Milner Primitive path fluctuations

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. 

The advanced molecular models described in this chapter, namely the Milner-McLeish model and the hierarchical model, involve combinations of multiple relaxation mechanisms reptation, primitive path fluctuations, and constraint release described by both constraint release Rouse motion and dynamic dilution. However, all these mechanisms can be captured in algorithms in which entanglements are viewed as slip links between two chains see for example Fig. 9.22. 

---