Resolving the inconsistency between classical diffusion and adsorption

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, lambda, and diffusion coefficient, D = (1/3)lambdaV, where Vis the characteristic velocity. Using the limit of lambda --> 0 in the flux term gives classical diffusion equation, that is, Fick's law. However, imposing the limit of lambda --> 0 reduces two independent parameters, V and lambda, to one parameter, D = (1/3)lambdaV. This is equivalent to reducing two independent length scales, lambda and Vt, to only one length scale, (Dt)1/2, where t is time. Since the lambda length scale determines density profiles near surfaces, the classical diffusion model "loses" adsorption phenomena after applying the limit of lambda --> 0 and classical solutions are in conflict with adsorption at surfaces. Here, we show that relaxing the requirement of lambda --> 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.