# Characterization of Self-Assembled 2D Patterns with Voronoi Entropy

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## Abstract

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## 1. Introduction

## 2. Topological and Scaling Properties of Voronoi Diagrams and Entropy

_{n}is the fraction of polygons with n sides or edges (also called the coordination number of the polygon) in a given Voronoi diagram [10,11,12]. The summation in Equation (2) is performed from n = 3 to the largest coordination number of any available polygon, e.g., to n = 6 if a polygon with the largest number of edges is a hexagon.

_{n}= 1 and ln P

_{n}= 0. For a typical case of a fully random 2D distribution of points (i.e., with a uniform probability distribution of seed points on a plane), the value of ${S}_{vor}=1.71$ has been reported [14]. Therefore, it is expected that for a self-organizing structure, the value of S

_{vor}decreases. Note that the Voronoi entropy is an intensive property, unlike the thermodynamic entropy, which is an extensive property. Therefore, the value does not depend on the number of seeds, which makes it appropriate to study processes where the number of seeds increases.

_{n}that a point has a n-sided Voronoi cell is given, for large n, by

## 3. Analysis of 2D Self-Assembled Surface Patterns with 2D Voronoi Diagrams

## 4. Droplet Clusters and Their Analysis with Voronoi Diagrams.

_{vor}= 0.335. Figure 8 depicts the self-assembly stages of a small droplet cluster [56].

## 5. The Relation between Voronoi Entropy and Thermodynamic Entropy

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Porous ordered polycarbonate honeycomb structures obtained with breath-figures self-assembly is shown. (

**a**) Scale bar is 2 µm. (

**b**) Scale bar is 1 µm. (

**c**) Scale bar is 10 µm. (

**d**) Voronoi diagram for the case (c), S

_{vor}= 1.0131, is depicted.

**Figure 3.**Example of the Voronoi tessellation on a set of points. Red points represent seeds or nuclei.

**Figure 4.**The main stages of the breath figures self-assembly, resulting in creation of ordered honeycomb microporous topographies, are depicted.

**Figure 5.**Self-organization of a droplet cluster is demonstrated. (

**a**) The image of the cluster and (

**b**) the Voronoi tessellation of the cluster. The scale bar is 200 µm. Yellow (1,9), gray (4–8), and blue (3,2) polygons have five, six, and seven neighbors (edges), respectively [56].

**Figure 6.**(

**a**) Self-assembly of a droplet cluster over a heated water and (

**b**) the Voronoi entropy, S (blue), correlated with the number of droplets, N (red) [56]. The scale bar is 200 µm.

**Figure 8.**A small droplet cluster during self-assembly and its corresponding Voronoi diagrams are shown [57]. The scale bar is 200 µm. Yellow, gray, and blue polygons have five, six, and seven neighbors (edges), correspondingly.

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**MDPI and ACS Style**

Bormashenko, E.; Frenkel, M.; Vilk, A.; Legchenkova, I.; Fedorets, A.A.; Aktaev, N.E.; Dombrovsky, L.A.; Nosonovsky, M.
Characterization of Self-Assembled 2D Patterns with Voronoi Entropy. *Entropy* **2018**, *20*, 956.
https://doi.org/10.3390/e20120956

**AMA Style**

Bormashenko E, Frenkel M, Vilk A, Legchenkova I, Fedorets AA, Aktaev NE, Dombrovsky LA, Nosonovsky M.
Characterization of Self-Assembled 2D Patterns with Voronoi Entropy. *Entropy*. 2018; 20(12):956.
https://doi.org/10.3390/e20120956

**Chicago/Turabian Style**

Bormashenko, Edward, Mark Frenkel, Alla Vilk, Irina Legchenkova, Alexander A. Fedorets, Nurken E. Aktaev, Leonid A. Dombrovsky, and Michael Nosonovsky.
2018. "Characterization of Self-Assembled 2D Patterns with Voronoi Entropy" *Entropy* 20, no. 12: 956.
https://doi.org/10.3390/e20120956