Primitive with reptation

Despite these complications, there are now numerous evidences that the tube model is basically con-ect. The signatory mark that the chain is trapped in a tube is that the chain ends relax first, and the center of the chain remains unrelaxed until relaxation is almost over. Evidence that this occurs has been obtained in experiments with chains whose ends are labeled, either chemically or isotopically (Ylitalo et al. 1990 Russell et al. 1993). These studies show that the rate of relaxation of the chain ends is distinctively faster than the middle of the chain, in quantitative agreement with reptation theory. The special role of chain ends is also shown indirectly in studies of the relaxation of star polymers. Stars are polymers in which several branches radiate from a single branch point. The arms of the star cannot reptate because they are anchored at the branch point (de Gennes 1975). Relaxation must thus occur by the slower process of primitive-path fluctuations, which is found to slow down exponentially with increasing arm molecular weight, in agreement with predictions (Pearson and Helfand 1984). 

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... 

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 ] ... 

These authors studied how tube models with reptation dynamics could be turned into a full theory of viscoelasticity. To do this one needs to describe the dynamics of the primitive path. Let the primitive chain make one step in time At. Define a random variable, (t) which is + 1( — 1) if the primitive chain moves backward (forward). Let p be a random vector of length a which is the new position of one of the ends of the primitive chain after one move (see Figure 23). Since the primitive chain is assumed to be made of N points R, . Rjy, connected by bonds of constant length a, the Langevin equation of the primitive chain is given by... 

At times t >Teq, the wriggling motion results merely in a fluctuation around the primitive path, so the ch moves coherently in a one-dimension diffusion process, keeping its arc length constant. The macroscopic diffusion coefficient of a reptating chain scales with chain length (molecular weight) as ... 

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. 

The center of gravity of a linear chain now moves by two uncorrelated processes, reptation and constraint release, so the diffusion coefficient is just the sum of the individual contributions. Equation 21 gives the reptation contribution. Equation 18 gives the constraint release contribution with

primitive path steps ... 

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... 

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. 

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. 

Fig. 44. Schematic of the stress relaxation process after a large step in deformation, (a) Equilibrium conformation of the tube prior to deformation, (b) Immediately after deformation, the primitive chain has been affinely deformed, (c) After the time Xr, the primitive chain retracts along the tube and recovers its equilibrium contour length (t=XR). (d) After the time Xd the primitive chain leaves the deformed tube by reptation (t=Xa). After Doi and Edwards (56), with permission.

Now we move the primitive chain. The chain reptates along the tube with a diffusion coefficient The point at i = s(0) moves with the same diffusion coefficient D. We record the slithering motion of the point by the curvilinear distance measured from the end of the primitive chain at i = 0 (gray line), as shown in Figure 4.36. Let the point on the primitive chain slide to s(t) in time t. Then, the one-dimensional displacement s(t) - x(0) along the contour of the primitive chain... 

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