Predictions of Reptation Theories

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). 

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.)...

These should he compared to the dependencies on molar mass predicted hy the Rouse model (Eq. 2.34). As indicated in Fig. 2.21, experiments reveal that for entangled polymers the zero-shear viscosity scales with The subtle difference in exponent compared to the predictions of reptation theory indicates additional effects not considered in the original model. These include the release of the entanglement constraints and fluctuation-driven stretchings and contractions of the chain along the tube. 

The self-diffusion of PDMS chains in melts satisfied Eq. (1) according to the theoretical predictions of reptation theory to a degree of accuracy... 

By extrapolating to zero field strength it is possible to find an inverse relationship between electrophoretic mobility and molecular weight (14). This relationship contrasts with that obtained with a Ferguson plot (15) normally used for proteins, but also frequently and often erroneously used for DNA. However, the inverse relationship is fully predicted by reptation theory (3, 9). In summary, determining precise DNA molecular weights using gel electrophoresis requires careful attention to experimental detail and Judicious choice of data workup. 

Recent research has addressed the shortcomings of the original Doi-Edwards exposition of reptation theory, firstly for low shear rate linear behaviour and secondly for non-linear behaviour at large shear rates. Doi-Edwards linear reptation theory predicts thatar scales with A as A, whereas there is a large body of experimental evidence that viscosity scales with molecular mass M as M -. Secondly, linear theory predicts that the dynamic loss modulus G2 o)) is proportional to in the intermediate frequency range whereas experiment... 

Particular emphasis will be laid on the experimental verification of features of standard theories of polymer dynamics such as the Rouse model, the tube/reptation model [1] or the renormahzed Rouse models. Based on the special conditions under which predictions of these theories match experimental findings well, the application limits and the deficiencies of these models for more general scenarios become obvious and help to ameliorate the underlying model ansatz for chain dynamics. 

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. 

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. 12. The rheological functions G ((o) and G"(co) for an H-shaped PI of arm molecular weigh 20 kg mol and backbone 110 kg mol" [46]. The high-frequency arm-retraction modes can be seen as the shoulder from co 10 to co 10 together with a low-frequency peak due to the cross-bar dynamics at co 10. The smooth curves are the predictions of a model which takes Eq. (33) as the basis for the arm-retraction times and a Doi-Edwards reptation spectrum with fluctuations for the backbone. The reptation time is correctly predicted, as is the spectrum from the arm modes, though the low frequency form is more polydisperse than the simple theory predicts...

Wagner et al. (63-66) have recently developed another family of reptation-based molecular theory constitutive equations, named molecular stress function (MSF) models, which are quite successful in closely accounting for all the start-up rheological functions in both shear and extensional flows (see Fig. 3.7). It is noteworthy that the latest MSF model (66) is capable of very good predictions for monodispersed, polydispersed and branched polymers. In their model, the reptation tube diameter is allowed not only to stretch, but also to reduce from its original value. The molecular stress function/(f), which is the ratio of the reduction to the original diameter and the MSF constitutive equation, is related to the Doi-Edwards reptation model integral-form equation as follows ... 

The spectroscopic data have been compared with the theoretical predictions of the Doi- Edwards model. In the time scale of our experiments, a quantitative agreement between experiment and theory is obtained if chain length fluctuations, retraction and reptation are taken into account. In the case of star polymers, the large scale fluctuation mechanism as proposed by Pearson and Helfand associated with the retraction process is accounting for... 

The prediction of the above equation is the closest of any molecular theory to experimentally observed 3.4 power. A constitutive equation based on the reptation model can be written as ... 

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... 

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 3.35 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.)...

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... 

The reptation theory (22-23) for polymer solutions predicts that the viscosity of the solution, T, varies with the polymer molecular weight, M, as... 

Since it emeiged, the idea of reptation motion has rapidly gained great popularity in the polymer community. In fact, it makes testable predictions about a wide range of dynamic properties of polymer concentrates, and also the reptation motion is appealing to physical intuition. Details of the reptation theory and its applications can be found in the book of Doi and Edwards [4]. 

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