Characterization of Self-Assembled 2D Patterns with Voronoi Entropy

Abstract

The Voronoi entropy is a mathematical tool for quantitative characterization of the orderliness of points distributed on a surface. The tool is useful to study various surface self-assembly processes. We provide the historical background, from Kepler and Descartes to our days, and discuss topological properties of the Voronoi tessellation, upon which the entropy concept is based, and its scaling properties, known as the Lewis and Aboav-Weaire laws. The Voronoi entropy has been successfully applied to recently discovered self-assembled structures, such as patterned microporous polymer surfaces obtained by the breath figure method and levitating ordered water microdroplet clusters. Entropy 2018, 20, 956 11 of 13 functionals and Fourier analysis remains an open problem. The relation between the Voronoi entropy and thermodynamic entropy remains unclear and calls for future investigations.

Symmetry and Shannon Measure of Ordering: Paradoxes of Voronoi Tessellation

“…The other fundamental value playing the central place in modern science is the Shannon measure of information [9][10][11][12][13][14]. In the present article we demonstrate that the analysis of the Voronoi diagrams (or Voronoi tessellations) enables the synthesis of symmetry and the Shannon measure of ordering considerations [15][16][17][18]. Voronoi diagrams arise from the problems involving patterns with a surface distribution of spots [19].…”

“…Voronoi diagrams arise from the problems involving patterns with a surface distribution of spots [19]. A Voronoi tessellation (or diagram or mosaic) is a partitioning of the plane into regions based on the distance to a specified discrete set of points (called seeds, sites, nuclei, or generators) [17,18]. For each seed, there is a corresponding region consisting of all points closer to that seed than to any other.…”

“…The Voronoi polyhedron of a point nucleus in space is the smallest polyhedron formed by the perpendicularly bisecting planes between a given nucleus and all the other nuclei. The Voronoi tessellation divides a region into space-filling, non-overlapping convex polyhedrons [17][18][19][20]. It was already mentioned, that the idea of what is now called the Voronoi tessellation has been proposed already by Johannes Kepler and Rene Descartes in the 17th century [16,18,21].…”

“…The Voronoi tessellation divides a region into space-filling, non-overlapping convex polyhedrons [17][18][19][20]. It was already mentioned, that the idea of what is now called the Voronoi tessellation has been proposed already by Johannes Kepler and Rene Descartes in the 17th century [16,18,21]. Kepler used it to study the densest sphere packing problem, whereas Descartes employed these tessellations to verify that the distribution of matter in the Universe forms vortices centered at fixed stars [21].…”

Mesophasic organization of GABAAreceptors in hippocampal inhibitory synapse

“…6c) using scipy.spatial.Voronoi function in SciPy (https://scipy.org). Voronoi entropy was calculated using follow formula 32 : …”