I already expressed my interest in determining if the famous 3.4 exponent that characterizes the melt viscosity dependence on molecular weight (for entangled polymers) would be different when the rheological data are determined at constant free volume.
In particular, I wanted to know if de Gennes had been right in the first place, coining the exponent at 3.0 in his famous 1971 paper.
The analytical technique to analyze rheological data at constant free volume is presented in a WIZIQ lecture found at http://www.wiziq.com/NewSchoolPolymerPhysics977161
(class#2: On the incidence of Tg and Talpha on the formulation of rheolgical equations)
I have now addressed and completed a series of monodispersed polystyrene grades and the result is shown in the figure above:
THE EXPONENT IS NOT 3.0, NOR 3.4, IT IS 5.3
Besides, in the Vogel-Fulcher's expression of the temperature dependence of the friction factor, the famous T2 - for which viscosity becomes infinity- is raised from 55 oC to 123 oC, when the free volume is accounted for.
These results show how important it is to correctly describe the effect of the free volume on molecular mobility when analyzing dynamic rheological data. The mythical constants, 3.4 for the viscosity exponent, T2=Tg-52.5 for the WLF equation, are the results of cooperative contributions from free volume and conformer rotations. The influence of free volume is not separable the way it is traditionally presented: in our analysis, free volume influences both the T and M factors in the viscosity expression.
Interestingly, the temperature of 123 oC is 23 oC above the Tg of polystyrene, determined by DSC, for instance. And this temperature is precisely the temperature of compensation for this polymer for all the relaxation modes occuring below Tg. Refer to a previous blog page on "Interactive Coupling between Relaxation Modes". One knows that the coupling between the molecular motions below Tg, resulting in compensation, occurs in a very restricted free volume environment, compared to what is assumed to occur above Tg. It is, therefore, somewhat satisfying to find that the T2 obtained after removing the effect of free volume is the same as the compensation temperature found from a study of motions in the solid state.
It is also remarkable to see how the free volume is intrinsically coupled with the effect of molecular chain length above Tg: mobility is much more reduced (by a factor 100) than what one thought was only due to molecular weight alone. The influence of molecular weight is described by the exponent 5.3, an extraordinary large number. This is only because the free volume is interactively coupled with the configurational effect that we observe the 3.4 exponent. As said before, viscosity does not separate into a term that varies with T only and a term that is function of M only. This formulation is only a convenient approximate representation.
So, unfortunately, the brilliant demonstration by Prof. de Gennes to explain the restrictive mobility in polymer melts will remain, in my mind, incomplete, perhaps even "un faux pas".
