Reptation Combined with Primitive Path Fluctuations

3 Reptation Combined with Primitive Path Fluctuations 

For linear polymers, primitive path fluctuations (PPF or CLF for contour length fluctuations ) occur simultaneously with reptation. At short times (or high frequencies) the ends of the chain relax rapidly by primitive path fluctuation. But primitive path fluctuations are too slow to relax portions of the chain near the center, and these portions therefore relax only by reptation. However, the relaxation of the center by reptation is speeded up by primitive path fluctuations, because the tube remaining to be vacated by reptation is shortened, since its ends have already been vacated by primitive path fluctuations. As a result, the longest reptation time Tj (i.e., the terminal relaxation time) and zero-shear viscosity, are lower than in the absence of the fluctuations and can be approximated by the following equation [ 1 ]  

Primitive path fluctuations (PPF or CLF) also broaden the spectrum of relaxation times. Note in Fig. 6.10 that the inclusion of PPF results in a less steep decrease in G with frequency (at 

Prediction of the zero-shear viscosity rify normalized by the zero-shear viscosity of a polymer with molecular weight corresponding to one entanglement, as a function of normalized molecular weight Z=M/M for pure reptation (solid thin line), reptation with fluctuations, given by Eq. 6.33 with X = 1.3 (solid thick line), and the empirical formula t]q (dashed 

Although reptation and primitive path fluctuations together provide a nearly quantitative prediction of the linear viscoelasticity of monodisperse melts of linear chains, for polydisperse melts it is clear that these are not the only important relaxation mechanisms. To develop quantitative, or even qualitative, theories for polydisperse melts, constraint release must be taken into accoimt. 

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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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