Routes to fractality and entropy in Liesegang systems

Abstract

Liesegang bands are formed when solutions of co-precipitate ions interdiffuse in a 1D gel matrix. In a recent study [R. F. Sultan, Acta. Mech. Sin. 27, 119 (2011)], Liesegang patterns have been characterized as fractal structures. In addition to experimentally obtained patterns, geometric Liesegang patterns were constructed in conformity with the well-known empirical laws. Both mathematical fractal dimensions and box count dimensions for images of PbF2 and PbI2 Liesegang patterns have been calculated. Liesegang patterns can also be described by the entropy state function, and categorized as more or less ordered structures. We revisit the relation between entropy and fractal dimension, and apply it to simulated geometrical Liesegang patterns. We have resort to three different routes for the estimation of the entropy of a Liesegang pattern. The HarFA software enabled the calculation of the Hausdorff dimension and the topological entropy, then the information dimension and the Shannon entropy. In a third pathway, analytical calculations were carried out by estimating the probability of occurrence of a fractal element or coverage. The product of Shannon entropy and Boltzmann constant yields the thermodynamic entropy. The values for PbF2 and PbI2 Liesegang patterns attained the order of magnitude of the reported Third Law entropies, but yet remained lower, in conformity with the more ordered Liesegang structures.

Tunability of Self-Organized Structures Based on Thermodynamic Flux

“…These facts strongly suggest that the selection rule that selects the structure with the highest d S /d t , i.e., the MaxEPP, functions in the reaction–diffusion system. Kalash and Sultan discussed the role of entropy in Liesegang structure formation, where S was estimated from the fractality of the final structure . Because this analytical method relates to the selection rule F = U – TS , it is very useful for discussing systems that select their final structure from among the many ordered structures and need not consider a significant energy barrier for ordering .…”

“…Kalash and Sultan discussed the role of entropy in Liesegang structure formation, where S was estimated from the fractality of the final structure. 40 Because this analytical method relates to the selection rule F = U − TS, it is very useful for discussing systems that select their final structure from among the many ordered structures and need not consider a significant energy barrier for ordering. 1 However, a more comprehensive analysis is required for realistic chemical systems, including the Liesegang pattern formation, because ions, molecules, and particles can be assembled after the energy barrier has been overcome.…”

Random Spacing between Metal Tree Electrodeposits in Linear DLA Arrays

“…This formula has been used in different applications, notably to define the entropy of river networks [ 6 ]. In fractal ramification, the Shannon entropy has been associated with information fractal dimension [ 25 ], and was used for calculating the entropy of Liesegang patterns [ 26 ]. Although we are dealing with fractal systems, we do not adopt this approach here because we are focusing on the separation distances, and not the density of the ramification.…”

Fractal-Thermodynamic System Analogy and Complexity of Plant Leaves

“…From an information-theoretic point of view, the analysis of geometrical structures begins with Shannon's work [10], expressing complexity as the amount of information (entropy) required to describe a phenomenon at a particular scale [11]. Entropy measures effectively describe complex patterns and have many applications in science and technology, ranging from medical imaging [12,13], remote sensing [14,15], security [16,17] and materials science [18,19].…”