Revisiting Discrepancies in Diffusion Theory: Abstract and Introduction

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

In a paper on “the Brownian motion analog of the well-known Milne problem in radiative transfer theory” [J Stat Phys 25 (1981) 569–82], Burschka and Titulaer reported: “The value we find for this ‘Milne extrapolation length’ is, in the appropriate dimensionless units, approximately twice the value found in the radiative transfer problem.” A study by Ziff [J Stat Phys 65 (1991) 1217–33], concerned with the absorption of particles executing a Rayleigh flight (randomly directed displacements of equal length 𝑙) by a black sphere of radius 𝑅, led to a value for the extrapolation length 𝛾 about half as small as the benchmark result (for 𝑅 ≫ 𝑙). The first discrepancy is shown to result from the disparity of the two length scales; the second, from the zero variance of the jump lengths. Ziff’s finding that 𝛾 is independent of 𝑅 for 0 < 𝑙 ≤ 2𝑅, cannot be reconciled with studies based on the Lorentz-Boltzmann equation and the Klein-Kramers equation.

1 Introduction

The purpose of this article is to ask which if either of the discrepancies mentioned in the abstract is genuine, and to probe into their implications for the long-established (but illadvised) practice of equating the rate constant of diffusion-mediated bimolecular reactions in liquids to the flux found by solving the diffusion equation (DE) subject to the Smoluchowski boundary condition (SBC), and multiplying the calculated flux–if found to be smaller than the experimentally determined rate constant–by a factor smaller than unity [1–3].

The background can be best laid out by quoting the introductory lines from a 1981 article [4] of Burschka and Titulaer (B&T):

The flow of a reactant in a diffusion-controlled reaction can often be described in terms of Brownian motion of a particle in the presence of absorbing or partially absorbing boundaries. The simplest description is obtained through a diffusion or Smoluchowski equation for the probability density of the particle position with either absorbing or “radiative” boundary conditions. In the former case the density is put equal to zero at the boundary, while in the latter the outward normal flux is proportional to the density with a phenomenological proportionality constant. This traditional theory has often been criticized; in particular there seems to be no clear way of relating the proportionality constant in the radiative boundary conditions to a microscopic picture of the reaction kinetics.

The reasons for the inadequacy of the Smoluchowski equation can be exhibited by inspecting its derivation from a more detailed description of Brownian motion due to Klein and Kramers, in terms of the probability density for the velocity and position of the Brownian particle. The Smoluchowski equation can be recovered from this description via a procedure of the Chapman-Enskog type. This derivation breaks down, however, near a wall or at places where the potential varies rapidly; there a so-called kinetic boundary layer occurs. This breakdown is caused by the large deviations from the Maxwellian velocity distribution that must occur, e.g., near an absorbing boundary, whereas for validity of the Smoluchowski equation approximate local equilibrium is required. [Bibliographic indicators have been suppressed here.]

The two paragraphs quoted above provide an accurate account of the (then) prevalent paradigm. B&T could not have foreseen that new evidence, sufficient to overturn the paradigm, was in the offing. The evidence [5– 8], which became available soon after the publication of ref [4], failed to deflate the paradigm, and caused at most a few minute punctures, which appeared to have been repaired by the dust which accumulates on printed matter and memory cells, if not blown away every now and again by enquiring spirits. Some comments on the passage are therefore in order, and should be construed not as shooting the messenger but as an attempt to shoot down a paradigm that has proved to be particularly recalcitrant.

Although an additional and shorter excerpt from ref [4] is needed, a statement of the Milne problem (tailored for this note), a few nomenclatory notes, and a recapitulation of the results most pertinent for the present discussion must precede the last excerpt.