Critical chain length for entanglements

We saw in Sec. 2.9 that the viscosity of a bulk polymer is proportional to M, where a is either 1.0 or 3.4, depending on whether the polymer is below or above, respectively, the critical chain length for entanglement. For solutions, a similar result is obtained, only it is [r ] rather than r itself which is proportional to M ... 

O Driscoll has proposed that the auto-acceleration can be modeled by recognizing that the termination reaction is diffusion controlled but will also depend on the size of the chain involved. The critical chain length for entanglement then becomes an important parameter, and two termination rate constants can also be defined, one for chains smaller than and one for large entangled chains. These are, respectively, and k. If v is the kinetic chain length and Vp is the conventional steady-state polymerization, then the observed rate v is given by... 

In semidilute solutions, the polymer volume fraction is low compared to a melt but the polymer chains are overlapped. We are particularly interested in solution concentrations where the chains are entangled and not simply overlapped. We therefore require that the chain degree of polymerization be larger than the critical chain length for entanglements in a melt of pure... 

The critical chain length for entanglements in semidilute solution from Eqs. (32) and (51) scales with density according to ... 

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