Milner-McLeish theory

Figure 38 Master curves of elastic storage (S, ) and viscous loss (S", o) linear viscoelastic moduli of the 12-arm 12 828 (a) and 64-am 6430 (b) star-PBd polymers in the temperature range from 150 up to -103°C, with reference temperature-83 °C. Solid arrows represent the various transitions and corresponding crossover frequencies (cos. glass to Rouse-like transition cof.. transition to rubber plateau

The constraint-release models discussed above have been tested by comparing their predictions to experimental data, as shown in Figures 7.9 and 7.10. For linear polymers for which the molecular weight distribution is unimodal, and not too broad, dynamic dilution is not very important, and theories that account for constraint release without assuming any tube dilation are adequate. Such is the case with the version of the Milner-McLeish theory for linear polymers used to make the predictions shown in Fig. 6.13. The double reptation theory also neglects tube dilation. The dual constraint theory mentioned in Chapter 6 does include dynamic dilution, although its effect is not very important for narrowly dispersed linear polymers. As described above, dynamic dilution becomes important for some bimodal blends, and is certainly extremely important for branched polymers, as discussed in Chapter 9. 

Finally, we remark that the idea of self-consistent dynamic dilution was applied first by Marrucci [20] to the case of monodisperse linear polymers, and was then adapted by BaU and McLeish [11] to monodisperse stars. We also note that theories combining reptation, primitive path fluctuations, and constraint release by dynamic dilution have been applied successfully by Milner and McLeish and coworkers to monodisperse linear polymers [21], monodisperse stars [13], bimodal star/star blends [22], and star/linear blends [23], as well as H-branched polymers [24], and combs [25]. The approach taken for all these cases is similar at early times after a small step strain, the star arms and the tips of linear molecules relax by primitive path fluctuations and dynamic dilution. At some later time, if there are linear chains that reach their reptation time, there is a rapid relaxation of these linear chains. This produces a dilation of the effective tubes that surround any remaining unrelaxed star arms by constraint-release Rouse motion (see Section 7.3). Finally, after dilation has finished, the primitive path fluctuations of remaining portions of star arms begin again, in the dilated tube. We refer to this set of theories for stars, linears, and mixtures thereof as the Milner-McLeish theory . The details of the Milner-McLeish theory are beyond the scope of this work, but the interested reader can learn more from the original articles as well as from McLeish and Milner [26], McLeish [14], Park and Larson [27], and by Watanabe [19]. 

Comparison of Milner-McLeish Theory to Linear Viscoelastic Data 9.3.3.1 Monodisperse Stars... 

Figure 9.6 compares the predictions of the Milner-McLeish theory with a = 1 and a = 4/3 compared to the zero-shear viscosities for linear and star 1,4-polybutadienes from several different sources. Figures 9.7 and 9.8 show similar comparisons to G and G" data for nearly monodisperse linear and star 1,4-polybutadienes. These very accurate predictions were made using the same algorithm for both star and linear polymers. Also, the same parameter values (G and t ) were used in Figs. 9.6 through 9.8, except for a small shift in (see Table 7.1) to account for small differences in temperatures for the star polymers (28 °C) and linear ones (27 C). Furthermore the value for the parameter for a= 4/3 was set to 1650, which is rather close to the value, 1543, given in Fetters etal. [36], and calculated from the plateau... 

Figure 9.11 Zero-shear viscosities of 3-arm and 4-arm hydrogenated polybutadiene stars at 190 °C as functions of arm molecular weight. The line is the prediction of the Milner-McLeish theory with the same parameter values as in Fig. 9.10. From Park [44].

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. 

Figure 9.14 Storage and loss moduli of binary blends of a nearly monodisperse linear 1,4-polybutadiene (M = 23,600) with a four-arm star 1,4-polybutadiene (total = 1,367,000) at a star volume fraction of 0.025 at T = 27 °C. The lines are from the Milner-McLeish theory, modified by addition of a disentanglement relaxation process that occurs when Mg P < where...

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