Reptation model viscoelasticity

The reptation model for polymer diffusion would predict that the thickness of the gel phase reflects the dynamics of disentanglement. The important factors here are chain length, solvent quality and temperature since they affect the dimensions of the polymer coils in the gel phase. The precursor phase, on the other hand, depends upon solvency and temperature only through the osmotic force it can generate in the system and the viscoelastic response of the system in the region of the front. These factors should be independent of the PMMA molecular weight. 

The investigation of viscoelasticity of dilute blends confirms that the reptation dynamics does not determine correctly the terminal quantities characterising viscoelasticity of linear polymers. The reason for this, as has already been noted, that the reptation effect is an effect due to terms of order higher than the first in the equation of motion of the macromolecule, and it is actually the first-order terms that dominate the relaxation phenomena. Attempts to describe viscoelasticity without the leading linear terms lead to a distorted picture, so that one begins to understand the lack of success of the reptation model in the description of the viscoelasticity of polymers. Reptation is important and have to be included when one considers the non-linear effects in viscoelasticity. 

In the present paper, after a rapid presentation of the reptation model in its simplest version, in order to pinpoint the underlying hypothesis, we discuss the interest of complementary self diffusion and viscoelastic measurements, and present the currently available methods for measuring diffusion in entangled polymer systems. Then, results obtained on polydimethylsiloxane (PDMS), a model liquid polymer well above its glass temperature at room temperature will be described, and the consequences on the limits of the entangled regime as seen from diffusion measurements, compared to what is observed in rheometry, will be discussed. 

The question of the detailed limits of validity of the reptation model thus remains a pending question. What appears puzzling is the fact that, on one hand, the reptation model and the Doi - Edwards description of the linear viscoelasticity work so well both qualitatively and quantitatively for some experiments, while, on the other hand, they seem unable to account for all the existing data. This may suggest that the reptation model does not contain the whole story of linear polymer dynamics, and that one needs to learn more on other possibly competing processes. 

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... 

There is growing evidence that t-T superposition is not valid even in miscible blends well above the glass transition temperature. For example, Cavaille et al. [1987] reported lack of superposition for the classical miscible blends — PS/PVME. The deviation was particularly evident in the loss tangent vs. frequency plot. Lack of t-T superposition was also observed in PI/PB systems [Roovers and Toporowski, 1992]. By contrast, mixtures of entangled, nearly mono-dispersed blends of poly(ethylene-a/f-propylene) with head-to-head PP were evaluated at constant distance from the glass transition temperature of each system, homopolymer or blend [Gell et al, 1997]. The viscoelastic properties were best described by the double reptation model , viz. Eq 7.82. The data were found to obey the time-temperature superposition principle. 

Molecular Theory of Polymer Viscoelasticity — Entanglement and the Doi-Edwards (Reptation) Model... 

As shown by the above analyses of the viscoelasticity and diffusion data in terms of the ERT, the paradox between the scaling relations r]o oc M and Dg oc predicted by the pure reptational model, that occurs in their comparison with the experimental results, is resolved. Furthermore, the relation between viscoelasticity and diffusion as given by the ERT is quantitatively supported by the data of polystyrene. The analysis of the viscosity data at Me in relation to the Kd value obtained from the diffusion measurements also supports the ERT. Considering the different nature of the experiments of viscoelasticity and diffusion, the quantitative agreement between these two kinds of data as analyzed in terms of the ERT is remarkable and thus indeed significant. 

The reptation model is more powerful than you might think. You can get much more out of it than just the simplest basic laws for the viscosity, the longest relaxation time, and the diffusion coefficient of a chain in a polymer melt. This model allows you to describe, for instance, the relaxation of a pol mier after a stress has been released, or the response to a periodic force. As a result, you gain a fairly complete picture of the dynamics of polymer liquids, and of their viscoelasticity in particular. 

In summary, we conclude that the linear viscoelastic properties of entangled-polyelectrolyte solutions in the terminal regions can be well explained by the reptation model taking into account the electrostatic interactions evaluated by the Donnan equilibrium [10,11]. 

More evidence comes from the study of viscoelasticity, which has been done extensively in the past and established the characteristic aspects common to all flexible polymers. The reptation model has succeeded in explaining many of these features and also predicting some of the behaviour in nonlinear viscoelasticity. In this chapter we shall describe the reptation theory for viscoelasticity in detail, and discuss the validity of the reptation model in solutions and melts. 

The reptation model has been applied to various problems other than the problems of viscoelasticity and d sion that have been discussed. These... 

Entanglements of flexible polymer chains contribute to non-linear viscoelastic response. Motions hindered by entanglements are a contributor to dielectric and diffusion properties since they constrain chain dynamics. Macromolecular dynamics are theoretically described by the reptation model. Reptation includes fluctuations in chain contour length, entanglement release, tube dilation, and retraction of side chains as the molecules translate using segmental motions, through a theoretical tube. The reptation model shows favourable comparison with experimental data from viscoelastic and dielectric measurements. The model reveals much about chain dynamics, relaxation times and molecular structures of individual macromolecules. 

The double reptation model was used to evaluate viscoelastic behavior of metallocene-catalyzed polyethylene and low-density polyethylene blends by Peon et al. (2003). They compared their results with those obtained for HDPE/BPE blends prepared under similar conditions. Since this model assumes miscibility between the mixed species, the experimental viscosity of HDPE/BPE blends showed only small deviation compared to that expected according to the reputation miscible model. However, the model underestimated the compositional dependence of the zero-shear viscosity for mPE/LDPE blends, especially at intermediate levels. The enhanced zero-shear viscosity in immiscible blends such as PETG/EVA, PP/EVA, or EVA/PE blends was found to be more abrupt than it is for mPE/LDPE blends (Lacroix et al. 1996, 1997 Peon et al. 2003). 

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... 

Constitutive Description of Polymer Melt Behavior K-BKZ and DE Descriptions. Although there are many nonlinear constitutive models that have been proposed, the focus here is on the K-BKZ model because it is relatively simple in structure, can be related conceptually to finite elasticity descriptions of elastic behavior, and because, in the mind of the current author and others (82), the model captures the major features of nonlinear viscoelastic behavior of polymeric fluids. In addition, the reptation model as proposed by Doi and Edwards provides a molecular basis for understanding the K-BKZ model. The following sections first describe the K-BKZ model, followed by a description of the DE model. 

The dynamic properties for the reptation model just described are obtained by solving the equations 71,72,73,74 and the linear viscoelastic properties are solved from the relationship between the stress and the chain orientation function (63). Without going into detail, the results for times longer than the tube equilibration time but shorter than the reptation time are... 

In contrast to D, the prediction of other viscoelastic properties, such as the friction coefficient f or the zero-shear rate viscosity i/o, requires that the atomistic MD data be mapped upon a mesoscopic theoretical model. For unentangled polymer melts, such a model is the Rouse model, wherein a chain is envisioned as a set of Brownian particles connected by harmonic springs [25,28]. For entangled polymer melts, a better model that describes more accurately their dynamics is the tube or reptation model [26]. According to this model, the motion of an individual chain is restricted by the surrounding chains within a tube defined by the overall chain contour or primitive path. During the lifetime of this tube, any lateral motion of the chain is quenched. 

On the nano-scale, the discrete moleculai structure of the polymer has to be considered. Segmental immobilization seems to be the primary reinforcing mechanism in true polymer nanocomposites at temperatures near and above the Tg. Reptation model and simple percolation model were used to describe immobilization of chains near solid nanopaiticles and to explain the peculiarities in the viscoelastic response of polymers near solid surfaces of lar ge polymer-inclusion contact areas. The inteiphase in the continuum sense does not exist at the nano-scale when relaxation processes in individual discrete chains are taken into account and the chains with retarded reptation catr be considered forming the iirterphase analogue irr the discrete matter. For a common polymer, all the chains in the composite are immobilized when the internal filler-matrix interface area reached about 42 m per 1 g of the nanocomposite. 

II.3 Viscoelasticity. The Doi-Edwards model The reptation model is also an essential tool to understand the response of polymer materials to external solicitations. Its extension to viscoelasticity has been... 

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