Reptation model primitive path

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 reptation model assumes the contour length of the primitive path is fixed at its average value (L). In reality, the primitive path length... 

These g t) data do not favor the reptation model, but are not definitive enough to rule out the possibility of reptation motion. Thus, Kolinski et al. [57] stepped further by calculating the longitudinal and lateral displacements of the primitive path used in the Doi-Edwards theory. Such calculations should give us direct information on local chain motions. 

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. 

In contrast to D, the prediction of other viscoelastic properties, such as the friction coefficient f or the zero-shear rate viscosity i/o, requires that the atomistic MD data be mapped upon a mesoscopic theoretical model. For unentangled polymer melts, such a model is the Rouse model, wherein a chain is envisioned as a set of Brownian particles connected by harmonic springs [25,28]. For entangled polymer melts, a better model that describes more accurately their dynamics is the tube or reptation model [26]. According to this model, the motion of an individual chain is restricted by the surrounding chains within a tube defined by the overall chain contour or primitive path. During the lifetime of this tube, any lateral motion of the chain is quenched. 

The process of disentangling, as it is envisaged in the reptation model, is sketched in Fig. 6.11. The motion of the primitive chain , the name given to the dynamic object associated with the primitive path, is described as a diffusion along its contour, that is to say, a reptation . The associated curvilinear diffusion coefficient can be derived from the Einstein relation, which holds generally, independent of the dimension or the topology. Denoting it D, we have... 

How does this result change for an entangled melt The reptation model gives an answer. One has only to realize that the disentangling process is associated with a shift of the center of mass of a polymer molecule over a distance in the order of /pr along the primitive path and therefore leads to a mean-squared displacement... 

M. Rubinstein (Eastman Kodak Company) In the des Cloizeaux double reptation model which is similar to the Marrucci Viovy model, it is assumed that a release of constraint chain A imposes on chain B when chain A reptates away completely relaxes the stress in that region for both chains. This would imply that for a homopolymer binary blend of long and short chains would be completely relaxed after each of these K entanglements is released only once. But if an entanglement is released, another one is formed nearby. I believe that to completely relax this section one needs disentanglement events and that the Verdier-Stockmayer flip-bond model or the Rouse model is needed to describe the motion and relaxation of the primitive path due to the constraint release process, as was proposed by Prof, de Gennes, J. Klein, Daoud, G. de Bennes and Graessley and used recently by many other scientists. The fact that double reptation is an oversimplification of the constraint release process has been confirmed by experiments. 

Chapter 9 presents tube models for linear viscoelasticity in systems with long-chain branching. Reptation of the molecule as a whole is suppressed by branch points, and relaxation takes place primarily by primitive path fluctuation, a relatively slow process. An alternative to the tube picture, the slip-link approach, is examined in detail. 

The exponential increase of viscosity with M is consistent with the picture in which relaxation occurs primarily by means of primitive path fluctuations (sometimes called arm retraction). In Chapter 9 we will see that this effect can be explained quantitatively by a tube model. The exp onential increase of t]q with M results from the fact that the branch point prevents reptation, so that the principal mechanism of relaxation is primitive path fluctuation, which becomes exponentially slower with increasing arm length. The energy of activation for the zero-shear viscosity is little affected by star branching, except in the case of polyethylene and its close relative, hydrogenated polyisobutylene. 

For linear chains, only the portions near the chain ends relax by primitive path fluctuations, and we will use the symbol Tgariy( ) in our development of a model for this fast relaxation. The interior parts of the chain, however, require quite deep fluctuations to reach them, and the time required to do this is slower than the time at which these portions of the chain will have already relaxed by reptation. We will later use the symbol in our discussion of this... 

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