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GEOMETRY AND FLUCTUATIONS OF SURFACES
N. Rivier
To cite this version:
N. Rivier. GEOMETRY AND FLUCTUATIONS OF SURFACES. Journal de Physique Colloques, 1990, 51 (C7), pp.C7-309-C7-317. �10.1051/jphyscol:1990731�. �jpa-00231130�
COLLOQUE DE PHYSIQUE
Colloque C7, supplkment au n023, Tome 51, ler dkcembre 1990
GEOMETRY AND FLUCTUATIONS O F SURFACES
N. RIVIER
Blackett Laboratory, Imperial College, GB-London SW7 2BZ, Great-Britain
Resume - On d&crit les surfaces comme des mousses bidimensionelles aleatoires, duales de la representation en "filet de pCcheura'. GeomCtrie et fluctuations sont dues a des transformations topologiques ClCmentaires, qui sont aussi les "collisions"
responsables de leur Cquilibre statistique. Ce dernier est caract&risb par quelques relations observables (Aboav. Lewis). La surface et s a dynamique peuvent Ctre representees comme u n probleme a N corps avec interactions a courte portee (modele de H. Telley). Les N=C corps sont des paraboloides attaches aux C cellules de la mousse, mais dans un espace a une dimension de plus que l'espace physique.
Les transformations topologiques ClCmentaires correspondent alors a des mouvements simples et orthogonaux de ces paraboloides. Le modele de Telley a des syrnetries remarquables, en particulier la sterkologie (il est identique a s a propre coupure). I1 dCcrit aussi parfaitement l'evolution de diverses mousses naturelles (savon, frittage de mosaiques polycristallines, croissance de tissus biologiques ou dCposition de films amorphes). I1 autorise enfin des fluctuations locales de courbure, allant jusqu'a u n changement du genre de la surface.
Abstract - Surfaces are described a s a two-dimensional random froth. the dual of the
"fisherman's net". Its geometry and fluctuations are seen a s local, elementary topological transformations (dissociation and motion of dislocations and
disclinations). These transformations are the "collisions" responsible for statistical equilibrium of the structure, characterized by observable relations (Aboav. Lewis).
The surface and its dynamics can be represented a s a many-body problem with short-ranged interactions (Telley). The discrete bodies are paraboloids attached to the cells of the froth, with one additional degree of freedom beside their position in physical space. Elementary topological transformations are caused by simple and orthogonal motions of the bodies. Telley's model has remarkable syrnmetries, notably stereology (it is identical to its cut). It also describes quantitatively the evolution of many natural froths (soaps, sintering of polycrystalline mosaics, growth of biological tissues, deposition of amorphous films). As a Hamiltonian, it allows for local curvature fluctuations, leading to change of genus.
1 - INTRODUCTION
This paper gives a topological and metric description of a fluid surface, of its fluctuations and of its dynamics. This is a representation intermediate between the continuous Hamiltonian of Helfrich's (in which fluctuations in the Gaussian curvature are not penalized a s long a s the surface retains constant Euler-PoincarC characteristic X , or constant genus) / l / , and purely topological triangulations such a s the fisherman's net /2/.
We shall discuss first a topological representation of the surface as a random froth in two dimensions (2D). that is a space-filling cellular complex made of C cells bounded by E edges and V vertices. There are many natural examples of such a froth, whether physical (soap froth, metallurgical grain aggregate, etc.) or biological (tissues) /3/. At first sight.
these are all indistinguishable structurally. However, their evolution seems different: Soap froths and grain aggregates coarsen (large cells eat u p small ones), while biological tissues have steady state growth. These properties, characteristic of the statistical ensemble whose representative is the froth, can be explained by the fact that the structures are in statistical equilibrium, and that the "collisions" responsible for this equilibrium and for the structural disorder are local. elementary topological transformations.
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1990731
C7-3 10 COLLOQUE DE PHYSIQUE
Then, following Gaultier /4/, Laguerre /5/ and Telley /6/, we shall describe the froth a s a many-body problem, with discrete bodies (the cells) in a space with one additional dimension, i.e. with 3C degrees of freedom. This will enable us to compute simply the energy, and the forces on a given cell, thereby grafting a quantitative Hamiltonian and a metric on the topological froth. The construction can easily be generalized to space of arbitrary dimensions. The Laguerre froth has surprising symmetries: The (D-1)- dimensional cut of a D-dimensional Laguerre froth is also a Laguerre froth (stereological symmetry), exactly a s metallurgists had hoped that their polycrystalline aggregates cuts would be representative of the invisible bulk material. The froth also evolves a s a power law, whether in coarsening or in the steady state of biological tissues (local scale invariance).
The Laguerre froth is even self-similar / 6 / .
2 - TOPOLOGICAL REPRESENTATION OF A SURFACE
Consider a 2D cellular froth (random space-filling cellular structure). Ra~domness imposes (see below) z=3 edges incident on each vertex. Its topological dual (V*=C, E*=E, C*=V) is the fisherman's net, a triangulation (since it has exactly n*=3 (=z. by duality) edges bordering any cell C*) which is a classical topological representation of the surface, already discussed elsewhere in this conference /2/. Alternatively, the froth is the pore space of the close-packing of the V* vertices (disks, say) of the fisherman's net /3,7/.
Why triangulate a surface ? Because it offers an "engineering solution" to its structure and topology /B/. Because tt introduces automatically the notion of local topological defects which are mobile. And because it enables the Gaussian curvature to fluctuate locally
-
at a cost in energy - in a globally flat surface (e.g. the gut or an egg carton) or a surface of k e d genus. I t s dual is a topological froth. which is relevant for several reasons, the main ones being that it is the obvious representative of a statistical ensemble, in statistical equilibrium (detailed balance is here a local balance under elementary topological transformations). It has also a manifest local scale invariance, which, in 2D can be translated into conformal invariance (integrable Weyl geometry / g / ) , with its well-documented richness /10/. It can also be represented a s a many-body problem, involving a discrete set of objects behaving collectively. Finally, for the readers with botanical interests (and a referee to Nature) who may be tempted to ask why should one represent cells as (topological) polygons. "...there are aspects of tissue geometry so obvious that they can hardly escape the attention of any person who seriously considers the question a t all. The appreciation that cells are polyhedra came with the very first histological report ever published". (Ref./l l / , p.7-8).The key actors making the froth or triangulation dynamical representations instead of static pictures frozen once and for all, are elementary topological transformations, i.e. local topological fluctuations.
There are two, and only two types of elementary topological transformations in 2D /3/: neighbour switching (Tl) and cell disappearance (T2) and their inverses. (Fig.1).
Mitosis (cell division, the topological agent of growth in biological tissues) is the inverse (T2)-1, possibly composed with a few Tl's. Note that T l has a critical point with vertex coordination z=4 (Four-Corner Boundary). It is neither topologically nor dynamically stable.
and occurs with negligible probability in a random froth which has z=3: A small fluctuation either way splits the 4-coordinated vertex into two z=3 vertices. It is also easy to see that a z=4 vertex is unstable dynamically (e.g. by adding interface tensions vectorially. Stability requires 1200 between interfaces since cos 600= 1 /2).
Elementary topological transformations (ETT) act in three ways on the shape of the surface:
1) They change the topological character of the neighbouring cells, thereby creating local topological defects. If applied globally throughout the surface, they change its topological character: Bouligand /12/ shows here how one can transform a P- into a F- surface. and then into the gyroid (C-). by successive Tl's on the labyrinths. (In the P- surface. the vertices of the labyrinth are z=6, 3D critical points, which split by T1 into pairs of stable z=4 vertices and make the F-surface. But these z=4 vertices can then be regarded as 2D critical points, to be split by another T1 into normal z=3 vertices in a multiply- foliated 2D representation of the gyroid).
2) They make defects move (climb, glide) in response to external forces (evolution in biological tissues, applied shear, etc.).
3) They act as "collisions" to establish randomness, statistical equilibrium. and make the froth as the most probable (maximum entropy) archetype of a statistical ensemble
(characterized by "random avoidance of the niceties of adjustment", i.e. of critical points /13/).
Let us now discuss these points in some detail.
1) An elementary topological transformation changes the topological character of the cells involved, the number of their sides (Fig. l). Euler's relation, a topological conservation law.
where ~ = 0 ( 1 ) for a planar surface (X is negative and of order V for an infinite. periodic minimal surface), together with valencies relations, zV=2E, <n>C=2E (each edge joins 2.
z=3-valent vertices. and separates 2 cells, <n>-sided on average), imply that
<n> = 6 (planar surface)
<n> > 6 (IPMS)
Cells are hexagonal on average, for a large, planar froth (Gal). Also,
cn>C = 2E = 3V, (3)
so that cells are the least numerous topological elements, which makes them good candidates as independent degrees of freedom. For a planar surface, the number of independent degrees of freedom is 3C=E, which is interesting a s a representation of brittle fracture (see 54).
E'IT change the number of sides of the cells involved in the transformation. A 5- (resp. 7-) sided cell is a positive (negative) disclination (A pentagon is produced in a hexagonal lattice by cutting out a 27c/5 wedge and reglueing. The plane buckles into a cone.
and the pentagon is a source of positive curvature. Similarly, a heptagon is produced by adding a wedge, and is a source of negative curvature. The plane buckles into a saddle).
A dipole pentagon-heptagon is a (topological) dislocation (Fig.2) (conventionally imagined as produced by cutting out a semi-infinite rectangular strip. and reglueing. The plane remains flat (uncurved), but is has a step. The source of the step is the dislocation).
Fig. 1 - Elementary topological transformations in 2D: T1 (neighbour switching, left) and T2 (cell disappearance, right).
Fig. 2 - A pair of 5- and 7-sided cells is a topological dislocation. generated (for example) by cell division.
Fig. 4 - Climb of dislocation by successive cellular divisions.
COLLOQUE DE PHYSIQUE
Fig. 3
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Creation and glide of a pair of dislocations under shear.Fig. 5 - Growth of an appendix by successive mitoses. induced by growth or frustration.
21 Dislocations play all the parts which we learn in solid state courses and a few more, but do it locally, without cut or need for shoving material in from infinity (which should reassure those concerned with their intestinal growth). (a) A single T1 in a hexagonal froth generates a pair of dislocations, which glide apart upon applying shear stress, by successive T1, i.e., by a purely local process (Fig.3). (b) Similarly, one single cell division (Fig.4) generates a pair of dislocations. Further divisions in the neighbouring cells (Fig.4). make the two dislocations climb away from each other like a defective zipper, leaving behind a layer of new cells. Again, this is a mean of adding material (unlike the traditional cutting scenario above). It is the mechanism by which our intestine grows. (c) Successive mitoses can induce fluctuations in Gaussian curvature, as in Fig.5, which illustrates the growth of a n appendix. The mitoses may be induced by external forces, such a s growth in a tissue, or frustration in amphiphilic surfaces. In the latter, the fluctuations in Gaussian curvature induced by the process of Fig.5 will eventually provoke a transition between laminar and cubic phases / 16/, and a change of genus.
3) E'IT are local transformations, which. like collisions in gases, keep the cellular network in statistical equilibrium and guarantee its randomness (by maximizing the entropy /3.17/). This is done a t three levels. (i) Detailed, or rather local balance: The mean local structure and topological correlations are invariant under any ETT. The observable manifestation of this local balance is given by Aboav's relation (l) / 18.17/. (ii) The structure can then be in statistical equilibrium, and its structural equation of state (the pV=NkT of statistical crystallography) is a linear relation between cell shape and average size.
discovered empirically by Lewis in biological tissues / 19/.
Here. An is the average area of n-sided cells. AtOt is the total area of the froth. and h is an indeterminate multiplier, which describes the evolution of the froth. (iii) The structure evolves slowly, under external forces, while remaining in statistical equilibrium. There are two modes of evolution; a) coarsening, where dh/dt=cst>O (von Neumann evolution) and the multiplier h is proportional to the time, and, b) steady state in biological tissues, a n interplay of steady growth and discrete mitoses, where h is a constant characteristic of the 'Aboav's relation measures topological correlations, namely the average topological character (number of sides) m(n) of the neighbours to cell n,
n m(n) = 5n
+
(6+p2) ; p2 = <(n-<n>)2>.
steady state. Finally. ETT establish the froth as the archetype of a statistical ensemble containing soap. B&nard cells, biological tissues, cracked mud, etc. /3/, a t all stages of evolution (if any), as long a s it is not biased by external constraints.
3 - THE FROTH AS 'A MANY-BODY PROBLEM
The aim is to simulate the structure of the froth and its evolution by some "molecular dynamics" of discrete bodies. At first sight, this programme seems too ambitious, since every element of the froth depends on all the others, and a continuous description / l / may be the best we can hope for, even though it is inappropriate to describe large fluctuations.
coarsening. or shape transformations. We shall see that a discrete description of the froth with local forces is indeed possible. It relies on a simple representation, due to H. Telley in 1989, of a geometrical partition of space (Laguerre or radical froth) invented early in the 19th. century /4/. but forgotten until the beginning of the 1980, when it was uncovered by crystallographers /21,22/ and computer programmers /5.6/, apparently independently.
The question is not quite "how many bodies to have a problem?" /20/, but how many degrees of freedom ? (answer: (D+l)C, for C cells covering D-dimensional space). What are the bodies ? (the cells). What are the forces ? (vectors in (D+l)-dimensional space). We shall proceed in three steps: (i) represent a topological froth, (ii) capable of supporting ETT of both types. And only then. (iii) compute the energy. the forces, thus an Hamiltonian for the froth, which can be compared to the classic, but continuous Helfrich Hamiltonian.
The simplest way to generate a space-filling, random froth, is by Voronoi construction /3/. One begins with a Poisson distribution of points, which serve as seeds for *e cells. A Voronoi cell contains all points in space nearest to its seed. One obtains a froth (because the perpendicular bisectors of a triangle formed by three seeds are concurrent) very simply, but it has an unrealistic structure (the cells are very anisotropic). Worse for our purpose is the fact that the number of seeds, hence of cells, is fixed from the beginning, so there can be no T2, no coarsening or mitosis, and no evolution (only relaxation).
A simple generalization of perpendicular bisector produces a froth which is realistic and capable of evolution /6/. Consider circles instead of points as seeds, and define the distance d(X.T) of a point X to a circle (hypersphere in D>2 dimension) T(r,xo) as the length of the tangent to F through X (square root of the power of X with respect to circle F),
(Laguerre pointed out that the power of a point to a circle is the square of a distance /5,6/).
The locus of points a t equal distance between two circles is a straight perpendicular line, the radical axis (hyperplane if Dz-2). which generalizes perpendicular bisector to unequal circles. Like the perpendicular bisectors in a triangle, the three radical axes of three circles are concurrent (because the radical axis is an equivalence relation between two circles). They are the interfaces of a (radical or Laguerre) froth. The larger the circle seed, the larger its corresponding cell in the froth (2). Laguerre froth reduces to Voronoi's if all
&the seed circles are equal. Cell shapes are realistic already with little fluctuations in circle radii. Cell disappearance or division is easily accommodated. as we shall see. Generali- zation to 3 or more dimension is straightforward, a s is the analysis of a section of the froth (a Laguerre froth itself), important in the analysis of 3D polycrystalline aggregates in metallurgy (stereology)
.
Note that. while efficient algorithms exist for generating Laguerre froths on the plane /5.23/ or on any finite convex container, they are yet to be developed for froths on a torus (periodic boundary conditions) / 6 / . The dynamical evolution of the froth, on the other hand, is much more manageable than its construction /6/.
The radical axis between two circles of radii r l and r2 depends on r12-r22 only. This remark gave Telley the crucial idea of representing seeds as identical paraboloids (umbrellas) in one extra dimension, which intersect the physical space a s the Laguerre circles /6/. The altitude of the physical (hyper)plane is fixed but arbitrary. The umbrellas are the bodies of our problem, each specified by the 3 coordinates q=(x.z) of their apex.
q h e radical froth has been introduced into crystallography as a generalization of the Wigner-Seitz cell or Wirkungsbereich by Fisher and Koch /21/, and exploited in /22/.
C7-314 COLLOQUE DE PHYSIQUE
Here, X is the coordinate of the circle's centre in the 2-dimensional physical space, and z.
the height above physical space, measures the circle radius, z-r2, or, roughly, the cell size.
Now for the EIT. The set of seeds looks like a Yosemite or Rio de Janeiro mountain profile (Fig.6). A paraboloid below the horizon (a seed too small) does not generate any cell in the froth. Conversely, a paraboloid which is too high (a seed too large) obscures smaller paraboloids nearby and gobbles up their representative cell. So, a T2 topological transformation occurs when a paraboloid is pushed below the horizon whereas, when a new paraboloid rises above the horizon, it divides the cell containing its apex. T1 transfor- mations are produced by moving nearby seeds horizontally rather than vertically. ETT are therefore naturally induced by orthogonal motions of the bodies, T1 by horizontal (X)
motion, T2 or cell division by vertical (z) motion.
Fig. 6 - The "Yosemite" horizon of paraboloids. The horizontal axis represents the physical plane. qa is below the horizon and does not generate a cell in the froth. From Telley /6/.
There is a geometrical (and not only topological, as in 52) duality between cell seed paraboloids (umbrellas) and vertices of Laguerre froth, i.e., between cells and vertices. or between the vertices of the fisherman's net and those of the froth. Vertices can also be represented by the same paraboloid as the cell seeds, but inverted (opening upwards). The bottom q*=(xf.z*) of the inverted paraboloid lies on the intersection of its (3) adjacent paraboloids. The converse is also true, but more stringent: the umbrella apex q lies on the intersection of its 6 adjacent, inverted paraboloids. This non-random coincidence indicates clearly the restrictions necessary to produce a froth / 6 / , and confirms the fact that the independent bodies are the cells (they are the least numerous species constituting he froth (cf. eq.(3)). Any set of C seeds (bodies) produces a Laguerre froth, specified by 3C coordinates (x.z) in 2 dimensions. But the froth has V=2C vertices, which would have required 4C coordinates if chosen independently. Note that selecting a n independent orientation for every interface (E=3C) also gives the required number of degrees of freedom.
Physically, edge orientation (crack) is the relevant degree of freedom in fast brittle fragmentation / l4/.
Duality enables us to compute easily the driving forces on the cell paraboloids, if the energy E is proportional to the total interfacial length, as is clearly the case in soap froths (surface tension) and in metallurgical grain aggregates (grain boundary energy). Motion (q) of a seed paraboloid induces, geometrically, motion (q*) in the vertex paraboloids nearby, hence changes the interfacial length, thus the energy of the froth. This produces a force
on q and generates its dynamics. fill these motions are local. Neighbouring seeds are fixed.
so that only the n-sided central cell q is deformed, with its vertices sliding along the fixed directions of the n interfaces incident on the central cell (n degrees of freedom) to minimize the energy cost of moving q. Notably, vertical forces make a n-sided cell grow if n>6 and shrink if 1x6; the froth coarsens. Telley /6/ has simulated the coarsening of polycrystalline mosaics. The results of his simulations look realistic. After long times, he obtains "normal" growth (self-similar distributions of grain shapes and sizes, power-law growth of the linear grain dimension <L>-t* with mean-field /24/ exponent n=1/2).
We also obtain several symmetries hidden in the froth. First, coarsening of the froth while remaining in statistical equilibrium, is a local dilatation symmetry, a n example of conformal invariance. Second, a (D-l) dimensional cut of a D-dimensional Laguerre froth is also a Laguerre froth (invariance by stereology) (Fig.7). The cut is also a n example of the
"strip" or "cut-and-projection" method introduced recently to describe quasicrystals and incommensurate structures /25/: The froth is the "atomic surface", of current interest /26/. But here, the full symmetry survives intact the cut and projection! The acceptance region is local (compact) and the width of the strip is finite, albeit random.
Fig. 7
-
Stereology: Laguerre froth (dotted lines). dual of the fisherman's net (full lines) with paraboloid seeds (circles). One seed here does not generate a cell (it lies below the horizon in Fig.6). Cut (thick line) of a Laguerre froth is a Laguerre froth (symmetry-preserving cut- and-projection). The strip (shaded), made of cut cells. has random width.Remarks: (i) Irnai et al. /5/ were the Arst to hint a t a stereological symmetry. when they noted that a cut Voronoi froth is a Laguerre froth. The full symmetry is a new observation a s far as I am aware. (ii) Duality between accepted cell and projected seed is well known in quasicrystals as rivalry between strip and atomic surface methods. (iii) If one concentrates exclusively on the paraboloids, there is no need for a strip of finite width:
acceptance is defined by the horizon of Fig.6. (iv) In a random structure, any cut is irrational.
Laguerre's froth, with its paraboloid seeds, constitutes therefore an excellent model for understanding and simulating the local dynamics of a surface. It identifies the indepen- dent degrees of freedom (cells in one extra dimension), generates elementary topological transformation by simple, local motion, and computes the forces by a local geometrical algorithm. Interfaces are straight, and dynamics is obtained directly from the forces.
4
-
SOME APPLICATIONSTelley's construction can be generalized to different types of froth, corresponding to different physical situations. Note first that cones or cylinders also intersect the physical plane a s circles. But cylinders would never disappear or divide, and the intersection of two vertical cones is a hyperbola which is not perpendicular to the physical plane. The froth would then depend on the absolute level of the physical plane. Paraboloids, on the other hand. have vertical, parabolic intersections, whose projection on the physical floor is the Laguerre froth itself.
Instead of computing the forces and evolution by eq.(6), we can assume specific models for nucleation and growth of a circular grain.
1. Avrarni-Johnson-Mehl d e l of grain nucleation and growth The grain i is a circle of radius ri(t)=a(t-ti), which grows at the expense of a liquid matrix. The nucleation times {tJ are random. Where two circles meet, growth stops at that point. All growth stops when there is no liquid left and one remains with a frozen, polycrystalline mosaic /27/. It is easy to see that the interfaces between grains will be arcs of hyperbolas, as observed in calcospherites or tortoise shells (ref./28/, Figs 293-296). This is not a Laguerre froth.
C7-316 COLLOQUE DE PHYSIQUE
2. Froths with steady rate of growth and random nucleation time. Suppose now that growth does not stop when two circles meet. but that the circular grains generate a Laguerre froth. Their associated paraboloids rise a s dzi(t)=~ri(t)=~a(t-ti). The evolution of the froth is then controlled, as we have seen, by
where d=(tj-ti). The cellular network coarsens (larger cells (early ti) have their paraboloids rising faster than smaller cells (later ti)). The interfaces are straight (it is a Laguerre froth).
This seems to describe the model studied by Tang et at /29/ for the growth of amorphous metallic films by sputter deposition: A fluctuating initial surface evolves according to Huygens' principle, and one observes domed columns separated by caustics (cusplike singularities). like our paraboloids separated by Laguerre interfaces. There, not only the caustic froth, but the whole profile of Fig.6 is physically represented by the film. The froth coarsens with exponent n. However, there are now two contributions to growth: uniform rise of zi(t), as before, and ~Z=(t-ti)-q decreasing with time as the radius of curvature of the dome increases through normal growth. Self-similarity is obtained with n=(l+q)/2>1/2 (cf.
eq.(8)).
In fact, any power law growth ri(t)=a(t-ti)* in a Laguerre froth yields
i.e. coarsening for n>1/2, and refining for n<1/2 (the spread in z narrows with time, some paraboloids rise over the horizon and divide cells, a s happens in biological tissues. Their steady state is a balance between discrete refining by cellular division and continuous growth). The Laguerre froth remains identical (up to a length scale, corresponding to the level of the physical floor) for n=O (no evolution) and for the mean field exponent n=1/2 (self-similarity).
3. Soap froths. Construct from circle seeds a real soap froth, governed by Laplace's law Ap=2o/R. The interfaces are circle arcs, with curvature l/R=f(ri)-f(rj). Here o is the surface tension of the interfacial film, pi is the pressure inside bubble i, and f is any mono- tonically decreasing function of the radius ri of its seed. In order to make a froth, it is sufficient to define a distance (an interface is then the locus of points equidistant to both seeds). Laguerre's distance must be generalized (it gives straight interfaces). The (D+l)C independent degrees of freedom are still attached to the cells.
4. Stress, cracking patterns. "Goose flesh" is a familiar example of fluctuations on a surface. Uniform muscular contraction (in order to supply heat) results in a space partition by cells of roughly the same size, a packing of goose pimples (3). Similar patterns in an uniformly stressed material can be observed in cracked mud or basalt, a s long as the tip of the crack propagates more slowly than the stress is relaxed around it (4)/14/. Again, the regions of higher strain appear to repel each other, like the umbrellas of the Laguerre froth.
This observation is a direct consequence of Griffifth's /31/ energy accounting between interfacial energy cost due to the surface tension of the lips of the crack (work of fracture), and volumic energy credit due to stress relaxation about the tip of the crack.
I have benefited from conversations with R.F. Cook, H. Meinhardt. G. Oster. H. Gruler;
H.Telley, K. Strandburg. M. Peshkin, D. Weaire; J. Charvolin. J.F. Sadoc, R. Bruinsma; J.
Finney. D. Huson, and P. McMullin.
3I am grateful to H. Meimhardt for suggesting this example. I am however, responsible for any possible biological inaccuracy or oversimplification.
4Morphologically and dynamically, one distinguishes two regimes: The Up of the crack propagates either faster than the stress is relaxed ('T' junctions morphology between primary (horizontal bar of the 'T') and secondary (vertical) cracks, the degree of freedom is the orientation of the crack), or it goes more slowly (cellular network of cracks with 1200 junctions). In the latter case. the triangles of the fisherman's net are relaxed regions which tile the plane /14/. Then, the network of cracks may well be incomplete, either partially (showing some "damaged" cells about to divide) or totally (disconnected tripods, as can be seen in the early stages of cracking /30/).
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