Milner Monodisperse stars

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... 

It is not clear why this transition should occur at such a higher level of arm entanglement for polystyrene stars than for other star polymers. This observation is in direct conflict with the standard assumption that through a proper scaling of plateau modulus (Go) and monomeric friction coefficient (0 that rheological behavior should be dependent only on molecular topology and be independent of molecular chemical structure. This standard assumption was demonstrated to hold fairly well for the linear viscoelastic response of well-entangled monodisperse linear polyisoprene, polybutadiene, and polystyrene melts by McLeish and Milner [24]. 

The time evolution (decay) of ( /(f) has been calculated in several full-DTD models, and the corresponding G(t) and complex modulus G (o)) = G ((o) + iG"(o>) have been compared with the data (see Ball and McLeish, 1989 Marrucci, 1985 Milner and McLeish, 1998). Agreement of the model prediction and data was reported for monodisperse linear and star-branched polymer. 

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 9.6 Zero-shear viscosity vs. molecular weight of nearly monodisperse 1,4-polybutadienes at T=25°C for (a) linear molecules A Struglinski and Graessiey [29] Roovers [30, 31] Rubinstein and Colby [32] Baumgaertel etal. [33] and for (b) stars Raju etal. [34] Roovers [30] A Roovers [31 ] Struglinski, et al. [35]. The solid line is the prediction of the Milner-McLelsh theory using a = 1 with = 9.5 -JO" s,M = 2200 and the dashed line using a = 4/3 = 3.7...

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...

Figure 10 Linear viscoelastic behavior of monodisperse four-arm star PI at 25 °C. The numbers indicate 10" /I4rm- Data taken from Milner, S. T. McLeish, T. C. B. Macromolecules 9S8,31,7479 and Fetters, L. J. ...

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