Linear
and Nonlinear Rheology of Semi-Dilute Polymer Solutions and Melts
Ronald G. Larson,
Department of Chemical
Engineering,
Molecular rheological theories based on the tube model have, for the
most part, only been able to predict successfully the nonlinear rheological
properties of well entangled
polymers. For semi-dilute solutions that may not be well enough entangled for
the tube model to apply, the concept of a renormalized monomer called a blob
provides a basis for universal rheological scaling behavior,
with the concentration parameter c/ce
replacing the normalized molecular weight M/Me as the index of the
degree of entanglement. Raspaud et al. (1995) have
confirmed the prediction of the blob theory that there is scaling law
for polymer solutions at concentrations ranging from well
entangled to below the onset for entanglement, with n the excluded volume
exponent (whose value for good solvents is 0.589),
is the polymer contribution to the zero shear viscosity with 
the zero-shear viscosity of the solution and 
the solvent viscosity; 
is the hypothetical Rouse polymer viscosity. We here describe
vastly extended scaling laws based on a Rouse relaxation time
that predicts that all frequency dependent moduli for
semi-dilute solutions should fall onto master curves at each value of c/ce when plotted against frequency times the
Rouse time. Similarly, nonlinear rheological data should also collapse onto
master curves.We tested these predictions using
carefully chosen polystyrene, polybutadiene, and
other solutions in the linear viscoelastic regime, and to a limited extent in
the nonlinear regime, and find that general agreement with the predictions of
the scaling laws of the blob theory. In principle, the scaling laws should also
encompass the melt state, once a proper definition of the Rouse time is
settled. Our findings also carry implications for the value of the dilution
exponent a
relating entanglement density to concentration. Finally, we present recent
results from the pearl necklace model for weakly and modestly entangled polymer
melts that bear on the formation of the entanglement tube, the dilution
exponent, and mechanisms of relaxation in entangled polymers.
Raspaud, E., D. Lairez, and M. Adam,
Macromolecules 28, 927-933 (1995).