Relaxation time tube models

It is instructive to compare the system of equations (3.46) and (3.47) with the system (3.37). One can see that both the radius of the tube and the positions of the particles in the Doi-Edwards model are, in fact, mean quantities from the point of view of a model of underlying stochastic motion described by equations (3.37). The intermediate length emerges at analysis of system (3.37) and can be expressed through the other parameters of the theory (see details in Chapter 5). The mean value of position of the particles can be also calculated to get a complete justification of the above model. The direct introduction of the mean quantities to describe dynamics of macromolecule led to an oversimplified, mechanistic model, which, nevertheless, allows one to make correct estimates of conformational relaxation times and coefficient of diffusion of a macromolecule in strongly entangled systems (see Sections 4.2.2 and 5.1.2). However, attempts to use this model to formulate the theory of viscoelasticity of entangled systems encounted some difficulties (for details, see Section 6.4, especially the footnote on p. 133) and were unsuccessful. 

There were different generalisations of the reptation-tube model, aimed to soften the borders of the tube and to take into account the underlying stochastic dynamics. It seems that the correct expansion of the Doi-Edwards model, including the underlying stochastic motion and specific movement of the chain along its contour - the reptation mobility as a particular mode of motion, is presented by equations (3.37), (3.39) and (3.41). In any case, the introduction of local anisotropy of mobility of a particle of chain, as described by these equations, allows one to get the same effects on the relaxation times and mobility of macromolecule, which are determined by the Doi-Edwards model. 

Derived from linear approximation of the equations (3.37), the equilibrium correlation function (4.29), defines two conformation relaxation times r+ and r for every mode. The largest relaxation times have appeared to be unrealistically large for strongly entangled systems, which is connected with absence of effect of local anisotropy of mobility. To improve the situation, one can use the complete set of equations (3.37) with local anisotropy of mobility. It is convenient, first, to obtain asymptotic (for the systems of long macromolecules) estimates of relaxation times, using the reptation-tube model. 

It is not difficult to reproduce an expression for the correlation function Ma(t) and estimate times of relaxation due to the conventional reptation-tube model (see Section 3.5). Indeed, an equation for correlation function follows equation (3.48) and has the form... 

Each point is calculated as the asymptotic value of the rate of relaxation for large times (see examples of dependences in Fig. 6) for a macromolecule of length M = 25Me (x = 0.04, B = 429, ij) = 8.27) with the value of the coefficient of external local anisotropy ae = 0.3. The dashed lines reproduce the values of the relaxation times of the macromolecule due to the reptation-tube model. The labels of the modes are shown at the lines. Adapted from Pokrovskii (2006). 

The Pti samples (182) were prepared as colloids, protected by a PVP polymer film. Layer statistics according to the NMR layer model (Eqs. 28-30) for samples with x = 0,0.2, and 0.8 are shown in Fig. 63. The metal/ polymer films were loaded into glass tubes and closed with simple stoppers. The NMR spectrum and spin lattice relaxation times of the pure platinum polymer-protected particles are practically the same as those in clean-surface oxide-supported catalysts of similar dispersion. This comparison implies that the interaction of the polymer with the surface platinums is weak and/or restricted to a small number of sites. The spectrum predicted by using the layer distribution from Fig. 63 and the Gaussians from Fig. 48 show s qualitative agreement w ith the observed spectrum for x = 0 (Fig. 64a). 

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 reptation ideas discussed above will now be combined with the relaxation ideas discussed in Chapter 8 to describe the stress relaxation modiihis G t) for an entangled polymer melt. On length scales smaller than the tube diameter a, topological interactions are unimportant and the dynamics are similar to those in unentangled polymer melts and are described by the Rouse model. The entanglement strand of monomers relaxes by Rouse motion with relaxation time Tg [Eq. (9.10)] ... 

The longest relaxation time in this model is the reptation time required for the chain to escape from its tube... 

The simple reptation model does not properly account for all the relaxation modes of a chain confined in a tube. This manifests itself in all measures of terminal dynamics, as the longest relaxation time, diffusion coefficient and viscosity all have stronger molar mass dependences than the reptation model predicts. Tn Sections 9.4.5 and 9.6.2, more accurate ana-... 

On length scales larger than the correlation length but smaller than the tube diameter a, hydrodynamic interactions are screened, and topological interactions are unimportant. Polymer motion on these length scales is described by the Rouse model. The relaxation time Tg of an entanglement strand of monomers is that of a Rouse chain of N jg correlation volumes [Eg. (8.76)] ... 

On length scales larger than the tube diameter, topological interactions are important and the motion is described by the reptation model with the chain relaxation time given by the reptation time ... 

Recall that Fig. 9.3 showed the linear viscoelastic response of a polybutadiene melt with MjM = 68. The squared term in brackets in Eq. (9.82) is the tube length fluctuation correction to the reptation time. With /i = 1.0 and NjN = 68, this correction is is 0.77. Hence, the Doi fluctuation model makes a very subtle correction to the terminal relaxation time of a typical linear polymer melt. However, this subtle correction imparts stronger molar mass dependences for relaxation time, diffusion coefficient, and viscosity. 

Other computer simulations, such as the Evans Edwards model of a chain in an array of fixed obstacles (described in detail in Section 9.6.2) exhibit fluctuations of the tube length and also find stronger molar mass dependences of relaxation time t and diffusion coefficient... 

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... 

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