Length-Scale Dependent Viscosity in Semidilute Polyelectrolyte Solutions

Using optical microscopy and particle tracking algorithms, we measured the mean-squared displacements (MSDs) of fluorescent polystyrene particles with diameters ranging from 300 nm to 2 $\mu$m suspended in semidilute solutions of high molecular weight partially hydrolyzed polyacrylamide. The solutions had polymer concentrations ranging from 0.67 to 67c$^*$, where c$^*$ is the overlap concentration, and estimated correlation lengths of $\sim$ 100 to 900 nm. At short times, the particles exhibited subdiffusive behavior characterized by MSD $\sim t^\alpha$ with $\alpha < 1$. On long time scales, the particles transitioned to Fickian diffusion $( \alpha = 1 )$ and their diffusivity was calculated from the slope of the MSD. Whereas the large particles agreed with predictions using the Stokes-Einstein equation and bulk zero-shear viscosity, the smaller particles diffused much faster than predicted. The relative diffusivities do not collapse onto a single curve, but rather form a continuum that varies with particle size. This indicates that the particles experience a size-dependent effective viscosity mediated by the ratio of particle diameter to characteristic length scales in the polymer solution.