Milner Star-linear blends

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]. 

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 9.15 Zero-shear viscosity versus vol. fraction stars for bidisperse 1,4-poly butadiene star-linear blends as at 7= 25 "C.The symbols are for mixtures of a three-arm star of molecular weight 127,000 with linear polymers (M = 36,800) (M = 100,000) A(M = 68,000).The curves are the predictions of the theory of Milner etal. [23] using a = 4/3. The solid lines are for predictions with disentanglement relaxation and the dashed lines are without disentanglement relaxation.The data are from Struglinski etal. [35].The parameter values are the same as in Fig. 9.6. From Park and Larson [27].

Figure 9.13 Comparison of theory with data for the loss nrKxJuli of binary blends of nearly monodisperse, linear 1,4-polybutadiene (MW = 105,000) and three-arm star 1,4-polybutadiene (MW = 127,000) at r=25 °C.The star volume fractions, from right to left, are 0,0.2,0.5,0.75, and 1. The data are from Struglinski etal. [35]. The dashed lines are the Milner-McLeish model predictions, while the solid lines were obtained from the hierarchical model (see Section 9.5.2) both using a = 4/3.The parameter values are the same as in Rg. 9.6. From Park and Larson [49].

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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