Optimisation of rGO-enriched nanoceramics by combinatorial analysis

Jivkov, Andrey P.; Sheĭnerman, A. G.; Gutkin, M. Yu.

doi:10.1016/j.matdes.2021.110191

Cited by 11 publications

(9 citation statements)

References 57 publications

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“…We investigated the properties of patterns arising from the Voronoi tessellations stemming from the random initial distribution of points on the plane [ 11 , 12 , 13 , 14 , 15 , 16 , 17 ]. The following mathematical procedure was applied: at the first stage, the Voronoi tessellation generated by a random set of seeds was built.…”

Section: Discussionmentioning

confidence: 99%

Section: Methodsmentioning

confidence: 99%

“…The quantification of 2D ordering is crucial for understanding phase transitions [ 1 , 2 , 3 , 4 ], the characterization of attractors in non-linear systems [ 5 ], the treatment of images [ 6 ] and machine learning applied for study of physical systems [ 7 , 8 , 9 ]. Various measures and mathematical procedures were implemented for the quantification of ordering in 2D patterns, including Voronoi tessellations followed by the calculation of the information (Shannon) entropy of the distribution of the Voronoi polygons (which is also called the Voronoi entropy and abbreviated in the text as VE; VE below within the text denotes the Shannon entropy) [ 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 ], Minkovski functionals [ 18 , 19 , 20 ], method of correlation functions [ 21 , 22 , 23 ], and calculation of the recently introduced continuous and Shannon measures of symmetry [ 6 , 24 , 25 , 26 , 27 ]. It was demonstrated that the introduced measures of symmetry do not necessarily correlate [ 27 , 28 ].…”

Section: Introductionmentioning

confidence: 99%

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From Chaos to Ordering: New Studies in the Shannon Entropy of 2D Patterns

Legchenkova

Frenkel

Shvalb

et al. 2022

Entropy

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Properties of the Voronoi tessellations arising from random 2D distribution points are reported. We applied an iterative procedure to the Voronoi diagrams generated by a set of points randomly placed on the plane. The procedure implied dividing the edges of Voronoi cells into equal or random parts. The dividing points were then used to construct the following Voronoi diagram. Repeating this procedure led to a surprising effect of the positional ordering of Voronoi cells, reminiscent of the formation of lamellae and spherulites in linear semi-crystalline polymers and metallic glasses. Thus, we can conclude that by applying even a simple set of rules to a random set of seeds, we can introduce order into an initially disordered system. At the same time, the Shannon (Voronoi) entropy showed a tendency to attain values that are typical for completely random patterns; thus, the Shannon (Voronoi) entropy does not distinguish the short-range ordering. The Shannon entropy and the continuous measure of symmetry of the patterns demonstrated the distinct asymptotic behavior, while approaching the close saturation values with the increase in the number of iteration steps. The Shannon entropy grew with the number of iterations, whereas the continuous measure of symmetry of the same patterns demonstrated the opposite asymptotic behavior. The Shannon (Voronoi) entropy is not an unambiguous measure of order in the 2D patterns. The more symmetrical patterns may demonstrate the higher values of the Shannon entropy.

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Section: Discussionmentioning

confidence: 99%

Section: Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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From Chaos to Ordering: New Studies in the Shannon Entropy of 2D Patterns

Legchenkova

Frenkel

Shvalb

et al. 2022

Entropy

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“…A Voronoi tessellation of an infinite plane is a partitioning of the plane into regions based on the distance to a specified discrete set of points (called seeds or nuclei and shown with black squares in Figure 1). For each seed, there is a corresponding region, consisting of all points closer to that seed than to any other [10][11][12][13][14][15][16][17]. Next, the following iterative procedure was applied to the initial random Voronoi tessellation depicted in Figure 1 according to the following steps: the edges forming the Voronoi polygons were divided into two or three parts, equally or randomly.…”

Section: Procedures Of Generation Of Voronoi Diagramsmentioning

confidence: 99%

“…Quantification of 2D ordering is crucial for understanding of phase transitions [1-4], characterization of attractors in non-linear systems [5], treatment of images [6] and machine learning applied for study of physical systems [7][8][9]. Various measures and mathematical procedures were implemented for quantification of ordering in 2D patterns, including: Voronoi tessellations followed by calculation of the Voronoi entropy (abbreviated VE) [10][11][12][13][14][15][16][17], Minkovski functionals [18][19][20], method of correlation functions [21][22][23], and calculation of the recently introduced continuous and Shannon measures of symmetry [6,[24][25][26][27]. It was demonstrated that introduced measures of symmetry do not necessarily correlate [27][28].…”

Section: Introductionmentioning

confidence: 99%

From Chaos to Ordering: New Studies in the Voronoi Entropy of 2D Patterns

Legchenkova¹,

Frenkel²,

Shvalb³

et al. 2022

Preprint

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Properties of the Voronoi tessellations arising from the random 2D distribution points are reported. We applied the procedure of dividing the sides of Voronoi cells into equal or random parts to Voronoi diagrams generated by a set of randomly placed on the plane points. The dividing points were then used to construct the following Voronoi diagram. Repeating this procedure led to a surprising effect of positional ordering of Voronoi cells, reminiscent of the formation of lamellae and spherulites in linear semi-crystalline polymers and metallic glasses. Thus, we can conclude, that by applying even a simple set of rules to a random set of seeds we introduce order into an initially disordered system. At the same time, the Voronoi entropy showed a tendency to values typical for completely random patterns and did not distinguish the short-range ordering. The Voronoi entropy and the continuous measure of symmetry of the patterns demonstrated the distinct asymptotic behavior, while approaching the close saturation values with the increase of the number of the iteration steps. Voronoi entropy grew, with the number of iterations, whereas the continuous measure of symmetry of the same patterns demonstrated the opposite asymptotic behavior. The Voronoi entropy is not an unambiguous measure of order in the 2D patterns. The more symmetrical patterns may demonstrate the higher values of the Voronoi entropy.

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