A new scaling for the rotational diffusion of molecular probes in polymer solutions

Abstract

In the present work, we propose a new scaling form for the rotational diffusion coefficient of molecular probes in semi-dilute polymer solutions, based on a theoretical study. The mean-field theory for depletion effect and semi-empirical scaling equation for the macroscopic viscosity of polymer solutions are properly incorporated to specify the space-dependent concentration and viscosity profiles in the vicinity of the probe surface. Following the scheme of classical fluid mechanics, we numerically evaluate the shear torque exerted on the probes, which then allows us to further calculate the rotational diffusion coefficient D_r. Particular attention is given to the scaling behavior of the retardation factor R^rot ≡ D^r₀/D_r with D^r₀ being the diffusion coefficient in pure solvent. We find that R^rot has little relevance to the macroscopic viscosity of the polymer solution, while it can be well featured by the characteristic length scale r_h/δ, i.e. the ratio between the hydrodynamic radius of the probe r_h and the depletion thickness δ. Correspondingly, we obtain a novel scaling form for the rotational retardation factor, following R^rot = exp[a(r_h/δ)^b] with rather robust parameters of a ≃ 0.51 and b ≃ 0.56. We apply the theory to an extensive calculation for various probes in specific polymer solutions of poly(ethylene glycol) (PEG) and dextran. Our theoretical results show good agreements with the experimental data, and clearly demonstrate the validity of the new scaling form. In addition, the difference of the scaling behavior between translational and rotational diffusions is clarified, from which we conclude that the depletion effect plays a more significant role on the local rotational diffusion rather than the long-range translation diffusion.