Primitive paths

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.

As in Sect. 2.1, Dj is the curvilinear centre-of-mass diffusion constant of the chain, and is given in terms of the monomeric friction constant by the Einstein relation Dj =kT/Nl. L is as before the length of the primitive path, or tube length of the chain, which is Finally, we need the initial condition on p(s,t), which... 

The chain tension arises in a physical way at timescales short enough for the tube constraints to be effectively permanent, each chain end is subject to random Brownian motion at the scale of an entanglement strand such that it may make a random choice of exploration of possible paths into the surrounding melt. One of these choices corresponds to retracing the chain back along its tube (thus shortening the primitive path), but far more choices correspond to extending the primitive path. The net effect is the chain tension sustained by the free ends. 

In fact, without the inclusion of the chain-end tension, the equilibrium path-length of the chain is not maintained. We can write a potential U(z) for the length of the primitive path z by including both the (quadratic) curvilinear rubber-elastic term and the (linear) end-tension term as follows ... 

Henceforth we take the primitive path co-ordinate s=L-z from the free end inwards to the branch point so that t(s) is an increasing function of s. The prefactor Tq is an inverse attempt frequency for explorations of the potential by the free end, and may be expected to scale as the Rouse time for the star arm (in fact this is not quite true - the actual scaling is as [25,26]). The relaxation mod-... 

These two equations (one for each component) cannot be integrated simply as before because they are coupled by the dependence of the unrelaxed volume fraction on both fractional primitive path coordinates x, XpX2)=blend components. In this case, substitution of the variable into Eq. (37) removes all explicit dependence on... 

The primitive path is therefore much smaller than the contour length along the backbone of the chain. 

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.

The free energy and the elastic force for simple elongation or compression in the primitive path model is62)... 

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). 

The word W consisting of a sequence of letters corresponding to different entanglements (introduced in Sect. 2.21) plays a role of full topological invariant for the PCAO-model. It is closely connected with the concept of the primitive path obtained by means of roughing of the microscopic chain trajectory up to the scale of the lattice cell and by exclusion of all loop fragments not entangled with the obstacles (Fig. 5). 

Fig. 5. Subsequent roughing of microscopic chain trajectory. The final state of the process coincides with the primitive path...

The problem of determination of the partition function Z(k, N) for the iV-link chain having the fc-step primitive path was at first solved in Ref. [17] for the case a = c by application of rather complicated combinatorial methods. The generalization of the method proposed in Ref. [17] for the case c> a was performed in Refs. [19,23] by means of matrix methods which allow one to determine the value Z(k,N) numerically for the isotropic lattice of obstacles. The basic ideas of the paper [17] were used in Ref. [19] for investigation of the influence of topological effects in the problem of rubber elasticity of polymer networks. The dependence of the strain x on the relative deformation A for the uniaxial tension Ax = Xy = 1/Va, kz = A calculated in this paper is presented in Fig. 6 in Moon-ey-Rivlin coordinates (t/t0, A ), where r0 = vT/V0(k — 1/A2) represents the classical elasticity law [13]. (The direct Edwards approach to this problem was used in Ref. [26].) Within the framework of the theory proposed, the swelling properties of polymer networks were investigated in Refs. [19, 23] and the t(A)-dependence for the partially swollen gels was obtained [23]. In these papers, it was shown that the theory presented can be applied to a quantitative description of the experimental data. 

Equation (16) defines the probability density of the fact that two concrete end points of AMink chain are connected by the primitive path of length /i. The probability density for an arbitrary end points having the distance equal to fi is defined by ... 

In the course of investigation of polymer dynamics, all topological states of chains are equivalent due to the presence of their free ends and it is not necessary to construct the topological invariant. However, for every instantaneous chain configuration, its topological state is definite and is determined by the topological invariant - the primitive path. 

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 ... 

The tube itself is a random walk, each step of which has length a. This random walk is called the primitive path of the chain. The contour length of the tube, or the primitive path, is therefore Lj — aMfMg. For polymers of high molecular weight, the tube s contour length is much less than the contour length of the chain (see Fig. 3-24). Thus, the chain meanders about the primitive path. Some values for the tube diameter a for typical polymer melts are presented in Table 3-3. 

If both ends of the molecule are free to move, and so the chain can reptate, segments in the interior of the chain will relax faster by reptation than by primitive-path fluctuations, and so reptation will control the longest relaxation time of the chain. However, because primitive-path fluctuations are so much faster for the chain ends than for the chain center, the chain ends will still relax by primitive-path fluctuations. Only for very high molecular weights (MfMg 100) are the contributions of fluctuations confined to small enough portions of the chain ends that these effects can be neglected. 

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.)...

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