Reptation Model of Molecular Motion

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. 

In a number of experimental studies of polymer diffusion, molar mass exponents close to 2 have been found, though always with some deviations. For example, using radio-labelled molecules, the diffusion coefficient of polystyrene in dibutyl phthalate was found to follow the relationship 

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one in experimental and theoretical polymer physics, and we may expect that in time the simple reptation model will be superseded by more sophisticated models.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. 

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The BRM is a natural generalization of the original reptation model where the motion of the primitive chain in its "tube is considered to become biased when an electric field is applied. The electrophoretic properties of the DNA molecules are related in the BRM to the effect of the field on the conformation of the reptation tube since this conformation tends to orient in the field direction when the primitive chain creates new tube sections, the electrophoretic velocity becomes a nonlinear function of the electric field. This tube alignment also reduces the effectiveness of the entanglements in opposing the electrophoretic drift, with the consequence that, except for transient effects or very small molecules, the mobility becomes molecular-size independent in continuous fields. 

In 1971 DeGennes proposed a new model for molecular motion in concentrated polymer systems, one that now dominates all theoretical considerations (Lodge et al., 1990). His model is known as reptation because it describes macromolecular motion much like that of a snake moving in a contorted tunnel formed by the surrounding polymer molecules (Figure 11.5.7). The basic idea is... 

Many simulations attempt to determine what motion of the polymer is possible. This can be done by modeling displacements of sections of the chain, Monte Carlo simulations, or reptation (a snakelike motion of the polymer chain as it threads past other chains). These motion studies ultimately attempt to determine a correlation between the molecular motion possible and the macroscopic flexibility, hardness, and so on. 

The interdiffusion of polymer chains occurs by two basic processes. When the joint is first made chain loops between entanglements cross the interface but this motion is restricted by the entanglements and independent of molecular weight. Whole chains also start to cross the interface by reptation, but this is a rather slower process and requires that the diffusion of the chain across the interface is led by a chain end. The initial rate of this process is thus strongly influenced by the distribution of the chain ends close to the interface. Although these diffusion processes are fairly well understood, it is clear from the discussion above on immiscible polymers that the relationships between the failure stress of the interface and the interface structure are less understood. The most common assumptions used have been that the interface can bear a stress that is either proportional to the length of chain that has reptated across the interface or proportional to some measure of the density of cross interface entanglements or loops. Each of these criteria can be used with the micro-mechanical models but it is unclear which, if either, assumption is correct. 

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... 

Experimental values of the molar mass exponent close to 2 have been obtained. For example, for poly(methyl methacrylate), a value of 2.45 has found (see P. Prentice, Polymer, 1983, 24, 344—350). As with values of selfdiffusion coefficient, this has been regarded as close enough to 2 for reptation to be considered a good model of the molecular motion occurring at the crack tip. 

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