Doi fluctuation model

The stress relaxation modulus then decays exponentially at the reptation time [Eq. (9.22)]. The terminal relaxation time can be measured quite precisely in linear viscoelastic experiments. Hence, Eq. (9.82) provides the simplest direct means of testing the Doi fluctuation model and evaluating... 

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. 

Experimental verification of the Doi fluctuation model using data for polystyrene as open squares, from S. Onogi et al.. Macromolecules 3,... 

D. Shirvanyants. Open symbols are experimental data for the three polymers in Fig. 9.5, shifted parallel to the rjolM axis to coincide with the Repton model data. The curve is the Doi fluctuation model [Eq. (9.84)] with... 

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

Calculate the storage and loss moduli corresponding to the Doi fluctuation model with stress relaxation modulus... 

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... 

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... 

Single-chain models, such as the Doi Edwards reptation model [Eq. (9.21)] or the Doi tube length fluctuation model, assume a linear contribution to... 

Fig. 12. The rheological functions G ((o) and G"(co) for an H-shaped PI of arm molecular weigh 20 kg mol and backbone 110 kg mol" [46]. The high-frequency arm-retraction modes can be seen as the shoulder from co 10 to co 10 together with a low-frequency peak due to the cross-bar dynamics at co 10. The smooth curves are the predictions of a model which takes Eq. (33) as the basis for the arm-retraction times and a Doi-Edwards reptation spectrum with fluctuations for the backbone. The reptation time is correctly predicted, as is the spectrum from the arm modes, though the low frequency form is more polydisperse than the simple theory predicts...

The Doi-Edwards theory treats monodisperse linear chain liquids by a model which suppresses fluctuations and assumes a topologically invariant medium. Two parameters are required, the monomeric friction coefficient which characterizes the local dynamics and the primitive path step length a which characterizes the topology of the medium. The step length is related to the entanglement molecular weight of earlier theories, = cRqT/Gn, by Eqs. 1 and 37 ... 

The Doi-Edwords equation [Eq. (9.20)] is the first attempt at a molecular model for viscoelasticity of entangled polymers. It ignores tube length fluctuation modes that relax some stress on shorter time scales. These modes significantly modify dynamics of entangled polymers, as described in Section 9.4.5. 

The results of models that include tube length fluctuation modes [Fig. 9.23(b)] are in much better agreement with the experimentally measured loss modulus G" (ic) of monodisperse melts than the prediction of the Doi Edwards reptation model [Eq. (9.83)]. Tube length fluctuation corrections predict that the loss peak broadens with decreasing molar mass because the fraction of the stress released by fluctuations is larger for shorter chains. 

According to the Doi-Edwards theory, after time t = Teq following a step deformation at t = 0, the stress relaxation is described by Eqs. (8.52)-(8.56). In obtaining these equations, it is assumed that the primitive-chain contour length is fixed at its equilibrium value at all times. And the curvilinear diffusion of the primitive chain relaxes momentarily the orientational anisotropy (as expressed in terms of the unit vector u(s,t) = 5R(s,t)/9s), or the stress anisotropy, on the portion of the tube that is reached by either of the two chain ends. The theory based on these assumptions, namely, the Doi-Edwards theory, is called the pure reptational chain model. In reality, the primitive-chain contour length should not be fixed, but rather fluctuates (stretches and shrinks) because of thermal (Brownian) motions of the segments. 

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