Relationships among Mc, and

Each of the viscoelastic parameters G°, rj0, and Je° has associated with it a characteristic molecular weight which either measures an equivalent spacing of entanglement couples along the chain (Me, deduced from G with the kinetic theory of rubber elasticity), or marks the onset of behavior attributed to the presence of entanglements (Mc and AT, deduced from r/0 and Je° as functions of molecular weight). Table 5.2 lists Me, Mc, and M c for several polymers. Aside from certain difficulties in their evaluation, each is a rather direct and independent reflection of experimental fact. 

The Bueche and Chikahisa relations reduce to roughly MJMC = 2 and Mc/Me= 1 respectively, when typical values of S2/M and q are inserted and 

K(s) is taken to be approximately unity. Hayashi s result is MJMe = 1.52, while Graessley s expressions give values of 1.71 and 2.52 respectively. Thus, none of the friction factor theories is seriously inconsistent with the observed magnitude of MJMe. The Bueche and Chikahisa formulas predict some residual variation of MJMe with (S2/M)3,2q and the magnitude of Mc itself, while the Hayashi and Graessley results require MJMe to be the same for all linear polymers. None of these results account satisfactorily for the detailed variations among polymers noted earlier. 

A theoretical value for M JMC can be calculated by equating Graessley s expression for Je° [Eq. (6-60)] with the Rouse expression  

The result is = 1.798/0.4, giving M JMe = 4.5. With Mc/Me = 2.52 from the same theory, M JMC = 1.8 is obtained. The observed values are about twice this value part of the difference is probably attributable to residual polydispersity in the experimental samples. 

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