Hierarchical Dynamic Dilution Model

While the development of a general theory of linear viscoelasticity for arbitrarily branched polymers is an extremely ambitious imdertaking, many of the elements necessary to construct such a theory now appear to be in hand. Two of the major tasks yet to be completed are  

The derivation of expressions for branch point diffusivity that apply to general polymer branched architectures, and 

The construction of an algorithm that can cope with the relaxation of mixtures of molecules of many different architectures and molecular masses. 

The melt to be considered is therefore composed of an arbitrary mixture of such molecules, each having its own volume (or mass) fraction 0(i) and each having a potentially different distribution of arm lengths and arm positions. The volume fraction of material contained in arms of type i on molecules of type is therefore 

We can now account for dynamic dilution by inserting the unrelaxed melt fraction P(t) into Eq. 9.23, yielding 

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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. 

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. 

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