Pore Diffusion of Nonspherical Brownian Particles

A general theory is presented for calculating the effective diffusivity D of an arbitrarily shaped Brownian solute diffusing within a straight pore of arbitrary cross section. Hydrodynamic interactions with the walls as well as rotational diffusion of the solute are rigorously accounted for. Asymptotic analysis leads to the conclusion that D is given by a formula of the Brenner-Gaydos (1977) form -D/D_bulk = 1 + C₀λ ln λ + C₁λ + o(λ) in the limit λ≪ 1, irrespective of the particular geometry considered, where λ denotes the ratio of solute to pore sizes. The O (λ In λ) term derives from far-field hydrodynamic interactions between the solute and the local tangent to the pore wall and is universally quantified in terms of the ratio of the Stokes-Einstein equivalent radius of the solute to the hydraulic radius of the pore. The O (λ) term depends in a complicated way upon the pore and solute shapes, the reflected field due to a Stokeslet within the pore, and hydrodynamic interactions between the solute and a plane wall. Numerical evaluation of the O (λ) coefficient requires solution of a planar-wall translational-rotational diffusion problem. Application of the general theory is demonstrated via explicit derivation of the asymptotic formula giving D for a dumbbell-shaped solute diffusing within a circular cylindrical pore.