Reptation model viscosity

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... 

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. 

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 predicts that the viscosity of a melt scales with the chain length to the third power while the diffusion coefficient decreases with the second power of the chain length. 

These laws are in very good qualitative agreement with the large number of available experimental data on the dynamic behaviour of linear flexible entangled polymers, but quantitative departures still remain between the experiments and the predictions, as for example the fact that the exponent of the power law which characterises the variation of the zero shear viscosity with the molecular weight is observed to be 3.3 or 3.4 [3, 6] rather than the predicted value 3. These deviations have lead to a long controversy on the validity of the reptation model, and have stimulated a series of experimental and theoretical investigations to try to understand the limitations of this model and to propose the necessary modifications to obtain a better description of the dynamic properties of liquid polymers [7 to 22]. 

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... 

The storage and loss moduli, G and G", are obtained from the relaxation spectrum in the usual way—that is, using G = Gi[co rl/(l + co zl)] G — G,[mT /(l -P The longest relaxation mode of the relaxation modulus in Eq. (3-67) is the dominant one it accounts for 96% of the zero-shear viscosity. Thus, the reptation model predicts that for a nearly monodisperse melt, the relaxation spectrum is dominated by a single relaxation time, T = Ta. This is in reasonable accord with experimental data at low and moderate frequencies (see the dashed line in Fig. 3-29). As the frequency increases, however, there... 

Figure 7 Zero shear viscosity as a function of molecular mass. Mapping of atomistic MD data onto Rouse and reptation models are shown, as well as experimental values. Reprinted with permission from 387. Copyright 2003 American Chemical Society...

The reptation model prediction for the viscosity of an entangled polymer melt is determined by integrating Eq. (9.20) ... 

The viscosity of a polymer melt is predicted to be proportional to molar mass for unentangled melts (the Rouse model) and proportional to the cube of molar mass for entangled melts (the reptation model). 

The simple reptation model does not properly account for all the relaxation modes of a chain confined in a tube. This manifests itself in all measures of terminal dynamics, as the longest relaxation time, diffusion coefficient and viscosity all have stronger molar mass dependences than the reptation model predicts. Tn Sections 9.4.5 and 9.6.2, more accurate ana-... 

The entangled viscosity data in both good solvent and -solvent show stronger concentration dependences than predicted by the simple reptation model. The steeper experimental slopes are consistent with the additional relaxation modes discussed in Section 9.4.5 (see Problem 9.14). 

To understand this viscosity enhancement, it is easier to start with the theory for linear polymers. The behavior of linear polymers can be described by the reptation model.For a linear polymer of high molecular weight in the melt, chains can be modeled as a confined tube where the diffusion of the chain is restricted along the tube contour. Entanglements are formed between chains where the reptation of a chain along its contour becomes the dominant mode of movement. The addition of a branch point prevents reptation and other forms of movement must occur for the chain to change its configuration. In the case... 

The viscosity of the system is given by 7 = tE (Maxwell model) and according to the reptation model, the modulus E depends on the distance between obstacles and does not depend on the chain length. Therefore... 

Exact calculations based on this model are complex. Nevertheless, it is relatively easy to develop certain "scaling laws" that relate how various macroscopic properties might depend on molecular properties. We will briefly sketch the development of such a scaling law for viscosity and chain length (or molecular weight) based on the reptation model.23,24... 

This is the fundamental result of the reptation model the relaxation time is proportional to the eube of the chain length. The cube dependenee is not a precise match with the 3.4 exponent obtained from viscosity measurements of long chains, but it is acceptable, particularly as the model gives a satisfactory picture of how a polymer ehain ean overcome the restraining influence of entanglements and move within the matrix. 

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. 

Let s now use the reptation model to find the viscosity 77 of a polymer melt. We are going to use Equation (12.7) along with estimates (12.8) and (12.13) of Young s modulus E and the longest relaxation time r. This gives ... 

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. 

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). 

It must also be realized that for very large Ni, reptation itself is not the dominant mobility process for the long chain. For Nj - < we can think of the V chains as forming a solvent of comparatively small molecules, with a certain viscosity tjat- This viscosity is discussed in the next section, and, in the reptation model, it scales like tik = Then the diffusion constant is given by the Stokes-Einstein equation... 

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