Nonequilibrium self-assembly induced Liesegang rings in a non-isothermal system

We propose a model to show the formation of Liesegang rings under non-isothermal conditions. The model formulates reaction-diffusion equations for all components intervening in the process together with an evolution equation for the temperature. The reactive parts in these equations follow from the analysis of the non-equilibrium self-assembly (NESA) process undergone by the meso-particles which make up the patterns. The solution of these equations enables us to know the concentration of each component, the spherical structures diameter, and the system temperature as a function of time and radial position. The values found for the structures diameter and the rings position are in agreement with the experiments. The results for the system temperature with peaks at the rings positions suggest that heat accumulates at these positions as a consequence of the dissipation inherent to the NESA process. Our model enables us to rationalize how from non-homogeneous initial conditions a transient self-organization process involving formation of self-assembled structures may produce macroscopic patterns. It can, in general, be used to analyze pattern formation due to diffusion-reaction-precipitation processes with potential applications in the design of advanced materials.