Hierarchical tube model

On the basis of this success, the tube model has been extended to polymers having more complicated, multi-branched architectures such as H-shaped, pom-pom (with the stmcture shown later in Figure 16), and dendritic Cayley-tree polymers. " These extended models, assuming hierarchical relaxation starting from outer arms/branches toward iimer branches and adopting the DTD picture for the CR relaxation of the inner branches activated by outer branch motion, describe the data of these complicated polymers considerably well. For example, McLeish and Larson considered a tension balance between the arms and the trunk to develop a model for pom-pom chains, van Ruymbeke et developed models... 

The symbols in Fig. 9.18 are experimental data of Daniels etal. [25]. The solid and dotted lines are predictions of the hierarchical model with monodisperse and polydisperse arms and backbone molecular weights, respectively. The parameters are given in the caption of Fig. 10.6 with a= 4/3 the parameter value = 1/12 is used in Eq. 9.9 for the branch-point mobility, as suggested by Daniels et al. [25]. Once the arms relax, the backbone is assumed to reptate in a tube dilated by the dynamic dilution due to relaxation of the star arms. 

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