Aboav-Weaire Law Research Articles

The 3-D Aboav-Weaire relation from modified mean-field model

Grain-environment heterogeneity has been known to play a vital role in grain growth kinetics. In general, the topological correlation between grains and first nearest neighbors is described by the famous Aboav-Weaire law. However, its predictions in 3-D normal grain growth do not exactly match the results from experiments and simulations. In this work, with a ‘maximum interval’ assumption and an approximation of the MacPherson-Srolovitz law, an alternative approach of the Aboav-Weaire relation is developed from the modified mean-field model. The predictions are not only significantly close to the reality, but also successfully reflect the effect of grain-environment heterogeneity on the normalized grain growth rate.

3D cell neighbour dynamics in growing pseudostratified epithelia.

During morphogenesis, epithelial sheets remodel into complex geometries. How cells dynamically organise their contact with neighbouring cells in these tightly packed tissues is poorly understood. We have used light-sheet microscopy of growing mouse embryonic lung explants, three-dimensional cell segmentation, and physical theory to unravel the principles behind 3D cell organisation in growing pseudostratified epithelia. We find that cells have highly irregular 3D shapes and exhibit numerous neighbour intercalations along the apical-basal axis as well as over time. Despite the fluidic nature, the cell packing configurations follow fundamental relationships previously described for apical epithelial layers, that is, Euler's polyhedron formula, Lewis' law, and Aboav-Weaire's law, at all times and across the entire tissue thickness. This arrangement minimises the lateral cell-cell surface energy for a given cross-sectional area variability, generated primarily by the distribution and movement of nuclei. We conclude that the complex 3D cell organisation in growing epithelia emerges from simple physical principles.

The 3-D Aboav-Weaire Relation From Modified Mean-Field Model

Grain-environment heterogeneity has been known to play a vital role in grain growth kinetics. In general, the topological correlation between grains and first nearest neighbors is described by the famous Aboav-Weaire law. However, its predictions in 3-D normal grain growth do not exactly match the experiments and simulations. In this work, with an approximation of the MacPherson-Srolovitz law, an alternative approach of the A-W relation is developed from the modified mean-field model. The predictions are not only significantly closer to the reality, but also successfully reflects the effect of grain-environment heterogeneity on the normalized grain growth rate.

Generalized network theory of physical two-dimensional systems.

The properties of a wide range of two-dimensional network materials are investigated by developing a generalized network theory. The methods developed are shown to be applicable to a wide range of systems generated from both computation and experiment; incorporating atomistic materials, foams, fullerenes, colloidal monolayers, and geopolitical regions. The ring structure in physical networks is described in terms of the node degree distribution and the assortativity. These quantities are linked to previous empirical measures such as Lemaître's law and the Aboav-Weaire law. The effect on these network properties is explored by systematically changing the coordination environments, topologies, and underlying potential model of the physical system.

Vitreous 2D silica under tension: From brittle to ductile behaviour

Based on measured ring size distributions of real 2D silica, we numerically prepare sample sets of vitreous 2D silica with varying network topology. We use a Monte Carlo bond-switching method to apply a sequence of topological flip transformations to initially hexagonal 2D silica. In doing so, we are able to control the network heterogeneity by means of the ring size distribution. This method also enables physically meaningful ring neighbourhood statistics according to the empirical Aboav-Weaire law. The obtained vitreous 2D silica sample sets undergo athermal quasistatic tensile deformation in order to investigate the changes of the mechanical behaviour as a function of the underlying network heterogeneity. The molecular statics simulations reveal that the ductility and tensile strength of 2D silica can significantly be tuned varying the network ring statistics. Furthermore, the stress-drop statistics of the respective sample sets are evaluated and shown to display a self-organised critical behaviour.

Topology of dividing planar tilings: Mitosis and order in epithelial tissues.

We investigate a range of rule-based models of the in-plane structure of growing single-cell-thick epithelia represented by the distribution of frequencies of polygon classes. Within the Markovian framework introduced by Gibson etal. [Nature (London) 442, 1038 (2006)10.1038/nature05014], we discuss various topologically allowed cell division schemes assumed to control the structure of the tissue as well as a phenomenological Gaussian scheme, and we compute the stationary distributions for all of them. Some of the distributions reproduce those seen in tissues characterized by unbiased mitotic events but also in certain tissues with a preferred orientation of the mitotic plane or a cell-rearrangement process such as neighbor exchange. In addition, we propose the asynchronous-division variant of the model, which builds on the Lewis law and on the Aboav-Weaire law as well as on the fact that the dividing cells are larger than the resting cells. This generalization a posteriori validates the original model.

Ellipse packing in two-dimensional cell tessellation: a theoretical explanation for Lewis's law and Aboav-Weaire's law.

BackgroundLewis’s law and Aboav-Weaire’s law are two fundamental laws used to describe the topology of two-dimensional (2D) structures; however, their theoretical bases remain unclear.MethodsWe used R software with the Conicfit package to fit ellipses based on the geometric parameters of polygonal cells of ten different kinds of natural and artificial 2D structures.ResultsOur results indicated that the cells could be classified as an ellipse’s inscribed polygon (EIP) and that they tended to form the ellipse’s maximal inscribed polygon (EMIP). This phenomenon was named as ellipse packing. On the basis of the number of cell edges, cell area, and semi-axes of fitted ellipses, we derived and verified new relations of Lewis’s law and Aboav-Weaire’s law.ConclusionsEllipse packing is a short-range order that places restrictions on the cell topology and growth pattern. Lewis’s law and Aboav-Weaire’s law mainly reflect the effect of deformation from circle to ellipse on cell area and the edge number of neighboring cells, respectively. The results of this study could be used to simulate the dynamics of cell topology during growth.

Topological and geometrical quantities in active cellular structures.

Topological and geometrical properties and the associated topological defects find a rapidly growing interest in studying the interplay between mechanics and the collective behavior of cells on the tissue level. We here test if well studied equilibrium laws for polydisperse passive systems such as Lewis' and Aboav-Weaire's law are applicable also for active cellular structures. Large scale simulations, which are based on a multiphase field active polar gel model, indicate that these active cellular structures follow these laws. If the system is in a state of collective motion, quantitative agreement with typical values for passive systems is also observed. If this state has not developed, quantitative differences can be found. We further compare the model with discrete modeling approaches for cellular structures and show that essential properties, such as T1 transitions and rosettes, are naturally fulfilled.

Phase field crystal simulations of the kinetics of Ostwald ripening in two dimensions.

The kinetics of Ostwald ripening of solid domains in the liquid phase of one-component systems in two dimensions is investigated numerically via the phase field crystal model. The simulations, which are performed systematically as a function of volume fraction of the solid phase, show that dynamical scaling is reached during late times, and the growth law is in good agreement with the classical theory of Lifshitz, Slyozov, and Wagner (LSW), i.e., R[over ¯]∼t^{1/3}, an indication that domain growth is mediated by the long-range interdomain diffusion of atoms. In contrast to the LSW theory, however, the domain size distribution is symmetric, and can be fit with a Gaussian. The investigation of the topological domain structure, through the Voronoi tessellation of the domains' centers of mass shows that both the Lewis law and the Aboav-Weaire law of two-dimensional cellular patterns are satisfied, implying that the kinetics proceed such that the conformational entropy of the domain-containing Voronoi cells is maximized. These results are in very good agreement with an earlier experimental study of a phase-separating phospholipid-cholesterol Langmuir film.

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

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.

Ring correlations in random networks.

We examine the correlations between rings in random network glasses in two dimensions as a function of their separation. Initially, we use the topological separation (measured by the number of intervening rings), but this leads to pseudo-long-range correlations due to a lack of topological charge neutrality in the shells surrounding a central ring. This effect is associated with the noncircular nature of the shells. It is, therefore, necessary to use the geometrical distance between ring centers. Hence we find a generalization of the Aboav-Weaire law out to larger distances, with the correlations between rings decaying away when two rings are more than about three rings apart.

Multi-multifractality, dynamic scaling and neighbourhood statistics in weighted planar stochastic lattice

The dynamics of random sequential partitioning of a square into ever smaller mutually exclusive rectangular blocks, which we call weighted planar stochastic lattice (WPSL), is governed by infinitely many conservation laws. Recently, we have shown one of the infinitely many conservation laws can be used as multifractal measure. In this article, we show that except the conservation of total area, each of the infinitely many non-trivial conservation laws is actually a multifractal measure and hence WPSL is a multi-multifractal. We then look at the block size distribution function and find that it exhibits dynamic scaling revealing that the spatial patterns of WPSL of different sizes are statistically self-similar. Besides, we investigate how the mean area ⟨A⟩k of blocks with k neighbours and the average number of neighbours mk of a typical cell that neighbours a k-sided block behaves with k. These results suggest that the Lewis law, ⟨A⟩k∝k, is obeyed for up to k ≈ 8 and the Aboav–Weaire law, kmk∝k, is violated for the entire range of k.

Long-range topological correlations of real polycrystalline grains in two dimensions

The topological and geometric properties of real polycrystalline grains in two dimensions are investigated on the basis of a large dataset (14,810 grains). The distribution of grain edges, the grain topology–size relationship and the short- and long-range topological correlations between neighboring grains are characterized quantitatively. The results show a strong short-range topological correlation between a center grain and its first nearest neighbors, and a trivial long-range topological correlation beyond the first nearest neighbors (on average). Both the short- and long-range relations are well described by a generalized Aboav–Weaire law reported recently. Meanwhile, it is the perimeter law, rather than the Lewis law, that describes appropriately the relationship between the topology and geometry of metallurgical grains.

Ring statistics of silica bilayers

The recent synthesis and characterisation of bilayers of vitreous silica has produced valuable new information on ring sizes and distributions. In this paper, we compare the ring statistics of experimental samples with computer generated samples. The average ring size is fixed at six by topology, but the width, skewness and other moments of the distribution of ring edges are characteristics of particular samples. We examine the Aboav-Weaire law that quantifies the propensity of smaller rings to be adjacent to larger rings, and find similar results for available experimental samples which however differ somewhat from computer-generated bilayers. We introduce a new law for the areas of rings of various sizes.

Study on Grain Topology with Potts Model Monte Carlo Simulation

Three-dimensional normal grain growth has been simulated in scale 300×300×300 using the generally accepted Potts model Monte Carlo method. The studies of the topology of grains indicate that the mean number of faces in the grain network <f>=13.91 is similar to other simulation results, but higher than most of the experimental data which containing a wide range of values, i.e., <f>=11.16~13.93. The three-dimensional AboavWeaire law and Liu-Yu law are observed to hold, but the fit coefficient is different from the theory models.

Verification of a generalized Aboav-Weaire law via experiment and large-scale simulation

Topological correlations in grain boundary networks are investigated on the basis of more than 14000 experimental grains and 9000 Monte Carlo-Potts model simulation grains. A generalized Aboav-Weaire law which serves as a description of the short- and long-range nearest-neighbor topological correlations, is proved to hold both in 2D grain structures and 2D cross-section structures. However, the nearest-neighbor topological correlations have no obvious influence on the rate of 2D grain growth, which is explicitly different from the case in three dimensions that was previously reported in Wang H., Liu G. Q., Song X. Y. and Luan J. H., EPL, 96 (2011) 38003.

Quantifying microstructures in isotropic grain growth from phase field modeling: Topological properties

In a previous work [Acta Materialia 2012;60:4787], we developed a new method that utilizes discrete, voxel-based data for microstructure quantification. We successfully calculated some relatively simple microstructural quantities and relations. In this paper, we apply and extend this method to compute more complex microstructural quantities and, in particular, map out the connection between grain growth rate and various topological properties. We present detailed results for several local and average topological and geometric properties of the microstructures during grain coarsening, including the curvature of grain boundaries and triple junction lines, grain cell shape, and their relations with growth dynamics. We also examine several well-known topological relations, i.e. Euler relations, the Lewis rule and the Aboav–Weaire law. These quantities and relations are the centerpiece of the grain growth models and theories developed so far. We also compare our results with some existing results in three dimensions. The quantitative description of the dynamic behaviors of the microstructural attributes adds a valuable data set to grain growth that can be used for benchmarking for phase field modeling and comparison with other approaches.

Generalization of the Aboav-Weaire law

How cells are arranged in a three-dimensional celluar network is one of the classic problems of physics, biology and materials science. Based on the topological analyses on steady-state grain-growth structure, a generalization of the Aboav-Weaire law to layers beyond the first is suggested that the average number of faces per grain, mj(f), is related to the number of grains, qj(f), in thej-th layer of a center grain with f faces (on average). This result is verified by the data from the curvature flow simulation and assists in better understanding the arrangement of cells in three dimensions.

Coarsening of a two-dimensional foam on a dome

In this paper we report on bubble growth rates and on the statistics of bubble topology for the coarsening of a dry foam contained in the narrow gap between two hemispheres. By contrast with coarsening in flat space, where six-sided bubbles neither grow nor shrink, we observe that six-sided bubbles grow with time at a rate that depends on their size. This result agrees with the modification to von Neumann's law predicted by J. E. Avron and D. Levine [Phys. Rev. Lett. 69, 208 (1992)]. For bubbles with a different number of sides, except possibly seven, there is too much noise in the growth rate data to demonstrate a difference with coarsening in flat space. In terms of the statistics of bubble topology, we find fewer three-, four-, and five-sided bubbles, and more bubbles with six or more sides, in comparison with the stationary distribution for coarsening in flat space. We also find good general agreement with the Aboav-Weaire law for the average number of sides of the neighbors of an n-sided bubble.

Topological correlations of three-dimensional grains

Topological correlations of three-dimensional grains were investigated by Monte Carlo-Potts model simulation. The result shows that, unlike first nearest neighbors (the Aboav-Weaire law [D. Aboav, Metallography 3, 383 (1970) and D. Weaire, Metallography 7, 157 (1974)] holds), there appears to be very little correlation between grains and their second and third nearest neighbors (on average), i.e., the average number of faces of second nearest neighbors, m2, and third nearest neighbors, m3, are independent of faces f of the center grain (nearly m2 = 14.984 and m3 = 14.489). This result indicates that long range correlations beyond first nearest neighbors may have negligible effect on the growth of center grains and thus provides support to a topologically averaged growth law that only considered the non-random first nearest-neighbor interactions.