We present a novel method for estimating the charge fraction of polyelectrolytes in solution based on shear viscosity measurements. The viscosity of polyelectrolyte solutions decreases with increasing salt concentrations due to charge screening effects, and the salt concentration at which the solution viscosity starts to decrease provides insight into the number of charged groups on a polyelectrolyte chain. Accordingly, the charge fraction of a polyelectrolyte chain can be evaluated by analyzing the dependence of the specific viscosity on the salt concentration. To validate our viscosity-based method, we estimated the charge fraction, f, of poly(sodium styrenesulfonate) in deionized water at 25 °C. Our result closely matched with literature values, demonstrating that our viscosity-based method can serve as a practical tool for determining the charge fraction of polyelectrolytes in solution. Moreover, by using our viscosity-based method, we demonstrate that the effect of the counterion condensation on the charge fraction of polyelectrolytes in solution is more complicated than predicted by the classical Manning model.

We analyze the stress tensor and the gyration tensor of an unentangled polymer melt under flow by using a Rouse-type single chain model. We employ the bead-spring type single chain model, in which beads interact each other via nonlinear potentials such as the finite-extensible nonlinear elasticity (FENE) potential. Beads are assumed to obey the Langevin equation with a constant friction coefficient. We derive simple yet general relations between the stress tensor and the gyration tensor for this Rouse-type model, without any additional approximations. Various formulae for rheological quantities in terms of the gyration tensor can be derived from the general relations. For example, the steady shear viscosity is governed by the gyration radius in the shear gradient direction.