Crossover length scale for viscosity experienced by Gold nanoparticles in semi dilute polymer solution

Gold nanoparticles (Au NPs) were used as a probe to characterize the crossover length scale for viscosity in semi dilute poly (ethylene glycol) (PEG)-water solutions. Fluctuation correlation spectroscopy (FCS) was used to measure the diffusion of these NPs as a function of their size (5-20 nm), PEG concentration (0 to 40\%w/w) and PEG molecular weight (5 kDa to 35 kDa). Our results indicate that for particles with radius R $>$ R$_{g}$, usual hydrodynamic conditions can be applied, but for particles with radius R $\leq$ R$_{g}$, the diffusion is approximately an order of magnitude faster than that predicted by Stokes Einstein (SE) relation. The results imply that radius of gyration R$_{g}$ of the polymer gives the crossover length scale from nanoviscosity to macroviscosity. The relative viscosity experienced by the particles was scaled as $\eta$/ $\eta$$_{0}$ = exp (b (R$_{g}$/$\xi$) $^{a}$), where $\eta$$_{0}$ is the water viscosity, $\xi$ is the correlation length, a = 0.70 $\pm$ 0.03 and b = 1.59 $\pm$ 0.07.