Reptation concept

Introduction of the reptation concept by De Gennes [43] led to further essential progress. Proceeding from the notion of a reptile-like motion of the polymer chains within a tube of fixed obstacles, De Gennes [43-45], Doi [46,47] and Edwards [48] were able to confirm Bueche s 3.4-power-law for polymer melts and concentrated polymer solution. This concept has the disadvantage that it is valid only for homogeneous solutions and no statements about flow behaviour at finite shear rates are analysed. 

The simple reptation concept proposed by de Gennes and developed by Doi and Edwards deals with permanent entanglements creating a fixed tube around each chain. 

Figure 4-14 Illustration of Reptation Concept, Credited to de Gennes and Doi and Edwards, of a Polymer Chain in a Concentrated Polymer Solution. The cross sections of the constraining chains and the tube for reptation of the polymer chain are shown.

A molecular theory of viscoelasticity of molten, high molecular weight polymers that makes use of the reptation concept has been developed by... 

We will focus in this paper on the rheological properties, at room temperature, of styrene-isoprene block copolymers, particularly Triblock [SISj-Diblock [SI] copolymer blends. We will describe the effect of the molecular parameters of the copolymers on the rheological behavior, and wiU propose, on the basis of molecular dynamics models derived from the reptation concept and the analysis of the dynamic behavior of the blend [SIS-SI], a model which allows calculation of the variation of the complex shear modulus as a function of frequency. Different types of macromolecules have been designed from calculations using this molecular model in order to improve the processing and end-user properties of the full formulations (HMPSAs). 

The linear viscoelastic behavior of the pure polymer and blends has already been described quantitatively by using models of molecular dynamics based on the reptation concept [12]. To describe the rheological behavior of the copolymers in this study, we have selected and extended the analytical approach of Be-nallal et al. [13], who describe the relaxation function G(t) of Hnear homopolymer melts as the sum of four independent relaxation processes [Eq. (1)]. Each term describes the relaxation domains extending from the lowest frequencies (Gc(t)) to the highest frequencies (Ghf( )), and is well defined for homopolymers in Ref [13]. 

One of the most important and useful applications of the reptation concept concerns crack healing, which is primarily the result of the diffusion of macromolecules across the interface. This healing process was studied particularly by Kausch and co-workers [76]. The problem of healing is to correlate the macroscopic strength measurements to the microscopic description of motion. The difference between self-diffusion phenomena in the bulk polymer and healing is that the polymer chains in the former case move over... 

The local dynamics is naturally strongly dependent on the exact chemical nature and structure of the polymer one studies. The large scale dynamics, however, is largely universal and is described with the Rouse model whereas for longer chains the tube model and reptation concept is believed to describe the chain dynamics [2]. It is easy to see that no single simulation method can capture the physics of polymer dynamics on all these length and time scales [3]. For situations where we can ignore quantum effects (which can, however, be important in polymer crystals [4]) MD simulations with chemically realistic force fields are the method of choice to study local relaxation. 

A different approach, based on the reptation concept, was initiated by the present author and was recently augmented by Edwards and Doi. As explained in the next section, it leads to tj — N , and there is no... 

Finally, notice how large the leptation time [eq. (VII. 13)] can he. Assuming that our melt is far above any glass transition temperature T , Ti may be of order 10 sec. If we have a long chain (N = 10 ), this leads to T( 10 sec— i.e., to very macroscopic times. Thus the reptation concept does give us a plausible feeling for the viscoelastic behavior of polymers. 

From the considerations presented in the previous section, it is evident that it is desirable to choose MC moves X X such that the relaxation time resulting from a Markov chain of such moves for the configurations of the polymer chains is as small as possible. This is particularly important for dense melts of long (and hence mutually entangled ) polymer chains, where the reptation concepts imply an asymptotic scaling rocN , that is, the relaxation is distinctly slower than for isolated chains (cf. eqns [13]-[16]).The situation would be even worse for dense melts of star polymers (or other branched polymers) where even an exponential scaling (lnr°=N) may result. ... 

The reptation concept has subsequently led to the formulation of a detailed theory of viscosity, due to Doi and Edwards, which seems to explain a sizable part of the experimental data, not only for small deformations but even for non-linear phenomena. Today, the liquid linear polymers are becoming model systems for rheology. 

Masao Doi and Sam F. Edwards (1986) developed a theory on the basis of de Genne s reptation concept relating the mechanical properties of the concentrated polymer liquids and molar mass. They assumed that reptation was also the predominant mechanism for motion of entangled polymer chains in the absence of a permanent network. Using rubber elasticity theory, Doi and Edwards calculated the stress carried by individual chains in an ensemble of monodisperse entangled linear polymer chains after the application of a step strain. The subsequent relaxation of stress was then calculated under the assumption that reptation was the only mechanism for stress release. This led to an equation for the shear relaxation modulus, G t), in the terminal region. From G(t), the following expressions for the plateau modulus, the zero-shear-rate viscosity and the steady-state recoverable compliance are obtained ... 

The tube model and reptation concept have become the cornerstones of most of our understanding of the mechanical and transport properties of dense polymer systems. In 1967, S.F. Edwards originated the tube model while studying rubber elasticity. The concept of a tube was introduced in order to describe the topological constraints imposed on a polymer chain by the neighbouring polymer strands in a lightly crosslinked rubber. The... 

Jaan Noolandi Ph.D., is a senior research fellow at the Xerox Research Center of Canada. He is active in the areas of polymer blends, polymer dynamics and the application of reptation concepts to gel electrophoresis of DNA macromolecules. 

Fig. 4.1 Sketch of the historical development of the tube constraint and reptation concept. Starting from a network Edwards in 1967 defined the confinement to the tube, while deGennes in 1971 realized that for long chains the ends only play a small role for intermediate times.

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