Revisiting Discrepancies in Diffusion Theory: Molecular Motion in Liquids

Table of Links

Abstract and 1 Introduction

2 Preliminary material

3 Comments elicited by the solution to Milne’s problem for B-particles

4 Flux to a spherical trap

5 Molecular motion in liquids

6 Concluding remarks and References

5 Molecular motion in liquids

One of the most important and keenly controverted issues dicussed by Hildebrand in his thoughtful and purposely provocative note is the question whether concepts borrowed from the kinetic theory of gases or classical hydrodynamics can be legitimately applied to describe solute diffusion in liquid [30]. The bulk of his criticism is to be found in the last two paragraphs, excerpts from which appear below:

Attempts to calculate absolute values of diffusivity have been made by starting from Stokes’ law for a particle settling under the pull of gravity, and extrapolating over the long path to a molecule participating with its neighbors in aimless thermal motions that are never as long as the molecular diameter. Individual molecules are not impelled by a vector force that can serve to measure a “coefficient of friction.” Any such coefficient is fictitious.

If molecules were hard spheres, instead of electron clouds with imbedded nuclei, and were sufficiently far apart to justify speaking of binary collisions with linear free paths between them, the probable distance they could be expected to wander from their initial positions could be computed by the formula for a “random walk.” But polyatomic molecules that move less than 10 percent of their diameter require a more sophisticated mathematical formulation. They are in a continual state of soft, slow collision, with constant exchange between translational and internal energy. The random walk in this case is a slow, tipsy reel, without sudden changes of direction. The mathematical problem involved . . .

If one is content with the macroscopic description provided by the DE, concepts such as the mean free path and friction coefficient do indeed recede into the background, but not until we have derived a satisfactory boundary condition. If one wants to ascertain the validity of the macroscopic equations, for example the boundary condition for a particular problem, one is obliged to take a microscopic look. Whether one uses the LBE (which speaks of trajectories between separable collisions) or the KKE (which envisages a medium with a given friction coefficient) for delimiting the domain of validity of the macroscopic description is quite immaterial, so long as one can convince oneself that one’s conclusion does not rest on some idiosyncracy of the model; the means would justify the end provided that the end can be reached by some other means as well.