Assessment of Image-Derived Fractal Dimension for Diffusion-Controlled Periodically Precipitating Bands in a One-Dimensional Configuration

Abstract

Periodic deposition of a precipitate in a test tube, formed during a suitable chemical reaction under a non-equilibrium regime, is known as Liesegang bands or patterns. Periodicity of a Liesegang system is confirmed by establishing generic laws, including spacing, width, time, and Matalon-Packter (MP) laws. These generalizations are based on determining the precise location of the bands, which is a challenge in itself, despite the advent of recent image analysis techniques. This study demonstrates that the fractal dimension (FD) of each formed band varies from that of others and can be modeled without explicitly identifying the band location. The FD can account for the interband spacing and widths and could serve as a universal parameter for characterizing the Liesegang banding pattern. In this study, since FD is a property associated with individual bands, unlike other generic laws, it can serve as a link to understand nucleation, morphology, and materials science via reaction-diffusion processes. Fractal dimensions were determined using image analysis, and triangulation was found to be a suitable method for evaluating fractal dimensions in the considered Liesegang systems. We also assessed the FD from images of pure gel, water, and spaces between two successive bands as baseline values. It was observed that FD values for each band relate to the process of aggregation or crystallization of the reaction products deposited as a band.