Dynamic light scattering and mutual diffusion in non-ideal systems. One- and multi-component spherical solutes

Abstract

Starting from the dynamic structure factor, the diffusion equations for both local and non-local mutual diffusion of a one-component spherical solute are derived. The initial local and non-local mutual diffusion coefficients are expressed in terms of the equilibrium radial distribution function (or direct correlation function), hydrodynamic interaction tensors, and the isolated-molecule diffusion coefficient. This treatment is generalized to derive the coupled local and non-local mutual diffusion equations for multicomponent spherical solutes. Initial values of the diffusion coefficient matrices for both local and non-local mutual diffusion are expressed exactly in terms of equilibrium radial distribution functions (or direct correlation functions), hydrodynamic interaction tensors, and isolated-molecule diffusion coefficients. Provided that concentration fluctuations are not significantly coupled to fluctuations in other slowly relaxing conserved quantities, the dynamic structure factor of a one-component solute in the k 2 → 0 limit always decays as a single exponential with the initial, or first-cumulant, value of the dynamic light scattering diffusion coefficient. Similarly, in the k 2 → 0 limit the dynamic structure factor matrix for a multicomponent solute undergoes a first-order decay governed by a matrix of initial diffusion coefficients that remain constant throughout the decay. In that same case mutual diffusion is governed by a somewhat different diffusion coefficient matrix that likewise remains constant throughout the relaxation. A rigorous proof is given that the thermodynamic driving force for mutual diffusion of a one-component solute in an incompressible solution in the k 2 → 0 limit is (1 - ø) −1 (▿μ) T,P , as conjectured by Batchelor. The connection between the time-dependent diffusion coefficients treated here and the time-delayed diffusion coefficients recently employed by others, and the relation of both to autocorrelation functions of the fluctuating current densities are rigorously established. Polyelectrolyte solutions are shown to always exhibit a single charged mode that relaxes at a rapid rate independent of k 2, as well as diffusive modes which relax at rates proportional to k 2 in the k 2 → 0 limit. An assumption implicit in the derivation of Belloni et al. for the light scattering diffusion coefficient after the charged mode has decayed away is noted, and shown to be valid in general. The general treatment of polyelectrolytes is shown to reduce exactly to the simple coupled mode theory for the case of dilute point charges in the absence of hydrodynamic interactions.