On the Keller-Rubinow model for Liesegang ring formation

We study the model of Keller & Rubinow (Keller & Rubinow 1981 J. Chem. Phys 74, 5000-5007. (doi:10.1063/1.441752)) describing the formation of Liesegang rings due to Ostwald's supersaturation mechanism. Keller and Rubinow provided an approximate solution both for the growth and equilibration of the first band, and also for the formation of secondary bands, based on a presumed asymptotic limit. However, they did not provide a parametric basis for the assumptions in their solution, nor did they provide any numerical corroboration, particularly of the secondary band formation. Here, we provide a different asymptotic solution, based on a specific parametric limit, and we show that the growth and subsequent cessation of the first band can be explained. We also show that the model is unable to explain the formation of finite width secondary bands, and we confirm this result by numerical computation. We conclude that the model is not fully posed, lacking a transition variable which can describe the hysteretic switch across the nucleation threshold.