Pure reptation

The two situations are displayed in Fig.3.13b and c. The first process, where the chain performs Rouse motion along the tube, is called local reptation the creeplike diffusion along the tube which eventually leads to a complete tube renewal is also termed pure reptation. 

Outside the experimental error these data show that, similar to the viscosity result, the pure reptation prediction is violated (i) JV -2.3o o.o5 instead of... 

Fig. 3. 30 Apparent tube diameters from model fits with pure reptation (filled squares) and reptation and contour-length fluctuations (open squares) as a function of molecular weight. The dotted line is a guide for the eye (Reprinted with permission from [71]. Copyright 2002 The American Physical Society)...

One can try to locate a critical polymerisation index above which the data are no longer compatible with a Rouse-like dynamics, Ng = 500, lager than the Ng= 100 value determined from the diffusion measurements in a frozen matrix. This is an illustration of the fact that the two processes. Rouse motion and entangled motion are in competition the slowest process is the one which is indeed observed.When the matrix chains are mobile, the entangled dynamics becomes more rapid than pure reptation, and the Rouse motion can dominate the dynamics for larger molecular weights than when the matrix chains are immobile. 

Both expressions predict that x is higher than Xg for monodisperse samples with N higher than about 12 [15] or 4 [17]. This means that a pure reptation description is correct for highly entangled polymers as shown in section 3.1. 

Figure 29 Average relaxation time of a high molecular weight polystyrene (M = 900 000) in the presence of short chains (M = 8 500). The dotted line represents pure reptation and the full line stands for the contribution of tube renewal according to relation (6-8).[from ref. 28]...

We sume that disengagement by pure reptation is negligible for star molecules with diffidently long arms in an entangled medium. (For a contrary opmion, however, based on computer simulation of star molecule motions, see Ref. 33). Relaxation for an f-arm star in a topologically invariant medium is then equivalent to the relaxation of f tethered chains, where is the tethered chain relaxation time (Eq.66) for individual aruK (Rg.l2). 

Path length fluctuations (breathing or tube leakage) will compete with pure reptation in the case of unattached linear chains, although the effect will die cwt fta- suffidentty long chains. Suppose all chains occupy N steps initially. For pure reptation the fraction of still surviving steps at position j, measured from the center of the initial path (j = o, 1, 2,—, N/2), is... 

The deviations from the 3.4 power law at low molar masses (M < MJ are because those chains are too short to be entangled (see Section 8.7.3). The deviations at very high molar mass are consistent with a crossover to pure reptation (see Section 9.4.5),... 

As described above, we can clearly observe three distinct relaxation processes. It is obvious that to fully describe the relaxation modulus, G t) or G t, A), we need to consider other processes in addition to the reptational process. These additional processes may suggest the answer for the deviation of the observed scaling, t] oc M , from the prediction of the pure reptational chain model, p oc M . 

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. 

As shown by the above analyses of the viscoelasticity and diffusion data in terms of the ERT, the paradox between the scaling relations r]o oc M and Dg oc predicted by the pure reptational model, that occurs in their comparison with the experimental results, is resolved. Furthermore, the relation between viscoelasticity and diffusion as given by the ERT is quantitatively supported by the data of polystyrene. The analysis of the viscosity data at Me in relation to the Kd value obtained from the diffusion measurements also supports the ERT. Considering the different nature of the experiments of viscoelasticity and diffusion, the quantitative agreement between these two kinds of data as analyzed in terms of the ERT is remarkable and thus indeed significant. 

Pure reptation is possible only under the very strong topological constraint that puts the instantaneous orientations of all segments of a primitive chain, except ones at its ends, in a complete correlation. In actual entangled systems, since the constraint may not be that strong, the chains are likely to wriggle in modes other than reptation. We have no a priori reason to deny the possibility of such non-reptative chain motions. 

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