Statistical properties and shell analysis in random cellular structures

Aste, Tomaso; Szeto, Kwok Yip; Tam, Wing Yim

doi:10.1103/physreve.54.5482

Cited by 55 publications

(47 citation statements)

References 40 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In soap froth we have shown certain universal features such as the generalized Aboav-Wearie law [16] for the average number M͑i, n͒ of edges of cells in the ith shell of the central cell with n sides. We also found that the number of cells K͑i, n͒ in the ith shell is linear in i with slope related to the Aboav parameter [17]. In this Letter, we study the generalized Aboav-Weaire law, which relates the correlation between the ith shell neighbors with the disorder of the i-shell perimeter.…”

mentioning

confidence: 89%

“…In this Letter, we study the generalized Aboav-Weaire law, which relates the correlation between the ith shell neighbors with the disorder of the i-shell perimeter. We find new universal features for shells beyond the first.Inspired by ideas from the renormalization group, we consider the generalization of the Aboav-Weaire law based on the shell model analysis [17,18]. We propose a generalized Aboav-Weaire law in the form…”

mentioning

confidence: 99%

“…These experimental studies have been supplemented by detailed theoretical analysis using the shell model [16][17][18][19] and numerical simulations [20,21]. The structural characterization of the soap froth in the shell model can be applied to all twodimensional trivalent cellular patterns.…”

mentioning

confidence: 99%

“…Inspired by ideas from the renormalization group, we consider the generalization of the Aboav-Weaire law based on the shell model analysis [17,18]. We propose a generalized Aboav-Weaire law in the form…”

mentioning

confidence: 99%

See 3 more Smart Citations

Universal Topological Properties of Two-Dimensional Trivalent Cellular Patterns

Szeto

Tam

2002

Phys. Rev. Lett.

Self Cite

View full text Add to dashboard Cite

Universal topological properties of two-dimensional trivalent cellular patterns are found from shell analysis of soap froth and computer-generated Voronoi diagrams. We introduce a cluster analysis based on the shell model and find the universal relation ln͑a͞m 2 ͒ A 1 B ln͑m 2 ͒, with the generalized Aboav parameter a and second moment of the number of cell edge distribution m 2 . For the second, third, and fourth shells of the cluster, A and B are the same for all samples. Furthermore, A is increasing with shell number while B is a universal number, 20.90. For the first shell, the slope B is the same for soap froths, but slightly different from Voronoi graphs. DOI: 10.1103/PhysRevLett.88.138302 PACS numbers: 82.70.Rr, 02.50. -r, 05.70.Ln Two-dimensional cellular structure constitutes a large class of patterns with important technological and scientific applications. Soap froth, polycrystalline grain mosaics, and biological tissues are natural examples of random, space-filling cellular networks [1]. Since cellular structures exist on scales ranging from microscopic to geological, much work has been devoted to the search for universal geometrical characteristics. Despite the enormous difference in length scales and different physical forces driving the evolution of the networks, there exist certain universal topological laws governing their similarity. These laws leave aside metrical properties (e.g., sizes of cells) and address the probability distribution P n of the number n of edges of a given cell, or correlations between the numbers of edges of adjacent cells. One of the best-obeyed empirical laws is the Aboav-Weaire law [2,3] which states that on average the sum [M͑n͒n] of the number of sides of the cells immediately adjacent to an n-sided cell is linear in n:with the second moment m 2 Pǹ 3 P n ͑n 2 6͒ 2 being commonly used as a measure of the disorder. The notation M͑n͒ denotes the average number of edges of the adjacent cells to a given n-sided cell. The Aboav parameter a is a measure of nearest neighbor correlation and for soap froth a is approximately 1.There have been several important works published after the recognition of Aboav-Weaire law. Godreche et al. [4] have used planar graph theory in the context of counting planar Feynman diagrams with a cubic interaction to give an analytic expression of the Aboav-Weaire law in addition to the calculation of P n . However, these exact results are based on a particular model and cannot explain the vast diversity of natural cellular patterns with different Aboav parameters and second moments. Delannay and Le Caer [5,6] have studied the stationary topological properties of 2D cellular structures generated by random fragmentation in computer simulations, with very different a and m 2 that falls on a universal curve that is valid for many systems, both natural and computer generated [5 -8]. There are also many papers published on the two-cell correlation using maximum-entropy argument [9], but none of these theoretical efforts succeed in explaining the uni...

show abstract

mentioning

confidence: 89%

mentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

See 2 more Smart Citations

Universal Topological Properties of Two-Dimensional Trivalent Cellular Patterns

Szeto

Tam

2002

Phys. Rev. Lett.

Self Cite

View full text Add to dashboard Cite

show abstract

“…Our edge scaling hypothesis is simple and can be used to provide a phenomenological description of area scaling function, without going through the stochastic equation using von Neumann dynamics. However, we should remark here that any e v olution theory for froth without correlation e ects properly incorporated is not realistic, as experimental evidences for strong and long range correlation exist for real soap froth 12,13]. Our observation of edge scaling for individual topological class therefore poses new challenge for the theoretical understanding of evolving soap froth.…”

mentioning

confidence: 99%

Edge scaling of soap froth

Szeto

Tam

1998

Physica A: Statistical Mechanics and its Applications

View full text Add to dashboard Cite

We i n vestigate the scaling properties of the edge distribution of soap froth and compare it with the area scaling law. The data collapse for edge scaling of cells belonging to the same topological class is explained by a harmonic expansion of the energy of the froth with area-perimeter constraint. The edge scaling hypothesis is found to be comparable to area scaling. The implication on existing theories for the evolution of soap froth is discussed. The scaling behavior of evolving soap froth has recently been the subject of many t h eoretical works 1,2]. There are two scaling behaviors that are observed in experiments.Stavans 3] has reported that not only the probability distribution (P n (t)) of the cells of given number of sides verses the number of edges (n) is independent of time over a long period of time, but also the distribution function of area is invariant i n f o r m o ver the same period: P(A t) = f(x)= A with x = A= A and the time dependence only enters in the mean area A(t). These two t i m e i n variant distribution functions (P n and f(x)) de ne the scaling regime in the literature 4]. According to the theoretical basis of area scaling 2], one also expects that similar scaling laws apply to the subset of cells with same number of edges. However, detailed experimental test of area scaling for individual topological class (i.e. cells with same n) has not been made. In this short note, we compare the quality of data collapse for soap froth using the area scaling with edge scaling for individual topological class. We nd that edge scaling is at least as good a description for experiments on soap as area scaling. Furthermore, there are physical reasons supporting the edge scaling description.We will examine three forms of scaling function. Firstly, one uses the original area scaling hypothesis. We test area scaling of individual topological class by assuming that the probability P A n (A n t ) of nding an n;sided cell with area A n at time t among all n;sided cells to have the following form where P A n (A n t ) = F n (x 0 = An A ) A with Z 1 0 dA n P A n (A n t ) = 1 :0Note that P A n (A n t ) is a function extracted from the histogram of area distribution of the n-sided bubbles at time t and the normalization is 1. This form of scaling function implies that a data collapse on a universal curve o f F n (x 0 ) = A(t)P A n (A n t ) v s x 0 = An A(t) will be observed. The quality of the data collapse with this form is shown in Fig.1. We see that F n (x 0 ) is far from a universal curve.Secondly, w e test area scaling of a given topological class n using the mean area of the n-sided cells. Here we assume P A n (A n t ) = f n (x = An An ) A n with Z 1 0 dA n P A n (A n t ) = 12 here A n is its average area of all n;sided cells at time t. This form of scaling function implies that a data collapse on a universal curve o f f n (x) = A n (t)P A n (A n t ) v s x = An An(t) will be observed. This form di ers from Eq.1 in that A n , the mean area of the n;sided cells, replaces A, the mean area of all ce...

show abstract

References

2013

Stochastic Geometry and Its Applications

View full text Add to dashboard Cite

Statistical properties and shell analysis in random cellular structures

Cited by 55 publications

References 40 publications

Universal Topological Properties of Two-Dimensional Trivalent Cellular Patterns

Universal Topological Properties of Two-Dimensional Trivalent Cellular Patterns

Edge scaling of soap froth

References

Contact Info

Product

Resources

About