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Control of phase transition patterning in the wake of A+B→C reaction fronts

N. Maton1, G. Facchini2, and A. De Wit1

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Phys. Rev. E 114, 044218 – Published 22 October, 2026

DOI: https://doi.org/10.1103/5v7k-w2k3

Abstract

When two initially separated reactants A and B are put in contact, a chemical front can develop if a bimolecular A+B→C reaction interplays with diffusion. In the case of a precipitation reaction, a phase transition may occur in the wake of this reaction front provided that the concentration of C is larger than a critical threshold. We analyze the scalings of resulting phase transition patterns in both one-dimensional (1D) and two-dimensional (2D) rectilinear and radial geometries by numerical simulation of the Cahn-Hilliard equation with a reactive source term, solution of reaction-diffusion-advection equations for the concentrations of A and B. We find that the temporal and spatial scalings of the precipitate patterns depend on the properties of the A+B→C fronts and can be controlled by the injection flow rate Q. In particular, we show that the Matalon-Packter law classically used to characterize Liesegang patterns in one-dimensional rectilinear reaction-diffusion systems can be generalized to 1D radial geometry with injection. We discuss the further generalization of pattern selection between stripes and spotted patterns in 2D systems.

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Phys. Rev. E 114, 044218– Published 22 October, 2026
  • Received 25 June 2026
  • Accepted 25 September 2026
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