Pure reptational time

Lin Y-H (1985) Comparison of the pure reptational times calculated from linear viscoelasticity and diffusion motion data of nearly monodisperse polymers, Macromolecvles 18 2779-2781. 

Figure 29 Average relaxation time of a high molecular weight polystyrene (M = 900 000) in the presence of short chains (M = 8 500). The dotted line represents pure reptation and the full line stands for the contribution of tube renewal according to relation (6-8).[from ref. 28]...

We sume that disengagement by pure reptation is negligible for star molecules with diffidently long arms in an entangled medium. (For a contrary opmion, however, based on computer simulation of star molecule motions, see Ref. 33). Relaxation for an f-arm star in a topologically invariant medium is then equivalent to the relaxation of f tethered chains, where is the tethered chain relaxation time (Eq.66) for individual aruK (Rg.l2). 

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. 

One disadvantage of the slithering snake model is that the time step cannot be controlled in one step the chain moves exactly one tube segment. This in particular leads to the unphysical oscillations in i,mid(t) at early time in Figure 17. In order to resolve the motion on smaller timescales (and more importantly to account for fluctuations of the chain inside the tube, see below), one has to distinguish between the tube and chain coordinates. Now we introduce the main set of variables of the tube model the 3D tube coordinates V),(t), fe = 0...Z as in the previous section plus the one-dimensional (ID) chain coordinates inside the tube Xj( ), i = 0...N. In total, we have 3(Z-r 1) -r (N-f 1) variables, and their equations of motion are coupled. The main idea of the tube theory is that the chain inside the tube moves independent of the tube coordinates, whereas the tube segments are deleted at the ends when the chain does not occupy them any more, and are created when the chain sticks out of the tube. In the pure reptation model, only the center-of-mass of X coordinates moves according to... 

Figure 21 Mean-square displacement for pure reptation model (lines) and reputation + CLF model (points). Three characteristic times are shown by arrows for A/=64.

Finally, we discuss the dependence of the diffusion coefficient D on the molecular weight N. This dependence is often used to evaluate the tube theory, which originally predicted D N scaling resulting from pure reptation motion (no CLF or CR). This is easy to obtain by a simple argument that at long timescales the center-of-mass diffusion might be approximated by random jumps of size Ar with frequency of reptation time Tdo The diffusion coefficient of this simple jump... 

Decorrelated subunits In the first approach, de Gennes suggested that was just the jump frequency of a free subunit of size d, 1/X j(d), times the probability that an extremity of a matrix chain was located within the distance d from the test chain (a necessary condition for the corresponding constraint to be released), i.e., wd ll R(d) (2NJP) If P is the polymerization index of the matrix chains. This makes it possible to evaluate this description, for the particular case a contribution comparable to that of pure reptation (for independent processes, the inverses of the characteristic times add) and has the same scaling behaviour with molecular weight and concentration. 

The case of star/linear blends is a challenging one, because the description of constraint release that works best for pure star polymers is dynamic dilution, while for pure linear polymers, double reptation , or some variant of it, seems to be the better description. However, Milner, McLeish and coworkers [23] have developed a rather successful theory for the case of star/ linear blends. In the Milner-McLeish theory, at early times after a step strain both the star branches and the ends of the linear chains relax by primitive-path fluctuations combined with dynamic dilution, the latter causing the effective tube diameter to slowly increase with time. Then, at a time corresponding to the reptation time of the linear chains, the tube surrounding the unrelaxed star arms increases rather quickly, because of the sudden reptation of the linear chains. The increase in the tube diameter would be very abrupt, if it were not slowed by inclusion of the constraint release-Rouse processes, which leads to a square-root-in-time decay in the modulus (see Section 7.3). With this formulation, the Milner-McLeish theory yields very favorable predictions of polybutadiene data for star/linear blends see Fig. 9.13, where the parameters have the same values as were used for pure linears and pure stars. 

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