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.
Highlights
Many scientific and technological problems involve patterns with a surface distribution of spots.A common example is microscaled porous honeycomb patterns on a polymer’s surface arising from the so-called breath-figures self-assembly, which will be described in detail below [1,2,3,4] (Figure 1).Intuitively, the images of the pores in Figure 1a,b look ordered, whereas the pattern presented in Figure 1c seems to be disordered
How this intuitive feeling can be quantified? Quantitative parameters of self-organization can be obtained by building the Voronoi diagram and calculating the appropriate Voronoi entropy, which is the topic of the present paper [5]
A comparison of the effectiveness of the Fourier transform, Minkowski functionals, and Voronoi diagrams for characterization of ordering in point patterns still remains an open problem. Another method enabling characterization of patterning in 2D self-assembled patterns with the correlation functions was reported in Reference [47], in which porous honeycomb structures arising from breath figure self-assembly [1,2,3,4,14,25,26,27], depicted schematically in Figure 4, were studied
Summary
Many scientific and technological problems involve patterns with a surface distribution of spots. How this intuitive feeling can be quantified? Snow identified infected wells by superposing the map of tessellations during the 1854 London cholera outbreak [7,8]. Snow identified infected wells by cholera casesthe andmap the Voronoi diagram of the sources sites [7,8], proving that. The idea was revived by Dirichlet in the context of his works proving that Voronoi diagrams can even save lives. The tessellation shownin [6,7]
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