On the Keller-Rubinow model for Liesegang ring formation

Area of Science:

• Physical Chemistry
• Chemical Dynamics
• Pattern Formation

Background:

• Liesegang rings form via Ostwald's supersaturation mechanism.
• The Keller & Rubinow (1981) model offers an approximate solution for Liesegang ring formation.
• Previous work lacked parametric basis and numerical validation for secondary band formation.

Purpose of the Study:

• To provide a new asymptotic solution for the Keller & Rubinow model.
• To investigate the model's ability to explain primary and secondary Liesegang band formation.
• To identify limitations in the existing model.

Main Methods:

• Development of a novel asymptotic solution based on a specific parametric limit.
• Numerical computation to corroborate theoretical findings.
• Analysis of model's capacity to explain band growth, cessation, and secondary band characteristics.

Main Results:

• The new asymptotic solution explains the growth and cessation of the first Liesegang band.
• The Keller & Rubinow model, even with the new solution, cannot explain the formation of finite-width secondary bands.
• Numerical computations confirm the inability to model secondary band formation.

Conclusions:

• The Keller & Rubinow model is incomplete for fully describing Liesegang ring phenomena.
• A critical transition variable is missing to account for hysteretic switching across the nucleation threshold.
• Further model refinement is needed to capture complex band formation dynamics.