In this work, a framework for deriving theoretical equations for mean squared displacement (MSD) and fractional Fokker-Planck (FFP) is developed for any arbitrary rheological model. The obtained general results are then specified for two fractional rheological models. To test our framework and bridge the gap between microrheology and fractional rheological models, the microrheology of polystyrene (PS) in tetrahydrofuran (THF) solutions at several polymer concentrations is measured. By comparing the experimental and theoretical MSDs, we find the fractional rheological parameters and demonstrate that the polymer concentration regimes can be distinguished using the fractional exponent and relaxation time data because of the existence of a distinct behavior in each regime. We suggest simple approximations for the critical overlap concentration and the shear viscosity of weakly subdiffusive solutions with small relaxation times. This work provides an alternative and more sensitive approach for distinguishing different polymer concentration regimes and measuring the critical overlap concentration and shear viscosity of polymeric solutions, which is useful when conventional rheological characterization methods are unreliable due to the volatility and low viscosity of the samples.
Comment: 23 Pages, 6 Figures, The following article is submitted to the Journal of Rheology. Once accepted, it will be replaced accordingly