Rheology reptation model

A (rheology) [7]. Obviously the local freedom for segmental motion is larger than anticipated so far from rheology. The result casts some doubts on the determination of the plateau modulus from rheological data in terms of the reptation model. 

Wagner et al. (63-66) have recently developed another family of reptation-based molecular theory constitutive equations, named molecular stress function (MSF) models, which are quite successful in closely accounting for all the start-up rheological functions in both shear and extensional flows (see Fig. 3.7). It is noteworthy that the latest MSF model (66) is capable of very good predictions for monodispersed, polydispersed and branched polymers. In their model, the reptation tube diameter is allowed not only to stretch, but also to reduce from its original value. The molecular stress function/(f), which is the ratio of the reduction to the original diameter and the MSF constitutive equation, is related to the Doi-Edwards reptation model integral-form equation as follows ... 

They have been developed based on either molecular structure or continuum mechanics where the molecular structure is not considered explicitly and the response of a material is independent of the coordinate system (principle of material indifference). In the former, the polymer molecules are represented by mechanical models and a probability distribution of the molecules, and relationships between macroscopic quantities of interest are derived. Three models have found extensive use in rheology the bead-spring model for dilute polymer solutions, and the transient net work and the reptation models for concentrated polymer solutions and polymer melts. 

The concept of polymer entanglements represents intermolecular interaction different from that of coil overlap type interaction. However, it is difficult to define the exact topological character of entanglements. The entanglements concept was aimed at understanding the important nonlinear rheological properties, such as the shear rate dependence of viscosity. However, viscoelastic properties could not be defined quantitatively as is possible with the reptation model. Because an entanglement should be... 

Radical initiator 127 Ramified clusters 219 Random walk 191, 214, 263 Random phase approximation (RPA) 191-226, 245, 271, 274, 284, 286, 290 Reprotonation 95 Reptation model 202, 204 Rheology 280... 

Yu et al. (2011) studied rheology and phase separation of polymer blends with weak dynamic asymmetry ((poly(Me methacrylate)/poly(styrene-co-maleic anhydride)). They showed that the failure of methods, such as the time-temperature superposition principle in isothermal experiments or the deviation of the storage modulus from the apparent extrapolation of modulus in the miscible regime in non-isothermal tests, to predict the binodal temperature is not always applicable in systems with weak dynamic asymmetry. Therefore, they proposed a rheological model, which is an integration of the double reptation model and the selfconcentration model to describe the linear viscoelasticity of miscible blends. Then, the deviatirMi of experimental data from the model predictions for miscible... 

The rheological properties of miscible blends tmder different temperatures can be obtained from some theoretical models. One such model is the double reptation self-concentration. The DRSC (double reptation self-concentration) model actually includes the temperature dependency and concentration dependency through a complex mixing mle given by the double reptation model and self-concentration model, which helps to exclude the complex contribution from miscible components under different temperatures in the experimental data and only illustrate the effect of the concentration fluctuation and interface formation. This model is applied to study PMMA/SMA (Wei 2011). 

Relaxation 67,70,96,99,111,155 Reptation model 1,24,42 Resolution 14 Resonance NSE 20 Rheology 35,55 Rotational isomeric state 118 Rotational transitions 117 Rouse diffusion coefficient 28,42, 175 Rouse model 24-26,30-35,38, 117, 119, 142, 193,200 —, generalized 47 Rouse time 27 RPA 162, 163, 199... 

These equations were used by Doi and Edwards2-4 to derive the predictions of the reptation model concerning the rheological properties of polymer melts. In the next section, we show how we have modified these equations and concepts to include the effects on an electric field. 

This model, which was among the very first proposed to explain some aspects of entangled polymers, is mainly aimed at understanding some of the important and dramatic, nonlinear rheological properties, particularly the shear rate dependence of viscosity (Graessley, 1974). The approach was difficult to extend to viscoelastic effects quantitatively and has been abandoned in favor of the reptation model. 

The plateau modulus is determined from rheological measurements. In the reptation model, it is related to the tube diameter dr —... 

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]). 

In general, there are multiple relaxation processes in polymers, many of which are much too complex to be described by simple rheological theories (such as the double reptation model presented below), and it is not our objective to describe all such processes in detail. The interested reader can find the details in the book by Doi and Edwards [ 1 ], and in the review article by Watanabe [2]. Nevertheless, in Chapter 9 we will present some advanced theories for polymer melts, including theories of McLeish, Milner, and coworkers, that include all the known important mechanisms of polymer relaxation, and in Chapter 11, we will combine... 

To illustrate this, in Figs. 6.17a and b, we compare the predictions of the double reptation model against data for both a monodisperse and a polydisperse polystyrene of nearly the same molecular weight at 150 °C [19]. The sample of Fig. 6.17a is a nearly monodisperse polystyrene, with M = 363,000, and M /M = 1.03. The linear rheology for this sample has been fitted by the double reptation theory in the terminal region, yielding a value of K = 2.275 10 at 150 °C for the double reptation constant. Using this value of K, we make the a priori predictions shown in Fig. 6.17b (at 150 °C) of the linear moduli of the polydisperse sample = 2.3) with = 357,000, which is very close to the molecular... 

In this chapter and Chapter 9, we wish to introduce more advanced constraint-release concepts, which can be applied to cases for which the double reptation model works poorly, including monodisperse and bidisperse, linear polymers. We will show that when the advanced concepts of constraint release Rouse relaxation and dynamic dilution are introduced into the tube model, then successful predictions of the linear rheology of bidisperse melts can be achieved. While bidisperse melts are not of great commercial interest, the concepts we will introduce in this chapter are also important for polymer with long side branches, which are of great commercial interest, and are discussed in Chapter 9. The reader not interested in the details of advanced tube theories may want to focus on the comparisons of predictions of these models with experimental data in Figures 7.9 through 7.13. However, where needed, results from this chapter will be used in Chapter 9, which covers branched polymers. 

Figure 16 Tube model for reptation of a branched polymer molecule from the work of Blackwell et al. [124]. Reproduced with permission from Blackwell et al. [124]. Copyright 2000, The Society of Rheology, Inc.

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...

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