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

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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. © 1979 American Institute of Physics.