Reptation in a tube

While the flow and creep of a polymer are necessarily associated with translational motion, the internal rotational shape change is taking place at the same time. So the total movement is a combination of the two. As before, when polymers approach the ideal flexible model with copious internal rotation, then their behaviour ceases to be sensitive to local conformational constraints and can 

From the fact that the high viscosity of a molten polymer varies as almost the power to 3.4 of the molar mass, reptation must have a similar high dependence on molar mass. But why is the process also so very dependent on the time available for it  

Reptation consists of two coupled processes (shape change and translation), both of which can be considered as diffusion random walks. Now the statistics of 

So the distance diffused is proportional to the fourth root of time, or the time required is proportional to the fourth power of the distance to be moved. Now the time required is proportional to the viscosity, and the distance to escape is proportional to the chain length, or the molar mass. So the viscosity (and time for flow) of entangled chains should be proportional to the fourth power of the molar mass. 

In conclusion, the important difference between the flow behaviour of short and long chains is the influence of chain entanglement, and this factor influences many of the physical properties, both in the melt and in the solid state. 

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The crucial assumption of the Doi-Edwards theory is that the primitive chain reptates in a tube fixed in space. However, in monodisperse solutions, all chains are wriggling simultaneously, so that the tube around each chain is never fixed but successively renewed by different chains. Hence, the Doi-Edwards theory is not self-consistent. This fact has given rise to recent measurements of the tracer diffusion coefficient as a function of the molecular weight and concentration of the matrix component. 

The symbols in Fig. 9.18 are experimental data of Daniels etal. [25]. The solid and dotted lines are predictions of the hierarchical model with monodisperse and polydisperse arms and backbone molecular weights, respectively. The parameters are given in the caption of Fig. 10.6 with a= 4/3 the parameter value = 1/12 is used in Eq. 9.9 for the branch-point mobility, as suggested by Daniels et al. [25]. Once the arms relax, the backbone is assumed to reptate in a tube dilated by the dynamic dilution due to relaxation of the star arms. 

Fig. 23.4. Each molecule in o linear polymer con be thought of as being contained in a tube made up by its surroundings. When the polymer is loaded at or above Tg, each molecule can move (reptate) in its tube, giving strain.

A second appealing feature of tube model theories is that they provide a natural hierarchy of effects which one can incorporate or ignore at will in a calculation, depending on the accuracy desired. We will see how, in the case of linear polymers, bare reptation in a fixed tube provides a first-order calculation more accurate levels of the theory may incorporate the co-operative effects of constraint release and further refinements such as path-length fluctuation via the Rouse modes of the chains. 

Most theoretical treatments of the gel electrophoresis of DNA molecules are based upon de Gennes (1971) concept of reptation. The migration of the nucleic acid is considered to occur in a tube formed by the polymer matrix of the gel, through which the DNA migrates rather like a... 

Figure 5 The basic concept of P.G. De Gennes reptation of a chain trapped in a tube-like region by migration of "defects" along the nhain.

According to Doi-Edwards theory, the reptation of the long chains will occur in a tube whose diameter a veuies as Thus the number of monomers between entanglements will scale as < ) . Accordingly, the reptation time x (relation 3-14) should be proportional to (]) as a first approximation, the zero-shear viscosity tio and the steady-state compliance J should respectively scale as [Pg.133]

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). 

Consider a chain in a tube at time zero. Analysis of the reptation process (de Gennes 1971) shows that after a time t, only a fraction P (t) of the original tube remains unvacated, namely... 

Figure 1 (a) Reptation of a linear polymer molecule in a tube, (b) Arm retraction mechanism in the tube model for a star polymer... 

Curtiss and Bird introduce reptation in a maimer which does not involve the tube concept, at least not in an explicit way. Their model leads to a constitutive equation in which the stress is the sum of two contributions. On contribution is exactly 1/3 the expression obtained by Doi and Edwards when those workers invoke the independent alignment approximation, i.e., that contribution is a special case of the BKZ relative strain equation. The othCT contribution depends on strain rate and is proportional to a link tension coefficent c (0 < e < 1) which diaracterizes the forces along the chain arising from the continued displacements of chain relative to surroundings. 

The stress relaxation modulus G t) for the reptation model was calculated by Doi and Edwards in 1978 by solving the first-passage problem for the diffusion of a chain in a tube (see Problem 9.6) ... 

The simple reptation model does not properly account for all the relaxation modes of a chain confined in a tube. This manifests itself in all measures of terminal dynamics, as the longest relaxation time, diffusion coefficient and viscosity all have stronger molar mass dependences than the reptation model predicts. Tn Sections 9.4.5 and 9.6.2, more accurate ana-... 

These tube length fluctuation modes (see Section 9.4.5) of the neighbouring chains affect the constraint release modes of a given chain. If entanglements between chains are assumed to be binary, there should be a duality between constraint release events and chain in a tube relaxation events. A release of an entanglement by reptation or tube length fluctuation of one chain in its tube leads to a release of the constraint on the second chain. If this duality is accepted, the distribution of constraint release rates can be determined self-consistently from the stress relaxation modulus of the tube model. 

There is a conceptual difference between tube dilation and constraint release. The motion of the tube in constraint release does not affect singlechain modes inside the tube (reptation and tube length fluctuation). The tube diameter for these single chains in a tube processes is, on average, constant. In contrast, the tube diameter increases with time in the tube dilation process and dramatically affects the chain motion within the tube. 

Consider an isolated long probe P-mer entangled in a melt of shorter Wmers. Tube dilation assumes that as soon as short chains relax, stress in the long P-mer drops to zero. In particular, a version of tube dilation called double reptation imposes an exact symmetry between single chains in a tube and multi-chain processes. As one chain reptates away, stress at a common entanglement (stress point) is relaxed completely. In constraint release models, this stress relaxes only partially due to connectivity of the P-mer. 

Figure 3-24. Reptation model view of a polymer chain (a) with obstacles (dots) (b) in a tube.

Another physical representation of the effect in question is the use of the reptation model [42], in which a macromolecule is assumed to be encapsulated in a tube of a finite... 

Self-diffusion of random coiled chains in the bulk was studied by Edwards [32] and de Gennes [33], who developed a reptation model of a chain confined to a tube. As shown in Figure 8.2, a single chain of TP with a contour length of L is diffused in a tube, which represents the topological constraints on its motion imposed by the matrix and by other chains in the bulk. At the same time, the tube contour represents another random walk. At a given time the... 

In the previous chapter, we discussed the dynamics of a polymer in a fixed network. We shall now discuss the polymer dynamics in concentrated solutions and melts. In these systems, though aU polymers are moving simultaneously it can be argu that the reptation picture will also hold. Consider the motion of a certain test polymer arbitrarily chosen in melts. If the test polymer moves perpendicularly to its own contour, it drags many other chains surrounother hand the movement of the test polymer along its contour will be much easier. It will be thus plausible to assume that the polymer is confined in a tube-like region, and the major mode of the dynamics is reptation. 

The mutual steric restrictions of entangled chains at deformation are accotmted for in a tube model considering the reptation motion of network subchains. This approach was proposed by Edwards and Vilgis ° and Heinrich et al. Later the tube model was further developed (see References 42 and 43). 

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