Maximum-Correlation Mode-Coupling Approach to the Smoluchowski Dynamics of Polymers

Abstract

The Smoluchowski generalized diffusion equation with hydrodynamic interactions is used to derive the dynamics of polymers in solution. The time correlation functions (TCF) of bond vector variables are calculated using a mode-coupling expansion. While the first-order expansion represents the optimized Rouse−Zimm (ORZ) theory, higher orders result in an explosive number of elements in the basis set adopted to expand the eigenfunctions of the dynamic operator. An optimum approximation is generated by adding to the ORZ basis set, linear in the chosen slow variables, selected nonlinear terms formed by increasing powers of only those slow variables having maximum correlation with the observed relaxing variable. When this maximum-correlation mode-coupling approximation (MCA) is applied to the generalized diffusion equation with full hydrodynamic interaction, a great improvement to the ORZ theory is obtained which is easily amenable to more reliable computations of local dynamics in polymers and proteins in solution. Applications to freely jointed chains, broken rods, and rods are discussed to show the usefulness of the MCA concept in deriving the dynamics of simple model chains with small or strong correlation between bond variables. The MCA theory with a basis set twice larger than that of the ORZ theory gives exact rotational correlation times for the rod with and without hydrodynamic interactions.