Reptation model constraint release

A second appealing feature of tube model theories is that they provide a natural hierarchy of effects which one can incorporate or ignore at will in a calculation, depending on the accuracy desired. We will see how, in the case of linear polymers, bare reptation in a fixed tube provides a first-order calculation more accurate levels of the theory may incorporate the co-operative effects of constraint release and further refinements such as path-length fluctuation via the Rouse modes of the chains. 

In interpreting the relaxation behavior of polydisperse systems by means of the tube model, one must consider that renewal of the tube occurs because the chain inside it moves thermally, either by reptation mode, by fluctuation of the tube length in time (breathing motion), or in both ways (13,14). Moreover, the tube wall can be renewed independently of the motion of the chain inside the tube because the segments of the chains of the wall are themselves moving. The relaxation mechanism associated with the renewal of the tube is called constraint release. 

The constraint release process for the P-mer can be modelled by Rouse motion of its tube, consisting of P/A e segments, where is the average number of monomers in an entanglement strand. The average lifetime of a topological constraint imposed on a probe P-mer by surrounding A -mers is the reptation time of the A -mers Trep(A ). The relaxation time of the tube... 

Constraint release controls the terminal relaxation in the reptation model if PjN > (NlNe) and in the Doi fluctuation model if PIN,. >... 

Thus, a finite fraction of the stress relaxes by constraint release at time scales of the order of the constraint lifetime in the Rouse model of constraint release. This is also the time scale at which the stress relaxes by reptation in monodisperse entangled solutions and melts. Both processes simultaneously contribute to the relaxation of stress. Therefore, constraint release has to be taken into account for a quantitative description of stress relaxation even in monodisperse systems. The contribution of constraint -release to stress relaxation in pefydispcrse solutions and melts is even more important as will be discussed below. 

These tube length fluctuation modes (see Section 9.4.5) of the neighbouring chains affect the constraint release modes of a given chain. If entanglements between chains are assumed to be binary, there should be a duality between constraint release events and chain in a tube relaxation events. A release of an entanglement by reptation or tube length fluctuation of one chain in its tube leads to a release of the constraint on the second chain. If this duality is accepted, the distribution of constraint release rates can be determined self-consistently from the stress relaxation modulus of the tube model. 

Consider an isolated long probe P-mer entangled in a melt of shorter Wmers. Tube dilation assumes that as soon as short chains relax, stress in the long P-mer drops to zero. In particular, a version of tube dilation called double reptation imposes an exact symmetry between single chains in a tube and multi-chain processes. As one chain reptates away, stress at a common entanglement (stress point) is relaxed completely. In constraint release models, this stress relaxes only partially due to connectivity of the P-mer. 

In panels (a)-(c), comparison of the blend moduli calculated from the model (curves Equations 3.69 through 3.73) with the moduli data (symbols) for various high-M polyisoprene/ poly(p-tert butyl styrene) (PI/PtBS) blends as indicated. The sample code nmnbers of the blends indicate 10 M of the components. The model considers the cooperative Rouse equilibration and successive constraint release (CR)/reptation relaxation of the component chains, and the model parameters summarized in Table 3.1 were determined experimentally. (Redrawn, with permission, from Watanabe, H., Q. Chen, Y. Kawasaki, Y. Matsumiya, T. Inoue, and O. Urakawa. 2011. Entanglement dynamics in miscible polyisoprene/poly(p-fert-butylstyrene) blends. Macromolecules 44 1570-1584). 

Each of these entanglements has a mean lifetime corresponding to the reptation time of the LP. In modeling the constraint release (CR) relaxation of the CP, we visualize the entanglements as Rouse beads with a frictional drag proportional to T[,. 

The molecular theory of extensional viscosity of polymer melts is again based oti the standard tube model. It considers the linear viscoelastic factors such as reptation, tube length fluctuations, and thermal constraint release, as well as the nonlinear viscoelastic factors such as segment orientations, elastic contractimi along the tube, and convective constraint release (Marrucci and lannirubertok 2004). Thus, it predicts the extensional stress-strain curve of monodispersed linear polymers, as illustrated in Fig. 7.12. At the first stage, the extensional viscosity of polymer melts exhibits the Newtonian-fluid behavior, following Trouton s ratio... 

S Diffusion of Polymer Chains in a Fixed Network Although the tube model and the reptation model were originally developed to explain the diffusion of polymer chains in concentrated solutions and melts, we can use it more naturally for the motion of polymer chains in a fixed network, for instance, a cross-linked network of polymer swollen in a good solvent. In the fixed network, the constraint release is absent. Therefore, we should be able to observe the reptation without being compromised. 

Removing the hypothesis of fixed obstacles the models In the context of the reptation model, the finite lifetime of the tube constraints leads to an extra relaxation process called tube-renewal or constraint-release. 

The subject of this paper is limited to the dynamics of a single entangled polymer chain, just as in the original de Gennes paper on reptation. In a melt or concentrated solution, the dynamics of any chain would be affected by the motion of surrounding polymers (by constraint release), and this effect has to be self-consistently taken into account. In order to do that, one has to start from a reliable model of the single-chain dynamics, such the reptation model, Doi s fluctuation theory,or the repton modeldescribed in the present paper. 

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. 

Hess derived a similar expression from his microscopic model by explicitly considering the effective entanglement as a dynamic effect. Hess included the important many chain cooperative effects of constraint release and tube renewal, which are necessary in order to get quantitative predictions for the stress relaxation functions. Ultimately this does not affect the N dependence of the relaxation time. He found that after an initial fast Rouselike decay up to time r, Tp Hess = / irp Rep- Both models describe essentially the same physical picture. For the generalized Rouse model, Kavassalis and Noolandi found that Tp rm N /p. MD simulation results of Kremer and Grest could not distinguish between the standard reptation and Hess models but could rule out the generalized Rouse model. 

Tj represents some relaxation time, of component i, which in terms of the tube model, is related to the idealised Doi-Edwards relaxation time for component i in a matrix of fixed obstacles, tde. by, Xi = (1/2)tde. Hence, in the double reptation model, the effect of constraint release is to half the relaxation time (if single exponential decay is assumed), from that predicted for a polymer in a fixed matrix. In the heterogeneous blends considered here, the tj are the tube survival times for chains of species i in an idealised environment, in which the chemical heterogeneity matches that of the blend, but all chains share the same relaxation time. That is, double reptation accounts for mutual effects in topological stress relaxation, but not for direct effects of local composition on the monomeric friction factors. The parameters of the double reptation model should be treated as phenomenological, to be determined from independent linear rheology experiments in the one phase region (see for example reference [61]). 

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