On the equivalence of the Kirkwood diffusion equation to the coupled polymer-solvent Langevin dynamics

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Abstract
The role of solvent velocity fluctuations on the transport properties of polymer solutions is studied using renormalisation group expansions to order in 2 of the coupled Langevin equation model within a Fokker-Planck equation formulation. Introduction of the timescale separation approximation between polymer and solvent characteristic relaxation times leads to an additional expansion in powers of a small dimensionless parameter which may heuristically be interpreted as the ratio of characteristic polymer (rouse-Zimm) to solvent relaxation rates. Retaining zeroth-order terms in the latter expansion reduces the order in 2 solution of the Fokker-Planck equation identically to the corresponding expanded solution of the Kirkwood diffusion equation representation of the Rouse-Zimm model. Explicit expressions are derived for corrections to the Kirkwood diffusion equation due to solvent velocity fluctuations. The bare leading corrections for the intrinsic viscosity of preaveraged Gaussian chains are analysed and are shown to have a magnitude consistent with heuristic timescale separation arguments and to be negligibly small for typical polymer-solvent systems. It is therefore conjectured that the renormalised parameter is also sufficiently small to validate its neglect for these polymer systems.