Resolving the Inconsistency between Classical Diffusion and Adsorption

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Abstract
The inconsistency between density profiles of fluids near surfaces and predictions of classical diffusion model is analyzed. A new diffusion equation and its solutions are proposed to reconcile adsorption behavior with predictions of the diffusion equation at the equilibrium limit. The classical phenomenological model of diffusion in fluids is based on the concepts of the mean-free-path, λ, and diffusion coefficient, D = (1/3)λV̅, where V̅ is the characteristic velocity. Using the limit of λ → 0 in the flux term gives classical diffusion equation, that is, Fick’s law. However, imposing the limit of λ → 0 reduces two independent parameters, V̅ and λ, to one parameter, D = (1/3)λV̅. This is equivalent to reducing two independent length scales, λ and V̅t, to only one length scale, (Dt)1/2, where t is time. Since the λ length scale determines density profiles near surfaces, the classical diffusion model “loses” adsorption phenomena after applying the limit of λ → 0 and classical solutions are in conflict with adsorption at surfaces. Here, we show that relaxing the requirement of λ → 0 by using an exact (finite-difference) functional for the flux term fixes the problem. Solution of the finite-difference diffusion equation is analyzed. This solution allows boundary conditions consistent with density profiles in fluids near surfaces.

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