Polymer melts reptation model

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. 

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. 

Diffusion of flexible macromolecules in solutions and gel media has also been studied extensively [35,97]. The Zimm model for diffusion of flexible chains in polymer melts predicts that the diffusion coefficient of a flexible polymer in solution depends on polymer length to the 1/2 power, D N. This theoretical result has also been confirmed by experimental data [97,122]. The reptation theory for diffusion of flexible polymers in highly restricted environments predicts a dependence D [97,122,127]. Results of various... 

Fig. 3.8 Single-molecule bead-spring models for (a) a dilute polymer solution, and (b) an undiluted polymer (a polymer melt with no solvent). In the dilute solution, the polymer molecule can move about in all directions through the solvent. In the undiluted polymer, a typical polymer molecule (black bead) is constrained by the surrounding molecules and tends to execute snakelike motion ( reptation ) by sliding back and forth along its backbone direction (46). [Reprinted by permission from R. B. Bird, W. E. Stewart and E. N. Lightfoot, Transport Phenomena, 2nd Edition, Wiley New York (2002).]...

The reptation model tests through diffusion measurements in linear polymer melts... 

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. 

In the subsequent 20 years (1960-80), the main principles of modern polymer physics were developed. These include the Edwards model of the polymer chain and its confining tube (Chapters 7 and 9), the modern view of semidilute solutions established by des Cloizeaux and de Gennes (Chapter 5), and the reptation theory of chain diffusion developed by de Gennes (Chapter 9) that led to the Doi-Edwards theory for the flow properties of polymer melts. 

The reptation ideas discussed above will now be combined with the relaxation ideas discussed in Chapter 8 to describe the stress relaxation modiihis G t) for an entangled polymer melt. On length scales smaller than the tube diameter a, topological interactions are unimportant and the dynamics are similar to those in unentangled polymer melts and are described by the Rouse model. The entanglement strand of monomers relaxes by Rouse motion with relaxation time Tg [Eq. (9.10)] ... 

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). 

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. 

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... 

High-resolution proton DQ MAS NMR is used as a new technique that is capable of revealing complex motional processes in entangled polymer melts. Theoretical analysis shows the connection of quantities relating anisotropic polymer dynamics to data obtained from our DQ-MAS NMR experiment. With this technique, dynamic chain ordering as well as scaling laws consistent with the reptation model was previously observed for polybutadiene (PB). 

Theories of gel electrophoretic mobility are usually based on the reptation theory introduced for polymer melts by de Gennes, as well as Doi and Edwards [32,33,7]. Their approach succeeded in explaining the inverse relationship between mobility and chain length for short fragments of DNA [7d], By replacing the tube in the reptation model by open spaces and lakes connected by straights, Zimm explained the antiresonance phenomena observed in the field inversion experiments that occur when the time scale for the formation of the conformational change of the DNA coincides with the time scale for the field cycle [7c]. 

It is interesting to see what we can learn from the reptation model. The best way to see it is to look at a simple experiment. Take a polymer melt or a concentrated solution and place it between two plates, similar to the geometry shown in Figure 12.1 (in practice it can also be a gap between two co-axial cylinders). At time t = 0, apply a constant shear stress a, and measure the relative deformation, or strain 7, as it develops in time after the stress is switched on at f = 0. If a is small, the deformation will be proportional to the stress ... 

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