Reptational motion

As mentioned in Section 3.1, the incoherent dynamic structure is easily calculated by inserting the expression for the mean square displacements [Eqs. (42), (43)] into Eq. (4b). On the other hand, for reptational motion, calculation of the pair-correlation function is rather difficult. We must bear in mind the problem on the basis of Fig. 19, presenting a diagrammatic representation of the reptation process during various characteristic time intervals. 

Fig. 18. Mean square displacement of a chain segment undergoing reptational motion as a function of time...

The model described by equations (3.42)-(3.45) is valid for equilibrium situations. For chain in a flow, one ought to define displacements of the particles under flow and to consider the average values (3.44) to depend on the velocity gradient (Doi and Edwards 1986). McLeish and Milner (1999) considered mechanism of reptation motion of branched macromolecules of different architecture. 

One can see from the comparison of equations (5.10) and (5.19) that the reptation motion of the macromolecules is revealed at the condition... 

As far as the orientation of the central block is concerned, a slower relaxation is also observed for the star as shown in Figure 12. At long times, the central block of the star remains more oriented than that of the linear chain. The differences are significant and cannot be explained by simple differences between length of chains and blocks. This effect must therefore be attributed to the "pinning" effect of the centre of the chain and subsequent suppression of the reptation motion. 

In that final relaxation process the molecule recovers its final isotropic configuration by a reptation motion. The characteristic time for the reptation process is (see so chapters 1.1 and L3) ... 

Callaghan, P. T., and Coy, A. (1992). Evidence for reptational motion and the entanglement tube in semi-dilute polymer solutions. Phys. Rev. Lett. 68, 3176-3179. 

The word reptation was created by De Gennes in 1971 (see De Gennes23). The term tube model is used to describe complete theories that incorporate Rouse and reptation motions within a tube-like constraint of the surrounding polymer chains. 

The viscosity associated with this reptating motion may now be calculated as... 

This result shows that the diffusion constant of a long polymer chain in a concentrated system, because of the constraint effect of entanglement, is inversely proportional to the square of the molecular weight. This molecular-weight dependence is distinctively different from the result, Dg oc M, given by the Rouse model (Chapter 3) and its observation is often regarded as the indication of the reptational motion. As shown in... 

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... 

Equation (12.5) contains two relaxation processes (v (t)v (t)) is to be randomized or relaxed by the chain reptational motion, and the change in with time represents the chain-tension relaxation. The former is much slower than the latter if the molecular weight is large. At t T q, the chain-tension relaxation process ends and ln t)) returns to its equihbrium value / the stress tensor T(t Teg) becomes... 

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