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Hierarchical interlaced networks of disclination lines in non-periodic structures
J.-F. Sadoc, R. Mosseri
To cite this version:
J.-F. Sadoc, R. Mosseri. Hierarchical interlaced networks of disclination lines in non-periodic struc- tures. Journal de Physique, 1985, 46 (11), pp.1809-1826. �10.1051/jphys:0198500460110180900�. �jpa- 00210132�
Hierarchical interlaced networks of disclination lines in non-periodic structures
J.-F. Sadoc and R. Mosseri (+)
Laboratoire de Physique des Solides, Bât. 510, 91405 Orsay, France
(+) Laboratoire de Physique des Solides, CNRS, 1, place A. Briand, 92190 Meudon-Bellevue, France (Reçu le 3 mai 1985, accepté le 8 juillet 1985 )
Résumé. 2014 Nous décrivons l’ensemble des défauts dans une structure non cristalline déduite d’un polytope
par une méthode itérative de décourbure. Les défauts apparaissent comme un ensemble hiérarchisé de réseaux de disinclinaisons entrelacés qui sont le lieu des sites où l’ordre local s’écarte de l’ordre icosaédrique parfait. La
méthode itérative est décrite à 2D et 3D. Nous discutons aussi de l’utilité du concept de défaut hiérarchisé pour décrire la structure microscopique des quasi-cristaux icosaédriques.
Abstract. 2014 We describe the defect set in non-crystalline structures derived from polytopes by an iterative flattening
method. Defects appear as a hierarchy of interlaced disclination networks which form the locus of sites where the local order deviates from a perfect icosahedral environment. The iterative procedure is fully described in 2D
and 3D. We also discuss the usefulness of introducing the concept of hierarchical defect structure for the micro-
scopic description of icosahedral quasicrystals.
Classification
Physics Abstracts
61.40D - 02.40
1. Introduction.
Amorphous systems generally present an appreciable
amount of Short-Range Order (SRO). For example amorphous metals can be well described by close packing tetrahedra [1]. A regular tetrahedron is the densest configuration for the packing of
four equal spheres. The dense random pack- ing of hard spheres problem can thus be mapped
on the tetrahedral packing problem. The dihedral
angle of a tetrahedron is not commensurable with 2 n,
consequently a perfect tiling of the Euclidean space R3 is impossible with regular tetrahedra. Note that,
at this local level, the o frustration » (deviation to perfectness) is of metrical rather than topological
nature. One of us (J.F.S.) has proposed to define an
ideal (unfrustrated) amorphous structure by allowing
for curvature in the space in order for the local confi-
guration to propagate without defects throughout
the whole space [2]. It is possible to pave a 3D manifold,
the hypersphere S3, by 600 regular tetrahedra arranged by five around a common edge. The obtained geo- metric object is called the polytope { 3, 3, 5 } using
the standard notation [3]. Note that the underlying
space S3 is 3 Dimensional although not Euclidean,
even if one often thinks of S3 as being imbedded in R4. Indeed S3 is the locus of points of R4 given by xi + x2 + x3 + x4 = R 2, which shows that only
JOURNAL DE PHYSIQUE. - T. 46, ? 11, NOVEMBRE 1985
3 coordinates are independent. The polytope model,
or o Constant Curvature Idealization » (CCI) has
been extended to several other kinds of disordered materials such as tetracoordinated covalent sys- tems [4]. A simple example is given by the packing of pentagonal dodecahedra which is forbidden in R3
(as in the tetrahedral case) because of the polyhedron
dihedral angle value. Packing these dodecahedra on
S3 leads to the regular polytope { 5, 3, 3 } which is
dual of the above mentioned { 3, 3, 5 } and thus
possesses the same symmetry group. However the values of the curvature associated with these two
polytopes are not identical (when scaled to the edge length) and this reflects the fact that the local angular
mismatch (in R3) are not equal. Polytope { 5, 3, 3 }
can be useful for modelling the « caged »-like tetra-
valent structures and those which are related like
amorphous ice for example. Several other CCI have been proposed, like the regular honeycombs in the
’
3D hyperbolic space H3 (with constant negative curvature) [2, 5] and the continuous double twisted
configuration of directors on S3 (as an ideal model for the cholesteric blue phase) [6].
The idea is that the disordered material contains
« ordered >> regions where the local order can be put in correspondence with the ideal model, the comple- mentary regions being the locus of defects. We expect that a suitable map of the ideal model onto
111
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0198500460110180900
R3 (minimizing the energy) will provide a realistic amorphous structure. The mapping introduces dis- tortions and topological defects, among which the disclination lines play an important role. The final structure consists in mixing of regions with positive
curvature (where the local order is that of the poly- tope) and regions with negative curvature (the locus
of defects), arranged such as to give zero curvature
on the average. In this Corrugated Space Approach [7],
also called «variable curvature idealization », the local order is still perfect within coherence regions.
As has already been mentioned, disclination lines have proved to be natural defects for non-crystalline
structure. S. N. Rivier [8] proposed that one type of linear defects, characterized by oddness rather than
intensity, is stable in real glasses as a result of the
double connectedness of the rotation group SO(3).
In the curved space approach, line defects can also be classified using the homotopy theory of defects [9].
Let us call Y the icosahedral group (subgroup of SO(3)) and Y’ its lift in SU(2) (the covering group of SO(3)). The full symmetry group G* of polytope {3,3,5}, with 14 400 elements, is described in
Appendix B. G, the subgroup of G* containing only
the direct symmetry operations (preserving orien- tation), is given by : G = Y’ x Y’/C2. C2 is the two-
element group. As shown by Nelson and Widom [10]
the defect lines that can be generated in the polytope { 3, 3, 5 } belong to the conjugacy classes of R = ni (SO(4)/G) = Y’ x Y’. More recently Trebin [11]
has proposed a « coarser » classification for the line defects in polytope { 3, 3, 5 } by considering the first homology group of the order parameter space. Since Y’ is a « perfect group » (it is isomorphic to its own
commutator subgroup), the homology group is trivial which expresses the fact that any line can be transform- ed into any other by a suitable combination process.
One might hope to generate increasing numbers
of disclination lines in the { 3, 3, 5 } in order to achieve
a complete flattening of the polytope. We have already shown [12] that it is possible to interlace two
such disclination lines and get a polytope containing
144 Z 12 vertices and 24 Z 14 vertices with less intrinsic curvature. We use standard notations [13] to label
the sites according to their coordination number. In order to annul the curvature, one should iterate this
procedure and incorporate the disclination lines step by step. There are up to now unsolved difficulties in doing this which are probably due to the non-
commutative character of the required operations (R is non-Abelian). On the other hand, this non- Abelian character is the key to understand why it is possible to model very complex disordered structures
starting from a regular polytope and using only a
finite collection of defect types. An « alphabet >> can
be defined whose elements, the «letters », denote each type of defect (the conjugacy classes of R).
A structural model, based on the polytope, is then represented by a « word », an ordered set of letters,
and its complexity is encoded in the information content of the word.
In a recent letter, hereafter referred to as 1 [14], we
have shown how it is possible to bypass the above-
mentioned difficulties and achieve the complete flat- tening of the polytope. The key idea is, at each step,
to introduce a disclination network (instead of a single disclination line) whose symmetry group is contained in G. In the present paper we shall give a complete description of how the method works. In order for this paper to be self-contained, we shall give a detailed description of the different tools which
are needed (symmetry group, orthoscheme, bary-
centric transformation...).
In section 2, a simple 2-Dimensional example is described, in which an icosidodecahedron is itera-
tively flattened and gives rise to an asymptotic non- periodic plane tiling. The main ideas that will be used later in the 3-Dimensional case are introduced and visualized.
Section 3 contains the application of this method to the polytope { 3, 3, 5 } case. At each step a « defect » subnetwork is generated which has the same symme- try properties as the structure itself. Also the matricial
description of the iterative procedure is introduced.
We show how to generate more disorder by combining
two different iterative procedures which are compa- tible since their associated defect structure share the
same symmetry group. In the last section (4) we
describe miscellaneous aspects and extensions of the model.
2. A 2D example : iterative flattening of an icosi-
dodecahedron.
The icosidodecahedron (Fig. 1) is a quasi-regular
or Archimedean polyhedron [15]. It is
noted 3 5 [3]
and shares the same symmetry group with the ico- sahedron { 3, 5 } and the dodecahedron { 5, 3 }.
Fig. 1. - The icosidodecahedron
Indeed it can be obtained by joining the mid-points
of the edges of { 3, 5 } or { 5, 3 }. Each vertex belongs
to two regular pentagons and two equilateral triangles.
The Iterative Flattening Method (I.F.M.) is much
more easily visualized in two dimensions than in three and almost all of the ideas that we present here can be simply generalized to one higher dimen-
sion. As will be seen below, the I.F.M. can be under- stood as an « inflation >> or as a « deflation >> method.
The second one is very easy to visualize (in some sense
it does not need a coordinate system). We shall
however insist on the inflation method since it is the
one which allows to focus on the symmetry relations between vertices and on the curvature aspect.
Z .1 GEOMETRICAL PROCEDURE. - In appendix B we
introduce the binary icosahedral group Y’ [16] which
is the lift in SU(2) of the icosahedral group Y (the covering map SU(2) -+ SO(3) is carried on the
discret polyhedral subgroups of SO(3)). Y is a pure subgroup of Y*, the full icosahedral group (including
indirect transformations) also called the triangle
group (2, 3, 5)* [17]. This group of order 120 allows for a division of S2 in a pattern of 120 spherical triangles, each of one being a fundamental region of
the sphere tesselation (Fig. 2). Any vertex configura-
tion on S2 having Y* as symmetry group is comple- tely defined by giving one fundamental region (the orthoscheme, Appendix C), and the distribution of vertices inside it. It is analogous to the description
of crystals by unit cells and the translation part of the symmetry group. In the case of spherical tessela- tions, the symmetry operations are reflections in the sides of the fundamental triangle. The pure rotations
belonging to the direct group Y are products of even
number of such reflections.
In the following we take as a basic icosidodeca- hedron
the 3
which tiles the equatorial >>5
Fig. 2. - Partition of S2 into 120 spherical triangles under
the action of the full icosahedral group Y*. Elements of the pure subgroup Y interchange triangles of the same colour (either white or shaded). A particular fundamental region (M6bius triangle) is distinguished.
sphere of polytope { 3, 3, 5 } (see next section). The
vertex coordinates are given by the 30 quatemions
in Y’ whose scalar part vanishes. Indeed to a unit pure quatemion q = a, i + a2 j + a3 k (see Appen-
dix A) there corresponds a point of coordinates
(a1’ a2’ a3) on the sphere S2 of unit radius. The three orthoscheme vertices Mo, M1, I M2 are vertices
of respectively the { 3, {3,5 5
3
}{
5 and 5 { 5,3 3 } } whichshare the symmetry group Y*. Indeed it can be veri- fied that the orbit of Mo (resp. M1, M2) generates a 3 5
resp. a 3 ,
a 5, 3 under successive{ 3,5
(
p(
5 {)
reflections in the sides of the spherical triangle Mo M1 M2. We shall consider that the polyhedra
under construction are either spherical, with bent
faces and geodesic edges, or Euclidean, with flat
faces and straight edges (chords). They are trivially related, the spherical polyhedron being the central
projection of the Euclidean one OIrto-tl1e-surfiice of
the sphere S2.
Let us now describe the method. In a first step the
original polyhedron is constructed. It will be called the «source polyhedron Po. It is characterized by
the orthoscheme MoMiM2 and by the location of a
point M (or several points) in it. Po vertices are the
orbit of M under the Y* symmetries. The second step consists in selecting a new triangle MoM1M2 which
shares an angle with the orthoscheme MoMiM2 (here
the angle at vertex M1, see Fig. 3). The triangle MoM1M2 is chosen as to contain an integral number
of orthoscheme replicas. Thus it contains several Po
vertices. Note that the two spherical triangles MOMlM2 and MoMiM2 do not have all their cor-
responding angles identical since they have different
areas (recall that the area of a spherical triangle is proportional to the sum of its interior angles minus n).
Fig. 3. - The two spherical triangles MOMIM2 and MoMlM2 which share the vertex Mi. Marks label vertices
on S2 where the symmetry is either 5-fold, 2-fold and 3-fold.
The 12 vertices (resp. 30, 20) which are the orbit of Mo (resp. M1, M2) are vertices of an icosahedron (resp. icosido- decahedron, dodecahedron).
The crucial point consists now in the identification of the two triangles. In flat space this could be easily
done by homothety when the triangles are similar.
In the curved space S2, homothety is not easy to define because of the presence of an internal length scale (the radius of curvature). It is possible to define
homotheties along the geodesic lines M1M0 and MiM2 which makes the points Mo and Mo, Mz and M2 coincide. But to ensure that all the points of the geodesic line M’M’ will lie on the geodesic line MOM2 requires a continuous set of homotheties. Instead there is a very simple and natural way to do this if
we work with an Euclidean polyhedron (with flat faces). Now the flat triangles MoMiM2 and MoMlM2
are not coplanar (Mo does not belong to the chord MiMo) and the underlying geometry is no more a
simple 2D geometry but a collection of bounded 2D flat regions (the faces) glued along the edges. It is
then easier to consider the homogeneous space in which the polyhedron is embedded, the 3D Euclidean
space E3. Now any point in E3 can be specified in a barycentric coordinate system once a particular
tetrahedron is given. Let us for instance take the tetrahedron with the flat triangle MoMl M2 as its
base and the centre 0 of the sphere as its apex. The four barycentric coordinates corresponding to the
tetrahedron OMoMiM2 are noted (Sl, ao, al, a2) (with the conditions + ao + ai + a2 = 1). A point belongs to the 3D sector bounded by the planes OMOMI, OMOM2, OMlM2 if and only if its last three coordinates ao, al, a2 are simultaneously posi-
tive or zero. The sign of Q says whether the point is
in the same side of MoMiM2 as the point 0 (Q > 0)
or in the opposite side (Q 0).
The identification of the two triangles proceeds as
follows : First, we calculate the coordinates (Q’, ao, ai, a2) of Po vertices in the barycentric system based
on the tetrahedron OM’M,M’. Then, we only keep
those vertices which have ao, ai and a2 simulta- neously positive or zero. Finally we re-interpret those
coordinates (Q’, ao, ai, a2) as being the coordinates
(Q, ao, ai, a2) in the barycentric system based on the tetrahedron OMoMIM2. The two basic tetra-
hedra are drawn in figure 4. The new points do not necessarily lie on the flat triangle MoMiM2 but they
can be projected on it by central projection. In terms
of barycentric coordinates it consists in putting equal to zero and rescaling ao, al and a2 (to insure
that their sum equals unity). Note that the points can
also be projected onto S2 and then belong to the spherical triangle MOMlM2’
The first iteration is almost finished, the first iterated
polyhedron P1 is obtained as the orbit under the group Y* operations of the vertices lying in the ortho-
scheme MOMlM2’ It is represented in figure 5a. The
local order in P1 presents similarities with that of Po.
Its description in terms of order and defects will be done in the next paragraph. Let us just say that the configuration around the point M 1 is identical
Fig. 4. - The two basic tetrahedra OMoMIM2 and OM’M,M’. 0 is the centre of the sphere S2. The points Mo, Mo, M1, M2 and M2 have the same location than in
figure 3. But now the edges are straight lines (chords) instead
of geodesic curves.
Fig. 5. - a) The polyhedron Pi (centrally) projected onto
the sphere S2. b) Tentative drawing of the underlying cor- rugated geometry of Pi. The radius of curvature (scaled to
the edge-length) near Mi and its replicas should be similar to that of Po.
to that of Po (M 1 belongs to 2 pentagons and 2 trian-
gles). In terms of the edge length, the radius R1 of
the P1 circumsphere is much larger than Ro, the
radius of the Po circumsphere. In some sense the ratio R1 I Ro is characteristic of the «bary centric homothety ». It is possible to recover some regularity
for the polygons around M1 by supposing that P1
vertices lie on the surface of a sphere of a radius R1
covered by small domes of radius Ro which centres
are located somewhere on the radii connecting 0
with M1 and its replica. This is tentatively drawn in figure 5b where the whole surface has been smoothed.
It is very easy to iterate again the procedure, build
a new polyhedron P2 by keeping the P1 vertices lying inside M0M1M2, identifying MoMiM2 and MoMiM2 and then getting P2 with the help of Y*
symmetries. As long as the two triangles keep specific
relations (they have an angle in common and the big one contains an integral number of replicas of
the first one), the successive polyhedron Pn will
present very interesting properties. For example, as
seen below, their defect set presents the same kind of
regularity as the polyhedra themselves. If the same
triangle M0M1M2 is used at each iteration, the
polyhedra Pn belong to the family of deterministic
recurrent sets [18]. From a topological point of view
the asymptotic polyhedron P 00 presents the same
kind of non-crystallinity as the Penrose tiling [19].
A simple algorithm to build iterative polyhedra is presented in Appendix D.
2.2 THE DEFECT SET. - By definition the source
polyhedron Po is called the «ordered (defect free) configuration ». It can be obtained by the local building rule : « put two pentagons and two triangles
around each vertex ». A specific value for the curva- ture of the underlying 2D manifold is necessarily
associated with such a rule. This curvature takes a
constant value at each site and is related to the so-
called deficit angle bM at the vertex M :
0 are the internal angles of the flat polygons sharing
site M. Different relations between local configura-
tions and curvature has been derived in connection with the Corrugated Space Approach to disordered
structures [7]. In 2-Dimensions, owing to the existence of two famous relations, the Euler-Poincare and the Gauss-Bonnet relations, simple and exact correspon- dence can be found between the geometry of the
underlying manifold and the defect density. Here
defects are disclination points whose weight is related
to the values of polygon angles at the vertices. For
example, when flat space is taken as the ordered
(undefected) state, one has the exact relation
where ( K, is the mean Gaussian curvature per unit area (R is the radius of curvature) and n the
number of disclination points (each assumed to carry the same angular deficit 6) per unit area. In the present
case where the ordered state is defined to be the
polyhedron Po, the defect intensity is measured with respect to the value of £ 0; at Po vertices (instead
of 2 x). I
The I.F.M. in its inflation >> form yields a very
simple way to locate and measure the defects. Indeed
a disclination point is generated at each vertex of the
orthoscheme MOMlM2 (and at all its replicas under
the Y* symmetries) whenever the angle is different
from that of the corresponding triangle M0M1M2.
Let us consider the polyhedra as being spherical. The
internal angles are evidently
In Po the point M2 is surrounded by a pentagon. At the first iteration and after identification of the two
triangles MOMlM2 and M0M1M2, the point M2 will
be surrounded in P1 by an hexagon (6 = 5 x eM2/eM2).
Sirnilarily, the point Mo in Po is surrounded by a triangle. Thus the point Mo in P1 is surrounded by
a pentagon (5 = 3 x 8Mo/9Mo). Note the important
following remark : in the polyhedron Po, the point Mo
was already surrounded by a pentagon, which was
not a defect polygon. At the first iteration this pentagon has been « pushed >> toward the point M,. The pen- tagon which now surrounds the point Mo can be
considered as a defect since it is a disclinated form of the triangle which surrounded Mo before the first iteration (in Po).
Let us call Dp the defect set of a polyhedron Pp.
Dp is the union of several subsets, each of one asso-
ciated with a particular kind of disclination, and all sharing the symmetry group Y*. For instance D1
contains two subsets D1 and D" 1
- D 1 is the set formed by point Mo and all its
replicas under Y*. This point set forms an icosahedral pattern. Each point is surrounded by a pentagon in P1. I
- D1 is the set formed by point M2 and all its
replicas under Y*. This point set forms a dodeca-
hedral pattern. Each point is surrounded by an hexagon.
Both D’ and D1 are represented in figure 6. In
the next iteration it is clear that the disclination
points in D’ and D1 will persist as defects. However they will be located at different places in the ortho-
scheme and the number of points after replication
will be different. It will happen that two points in D’ which belonged to two different fundamental
triangles in Pp will be in the same triangle Pp+1. So
the number of defects will increase indefinitely with
the iterations. Also each iteration generates two new
