Estimating the Rouse Time of an Entangled Chain

According to the Rouse model (Eq. 10.22) the viscosity is proportional to the molecular weight for an unentangled molecule, as was indicated by Eq. 6.13, which is repeated here as Eq. 10.23. 

Now we assume that this linear relationship is valid up to M = Mq, where Mq is the critical molecular weight for entanglement introduced in Section 5.2.1. This implies that  

Then we further assume that this equation describes the hypothetical, unentangled melt with M Mq, and we call the zero-shear viscosity of this hypothetical material 

For an entangled polymer, we know that the zero-shear viscosity is proportional to Af, where a = 3.4, and if this relationship holds at M = then  

Inserting this expression for %(M) into Eq. 6.12 in place of the tIq of the unentangled melt, we obtain  

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