Derivation of the Universal Scaling Equation of the Hydrodynamic Scaling Model via Renormalization Group Analysis

The Altenberger−Dahler positive-function renormalization group (PFRG) method is shown to yield the universal scaling equation D s = D o exp(−α c ν ) of the hydrodynamic scaling model for polymer self-diffusion. Here D o is the bare polymer self-diffusion coefficient at some low concentration, c is the (potentially high) polymer concentration, and ν and α are a scaling coefficient and scaling prefactor, respectively. To integrate the Lie equations of motion of the PFRG and obtain the universal scaling equation, the Kirkwood−Risemann model for polymer hydrodynamics is extended analytically to determine leading terms of the chain−chain and (for the first time) chain−chain−chain translation−translation hydrodynamic interaction tensors,,,, and, as well as many of their translational−rotational and rotational−rotational analogues.