Nicolas Rivier

Institut de Physique et Chimie des Matériaux de Strasbourg | IPCMS · DMONS

Phyllotaxis describes the arrangement of florets, scales or leaves in composite flowers or plants (daisy, aster, sunflower, pinecone, pineapple). As a structure, it is a geometrical foam, the most homogeneous and densest covering of a large disk by Voronoi cells (the florets), constructed by a simple algorithm: Points placed regularly on a generati...

A close packed organization with circular symmetry of a large number of small discs on a plane is obtained when the centres of the discs are distributed according to the algorithm of phyllotaxis. We study here the distributions obtained on surfaces of constant Gaussian curvatures, positive for the sphere or negative for the hyperbolic plane. We exa...

Dry granular matter, with infinite tangential friction, is modeled as a connected graph of grains linked by purely repulsive contacts. The degrees of freedom of a grain are non-slip rotation on, and disconnection from another. The material stability under shear (jamming) is ensured by odd circuits of grains in contact that prevent the grains from r...

Phyllotaxis, the search for the most homogeneous and dense organizations of small discs inside a large circular domain, was first developed to analyse arrangements of leaves or florets in plants. It has since become an object of study not only in botany, but also in mathematics, computer simulations and physics. Although the mathematical solution i...

Spiral lattices, beautifully illustrated in composite flowers such as daisies and discussed in a recent letter by Bursill, Peng and Fan, have concentric crystalline grains separated by quasicrystalline grain boundaries. They are generated by a simple algorithm parametrized by one golden (or noble) angle. The topology of the patterns and their cryst...

Dry granular matter, with infinite tangential friction, is modeled as a connected graph ofgrains linked by purely repulsive contacts. The degrees of freedom of a grain are non-slip rotationon, and disconnection from another. The material stability under shear (jamming) is ensured by oddcircuits of grains in contact that prevent the grains from rolli...

Dry granular matter, with infinite tangential friction, is modelled as a graph of grains linked by purely repulsive contacts. Its stability (jamming) is insured by odd circuits that prevent the grains from rolling on each other. A topological dynamical matrix is associated with the graph; it has a spectrum of low-energy excitations characteristic o...

Dry granular matter is modelled as a graph of grains linked by purely repulsive contacts. Its stability (jamming) is insured by odd circuits that prevent the grains from rolling on each other. A topological dynamical matrix is associated with the graph; it has a spectrum of low-energy excitations characteristic of dry, disordered granular matter. I...

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 Giant's Causeway.

Wetting with µm-sized Pb droplets on thin polycrystalline films of decagonal Al13Co4 is reported. The films were prepared under high vacuum conditions in order to have Pb droplets lying on a clean surface. The method used is sequential deposition and annealing of specific stackings of Al and Co layers of nanometric thicknesses. A 300 nm thick Pb sl...

We report the formation of stable two-dimensional clusters consisting of long-range-interacting colloidal particles with predefined magnetic moments. The symmetry and arrangement of the particles within the cluster are imposed by the magnetic frustration. By satisfying the criteria of stability, a series of magic number clusters is formed. The magi...

We show by decurving techniques that there are two possible atomic arrangements for the tetrahedrally close-packed Mg32(Zn, Al)49 structure, which have the same elastic energy to first order. One is the structure of Bergman, Waugh and Pauling, which is b.c.c. and has one single network of Frank-Kasper lines and 15-coordinated atoms. The other is si...

An ordered soap froth is created in a cylindrical tube. By dilating (up to 500%) and compressing the froth, we observe sequences of structural transitions. After dilation followed by compression, the froth recovers its initial structure, but the sequences of intermediate structures upon dilation and upon compression are different. This topological...

A soap froth with bubbles of equal size in a cylindrical tube tends to crystallise. We exhibit, experimentally and by simulation, two sequences of transitions between different crystalline structures. Each transition is induced by a single dislocation. The two different sequences of transitions are associated with different motions of the dislo...

We study the assembly of collagen molecules of the so-called fibrils, long, periodic bundle of finite collagen molecules. The appearance of three-dimensional periodic structures leads to very interesting geometrical questions similar to the problems of classification textures and defects in liquid crystals (smectics and discotics), lattices of defe...

A granular material is a network of fixed length edges linking two grains in contact, and flexible hinges if the grains can rotate without sliding on each other. The network is then dynamically unfrustrated: it can deform freely under shear and behaves like a dry fluid. A sufficient condition for non-frustration in 3D is that all circuits of grains...

The dynamics of two-dimensional cellular networks is written in terms of coupled population equations, which describe how the population of s-sided cells is affected by cell division and disappearance. In these equations the effect of the rest of the foam on the disappearing or dividing cell is treated as a local mean field. Under not too restricti...

Collagen is the principal constituent of extra-cellular, connective tissue. It is a tightly-packed but flexible bundle of proteins that constitutes a material. The collagen molecule is a triple helix made of three, intertwined polypeptide chains, associated with and its rational convergents. Accordingly, the collagen fibril, a long, periodic bundle...

Wetting of micron-sized Pb droplets on thin polycrystalline films of decagonal Al13Co4 and cubic crystalline AlCo phases is reported. The sample preparation is crucial to having Pb droplets lying on a clean surface. Decagonal and cubic films were prepared under high vacuum conditions, by sequential deposition and annealing of specific stackings of...

This paper is concerned with elasticity and plasticity of two-dimensional cellular structures. The deformation of continuous media is defined by a mapping from the actual, deformed state of the material, into a reference (natural) state, where all elastic deformations have been relaxed. In two dimensions, the two states can be represented by a comp...

A granular material is a bearing if the grains can roll on each other without sliding. It is dynamically unfrustrated and can deform freely under shear. It behaves as dry quicksand (Lohse et al. 2004). The sufficient condition for non-frustration in a three-dimensional granular material is that all circuits of grains in contact are even. A granular...

The three-dimensional space-filling sphere bearing in which an arbitrary chosen sphere rotate around any axis was discussed. The filling procedure of sphere with unit radius was initialized by placing seven spheres on the vertices and center of regular octahedron inside the unit sphere. The bichromatic topology revealed that each loop of spheres in...

We present the first space-filling bearing in three dimensions. It is shown that a packing which contains only loops with even number of spheres can be constructed in a self-similar way and that it can act as a three dimensional bearing in which spheres can rotate without slip and with negligible torsion friction.

A foam is a space-filling cellular pattern, that can be decomposed into successive layers or strata. Each layer contains all cells at the same topological distance to an origin (cell, cluster of cells, or basal layer). The disorder of the underlying structure imposes a characteristic roughening of the layers. In this paper, stratifications are desc...

An extension of the Voronoi tessellation, the Laguerre polyhedral decomposition, is introduced and applied to the analysis of the packing geometry of amino-acids in folded proteins. This method considers an ensemble of points with different weights and therefore it is well suited for a geometrical analysis of a set of objects with a wide size distr...

Amorphous silicon and covalent glass are represented structurally and physically as a random, regular graph of degree 4. Odd circuits form extended structures, called odd loops or R-loops, the topological defects surviving the absence of translation and rotation symmetries in the material. Odd loops are responsible for the topological entropy froze...

Geometrically, foams or covalent graphs can be decomposed into successive layers or strata. Disorder of the underlying structure imposes a characteristic roughening of the layers. Our main results are hysteresis and convergence in the layer sequences. 1) If the direction of construction is reversed, the layers are different in the up and down seque...

The elasticity of continuous media with topological defects is described naturally by differential geometry, since it relates metric to strain. We construct a geometrical field theory, identifying disclinations, dislocations and extra-matter defects with the curvature, torsion and nonmetricity tensors, respectively. Connection and metric are given...

A foam can be decomposed into successive layers of cells at the same topological distance to an origin, which is either an arbitrary cell or a basal plane. The shape of these layers (profile and thickening) indicates the degree of randomness of the cellular pattern. To support this idea, we analyse the layer shapes in 2D rectangular models of foam....

Erratum of Europhys. Lett. 54 (1) p. 112-117 (2001)

The dynamics of two-dimensional cellular networks (foams) is written in terms of coupled rate equations, which describe how the population of s-sided cells is affected by cell disappearance or coalescence and division. In these equations, the effect of the rest of the foam in statistical equilibrium on the disappearing or dividing cell is treated a...

The resistivity of a dilute alloy due to localized spin fluctuations (LSF) occurring on the impurity orbitals is given by a universal function of the temperature. The resistivity is finite at T=0, then decreases with temperature as T2, T, lnT and T-1 successively. The T2 and T behaviour are in good agreement with the experimental data on AIMn. The...

A resistance minimum is derived from a general theory of Adkins and the author, for spin glasses with narrow band metallic hosts and sizeable valence differences between host and impurity, as observed in PtMn by Sarkissian and Taylor (see ibid., vol.4, L243 (1974)). The resistivity follows the general T3/2 law associated with spin glasses at low te...

The resistance anomaly discovered by Coles in RhFe and occurring in several similar dilute alloys is shown to be of the same nature as the Kondo anomaly. The resistivity is due to localized spin fluctuations (LSF) and is given by a universal function of the temperature, increasing as T2, T and in T to tend finally to the unitarity limit. The charac...

The “staircase” (2,1,1) foam structure in an ordered foam under expansion in a cylindrical tube and the structural transition to the “bamboo” (1,1,0) structure are studied experimentally, analytically, and numerically using the Surface Evolver minimization program. An analytical expression of the oblique interface in the (2,1,1) structure, a minima...

The static and dynamical properties of the molecular (two sites) polaron are investigated. The problem is reduced to that of a succession of electron hops in imaginary time, interacting with each other by phonon exchange. It can be solved analytically in the two regimes h(cross) omega 0/kT>>1 (self trapped) and h(cross) omega 0/kT<<1 (nearly free)...

A phenomenological Ginzburg-Landau free energy is introduced to describe mixed magnets or spin glasses with two order parameters, the bulk magnetisation and the spin glass order parameter. The susceptibility has a cusp at the onset of the spin glass phase. To the two order parameters are associated two coherence lengths and, depending on their rati...

The dynamics of a random ferromagnet with infinite-range exchange interactions and Gaussian distribution is studied. The Green function for the propagation of magnetic excitations is obtained in an exponential form which permits the use of techniques associated with problems involving random matrices. The magnetic properties of this system are obta...

The main features of the resistivity of spin glasses as a function of temperature, in particular the T3/2 dependence at low temperature, are explained in terms of the scattering of the conduction electrons by elementary excitations of the system which are diffusive in character.

A first-order phase transition is predicted for several superconducting sandwiches as a function of the temperature or of the thickness of the normal layer. It is due to the competition between two configurations for the order parameter, both necessary consequences of Ginzburg-Landau superconductivity. This is an extension and a justification of th...

The authors show that longitudinal magnetic excitations are responsible for an additional T3/2 temperature dependent term in the decrease of the magnetisation of amorphous ferromagnets at low temperatures. This provides a possible explanation for the discrepancy on the values for the spin wave stiffness as measured by magnetisation and neutron scat...

The resistivity of an amorphous ferromagnet is calculated in the multiple scattering approximation. The scattering at each magnetic site has a static component represented by a real parameter V which characterises the amount of disorder in the system and a dynamic component whose dominant contribution is the inelastic scattering of conduction elect...

Helices and dense packing of spherical objects are two closely related problems. The Boerdijk–Coxeter helix (B–C), a linear stacking of regular tetrahedra, is a very efficient solution to some close-packing problems, including protein folding. The structure of biological helices (α-helix, collagen) is determined chiefly by steric repulsion. Thus, m...

The electronic density of states (DOS) of 1D quasicrystals looks like that of a highly doped, p-type semiconductor, the Fermi level lies in the impurity band which consists of localized states. At low temperatures, the conductivity of an electronic structure is by variable-range hopping, as observed in icosahedral-Al-Pd-Re. The electronic structure...

In these lectures, scaling is taken as a symmetry, the symmetry of inflation. Quasicrystals [cf. C. Janot, Quasicrystals. A Primer, Oxford Univ. Press (1992; Zbl 0838.52023)] are real materials with inflation symmetry. Their generic physical properties (electrical resistivity, nonstick, nonwet surface, etc.) are caused directly by inflation symmetr...

The authors identify the topological or configurational degrees of freedom in dense liquids as continuous line defects, labelled by the two-element group Z2. These defects are absent in perfect solids, and yield a topological contribution to the latent heat of melting. The topological entropy is universal (for dense atomic liquids) and equals R ln...

We give a review of the different models developed recently that describe the renewal of the epidermis. These models, based on concepts borrowed from statistical mechanics, geometry and topology, shed new light on the understanding of the organization and the dynamics of the system. We discuss in detail a topological model of the dynamics of the in...

: Helices and dense packing of spherical objects are two closely related problems. For instance, the Boerdijk-Coxeter helix, which is obtained as a linear packing of regular tetrahedra, is a very efficient solution to some close-packing problems. The shapes of biological helices result from various kinds of interaction forces, including steric repu...

Glass has a complicated structure (a random, regular graph of degree 4), but very simple elementary excitations (decoupled tunnelling modes between two valleys degenerate in energy). There is one tunnelling mode per odd loop (a necklace through odd rings in the network). Tunnelling is imposed by gauge invariance of the structure, the symmetry of di...

Inside a cylindrical tube, soap bubbles of equal size pack as ordered, crystallized structures. On the surface of the tube is a hexagonal honeycomb of bubbles. If each bubble in the honeycomb is labeled with an integer i in order of increasing altitude, then the six surrounding bubbles are specified by the integers i ± k, i ± l, and i ± m. The inde...

The statistical properties of two-dimensional, space-filling random cellular structures (foams, or their dual, random triangulations) in statistical equilibrium are obtained by maximum entropy inference and topological simulations. We show by maximum entropy inference that for a broad class of foams (shell-structured, including three-sided cell inc...

Froth is a random partition of a D-dimensional space by cells. This assembly of cells obeys two fundamental laws: Euler's relation and the condition of maximum vertex figure, imposed by geometry and by topological stability, respectively. These two conditions generate a set of relations between the variables that fully characterize the system topol...

It is shown that Lewis's 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.

Benard-Marangoni convection, in containers with large aspect ratio, exhibits space-filling cellular structures, highly deformable, but crystallized. They contain dislocations and grain boundaries generated and moved by elementary topological transformations, and are subjected to a weak shear stress due to the earth's rotation. The cellular structur...

Magnetic spheres repel each other in two dimensions and form a periodic lattice, when their weak magnetic moments are aligned along a unique axis perpendicular to the lattice plane. When, in addition, the spheres are subjected to an external force such as gravity or magnetic field gradient, they constitute an ordered structure described by a confor...

A general and introductory survey of foams, emulsions and cellular materials. Foams and emulsions are illustrations of some fundamental concepts in statistical thermodynamics, rheology, elasticity and the physics and chemistry of divided media and interfaces. They also give rise to some of the most beautiful geometrical shapes and tilings, ordered...

These lectures are divided into two parts.

A foam is a random partition of space into cells. In two dimensions, it is a planar, regular graph, consisting of trivalent vertices, edges and cells. The number n of neighbours of a cell (which is equal to the number of its edges) is the only, local random topological variable of the problem. One looks for the most probable distribution p n , subj...

Two-dimensional foams are used to model the evolution and the steady state of biological tissues. When only cell division occurs, we deduce the mode of division simply from the stationary distribution of the number of sides per cells, by inverting a system of coupled rate equations. Comparisons with experimental data confirm the method. We then dis...

Glasses have a complicated structure (a random, regular graph of degree 4), but very simple elementary excitations (decoupled tunnelling modes between two valleys degenerate in energy, or nearly so). These facts can be explained by two straightforward applications of combinatorics. Tunnelling modes arise directly from the local invariance of the st...

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 org...

Using a topological approach, we study the dynamics of the basement membrane of the mammalian epidermis when basal cells detach or divide. A theoretical characterization of the steady state of the tissue, in very good agreement with experimental data, includes for the first time the division and the disappearance of cells in a two-dimensional rando...

The authors model the structure of space-filling disordered cellular systems. These systems are cellular networks with minimum incidence numbers (D+1 edges incident on a vertex in D-dimension). In the literature such systems are known as froths since the soap froth is the archetype of these structures. They present a method where the structure of f...

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 trans...

Bénard-Marangoni convection displays a two-dimensional (2D) hexagonal pattern. A topological analysis of these structures is presented. We describe the elementary topological transformations involved in such patterns (neighbor switching process, cell disappearance or creation, and cellular division) and the typical defects [pentagon-heptagon pair,...

A cellular structure is a compromise between best local packing and filling space without gaps or overlap. The fact that soap bubble froths or the human epidermis are and remain steadily disordered, demonstrates that this compromise is essential. Two alternative methods are presented: decurving progressively and at random a cellular structure initi...

A tissue is a geometrical, space-filling, random cellular network; it remains in this steady state while individual cells divide. Cell division is a local, elementary topological transformation which establishes statistical equilibrium of the structure. We describe the physical conditions to maintain stationary the epidermis (of mammals or of the c...

Random cellular structures (froths, foams, undifferentiated biological tissues) are in statistical equilibrium thanks to elementary local transformations. They form a statistical ensemble, with universal properties (structural equation of state, and distribution of cell shapes, up to priors). Notably, all natural random cellular structures in two d...

The 'molecular dynamics' of cells in the epidermis of mammals is presented by generalizing a model of grain growth in polycrystals or coarsening foams due to Telley. The tissue remains in statistical equilibrium in spite of and because of constant renewal through local elementary topological transformations, which are division (mitosis), neighbour...

A tissue is a geometrical, space-filling, random cellular network; it remains in this steady state while individual cells divide. Cell division (fragmentation) is a local, elementary topological transformation which establishes statistical equilibrium of the structure. Statistical equilibrium is characterized by observable relations (Lewis, Aboav)...

Investigation of the geometry of thermodynamic state space, based upon the differential geometric approach to parametric statistics developed by Chentsov [Statistical Decision Rules and Optimal Inference (Nauka, Moscow, 1972)], Efron [Ann. Stat. 3, 1189 (1975)], Amari [Ann. Stat. 10, 357 (1982)], and others, provides a deeper understanding of the m...

Tunneling modes, elementary excitations of a large class of glasses, are local dynamical signatures of disorder. There is one tunneling mode per odd loop (a necklace through odd rings in the network), decoupled from other loops, and tunneling is forced by gauge invariance of the structure, the symmetry of disorder. Absence of tunneling modes in tri...

What are (topological1) defects in matter ? Mathematically, they can be defined as singularities of a function Φ(x) associating a quantity, number, vector, etc., Φ∈M to any point x inside the material. Φ(x) represents the configuration of matter in the material. It is better to speak of topological entanglement instead of singularity, since we will...

Weaire and Phelan's recent discovery of a clathrate structure (that of β-tungsten) as an ordered minimal froth (space-filling assembly of polyhedral bubbles) with less surface per unit volume than Kelvin's b.c.c. arrangement of truncated octahedra suggests searching among other known tetrahedrally close-packed crystalline phases for a structure wit...

Ground-state and elementary excitations (tunnelling modes) in glass are obtained from an analysis of its symmetry: a local gauge invariance. Glass is represented as a discrete fibre bundle. The base space is a continuous random network with tetravalent silicon atoms as vertices and chemical bonds as edges. The connection is given by the elasticity...

Besides their industrial and gastronomic applications, foams are also beautiful scaffoldings of bubbles, apparently disordered yet uniform, and alive through the sudden rearrangement of a few bubbles. Through these structural transformations, the foam spends its long life in statistical equilibrium, a symbol of liberty, equality and disorder.

Froths are random cellular networks in statistical equilibrium under elementary topological transformations. Natural, experimental and mathematical froths have many common features, which are direct and observable consequences of filling space at random with cells. Thus, froths constitute a simple statistical ensemble, characterized by maximum entr...

Polydisperse froths obtained from assemblies of différent size discs exhibit an Sshaped Aboav's relation (between thé total number of sides of thé neighbours to an n-sides cell, and n), rather than thé linear relation found in monodisperse or natural froths. One show that this Sshape is a result of maximum entropy inférence, and a signature of poly...

Ground state and elementary excitations (tunnelling modes) in glass are obtained from an analysis of its symmetry, a local gauge invariance. The configuration of glass is represented as a discrete fiber bundle. The base space is a continuous random network, standard model of the structure of covalent glasses. The connection is determined naturally...

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 ensemb...

Coatings of quasicrystalline AlCuFe, deposited by high temperature pulverisation, have ideal properties (hardness, stability, thermal conductivity, non-toxicity) for a frying pan. Surprisingly, they are also non-stick. Why? Sticking is related to wetting, and wetting is prevented if the surface tension of the coating is low. Thermodynamically, a ro...

We discuss the possibility of inducing superconductivity by proximity effect, from a high-Tc superconductor into a simple normal metal such as Ag. The characteristic length is the coherence length of Ag, which is much larger than that of the high-T, superconductor, so that a sizeable gap (larger than 10 K) can be induced into Ag films 50 nm thick....

Polycrystalline grain aggregates are cellular networks filling space at random, like soap froths. Their structure (in statistical equilibrium) and evolution (steady state coarsening) are universal, due to local elementary topological transformations (ETT), which are the “collisions” responsible for statistical equilibrium. The structure and its evo...

Diverse cellular systems evolve to remarkably similar stationary states. We therefore have studied and simulated a purely topological model. We use a maximum-entropy argument to predict that the average number of {ital l}-sided cells adjacent to an {ital n}-sided cell, {ital M}{sub {ital l}}({ital n}), will be linear in {ital n}. One consequence is...

Brillouin scattering from two classic glass‐forming materials, ZnCl 2 and glycerol, reveals a maximum in the ratio (γ) of specific heats C p and C v as a function of temperature. We propose that the temperature at which the maximum in γ occurs in our materials may be indicative of a ‘‘liquid‐like’’ to ‘‘solid‐like’’ transformation. As such it may b...

Compositae (daisies, pinecones, asters, sunflowers) have a structure shown in Fig.1. Understanding this structure -one aspect of the field of phyllotaxis (leaf or floret arrangement)1–6- is a problem of crystallography: A surface is tiled with florets (the “atoms”) of roughly the same size, the majority of which are hexagonal (daisy) or rhombus-sha...

On décrit les surfaces comme des mousses bidimensionelles aléatoires. duales de la représentation en "filet de pêcheur". Géométrie et fluctuations sont dues à des transformations topologiques élémentaires, qui sont aussi les "collisions" responsables de leur équilibre statistique. Ce dernier est caractérisé par quelques relations observables (Aboav...