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Structure of Bénard convection cells, phyllotaxis and crystallography in cylindrical symmetry
N. Rivier, R. Occelli, J. Pantaloni, A. Lissowski
To cite this version:
N. Rivier, R. Occelli, J. Pantaloni, A. Lissowski. Structure of Bénard convection cells, phyl- lotaxis and crystallography in cylindrical symmetry. Journal de Physique, 1984, 45 (1), pp.49-63.
�10.1051/jphys:0198400450104900�. �jpa-00209739�
Structure of Bénard convection cells,
phyllotaxis and crystallography in cylindrical symmetry
N. Rivier (*+), R. Occelli (+), J. Pantaloni (+) and A. Lissowski (* ~) (+) Dynamique des Fluides, Université de Provence, 13000 Marseille, France (*) Blackett Laboratory, Imperial College, London SW7 2BZ, U.K.
(~) Dept. of Psychology, Polish Academy of Sciences, Warsaw, Poland
(Reçu le 20 mai 1983, révisé le 21 septembre, accepté le 26 septembre 1983)
Résumé. 2014 Ceci concerne la cristallographie à deux dimensions en symétrie cylindrique. Des défauts (cellules
non hexagonales) sont introduits nécessairement par la symétrie, et, dans les structures de Bénard, des cercles de glissement de dislocations servent à dissiper un faible cisaillement du à la rotation terrestre.
Des cercles de glissement apparaissent naturellement en phyllotaxie (arrangement des florets dans les fleurs
composées) et la structure des marguerites, ananas, etc., constitue la première étape de la construction de structures de Bénard. Toutes ces structures sont engendrées par un algorithme élémentaire de Théorie des Nombres. Elles sont auto-similaires et localement homogènes, engendrées par un seul nombre irrationnel 03BB. L’homogénéité
demande que 03BB soit un nombre Noble, et explique la prolifération des nombres de Fibonacci en phyllotaxie.
Une fois la structure construite, il est élémentaire de simuler et d’analyser sa fonte.
Abstract. 2014 This paper is concerned with crystallography in two spatial dimensions, in the presence of cylindrical symmetry. Defects (non-hexagonal cells) are imposed by the symmetry and glide circles are necessary to dissipate
a weak, steady shear associated with the earth’s rotation.
Glide circles occur naturally in phyllotaxis (leaf or floret arrangement), and the structure of daisies represents the first stage of the construction of Bénard patterns. Both types of structures can be generated by an elementary algorithm, which constitutes a physical application of number theory. The structures are self-similar and locally homogeneous. They are generated by a single, irrational number 03BB. Homogeneity imposes 03BB to be a Noble number
and explains the pervasiveness of Fibonacci numbers in phyllotaxis.
Once the ideal structure is constructed, melting can be simulated and analysed.
Classification
Physics Abstracts
47.25Q - 61.00 - 87.45
1. Introduction.
Classical crystallography consists of an enumeration of the infinite, space-filling patterns made by repeti-
tion of identical cells. One then introduces defects
as local faults in the pattern, almost as an after- thought, despite their universal occurrence in real
crystals, and their overriding effect on the physical properties of crystalline matter. The perfect crystal corresponds to a strict energy minimum, however inaccessible experimentally.
The situation is very different if the system is finite.
The few, infinite pattern (space groups) of classical
crystallography are incompatible with most boundary conditions, and defects are a necessary ingredient of
the resulting pattern which minimizes the energy.
The classification and description of these finite patterns is still an open problem [1], which we shall
solve here for a particular class of systems.
This paper is concerned with crystallography in
two spatial dimensions, in the presence of cylindrical (or axial) symmetry. The symmetry is imposed by boundary conditions, by conditions of growth (as
in plant phyllotaxis), or by external agents like the earth’s rotation. In the absence of cylindrical sym- metry and for identical cells, the problem has been completely solved by bees and 19th century crystal- lographs, and nothing more need to be added to the
classic enumeration of space groups. (Nonperiodic,
or random patterns still await classification, but they require at least two different kinds of cells [1-3]).
We shall search here for crystalline structures filling
the space available, with as much homogeneity,
and made of as similar, isotropic cells, as is compatible
with the boundary conditions. The whole structure should be described by a single algorithm which
must be as simple as possible.
There are two direct fields of application. Cellular
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0198400450104900
patterns occur in Benard convection, whereby a
fluid heated from below exhibits convective motion above a certain temperature threshold. The centre of a cell corresponds to the hot, rising fluid, and the
vertices of the pattern to the cold, descending fluid.
A perfect infinite pattern would form a hexagonal (honeycomb) lattice. Cylindrical symmetry is imposed by the shape of the container, and, as we shall see,
by the earth’s rotation [4]. It is also believed that the solar granulation is also a result of convective motion [5]. There, too, some non-hexagonal cells
must appear for topological reasons (Euler’s theo- rem). A different manifestation of cylindrical crystal- lography can be found in the structure of daisies, sunflowers, pine cones, etc., called phyllotaxis (or leaf-arrangement), which constitutes a space-filling problem solved by some algorithm or code governing
the successive generation of cells or florets from the stem, and whose manifestation is through a sequence of numbers of opposite spirals (the Fibonacci sequence) [6, 7]. Here, cylindrical symmetry, and the resulting structure, are imposed by successive generation from a central axis, whereas in Benard
convection, the pattern appears more or less at once in the whole fluid.
As a consequence, this paper can be read at three
different, self-contained levels. It is first and foremost
a description of the structure of B6nard cells in cylin-
drical symmetry, that is, the solution of a problem
of fluid dynamics (sections 2, 4, 5 and Figs. 1, 2, 7).
The second level is that of phyllotaxis, where the emphasis lies on successive generation of cells, for
which coding replaces packing as the problem solved by nature (sections 3, 4 and Fig. 6). Finally, this paper is an essay on crystallography where defects are
necessary ingredients imposed by symmetry or boun-
dary conditions.
2. B6nard-Marangoni convection patterns, defects and
cylindrical symmetry.
Recent experiments [4, 8] on the Benard-Marangoni
convection in cylindrical containers with large aspect ratio, yield the following clues to the structure of
convective cells.
1) The cellular pattern is rotating. The trajectory
of an individual cell is a spiral, with angular period
that of the Foucault pendulum [4].
2) The cell rotates about itself with the same
period, but undergoes large, apparently random
fluctuation of magnitude 2 7r/5.
3) The pattern (Figs. 1, 2) always contains a finite
number of penta- and heptagonal cells. It can never
be made perfectly hexagonal. This remains so for
rectangular containers, but the number and position
of non-hexagonal cells varies with the geometry of the container. It also varies with the Marangoni
number : if the temperature of the bottom of the con-
tainer is increased further above To, the convective
Fig. 1. - Photograph of a Benard-Marangoni convective
structure [4] in a silicone oil layer (thickness about 1 mm).
threshold temperature, the pattern begins to melt [8].
Knowledge of the crystalline structure (with its
necessary number of defects, see below), is a pre-
requisite for understanding the dynamics of melting.
Similar patterns (Fig. 2) are obtained in nematic liquid crystals, with hydrodynamic instability induced by circular shear [8, 9]. The shear rate plays the part of the Marangoni number in the melting of the struc-
ture.
Observations (1) and (2) point at the Coriolis force of the earth’s rotation as the main agent in the evolution of the cellular structure (1). Benard cells
are convection patterns, the liquid flowing in the
free (upper) surface from the centre of the cell (hot point) to the vertices (cold points). Under Coriolis
force, both individual cells and the network as a
whole are subjected to a steady, weak rate of shear.
Crystalline patterns respond to shear by motion (glide) of dislocations which represents the most efficient means of dissipating shear energy.
To a first approximation, the cellular pattern is
clearly the solution of the purely geometrical problem
of filling an area with roughly isotropic and equal-
sized cells. The cells are deformable, and repel each (1) Whether the Coriolis force due to the earth’s rotation,
is entirely or partially responsible for the rotation of the cellular pattern, is still a matter for debate. It is not a pre-
requisite for the theory presented in this paper. The manifest
experimental facts on which the following discussion is
based, are that the pattern is rotating, and that there is
some dissipative mechanism. The advantage of assuming
Coriolis forces is that it suggests directly the presence of circles of dislocations (Coriolis --> shear dislocation
glide). But cylindrical symmetry, and the necessity to
screeri the strain due to the 6 pentagonal cells necessary from topology, leads to the same result. Of course, an alter- native solution is a random, polycrystalline mosaic.
Fig. z. - Keconsiruaion 01 ine cenuiar structure w igner- Seitz-Voronoi construction). a) In a nematic liquid crystal
with an instability induced by a circular shear [8, 9, 26].
b) In a standard Benard-Marangoni experiment [4, 8, 26]
0 = pentagon, + = heptagon, X = octagon, 0 + = dis- location.
other to maintain their correct size, which is given by hydrodynamics (a = 1.9 e, e = thickness [4]).
The repulsion is soft, more akin to the magnetic
ball model than to the bubble-raft model [10] popular
to visualize dislocations and grain boundaries. The Coriolis force and boundary conditions are small
perturbations on the main requirement of area- filling by isotropic cells of equal size. The specific
nature of the interaction between cells, which should
come out of the differential equations of hydrody- namics, is assumed to have an even smaller effect,
and will be neglected.
An area packed with convex objects of equal size produces a triangular pattern with hexagonal cells.
A cell can be defined geometrically as the region in
space closest to the centre of one particular object.
In this case the space is partitioned by Voronoi or Wigner-Seitz cells. The most obvious defects are
penta- or heptagonal cells (Figs. 1 and 2). There are
also a few vertices of coordination, 4 instead of 3, but these are not topologically stable (a small defor- mation splits them into two normal vertices of coor-
dination 3) and have a short lifetime in Benard patterns [11].
Penta- or heptagonal cells are positive or negative
disclinations (rotation dislocations), sources of posi-
tive or negative curvature. They are topologically
defined objects which are structurally stable, that is, they keep their identity under small deformations.
Opposite disclinations attract each other. A dipole
of disclinations, a pair pentagon-heptagon, is a (translation) dislocation, since two successive rota- tions of opposite sign about parallel but distinct
axes make up a translation. Such dislocations are
easily observed in figure 2. Exactly as isolated charges
are screened by polarizing the surrounding medium,
the strain energy of isolated disclinations is screened
by dislocations [12]. This is why a radiolaria has
more non-hexagonal cells than the 12 it needs topo-
logically [13]. Dislocations, as the element of pola-
rization of the cellular network, is the mechanism
whereby strain energy is locally screened.
Topologically, a finite cellular pattern must contain
6 isolated positive disclinations (pentagonal cells).
This results from Euler’s theorem, relating the number
of cells F, edges E and vertices V
for a finite, simply connected, two-dimensional net- work. (The cell at infinity is not counted.) Let Fn
be the number of n-sided cells, and Ep, Yp, the edges
and vertices on the perimeter of the network. We have the valence relations
because every edge, except those on the perimeter,
separates two cells, and
because every edge joins two vertices, and every vertex is trivalent (tetravalent vertices are not topo- logically stable : they can be split into 2 trivalent vertices by a small deformation), except those on the perimeter of the network, which are all divalent
(this defines perimeter vertices topologically). The
valence relations included in Euler’s formula yield
A natural method of closing peripheral cells is by replacing every rim segment by two perimeter edges
Fig. 3. - Natural closure of the cells-on the boundary
of the container. This convention eliminates the distinction between boundary and interior cells in Euler’s formula 2.4.
Ep and one perimeter vertex Vp (Fig. 3), thus Ep = 2 Vp
and, by (2.4) the network contains 6 positive discli-
nations. The natural method of closure does not
apply to containers with sharp corners, for which
a full analysis of the right-hand side of equation 2.4
is necessary. A more intuitive and geometrical deri-
vation is due to Dormer [14]. Consider a cigar-shaped surface, half of which contains our finite pattern.
The cigar, being homeomorphic to a sphere, must
contain 12 positive disclinations. This is easily obtained
from equation 2.1, with 2 instead of 1 on the right-
hand side (sphere), and equations 2.2, 2.3 with
Ep = Vp = 0 (no boundary). The cylindrical section
contains no disclinations, and the two hemispheres
6 each by symmetry.
In the experiments of Pantaloni et al. [4, 8], the edges of the peripheral cells are perpendicular to
the rim of the container. However, precise knowledge
of the boundary conditions (whose justification
remains an open problem), is not necessary for the argument of this paper, where only cellular patterns and an overall cylindrical symmetry need be assumed.
(The structure and stability of some convective
structures (rolls) under different conditions, have
been discussed recently in reference 30).
In conclusion, Euler’s theorem imposes the pre-
sence of 6 positive disclinations or pentagonal cells
in the midst of the network. Their curvature, or strain energy, must in turn be screened by dislocations,
which are pentagon-heptagon dipoles. Moreover,
the shear induced by the earth’s rotation is also dissi-
pated by dislocation glide. However, cylindrical symmetry requires the glide lines to be concentric
circles, instead of the straight glide lines or planes of
conventional crystals under shear. Disclinations and dislocations are an essential part of the structure,
imposed by boundary conditions and by topology.
They must not be regarded as defects which can
be annealed out by some heat treatment.
The key problem is to produce a structure with
concentric glide circles. It turns out that concentric circles of dislocations, forming boundaries between
defect-free grains, which can glide on each other,
are seen in large daisies and sunflowers, whose struc-
ture can also be regarded as the solution of a geo- metrical problem in cylindrical symmetry [15], which
we shall discuss in the next two sections.
3. Phyllotaxis (leaf or floret arrangement).
The inner florets of an aster, a daisy or a sunflower,
the scales of a pine cone or a pineapple, form an area-filling structure in which neighbouring « cells »
are arranged on spirals or parastichies. A hexagonal,
cell belongs to three parastichies, but most botanical
cells have the shape of a lozenge, and belong to only
two manifest parastichies of opposite chirality. In an overwhelming number of cases, the number of left- handed and right-handed parastichies are two conse-
cutive Fibonacci numbers. In the largest sunflowers (see, e.g. [13]), the structure stretches from one pair
of consecutive Fibonacci numbers to the next higher,
when one goes from the centre to the periphery. It is
also said that the number of parastichies can be
increased by intensive cultivation [16].
It seems, therefore, that a family of structures
can be generated by a simple code or algorithm, capable of dealing with affine (scaling) transforma- tions or with breeding, and that the Fibonacci series is the external manifestation of this code. Coding
operates upon generation of new florets from the stem or the centre. It fixes the angle of successive florets. The resulting structure is obtained simply by younger florets pushing the older ones from the
centre. The code is itself the practical translation of a biological variational principle, which may well be the most efficient sharing of horizontal space between florets or leaves in order to share the sun, rain or air [6]. As far as the physicist is concerned,
we have a genuine crystallography, with a few, area- filling structures, characterized by an algorithm or
code. Moreover, the structures obtained are self- similar radially, and, like the Benard convection pattern, dominated by the geometrical requirement
of filling space with roughly isotropic cells of equal
size (scaling due to growth notwithstanding). The symmetry is obviously axial and, as we have argued
in the preceding section, this automatically intro-
duces defects in the structure. Finally, one would
expect the code to be as simple as possible, in order
to generate the most probable structure consistent
with the geometrical constraints.
[The only structural information which is encoded (cf. algorithm 4.1) is the angle A between florets,
and the fact that successive florets appear at regular
time intervals. The shape of each floret (whether it
is a pentagon, hexagon, etc.) is not coded. It is simply
a consequence of filling an area (Voronoi construc- tion).- The fluidity of the structure must be empha-
sized. Indeed, nothing prevents the shape of a parti-
cular floret to vary with time.]
In the following section we shall construct such
structures from first principles. The agreement with
botanical structures is manifest (cf. Fig. 6). Moreover,
we shall associate the omnipresence of the Fibo- nacci series or the Golden section r in phyllotaxis,
with self-similarity and homogeneity of the structure.
The structure contains defects, is finite, and homo- geneity can no longer directly be associated with
global translational invariance. This justification of
the Golden section is new, although Coxeter [15], following Tait, has given a different reason (absence
of intermediate convergent) which is mathematically sufficient, but much less overriding structurally.
As a bonus, we shall find that the Golden section,
and Fibonacci series, are given by the simplest code.
Finally, because they have circular glide lines (Fig. 6),
the botanical structures must be very similar to their hydrodynamic counterpart.
4. The daisy : crystallography in axial symmetry.
The construction proceeds by steps. We shall end up with a genuine crystallography in cylindrical geometry, i.e. with only a few possible structures, identified by an elementary algorithm, or code, which
are structurally stable and sufficiently flexible to
accomodate elementary defects or excitations. Whether the physical system will take up such structures, or select a polycrystalline structure with polygonized grain boundaries, will depend on the relative strength
of the cylindrical perturbations (size of the circular
boundary relative to that of the liquid, strength of
shear stress, etc.). The same dilemma has been inves-
tigated by Farges in atomic clusters of n atoms.
Small clusters (n 200) take up polyicosahedral configurations which minimize (locally) the energy, but cannot fill space, and larger ones (n > 200)
have the closed-packed, space-filling configurations
associated with crystals.
4.1 Symmetry suggests polar coordinates for the cell centres, r(l), 0(l), where 1 = 1, ... labels the indivi- dual cells, proceeding outwards from the centre of symmetry. Once the centres are specified, the
cells are constructed by Voronoi partition of space around each centre (random Wigner-Seitz cells).
The construction is unique. Uniformity requires
that 0 is proportional to d, and homogeneity in cell densities, that r increases as 11/2, on average. Curva- ture of the cellular « substrate » (as occurs in some plants), or growth of the florets, can be accommo-
dated by any monotonic function r = .f(/). Thus, the algorithm reads
where a is the typical cell’s linear dimension (its
area is na2), and A is the only parameter characte-
rizing the structure. For definition, 0 A 1. The algorithm 4.1 introduces cells regularly on a gene-
rative, or genetic spiral in the reverse order of suc-
cessive leaves budding from the stem of a plant.
The continuous relation r - (J1/2 associated with
(4.1) is obviously radially self-similar. The discrete version (4.1) which corresponds to discrete cells will also be self-similar, in a remarkable connection with theory of numbers, as we shall see below.
The area can be regarded as partitioned into con- centric, quasi-circular shells containing an integer
number of cells, m say, such that 0(l + m) L-- 0(l) [17].
It follows immediately that A cannot be rational.
If it were, Â. = A/B say (A, B integers), all cells I > B would have their centres precisely on the same B
azimuths (lying on top of each other), and the resulting
cellular structure would resemble a spider’s web (Fig. 4), which is hardly desirable as a solution to the problem of filling area with isotropic cells. The shells
correspond, therefore, to successive rational approxi-
mations of A, or a division of 2 n by successive integers,
the number of cells pdr shell.
Fig. 4. - The spider-web, constructed from algorithm 4.1
with A = 13/21 rational.
4.2 Theory of numbers yields these successive rational
approximations of A elegantly and directly [18, 19].
Any number can be represented as a continued frac-
tion,
if A 1, with qi integers > 0. This decomposition is unique. Thus, A = { qi }, is given by the set of integers
qi’ which constitute a code. If A is rational, the continu-
ed fraction is finite. It is infinite for irrational A.
