Edwards model

In the limit that the number of effective particles along the polymer diverges but the contour length and chain dimensions are held constant, one obtains the Edwards model of a polymer solution [9, 30]. Polymers are represented by random walks that interact via zero-ranged binary interactions of strength v. The partition frmction of an isolated chain is given by... 

Figure 6.21 Comparison of Graessley and Doi-Edwards models for normalised viscosity versus normalised shear rate. Also shown is an estimate of the role of short time Rouse relaxation mechanisms within the tube...

We should note that Eq. (7.7) differs from the standard form of the Edwards model by a trivial factor of 1/2 in the first term, which can be absorbed in a... 

Optical measurements often have a greater sensitivity compared with mechanical measurements. Semidilute polymers, for example, may not be sufficiently viscous to permit reliable transient stress measurements or steady state normal stress measurements. Chow and coworkers [113] used two-color flow birefringence to study semidilute solutions of the semirigid biopolymer, collagen, and used the results to test the Doi and Edwards model discussed in section 7.1.6.4. That work concluded that the model could successfully account for the observed birefringence and orientation angles if modifications to the model proposed by Marrucci and Grizzuti [114] that account for polydispersity, were used. 

The Doi-Edwards, reptation based model makes specific predictions for the relaxation dynamics of different portions of a polymer chain. Specifically, the relaxation of the chain ends is predicted to be substantially faster than the relaxation of the center. This is a result of the reptation dynamics, which have the ends first leaving the confines of the tube. Using polymer chains that were selectively deuterated either at the ends or at the middle, Ylitalo and coworkers [135] examined this problem and found that the Doi-Edwards model was able to successfully predict the observed behavior once the effects of orientational coupling was included. The same group further explored the phenomena of orientational coupling in papers that focused on its molecular weight [136] and temperature [137]... 

The function F(t — t ) is related, as with the temporary network model of Green and Tobolsky (48) discussed earlier, to the survival probability of a tube segment for a time interval (f — t ) of the strain history (58,59). Finally, this Doi-Edwards model (Eq. 3.4-5) is for monodispersed polymers, and is capable of moderate predictive success in the non linear viscoelastic range. However, it is not capable of predicting strain hardening in elongational flows (Figs. 3.6 and 3.7). 

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. 

The rates of relaxation of the first and the third modes of macromolecule of length M = 25Me (x = 0.04, B = 429, ip = 8.27). The results calculated from analytical correlation function (4.29) are depicted by solid lines. By straight dashed lines, the values of the relaxation times due to the Doi-Edwards model are presented. The circles (for the first mode) and squares (for the third mode) depict the results of simulation for above values of parameters ip and B and values of parameters of local anisotropy ae = 0.3, cq = 0.06. Adapted from Pokrovskii (2006). 

One has no results for this case derived consequently from the basic equations (7.6) with local anisotropy. Instead, to find conformational relaxation equation, we shall use the Doi-Edwards model, which approximate the large-scale conformational changes of the macromolecule due to reptation. The mechanism of relaxation in the Doi-Edwards model was studied thoroughly (Doi and Edwards 1986 Ottinger and Beris 1999), which allows us to write down the simplest equation for the conformational relaxation for the strongly entangled systems... 

The molecular-level nature of our orientation relaxation data has led us to carry out an interpretation in terms of molecular motions. Therefore we have based our interpretation on the Doi-Edwards model [9]. We briefly recall the main qualitative features of the model and include other processes proposed by... 

The spectroscopic data have been compared with the theoretical predictions of the Doi- Edwards model. In the time scale of our experiments, a quantitative agreement between experiment and theory is obtained if chain length fluctuations, retraction and reptation are taken into account. In the case of star polymers, the large scale fluctuation mechanism as proposed by Pearson and Helfand associated with the retraction process is accounting for... 

M.H.Wagner, The nonlinear strain measure of pol3risobutylene melt in general biaxial flow and its comparison to Doi-Edwards model, RheolActa. 22 (1990), 594-603. 

Figure 3-31 Damping function hiy) obtained by vertically shifting the time-dependent nonlinear moduli in Fig. 3-30a into superposition at long times. The data are from Fukuda et al. (1975). The solid and dashed lines are the prediction of the Doi-Edwards model, respectively, with and without the independent alignment approximation. (From Doi and Edwards 1978a, reproduced by permission of The Royal Society of Chemistry.)...

In the original Doi-Edwards model, retraction is assumed to occur infinitely fast. The stress tensor is then given by the elastic, or Brownian, stress for rigid rods [see Eq. (6-36)] ... 

Although the expression for the stress tensor in the Doi-Edwards model is the same as that of the temporary network model, except for the coefficient [see Eq. (3-13)],... 

The uniaxial extensional viscosity rj(s) and the viscometric functions rj(y) and ki(y), predicted by the Doi-Edwards model for monodisperse melts, are shown in Fig. 3-32. The Doi-Edwards model predicts extreme thinning in these functions the high-shear-rate asymptotes scale as 17 oc oc y , and4 i oc The second normal... 

The Doi-Edwards equation predicts an overshoot in shear stress as a function of time after inception of steady shearing, but no overshoot in the first normal stress difference (Doi and Edwards 1978a). Typical overshoots in these quantities for a polydisperse melt are shown in Fig. 1-10. For monodisperse melts, the Doi-Edwards model predicts that the shear-stress maximum should occur at a shear strain yt = Yp, of about 2, roughly independently of... 

As already noted, the measured nonlinear shear relaxation modulus, for linear molecules with little polydispersity, is in excellent agreement with the Doi-Edwards model at long times. However, for melts or concentrated solutions of very high molecular weight (e.g., 10 for polystyrene, where 0 is the polymer volume fraction), the measuredfiamping function, h(y), is drastically lower than the Doi-Edwards prediction (Einaga et al. 1971 Vrentas and Graessley 1982 Larson etal. 1988 Morrison and Larson 1992). This anomalous... 

The Doi-Edwards model has been extended to allow processes of primitive-path fluctuations, constraint release, and tube stretching. These extensions of the theory allow accurate prediction of many steady-state and time-dependent phenomena, including shear thinning, stress overshoots, and so on. Predictions of strain localization and slip at walls... 

Problem 3.11 A theory incorporating convective constraint release and chain stretch into the Doi-Edwards model gives the constitutive equations below (Larson et al. 1998) ... 

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