Reptation model

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. 

We define a time as the time required for a chain to completely renew its configuration, that is, the time for the chain to diffuse one chain length L along s. We recognize that this will also be the longest relaxation time of the polymer liquid. From eq. 11.5.3 we see 

From which the molecular weight dependence follows directly 

This is to be compared with the prediction of the Rouse theory for the relaxation times A.Rouse (eq. 11.4.25) 

This molecular weight dependence of D has been seen experimentally in melts by Klein (1978) and in concentrated solutions by Legeretal. (1981). Reviews of diffusion behavior are available (Tirrell, 1984 Kausch and Tirrell, 1989). One can also deduce the molecular weight dependence of the viscosity by a nonrigorous but plausible argument. Suppose the entire fluid behaves as a simple viscoelastic solid (Maxwell element) then its relaxation time would be 

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In order to draw some conclusions about viscosity from the reptation model, it is again necessary to anticipate some results from Chap. 9 on diffusion. The... 

Figure 2.14 Reptation model for entanglements for (a) a linear molecule and (b) a branched molecule.

In connection with a discussion of the Eyring theory, we remarked that Newtonian viscosity is proportional to the relaxation time [Eqs. (2.29) and (2.31)]. What is needed, therefore, is an examination of the nature of the proportionality between the two. At least the molecular weight dependence of that proportionality must be examined to reach a conclusion as to the prediction of the reptation model of the molecular weight dependence of viscosity. 

There are three basic time scales in the reptation model [49]. The first time Te Ml, describes the Rouse relaxation time between entanglements of molecular weight Me and is a local characteristic of the wriggling motion. The second time Tro M, describes the propagation of wriggle motions along the contour of the chain and is related to the Rouse relaxation time of the whole chain. The important... 

We present here a simple experiment, conceived to test both the reptation model and the minor chain model, by Welp et al. [50] and Agrawal et al. [51-53]. Consider the HDH/DHD interface formed with two layers of polystyrene with chain architectures shown in Fig. 5. In one of the layers, the central 50% of the chain is deuterated. This constitutes a triblock copolymer of labeled and normal polystyrene, which is, denoted HDH. In the second layer, the labeling has been reversed so that the two end fractions of the chain are deuterated, denoted by DHD. At temperatures above the glass transition temperature of the polystyrene ( 100°C), the polymer chains begin to interdiffuse across the... 

Although the mathematics of the reptation model lie beyond the scope of this book, we should notice two important predictions. They are ... 

The beauty of the reptation model is that it is able to make predictions about molecular flow both in solution and at fracture by assuming that the molecules undergo the same kind of motions in each case. For both self-diffusion in concentrated solutions and at fracture, the force to overcome in pulling the polymer molecule through the tube is assumed to be frictional. 

The topic of molecular motion is an active one in experimental and theoretical polymer physics, and we may expect that in time the simple reptation model will be superseded by more sophisticated models. However, in the form presented here, reptation is likely to remain important as a semi-quantitative model of polymer motion, showing as it does the essential similarity of phenomena which have their origin in the flow of polymer molecules. 

In Chapter 5, we saw how de Gennes s reptation model could be used to give an indication of the relationship between fracture toughness and molar mass, and the following relationship was quoted ... 

The scaling results above all pertain to local segmental relaxation, with the exception of the viscosity data in Figure 24.5. Higher temperature and lower times involve the chain dynamics, described, for example, by Rouse and reptation models [22,89]. These chain modes, as discussed above, have different T- and P-dependences than local segmental relaxation. 

The reptation model thus predicts four dynamic regimes for segment diffusion. They are summarized in Fig. 18. 

Fig. 26. NSE spectra in polyethyleneo melts at 509 K in a Rouse scaling plot (o Q = 0.078 A-1 Q = 0.116 A-1 A Q = 0.155 A-1. Above Spectra in comparison to a fit of the generalized Rouse model [49]. Below comparison of the data with the predictions of the local reptation model [53] omitting the measurement points which correspond to the initial Rouse relaxation. The arrows indicate Q2/2V/Wxe. (Reprinted with permission from [39]. Copyright 1992 American Chemical Society, Washington)...

A comparison with the tube diameter derived from plateau moduli measurements on PEB-7 underlines this assertion. The coincidence of tube diameters determined macroscopically by application of the reptation model and direct microscopic results is far better than what could have been expected and strongly underlines the basic validity of the reptation approach. 

The deformation of polymer chains in stretched and swollen networks can be investigated by SANS, A few such studies have been carried out, and some theoretical results based on Gaussian models of networks have been presented. The possible defects in network formation may invalidate an otherwise well planned experiment, and because of this uncertainty, conclusions based on current experiments must be viewed as tentative. It is also true that theoretical calculations have been restricted thus far to only a few simple models of an elastomeric network. An appropriate method of calculation for trapped entanglements has not been constructed, nor has any calculation of the SANS pattern of a network which is constrained according to the reptation models of de Gennes (24) or Doi-Edwards (25,26) appeared. 

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. 

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