Reptation, theory

The five time regions are based on the reptation theory proposed by De Gennes [46,47] and Doi and Edwards [48,49] for bulk dynamics of polymer melts and concentrated polymer solutions, and are discussed in detail in Chapter 3 of Ref. [1]. 

The term aK2v", derived from reptation theory, describes the velocity-dependent energy necessary to fracture the bulk adhesive. K2 is the consistency which relates the viscosity to the shear rate for a non-newtonian fluid. a = TtraL fh", with r being the chain radius, L the chain length, a the density of chains crossing over the fracture plane, and h is the distance between the chain and reptation tube. 

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... 

So far we have invoked reptation theory to describe the behaviour in the melt state. We can use scaling theory in the form of Equations (5.122) and (5.123) to express the concentration dependence of the modulus and viscosity. By inspection of Equations (5.128) and (5.130) ... 

The rod is visualised as being constrained to a tube in a similar fashion to entanglements constraining a polymer in reptation theory. So for a finite concentration our diffusion coefficient and rotary Peclet number changes ... 

Polydispersity of the molecular weight is not so well described by the DE approach, even qualitatively. In reptation theory the blend of two compatible homopolymers A and B of different molecular weights is given by... 

Further, reptation theory asserts26) that the molecular-weight dependence of the diffusion coefficient in an entangled gel should have the form... 

From this point of view the Doi-Edvards reptation theory can be regarded as the most perfect network theoryS2). In a molten polymer, macromolecules can not move notable in lateral direction since that is impeded by other polymer chains. This circumstance in the Doi-Edvards theory is taken into account by means of introduction of a... 

In description of effects observed in extension of molten polymers, the determinant is the phenomenon of anisotropy of the mobility of macromolecules. In the Doi-Ed-vards reptation theory the anisotropy of the mobility of macromolecules is taken into account topologically by means of placing a macromolecule into a certain hypothetical tube. In this case large-scale movements are allowed only along the macromolecule and are totally inhibited in the lateral direction. This, indeed, is a limiting case of mobility anisotropy. 

The reptation theory of viscoelasticity developed by Doi Edvards has paractically predetermined the appearance of Curtiss-Bird s reptation theory55). The latter is constructed on the basis of general kinetic theory of polymer fluids in phase space. According to the Curtiss-Bird theory, the longitudinal viscosity may increase only by a factor of two compared to the initial value. Note that a more significant increase in longitudinal viscosity was observed in experiments. 

Reptation theory description of polymer structure is analogous to a bowl of live snakes (Teraoka et al., 1992). In this bowl reside a mesh of entangled, linear flexible polymer chains that continue to wriggle within a minimal range, effectively forming a tube-like structure. It is within this tube that the polymer chains move back and forth and over sufficient periods of time, the polymer chain can actually move along the tube to new interaction sites with fellow polymer chains or other media. 

One of the most successful models for gel electrophoresis is the reptation theory of Lumpkin and Zimm for the migration of double-stranded DNA (Lumpkin, 1982). An in-depth discussion can be found in Zimm and Levene (1992) for a synopsis see Bloomfield et al. (2000). The velocity v of a charged particle in a solution with an electric field E depends on the electrical force Fei = ZqE, in which Z is the number of charges and q is the charge of a proton, and the frictional force l fr = —fv, in which/is the frictional coefficient. At steady state, these forces balance and the velocity is v = ZqE/f. The electrophoretic mobility fi is the velocity relative to the field strength, fi = vE = Zq/f. 

Another prediction of reptation theory is that molecules move fastest when the entire chain is in the same tube. Partial unfolding or branching of the helix makes this less likely, and consequently impede migration, resulting in anomalous migration patterns that can be used to model helical junctions and bend angles (Lilley, 2008 Zinkel and Crothers, 1990). 

Pierre-Gilles deGennes (47) utilized this concept and coined the term in his work to explain why the relaxation times of entangled melts have a X A/34 dependence. Earlier, the lateral confinement of melt chains to a tubelike region had been postulated by Edwards (54). Since these early days of the reptation theory, a very significant volume of work has been dedicated to incorporating features that are physically reasonable and warranted in... 

The constitutive equations benefiting from the specific representations of reptation theory have the general form of the Lodge rubber-like liquid equation, since they are all... 

Kholodenko AL (1996) Reptation theory geometrical and topological aspects. Macromol Theory Simul 5 1031-1064... 

It can thus be deduced that the "dilution effect" of short chains is a dynamical process which does not act at short times. On the contrary, at longer times, as reptation theory postulates, the topological constraints which govern the chain dynamics appear looser as the concentration of short chains is increased. Indeed, even in an environment of equivalent local chain characteristics, as... 

For entangled systems, the two first conditions are fulfilled in the framework of reptation theories a comprehensive expression of the monodisperse relaxation modulus G(M,t) is given by expression 3-24 and the double reptation model generalized to a continous molecular weight distribution provides the integral relation between the MWD function P(M) and the polydisperse experimental... 

The Rouse model is the earliest and simplest molecular model that predicts a nontrivial distribution of polymer relaxation times. As described below, real polymeric liquids do in fact show many relaxation modes. However, in most polymer liquids, the relaxation modes observed do not correspond very well to the mode distribution predicted by the Rouse theory. For polymer solutions that are dilute, there are hydrodynamic interactions that affect the viscoelastic properties of the solution and that are unaccounted for in the Rouse theory. These are discussed below in Section 3.6.1.2. In most concentrated solutions or melts, entanglements between long polymer molecules greatly slow polymer relaxation, and, again, this is not accounted for in the Rouse theory. Reptation theories for entangled... 

The reptation theory has been controversial. In large part, this is because experimental data and computer simulations usually show some deviations from the behavior expected for pure... 

Despite these complications, there are now numerous evidences that the tube model is basically con-ect. The signatory mark that the chain is trapped in a tube is that the chain ends relax first, and the center of the chain remains unrelaxed until relaxation is almost over. Evidence that this occurs has been obtained in experiments with chains whose ends are labeled, either chemically or isotopically (Ylitalo et al. 1990 Russell et al. 1993). These studies show that the rate of relaxation of the chain ends is distinctively faster than the middle of the chain, in quantitative agreement with reptation theory. The special role of chain ends is also shown indirectly in studies of the relaxation of star polymers. Stars are polymers in which several branches radiate from a single branch point. The arms of the star cannot reptate because they are anchored at the branch point (de Gennes 1975). Relaxation must thus occur by the slower process of primitive-path fluctuations, which is found to slow down exponentially with increasing arm molecular weight, in agreement with predictions (Pearson and Helfand 1984). 

Figure 3.29 Linear moduli G and G" versus frequency shifted via time-temperature superposition to 27°C for a polybutadiene melt of molecular weight 360,000 and of low polydispersity. The dashed line is the prediction of reptation theory given by Eq. (3-67) the solid line includes effects of fluctuations in the length of the primitive path. (From Pearson 1987.)...

Figure 3.35 Steady-state values of the reduced shear stress <712/ and first normal stress difference N / as functions of dimensionless shear rate y Zr predicted by the equations of a constraint-release reptation theory (see Problem 3.10) for Xd/Zr — (a) 50, (b) 150, and (c) 500, where Zd is the reptation time and Zr is the Rouse retraction time. See also Marracci and lanniruberto (1997). (From Larson et al. 1998, with permission.)...

Problem 3.13(a) (Worked Example) You have a binary blend containing two different molecular weights. Ml and Ms, of the same polymer. Let the weight fraction of Ml be 0, where Ml corresponds to the high molecular weight. Approximate the linear relaxation moduli of the pure melts by Gi t) = Go exp(-t/rt) and Gs t) Go exp(-t/rs). Derive an expression for G(/) for the blend from double reptation theory. 

Doi and Edwards (1978) and Kuzuu and Doi (1980) have solved the Smoluchowski equation (6-47)-(6-48) for simple shearing and elongational flows, and they obtained predictions of rheological behavior that are similar to those of the reptation theory for concentrated flexible polymers discussed in Section 3.7.5.1. Figure 6-19, for example, shows the shear-rate-dependence of the shear viscosity and first and second normal stress coefficients predicted by the Doi-Edwards theory for semidilute rods these results are similar to those predicted by the Doi-Edwards theory for entangled flexible molecules. At... 

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