Molecular theory of translational diffusion: Microscopic generalization of the normal velocity boundary condition

Abstract

A simple molecular theory is presented for the diffusion constant D for a test hard sphere translating in a hard sphere solvent. It is argued that there is a breakdown of the applicability of hydrodynamics in the neighborhood of the test particle due to collisional effects. It is shown that, as a consequence, the traditional hydrodynamic boundary condition (BC) on the particle–solvent normal relative velocity is incorrect for molecular motion. An approximate replacement for this BC is constructed from collisional considerations. With this new BC and the usual hydrodynamic equations, D is found to have two additive contributions. The first is the microscopic, collisional Enskog diffusion constant; the second is of the hydrodynamic Stokes–Einstein form. It is shown how the standard hydrodynamic Stokes–Einstein relation for D can hold numerically to a good approximation despite the dominance of (or significant contribution to) the motion by microscopic collisional contributions. Observed trends of D with size and mass ratios which contradict the analytic Stokes–Einstein relation are reproduced. The predicted D values are compared with available results of renormalized kinetic theory and Boltzmann‐level kinetic theory. High density deficiencies of the new BC are discussed.