100+ years of colossal confusion on colloidal coagulation. Part I: Smoluchowski's work on absorbing boundaries

TL;DR

This paper reviews the historical development of boundary conditions in colloidal coagulation theory, highlighting the significance of Smoluchowski's absorbing boundary condition and its subsequent reinterpretations and applications.

Contribution

It clarifies the evolution of boundary conditions in colloidal coagulation, correcting past misconceptions and connecting classical theories with modern formulations.

Findings

The Smoluchowski boundary condition is fundamental to colloidal coagulation models.

Burger's alternative boundary condition was based on a fallacy, leading to historical misunderstandings.

Modern derivations confirm the equivalence of these boundary conditions with those from the Klein-Kramers equation.

Abstract

A report by Brillouin (from Perrin's laboratory) on the rate of adsorption of `granules' to a glass plate [\textit{Ann. Chim. Phys.} 27 (1912) 412--23] prompted Marian von Smoluchowski (MvS) to interpret the data in terms of his newly developed theory of restricted Brownian motion. Placing an adsorbing wall at $x=0$, he modelled the particle concentration $n(x,t)$ as that solution of the diffusion equation which vanished at the wall, a boundary condition (BC) hereafter called SBC. A gaping discrepancy between his theory and Brillouin's data elicited a suggestion from MvS (that a particle might not adhere to the wall on every impact), but no further action -- other than that of applying his theory to spherically symmtric systems. In a paper written before, but published shortly after MvS's untimely death [\textit{Proc. Roy. Acad. Amst.} 20 (1918) 642--58], H. C. Burger erected a new and sturdier framework, which led him to an alternative BC, $D(\partial n/\partial x)_{x=0}=\varkappa n(0,t)$, applicable to a surface with an arbitrary absorption probability ($1\leq\varepsilon\leq 0$); a fallacy (that subsequently claimed more victims, including the present author) prevented him from deducing the correct expression for $\varkappa$. Burger's approach became ``The Road Not Taken'', while the SBC became the cornerstone of colloidal coagulation and bimoleculer reaction kinetics. Burger's approach (but not the ABC) was partly rediscovered by Kolmogorov, and used by Sveshnikov and Fuchs.The emended version of Burger's BC is shown here to coincide with that deduced from the Klein-Kramers equation [\textit{Phys. Rev. Lett.} \textbf{49} (1982) 304--07; \textit{J. Chem. Phys.} \textbf{78} (1983) 2710--12] and the Lorentz model of random flights [\textit{J. Phys. Chem.} \textbf{86} (1982) 4750--56].