Nicolas Rivier

Soap, Cells and Statistics-Random Patterns in Two Dimensions

Random two-dimensional patterns crop up in a wide variety of scientific contexts. What do they have in common? How can they be classified or analysed? These questions are underlined, and partly answered, by a survey of such patterns, paying particular attention to soap cell networks, metallurgical grain structures and the Giants Causeway.

On the correlation between sizes and shapes of cells in epithelial mosaics

It is shown that Lewiss empirical, linear relationship between the average area of a cell and the number of its sides in two-dimensional mosaics corresponds to maximal arbitrariness in the cellular distribution. An expression for the distribution is given in the general case.

Hierarchy and disorder in non-crystalline structures

Abstract The structure of glasses and amorphous materials is constructed explicitly and described geometrically and algebraically in a very simple model of a succession of decurving operations on a crystal (polytope) in curved space. Two kinds of disorder can be introduced: (1) in the succession of decurvings—this yields an ensemble of hierarchical structures; (2) locally, by commuting the order of two decurving operations at different points in space. The structure can be patched up geometrically, and a connection denned, leaving behind defects, which are disclination lines (2π-disclinations in the Euclidean limit). The defect sizes and the barriers between configurations are themselves hierarchic, and depend at which stage in the succession of decurvings local commutation has been introduced. Two-level systems (tunnelling modes) are associated with the defects, and o⋅cur at all scales. Algebraically, local disorder is a gauge transformation, and a gauge-invariant energy of glass is given. The gauge tran...

Voronoi tessellation of packings of spheres: Topological correlation and statistics

Abstract We describe three-dimensional froths obtained by the Voronoi tessellation of monosize packings of spheres at different packing fractions C from C = 0 toC = 0·58. The packings are built numerically. The distribution p(f) of the number f of faces of a cell is well approximated by a Gaussian. The average number m(f) of faces of the neighbours of a f-faceted cell follows the three-dimensional equivalent of the Aboav-Weaire law. The Lewis and the Deschlaws can be generalized to the three-dimensional case to describe the metricproperties of the froths. The distribution of the volumes of the cells is wellfitted by a gamma law at low packing fractions but becomes narrower and more symmetric at higher concentrations.

From one cell to the whole froth: A dynamical map

We investigate two- and three-dimensional shell-structured-inflatable froths, which can be constructed by a recursion procedure adding successive layers of cells around a germ cell. We prove that any froth can be reduced into a system of concentric shells. There is only a restricted set of local configurations for which the recursive inflation transformation is not applicable. These configurations are inclusions between successive layers and can be treated as vertices and edges decorations of a shell-structured-inflatable skeleton. The recursion procedure is described by a logistic map, which provides a natural classification into Euclidean, hyperbolic, and elliptic froths. Froths tiling manifolds with different curvatures can be classified simply by distinguishing between those with a bounded or unbounded number of elements per shell, without any a priori knowledge on their curvature. A result, associated with maximal orientational entropy, is obtained on topological properties of natural cellular systems. The topological characteristics of all experimentally known tetrahedrally close-packed structures are retrieved. PACS number~s!: 82.40.Ck, 82.70.Rr

Arrangement of cells in Voronoi tesselations of monosize packing of discs

Abstract We describe a two-dimensional mosaic obtained by the Voronoi tesselation of a monosize assembly of discs at different packing fractions. The experimental device (hard discs moving on an air table) produces, for every concentration of the discs, a succession of mosaics in statistical equilibrium, which constitutes a statistical ensemble. This ensemble is large enough for fluctuations from the most probable distributions to be negligible. Both topological and metric properties show deviations from those of a totally random mosaic. These deviations can be ascribed to steric exclusions. In particular, distributions of the numbers of sides, of the perimeters and of the areas of the polygons differ from those observed in biological celi assemblies. The Aboav law holds, but with a slope which can be as Iow as 4–66. The Lewis law is obeyed only for smali packing fractions. The variance of the distribution of the number n of polygon sides is a universal function of p 6, the probability of finding a six-si...

The topological structure of 2D disordered cellular systems

Abstract:We analyze the structure of two dimensional disordered cellular systems generated by extensive computer simulations. These cellular structures are studied as topological trees rooted on a central cell or as closed shells arranged concentrically around a germ cell. We single out the most significant parameters that characterize statistically the organization of these patterns. Universality and specificity in disordered cellular structures are discussed.

Fragment-size distribution in disintegration by maximum-entropy formalism

Abstract By maximizing the information entropy under appropriate constraints, we obtain the distribution of fragment sizes when either the whole or part of the volume of a material disintegrates as the result of some dynamic process (for example, blasting, impact). The energy constraint involves a realistic fragment energy to which one adds, when the fragmented volume is fixed, a volume constraint. The prior probability distribution has been chosen to be a uniform function of the linear fragment size. Agreeing with a broad range of experiments, the fragment number distribution derived in this work follows an inverse power law, a −θ (a is linear in fragment size, θ∼2–5), except for large sizes when it decreases exponentially. The weight distribution is maximum at (5γ/ρe2)1/3 (γ is the surface energy density, ρ is the mass density and eέ is the volumetric dilution rate) in agreement with several experiments and Gradys prediction, and is independent of the stored elastic energy.

On the structure of random tissues or froths, and their evolution

AbstractVon Neumanns law, governing the evolution of two-dimensional soap bubble froths, is shown to be a direct consequence of the structural law of Lewis, and both express mathematically the most probable filling of space. Von Neumanns and Lewiss laws can therefore be generalized to a considerable variety of froths, tissues, metallurgical aggregates, etc., in two and three dimensions. Moreover, a hitherto arbitrary parameter in Lewiss law measures ageing of the structure

Size‐distribution in sudden breakage by the use of entropy maximization

We derive the number and weight distributions of fragment pieces (of linear size a) that are obtained in a sudden fragmentation process. The information‐entropic derivation is based on a fragment energy e(a) which incorporates the physical mechanisms and material properties responsible for breakage. For not too large fragments the number distribution follows a power law a−Θ, where Θ is between 2 and 5, depending on the size range and circumstances of the breakage. Our results are compared with other theories of size distributions.