Aboav-Weaire Law Research Articles

Ka-me: a Voronoi image analyzer

Summary: Ka-me is a Voronoi image analyzer that allows users to analyze any image with a convex polygonal tessellation or any spatial point distribution by fitting Voronoi polygons and their dual, Delaunay triangulations, to the pattern. The analytical tools include a variety of graph theoretic and geometric tools that summarize the distribution of the numbers of edges per face, areas, perimeters, angles of Delaunay triangle edges (anglograms), Gabriel graphs, nearest neighbor graphs, minimal spanning trees, Ulam trees, Pitteway tests, circumcircles and convexhulls, as well as spatial statistics (Clark–Evans Nearest Neighborhood and Variance to Mean Ratio) and export functions for standard relationships (Lewis's Law, Desch's Law and Aboav–Weaire Law).Availability: Ka-me: a Voronoi image analyzer is available as an executable with documentation and sample applications from the BioQUEST Library (http://bioquest.org/downloads/kame_1.0.rar).Contact: noppadon.khiripet@nectec.or.th

Spatial correlations in polydisperse, frictionless, two-dimensional packings

We investigate next-nearest-neighbor correlations of the contact number in simulations of polydisperse, frictionless packings in two dimensions. We find that disks with few contacting neighbors are predominantly in contact with disks that have many neighbors and vice versa at all packing fractions. This counterintuitive result can be explained by drawing a direct analogy to the Aboav-Weaire law in cellular structures. We find an empirical one parameter relation similar to the Aboav-Weaire law that satisfies an exact sum rule constraint. Surprisingly, there are no correlations in the radii between neighboring particles, despite correlations between contact number and radius.

Polygonal networks on the surface of Mars; applicability of Lewis, Desch and Aboav–Weaire laws

Verification of the applicability of Lewis, Desch and Aboav–Weaire laws to polygonal networks in terrains on the surface of Mars is reported. The networks analysed cover a great variety of small-scale polygonal terrains formed in periglacial regions of the planet. It is shown that these Martian terrains share some of the same geometrical and topological properties of other random networks built by materials and processes of diversified nature and origin.

Grain size distribution and topology in 3D grain growth simulation with large-scale Monte Carlo method

Three-dimensional normal grain growth was appropriately simulated using a Potts model Monte Carlo algorithm. The quasi-stationary grain size distribution obtained from simulation agreed well with the experimental result of pure iron. The Weibull function with a parameter β=2.77 and the Yu-Liu function with a parameter v =2.71 fit the quasi-stationary grain size distribution well. The grain volume distribution is a function that decreased exponentially with increasing grain volume. The distribution of boundary area of grains has a peak at S/< S>=0.5, where S is the boundary area of a grain and < S> is the mean boundary area of all grains in the system. The lognormal function fits the face number distribution well and the peak of the face number distribution is f=10. The mean radius of f-faced grains is not proportional to the face number, but appears to be related by a curve convex upward. In the 2D cross-section, both the perimeter law and the Aboav-Weaire law are observed to hold.

Cellular automata simulation of grain growth in three dimensions based on the lowest-energy principle

The microstructure and morphology evolution of grain growth were studied by 3D simulation using the cellular automata (CA) model based on the lowest-energy principle. In the present CA model, the transition of cells during the grain growth has a typical physical meaning due to the application of the lowest-energy principle. The results show that the kinetics of grain growth follows Burke equation with the growth exponent as 2. The average number of grain faces is 13.6 and the highest frequency of grain faces is 10 faces. The grain size distribution follows Weibull function. The relationship between the number of faces of a grain and the average number of faces of its adjacent grains follows the Aboav–Weaire law. There is a correlation between the topologies of the simulated 2D and 3D grain growth. The average number of sides per face for all grains is 5.65 and the average number of sides per face is about equal to 6 when the grain aces is larger than 35.

Linear relation on the correlation in complex networks

Correlation in complex networks follows a linear relation between the degree of a node and the total degrees of its neighbors for six different classes of real networks. This general linear relation is an extension of the Aboav-Weaire law in two-dimensional cellular structures and provides a simple and different perspective on the correlation in complex networks, which is complementary to an existing description using Pearson correlation coefficients and a power law fit. Analytical expression for this linear relation for three standard models of complex networks: the Erdos-Renyi, Watts-Strogatz, and Barabasi-Albert networks is provided. The slope and intercept of this linear relation are described by a single parameter a together with the first and second moment of the degree distribution of the network. The assortivity of the network can be related to the sign of the intercept.

Structure of random bidisperse foam

The Surface Evolver was used to compute the equilibrium microstructure of random soap foams with bidisperse cell-size distributions and to evaluate topological and geometric properties of the foams and individual cells. The simulations agree with the experimental data of Matzke and Nestler for the probability ρ ( F ) of finding cells with F faces and its dependence on the fraction of large cells. The simulations also agree with the theory for isotropic Plateau polyhedra (IPP), which describes the F-dependence of cell geometric properties, such as surface area, edge length, and mean curvature (diffusive growth rate); this is consistent with results for polydisperse foams. Cell surface areas are about 10% greater than spheres of equal volume, which leads to a simple but accurate relation for the surface free energy density of foams. The Aboav–Weaire law is not valid for bidisperse foams.

Short- and long-range topological correlations in two-dimensional aggregation of dense colloidal suspensions

We have studied the average properties and the topological correlations of computer-simulated two-dimensional (2D) aggregating systems at different initial surface packing fractions. For this purpose, the centers of mass of the growing clusters have been used to build the Voronoi diagram, where each cell represents a single cluster. The number of sides (n) and the area (A) of the cells are related to the size of the clusters and the number of nearest neighbors, respectively. We have focused our paper in the study of the topological quantities derived from number of sides, n , and we leave for a future work the study of the dependence of these magnitudes on the area of the cells, A . In this work, we go beyond the adjacent cluster correlations and explore the organization of the whole system of clusters by dividing the space in concentric layers around each cluster: the shell structure. This method allows us to analyze the time behavior of the long-range intercluster correlations induced by the aggregation process. We observed that kinetic and topological properties are intimately connected. Particularly, we found a continuous ordering of the shell structure from the earlier stages of the aggregation process, where clusters positions approach a hexagonal distribution in the plane. For long aggregation times, when the dynamic scaling regime is achieved, the short- and long-range topological properties reached a final stationary state. This ordering is stronger for high particle densities. Comparison between simulation and theoretical data points out the fact that 2D colloidal aggregation in the absence of interactions (diffusion-limited cluster aggregation regimen) is only able to produce short-range cluster-cluster correlations. Moreover, we showed that the correlation between adjacent clusters verifies the Aboav-Weaire law, while all the topological properties for nonadjacent clusters are mainly determined by only two parameters: the second central moment of number-of-sides distribution mu(2) = sumP (n) (n-6)(2) and the screening factor a (defined through the Aboav-Weaire equation). We also found that the values of mu(2) and a calculated for two-dimensional aggregating system are related through a single universal common form a proportional to mu2(-0.89), which is independent of the particle concentration.

Characterization of three-dimensional grain structure in polycrystalline iron by serial sectioning

The face number, the volume, and the surface area (boundary area) of grains are measured in recrystallized α-iron by serial sectioning coupled with quantitative microstructural analysis on two-dimensional (2-D) sections. The sampling volume contained approximately 1000 grains whose mean grain size was ∼15 µm. The grain volume distribution decreased monotonously with increasing grain volume, whereas the surface area had a peak around one-half of the average. The distribution of the sphere equivalent radii significantly deviated from the log-normal distribution and distributions predicted from mean field theories. The peak and the mean of the face number distribution were f=11 and 12.1, respectively. The linear relationship between the face number of a central grain and the mean face number of surrounding grains, known as the Aboav-Weaire law, was observed in three dimensions. The mean radius of f-faced grains was not proportional to the face number (perimeter law in 3-D), but appeared to be related by a curve convex upward.

On the Aboav–Weaire law

The most general form of the Aboav–Weaire empirical topological law of random cellular networks is derived from conditions on local averages in two dimensions. We assume the average number of edges in a local cluster depends on the number of actual edges and its moments together with the probability distribution function. Based on this constitutive assumption we show that the Aboav–Weaire law depends only on the first and second moments of the distribution function. We show that the Aboav–Weaire law is a direct consequence of the existence of the well-known microscopic topological transformations. We study the effect of the constitutive equation’s coefficients on the Aboav–Weaire law. The properties near to the equilibrium are also investigated to explain the role of the actual parameters of the network.

Voronoi tessellation of the packing of fine uniform spheres.

The packing of uniform fine spherical particles ranging from 1 to 1000 microm has been simulated by means of discrete particle simulation. The packing structure is analyzed, facilitated by the well established Voronoi tessellation. The topological and metric properties of Voronoi polyhedra are quantified as a function of particle size and packing density. The results show that as particle size or packing density decreases, (i) the average face number of Voronoi polyhedra decreases, and the distributions of face number and edge number become broader and more asymmetric; (ii) the average perimeter and area of polyhedra increase, and the distributions of polyhedron surface area and volume become more flat and can be described by the log-normal distribution. The topological and metric properties depicted for the packing of fine particles differ either quantitatively or qualitatively from those reported in the literature although they all can be related to packing density. In particular, our results show that the average sphericity coefficient of Voronoi polyhedra varies with packing density, and although Aboav-Weaire's law is generally applicable, Lewis's law is not valid when packing density is low, which are contrary to the previous findings for other packing systems.

THE USE OF THE ABOAV-WEAIRE LAW TO ESTIMATE A BIOLOGICAL CONSTRAINT

Developmental processes may impose limitations and directionality in the mode of development of a particular structure. The main problem is to determine the nature and the respective effects of physical and biological constraints in the development of organisms. The Aboav-Weaire law is a semi-empirical law developed to explain the topological structure of physical materials. In the present paper, we make a formal analysis of the quantitative relationships between physical and biological constraints in biological structures by using the inflorescence of the Araceae as a case study. The Aboav-Weaire law permits to obtain a quantitative estimate of the biological constraint acting on the inflorescences of this family. In the case of the Araceae, the empirical curve presents a constant deviation with respect to the Aboav-Weaire law. This deviation is due to the presence of a biological constraint as opposed to a physical constraint. The biological constraint tends to decrease the variance of the number of sides while it is the inverse situation for the physical constraint. The results obtained using the Araceae model can be used to study the interrelationships between biological and physical constraints in any organism or biological structure.

Two-dimensional simulation of grain growth based on an atomic jump model for grain boundary migration

The kinetics and the topological phenomena during two-dimensional grain growth are studied by computer simulations based on an atomic jump model for grain boundary migration. The grain boundaries are assumed to be straight. The kinetics show the 1/2-power growth law for the average grain size, and the size and the side distributions are time-invariant. In particular, the simulated side distribution is well consistent with the theoretical prediction. The present simulation follows the Aboav–Weaire law for entire topological classes, the Lewis law for intermediate topological classes and the von Neumann–Mullins law for intermediate and high classes. The deviation from the von Neumann–Mullins law for low topological classes is reduced by taking account of the effects of curved grain boundaries. The other distinctive results of the simulation are also shown and discussed.

Three-dimensional microstructural evolution in ideal grain growth—general statistics

The idealized grain growth in three dimensions (3D) was studied by the Surface Evolver method that models the process of boundary motion by curvature to minimize the boundary energy. Even starting from arbitrary packing of uniform grains, the boundary network reached steady structure after the incubation and transient period. The parabolic law in grain growth was observed only in a region where the steady structure was maintained. The more general von Neumann–Mullins law on kinetics of grain growth held in both transient period and normal grain growth period. The grain size distribution function and the distribution of the number of faces in steady structure were analyzed in 3D, and compared with the microstructure in cross-section. The perimeter law and Aboav–Weaire law in 3D on topological nature of boundary network structure held not only in the steady structure but also in transient structures.

Evolution of a colloidal soap-froth structure

We report the evolution of a quasi-two-dimensional cellular structure. This soap-froth-like structure is formed by 2.24 μ m colloidal particles trapped at the air/water interface. The froth evolves mainly through one of the modes of the T2 mechanism, the inverse mitosis process, and in minor proportion through the T1 or side switching mechanism. In addition, particle rearrangements can also be observed during the evolution. Eventually, the colloidal soap-froth evolves towards the formation of colloidal clusters, which detach from the edges of the colloidal froth. The cell side distribution is similar to distributions from other froth-forming systems and follows the Aboav–Weaire law. Other statistical comparisons are also made with other froth-forming systems and, in general, we found similar behaviour.

Interactions Between Physical and Biological Constraints in the Structure of the Inflorescences of the Araceae

Abstract A study of the inflorescences ofMonsteraandAnthuriumwas used to establish a relationship between biological and physical constraints for the structure of plant organs. The physical constraint between flowers in the compact inflorescences ofAnthuriumandMonsterais expressed by Aboav-Weaire's law. The application of this law to inflorescences indicates a linear relationship between the number of sides of a flower and the number of sides of neighbouring flowers. However, the slope of this straight line is significantly higher forAnthuriumandMonsterathan that expected in theory. This deviation from the law is attributable to a biological cause that can be estimated using Aboav-Weaire's law. Acting alone, the biological constraint tends to produce four-sided flowers. The equilibrium between biological and physical constraints reduces the number of sides per flower from six (theoretical value) to 5.9 (inAnthurium) or 5.8 (inMonstera) with a variance of the measures less than that expected in theory. Furthermore, when flower density in an inflorescence increases (towards the middle of the inflorescence inMonsteraand towards the lower section forAnthurium) the number of sides approaches six (i.e. the physical constraint dominates). When flower density decreases (towards the top of the inflorescence) the number of sides approaches 5.5 (i.e. the biological constraint dominates). The geometry of the inflorescences ofAnthuriumandMonsterais the result of the joint action of biological and physical constraints.

On some natural and model 2D bimodal random cellular structures

The topological and metric properties of a few natural 2D random cellular structures, namely an armadillo shell structure and young soap froths, which are formed from two classes of cells, large and small, have been characterized. The topological properties of a model generated from a Kagome tiling, which mimics such random binary structures, have also been exactly calculated. The distribution of the number of cell sides is bimodal for the structures investigated. In contrast to the classical Aboav-Weaire law for homogeneous 2D random cellular structures, nm(n), the mean total number of edges of neighbouring cells of cells with n sides does not vary linearly with n. Only the nm(i, n) (i = 1, 2) determined separately for every class of cells are linear in n for all investigated structures. Topological properties and correlations between metric and topological properties are finally compared with the predictions of various literature models.

Stochastic geometry of polygonal networks: An alternative approach to the hexagon-square transition in Bénard convection

We apply stochastic geometry to the transition from hexagonal to square cells recently observed in surface- tension-driven Benard convection. In particular we study the metric and topological evolution of Benard patterns as a function of the temperature difference, DT, across the layer. The preference of square Benard cells at higher DT is a consequence of both a higher efficiency in heat transfer and more favorable metric properties. Most notably, the perimeter-area ratio of a square cell exceeds that of a hexagonal cell by an unexpectedly high value. From a topological point of view, the Benard pattern obeys the Aboav-Weaire law at all times, even in the presence of threefold and fourfold vertices. The regimes above and below the transition are characterized by different topological correlations between neighboring cells. With the appearance of fourfold vertices, the topological correlation changes from negative to positive. @S1063-651X~98!14709-8#

Topological and geometrical properties of crack patterns produced by the thermal shock in ceramics

We study the crack patterns produced by thermal shock in ceramic tableware. Cracks produced at sufficiently high-temperature gradients partition the surface plane of a ceramic material into cells forming a random two-dimensional space-filling cellular structure. The topological and geometrical properties of these structures are described and analyzed. The distribution of the number of cell sides, the topological correlations between adjoining cells, the probability distributions of a cell area and side length, the average area and perimeter of $n$-sided cells, and the distribution of vertex angles are determined. The results show that the studied cellular structures obey the Aboav-Weaire law [Metallography 3, 383 (1970); 7, 157 (1974)] and Desch's law [J. Inst. Metals 22, 241 (1919)]. The scaling behavior of cellular structures obtained at different temperature gradients of the thermal shock is also presented.

Two-dimensional magnetic liquid froth: Coarsening and topological correlations

We present an experimental study of a cellular system which may be formed in a two-dimensional ~2D! layer made of an immiscible mixture of a magnetic fluid and an oil. We obtain a wet or a dry 2D froth, the characteristics of which are determined by the strength of an externally applied magnetic field. The froth is formed in an equilibrium state; the topological and geometrical features of the froth are fixed by the value of the applied field. The cellular pattern of the froth is stable in time, and a coarseninglike behavior is observed on decreasing the amplitude of the field. This cellular system can be used as a model to study topological processes in the equilibrium state, contrary to soap froths which are nonequilibrium systems because gas diffusion occurs. We show that the topological characteristics are statistically reversible after a magnetic-field cycle, and we present a statistical study of the froth features as a function of the amplitude of the field. The topological correlations between neighboring cells are well described by the Aboav-Weaire law. Finally, we consider the area of an n-sided cell, and show that its growth rate ~magnetic field dependent! is, in a surprising manner, proportional to n26, similarly to the Von Neumann law. @S1063-651X~97!10009-5#