Primitive chain reptational motion

Figure 5.26. Running along the centre of the tube is a primitive chain. This is the shortest path down the tube. The deviations of the polymer chain from this path can be considered as defects. The motion of these kinks or defects in the chain away from the primitive path allows the chain to move within the tube. The polymer creeps through the tube, losing its original constraints and gradually creating a new portion of tube. This reptilian-like motion of the chain was named by de Gennes from the Latin reptare, to creep, hence reptation.

According to the Doi-Edwards theory, after time t = Teq following a step deformation at t = 0, the stress relaxation is described by Eqs. (8.52)-(8.56). In obtaining these equations, it is assumed that the primitive-chain contour length is fixed at its equilibrium value at all times. And the curvilinear diffusion of the primitive chain relaxes momentarily the orientational anisotropy (as expressed in terms of the unit vector u(s,t) = 5R(s,t)/9s), or the stress anisotropy, on the portion of the tube that is reached by either of the two chain ends. The theory based on these assumptions, namely, the Doi-Edwards theory, is called the pure reptational chain model. In reality, the primitive-chain contour length should not be fixed, but rather fluctuates (stretches and shrinks) because of thermal (Brownian) motions of the segments. 

To sort out such a complicated dynamic situation, we first assume that the primitive chain is nailed down at some central point of the chain, i.e. the reptational motion is frozen only the contour length fluctuation is allowed. This is equivalent to setting rg —> oo while allowing the contour length fluctuation 5L(t) to occur with a finite characteristic relaxation time Tb- In this hypothetical situation, the portion of the tube that still possesses tube stress tt fa tb is reduced to a shorter length Lq, because of the fluctuation SL(t). Then, tt tube length that still possesses tube stress can be defined by... 

While the [t, E) relaxation is going on, the relatively slow relaxation of (v v ) by the reptational motion also gets under way. At t when the primitive chain has recovered its equilibrium contour length, the effect of the contour length fluctuation on the terminal relaxation should basically be the same as that in the linear region. In other words, the relaxation of (v (t)vn(t)) should be described by Eqs. (9.11) and (9.12). Thus from Eq. (12.24), we write the stress relaxation after t as... 

Pure reptation is possible only under the very strong topological constraint that puts the instantaneous orientations of all segments of a primitive chain, except ones at its ends, in a complete correlation. In actual entangled systems, since the constraint may not be that strong, the chains are likely to wriggle in modes other than reptation. We have no a priori reason to deny the possibility of such non-reptative chain motions. 

These g t) data do not favor the reptation model, but are not definitive enough to rule out the possibility of reptation motion. Thus, Kolinski et al. [57] stepped further by calculating the longitudinal and lateral displacements of the primitive path used in the Doi-Edwards theory. Such calculations should give us direct information on local chain motions. 

The first assumption corresponds to neglecting the fluctuations of the contour length. The second states that the motion of the primitive chain is reptation. The third guarantees that the conformation of the primitive... 

According to the three assumptions of the tube model given earlier, the primitive chain moves only along its contour. Its motion is like a snake slithering on earth. This motion is called reptation. The end of the primitive chain can explore its next direction, but the rest follows its own existing path. 

S Disengagement The tube renewal is made possible by reptation of the primitive chain along its own contour. To estimate the time necessary for the tube renewal, we need to know the nature of the reptation motion. The motion is a one-dimensional diffusion along a curve in three dimensions. Whether the primitive... 

Now we move the primitive chain. The chain reptates along the tube with a diffusion coefficient The point at i = s(0) moves with the same diffusion coefficient D. We record the slithering motion of the point by the curvilinear distance measured from the end of the primitive chain at i = 0 (gray line), as shown in Figure 4.36. Let the point on the primitive chain slide to s(t) in time t. Then, the one-dimensional displacement s(t) - x(0) along the contour of the primitive chain... 

The process of disentangling, as it is envisaged in the reptation model, is sketched in Fig. 6.11. The motion of the primitive chain , the name given to the dynamic object associated with the primitive path, is described as a diffusion along its contour, that is to say, a reptation . The associated curvilinear diffusion coefficient can be derived from the Einstein relation, which holds generally, independent of the dimension or the topology. Denoting it D, we have... 

Fig. 6.11. Reptation model Decomposition of the tube resulting from a reptative motion of the primitive chain. The parts which are left empty disappear...

IV.6.3 Tube relaxation In the tube model, relaxation occurs solely from random Brownian motion of the chain, i.e., unbiased reptation. This motion slowly brings the primitive chain out of its oriented tube in a... 

The BRM is a natural generalization of the original reptation model where the motion of the primitive chain in its "tube is considered to become biased when an electric field is applied. The electrophoretic properties of the DNA molecules are related in the BRM to the effect of the field on the conformation of the reptation tube since this conformation tends to orient in the field direction when the primitive chain creates new tube sections, the electrophoretic velocity becomes a nonlinear function of the electric field. This tube alignment also reduces the effectiveness of the entanglements in opposing the electrophoretic drift, with the consequence that, except for transient effects or very small molecules, the mobility becomes molecular-size independent in continuous fields. 

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 ... 

Let us now consider the dynamics of a primitive chain. Within the concept of reptation, in the short timescale the motion of the polymer can be regarded as wriggling around the primitive path. On a longer timescale, the conformation of the primitive path changes as the polymer moves, creating and destroying the ends of the primitive path. In the absence of an external potential, the time evolution (i.e., the dynamics) of the primitive... 

Figure 4.53 Schematic illustration of the reptation process in polymer melts, showing chain entanglements (light arrows), the wriggling motion of the polymer chain (darker arrows), and the primitive path of the polymer chain (dark Une). Reprinted, by permission, from G. Strobl, The Physics of Polymers, 2nd ed., p. 283. Copyright 1997 by Springer-Verlag.

Motion within the tube is achieved by a random walk ( primitive path ) of unit steps of the order of the tube diameter, a. When a straight reptation tube is considered, for simplicity, reptation diffussional motion of the chain out of the tube is represented schematically in the steps depicted from Fig. 3.9(d)(i) to Fig. 3.9(d)(v). 

At times t >Teq, the wriggling motion results merely in a fluctuation around the primitive path, so the ch moves coherently in a one-dimension diffusion process, keeping its arc length constant. The macroscopic diffusion coefficient of a reptating chain scales with chain length (molecular weight) as ... 

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