Tube diameter reptation model

Fig. 3. 30 Apparent tube diameters from model fits with pure reptation (filled squares) and reptation and contour-length fluctuations (open squares) as a function of molecular weight. The dotted line is a guide for the eye (Reprinted with permission from [71]. Copyright 2002 The American Physical Society)...

A comparison with the tube diameter derived from plateau moduli measurements on PEB-7 underlines this assertion. The coincidence of tube diameters determined macroscopically by application of the reptation model and direct microscopic results is far better than what could have been expected and strongly underlines the basic validity of the reptation approach. 

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 tube model avoids dealing with specification of an average molecular weight between entanglement points (Mg), but the diameter of the tube d) is an equivalent parameter. Further, the ratio of the contour length to diameter L/d) is taken to be approximately the magnitude of the entanglements per molecule (M/Me). Some of the important relationships predicted from the reptation model include (Tirrell, 1994) ... 

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)] ... 

On length scales larger than the tube diameter, topological interactions are important and the motion is described by the reptation model with the chain relaxation time given by the reptation time ... 

The above equation shows that as a time period equal to the reptation time elapses, the tube diameter goes to infinity, and hence reptation no longer becomes the mode of diffusion for diat chain. This completes the self-consistent formulation for the tube model. 

The first discrepancy at large q can be suppressed within the framework of the reptation model, but with a realistic value of the tube diameter corresponding to Eq. (7.24). The result will be the disappearance of the plateau in the calculated curve q S(q). A finite value of D is in fact the result of a Rouse motion during the period (0, T, where T. cx M is the Rouse time of a chain of mass M. Thus it resembles the Rouse model at this short T. At longer t the resemblance remains but the time decay is now slower and ind mdmt of q at large q, which is different from the Rouse model. 

The plateau modulus is determined from rheological measurements. In the reptation model, it is related to the tube diameter dr —... 

So far all the evidence from simulations supports the reptation model or its variants very strongly. Although the chains are relatively short, we can still ask to what extent do the chains move along the tube given by the coarse-grained contour of the chain. It is clear that one here is only able to observe the very onset of this motion. For the original chain of iV = 200 for the MD data the tube diameter dj is only N/Ne) 2.4 times smaller than the mean end-to-end distance of the chain itself. To see the confinement directly, one ean construct a primitive chain (PC) as ... 

One of the examples of the scaling representation of macromolecules is the reptation model [48], according to which the tube diameter O, in which the macromolecule is confined (equal to the distance between entanglements nodes), can be estimated according to the relationship [49] ... 

The reptation model outlined here has been studied in great detail by Doi and Edwards, who considered in particular the influence of the tube diameter (or the so-called entanglement mass) and the viscoelastic behavior of polymer melts this is the subject of Chapter 8. 

We have already mentioned that, for most purposes, a semi-dilute solution can be viewed as a melt of blobs of size (, each containing g = c monomers. It is thus reasonable to assume that the reptation model that we just described can be applied to a semi-dilute solution. The tube diameter is proportional to the blob size in this model. The overall contour length of the tube is (N )/g. The local mobility Pq is the blob mobility Pq = l/ 6nrjQ ) and the tube diffusion constant is — (Tg)/ (6nrjQ N). This gives a reptation time for a polymer chain in a semi-dilute solution... 

Fig. 7. Illustration of a polymer chain (the tagged chain) confined in the fictitious tube of diameter d formed by the matrix. The contour line of the tube is called the primitive path having a random-walk conformation with a step length a=d. The four characteristic types of dynamic processes (dotted arrow lines) and their time constants Zs, Zg, Zr, and za defined in the frame of the Doi/Edwards tube/reptation model are indicated...

Below we will come back to the reptation model in context with the dynamics of polymers confined in tube-like pores formed by a solid matrix. For a system of this sort the predictions for limits (II)de and (III)de (see Table 1) could be verified with the aid of NMR experiments [11, 95] as well as with an analytical formalism for a harmonic radial potential and a Monte Carlo simulation for hard-pore walls [70]. The latter also revealed the crossover from Rouse to reptation dynamics when the pore diameter is decreased from infinity to values below the Flory radius. 

The theoretical background of the confinement effect in (artificial) tubes was recently examined in detail with the aid of an analytical theory as well as with Monte Carlo simulations [70]. The analytical treatment referred to a polymer chain confined to a harmonic radial tube potential. The computer simulation mimicked the dynamics of a modified Stockmayer chain in a tube with hard pore walls. In both treatments, the characteristic laws of the tube/reptation model were reproduced. Moreover, the crossover from reptation (tube diameter equal to a few Kuhn segment lengths) to Rouse dy-... 

Fig. 47. Spin-lattice relaxation dispersion for a chain of 1 =1,600 Kuhn segments (of length b) confined to a randomly coiled tube with a harmonic radial potential with varying effective diameters d. The data were calculated with the aid of the harmonic radial potential theory [70]. c is a constant. At low frequencies the curves visualize the crossover from Rouse dynamics depending on the effective tube diameter. The latter case is described by a Tj dispersion proportional to characteristic for limit (II)de of the tube/ reptation model...

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