immediately after deformation, the primitive chain is in the afiindy deformed conformation (t = -1-0). (c) After time Tj, the primitive chain contracts along the tube and recovers the eqi brium contour length (t Tj,). (d) After the time Xj, the primitive chain leaves the deformed tube by reptation (t Xa). The oblique lines indicates the deformed part of the tube. Reproduced from ref. 107. 

The basic idea, proposed by de Gennes [23], is that relaxation mechanism of linear pendant chains is governed by the reptation or snake-like motion of the chains retracting along their primitive path from the free end to the fixed one. This model proposed that the relaxation time of pendant chains should increase exponentially with the number of entanglements in which it is involved. Pendant chains must then contribute to viscoelastic properties for frequencies greater than the inverse of reptation times. Tsenoglou [26], Curro and Pincus [27], Pearson and Helfand [24] and Curro et al. [25] developed models for the relaxation of pendant chains in random cross-linked networks. 

The test chain would follow the dynamics of the unrestricted Rouse chain if the entanglements were absent, as would the primitive chain at f > t. In Section 3.4.9, we considered the mean square displacement of monomers on the Rouse chain. We found that the dynamics is diffusional at r < and Tj < t, where % is the relaxation time of the Mth normal mode but not in between. When the motion of the Rouse chain is resnicted to the tube, the mean square displacement of monomers along the tube, ([ (t) - x(0)] ), will follow the same time dependence as the mean square displacement of the unrestricted Rouse chain in three dimensions. Thus, from Eqs. 3.240 and 3.243,... 

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