ORIENTATIONAL ORDER IN DISORDERED SYSTEMS

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ORIENTATIONAL ORDER IN DISORDERED

SYSTEMS

J. Sadoc

To cite this version:

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JOURNAL

DE PHYSIQUE

Colloque C9, supplément au n012, Tome 46, décembre 1985

ORIENTAT IONAL ORDER I N D ISORDERED SYSTEMS J.F. Sadoc

" Laboratoire de Physique des Solides, Université de Paris Sud, Bât. 510, 91405 Orsay Cedex, France

Résumé

-

Les structures désordonnées ne sont pas des structures aléatoires. Il y a encore beaucoup d'ordre dans un amorphe ou un liquide. Le point important est l'absence de périodicité qui était la référence pour la définition de l'ordre.

L'ordre cristallin suppose trois types d'ordre :

-

L'Ordre local

-

L'Ordre de position

-

L'Ordre d'orientation

Ces trois types d'ordre interagissant l'un avec l'autre.

Dans les structures liquides ou amorphes, on suppose habituellement que seul l'ordre local persiste. Mais l'observation récente de quasi-cristaux ayant un ordre d'orientation sans périodicité démontre l'existence d'état intermédiai- re entre le cristal et le liquide. Le concept d'ordre d'orientation est pré- senté à partir de l'exemple des phases hexatiques.

Les pavages non périodiques de Penrose sont aussi des structures qui repous- sent les frontières de la cristallographie. Des exemples à 2 et 3 dimensions sont présentés avec le calcul de leur transformée de Fourier (Duneau et Katz,

à paraître dans PRL).

Une propriété importante de ces structures est l'auto-similarité qui conduit au concept de hiérarchie. Des modèles avec un environnement local icosahédri- que et une structure hiérarchique de défauts sont aussi une approche efficace des structures ayant un ordre intermédiaire entre le cristal et le liquide. Abstract

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Disordered structures are not random structures. There is still order i< an amorphous or a liquid material. The main point is the lack of periodicity which was the reference for the definition of the order. Crystalline order supposes three types of order :

-

The local order - The positional order

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Tbe orientational order

Al1 these orders interact one with the other.

In amorphous or liquid structures we suppose usually that only the local order remains. But the recent ex~erimental observation of quasi-crystal with a clear orientational order but without any periodicity indicates that there are intermediate cases between the crystalline and simple liquid structure. The concept of bond-orientational order is presented on the simple example of hexatic phase. The Penrose non-periodic tiling are also important structures which extend the frontier of the crystallography. Examplesin 2D and 3D are presented with the recent calculation of their Fourier transform (Duneau and Katz, to appear in P R L).

One important property of these structures is the self-similarity which leads to the concept of hierarch~. Models with icosahedral local configuration and with a hierarchy of defects are also fruitful ways to approach structures which have an intermediate order between the crystal and the simple liquid order.

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1

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INTRODUCTION

In dense matter electron interaction necessarily imposes a local order. This can be described by steric relation and bonds, or by more sophisticated techniques, like ab initio calculations used in theoretical physics. In the metallic structure with isotropie interaction, the local order is similar to the local order obtained in a packing of soft balls. This is a very simple example of local order. In covalent structures like silicon there are oriented interactions leading to the tetracoordi- nated local order which is a more complex example, but in al1 cases a local order can be defined. So, a dense structure cannot be completely disordered like a gas.

The disorder appears as a long range property. It can be related to an

entropie

effect, as for example thermally actived dislocations or vacancies in a crystal, or due to geometrical frustrations in glasses or quasi-crystal. In the present paper, we are mainly interested in the disorder due to geometrical frustrations even in the 2-D hexatic phases which are presented as an example of orientational order, result of a thermally actived process.

In al1 cases, the disorder can be in a systematic way defined as defectsin a perfect structure, but in the case of a disorder due to frustration, the perfect structure is described in non-physical curved spaces. The defects are localized variations of the local order, which perturb drastically the long range order. Vacancies do not change atom positions on a long range scale and it is only the inter-vacancy

correlation function which involves a complete disorder : a gas of vacancies.

Dislocationsand disclinationsare defects which strongly change the long range order. They are the important defects in the present description of the disorder.

II

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THE HEXATIC PHASE : A 2-D EXAMPLE OF THE DISLOCATION AND DISCLINATION EFFECT

ON LONG RANGE ORDER

The hexatic phase has been described as a result of the melting theory of Kosterlitz

and Thouless (1) and Halperin and Nelson (2), and observed in some examples-of 2-D

hexagonal structures (3).

In an hexagonal structure, thermal energy can rotate a bond (fig. l ) , and consequen-

tly changes two hexagons into two pentagons and two hexagons into two heptagons

Fig. 1 : The flipping af a bond in the hexagonal network leads to a disclination

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Due to the conservation of the mean number of side in each ce11 for a 2-D tri-coor- dinated network, the number of pentagons must be equal to the number of heptagons So, it is easy to compare a pentagon to a positive unit electric charge, and a heptagon to a negative unit electric charge. Within this language, the rotation of a bond leads to a quadrupole It is also possible to use a disclination description The local effect of a positive disclination is to change a hexagon into a pentagon, when a negative disclination changes a hexagon into a heptagon (fig. 2). The figure

3 shows how a dislocation is a dipole of disclination.

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Fig. 3. : A dipole of disclinations and the Burger circuit used to define the equivalent dislocation and its Burger vector.

The quadrupole of defects due to a bond rotation does not perturb the long range order. But at a given transition temperature, there is a decomposition of the quadrupole into two dipoles. The quadrupole gives two dislocations. Dislocation changes the position of atom on a long range scale. So if abovea given temperature, a large number of dislocation appears,the positional (or translational) order disappears. But a dislocation does not change drastically the orientation of a bond as it is only related to translation displacement. The system has still an orientational order, it is this new phase which is called the 'hexatic phase. When the disclination dipoles (dislocations) are decomposedabove an other given temperature, there is a new transition in which the orientational order disappears : it is the rnelting point between the hexatic phase and the liquid.

It is possible to quantify orientational and translational order. Translational order is related to the structure factor S(q). A translational correlation length

5

is defined by the inverse of the first peak in S ( q ) . An orientational order can T be defined by the quantity

+

dJ,

(r) = exp [ 6 i 8

(f)]

where 8(r) is the angle a bond makes with some reference axis. The six appears in the exponent because of the predominance of hexagonal configurations. A direct mea- sure of the local order is the function,

This function is expected to decrease exponentially in a liquid, but decreases only with a power law under the hexatic

-

liquid transition.

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III

-

GEOMETRICAL FRUSTRATION : SPACE CURVATURE AND DISCLINATIONS

The simple example of geometrical frustration is the tetrahedral packing (5). If one tries to pack spheres in a dense way by a discreteaggregationprocess, one easily finds that the regular tetrahedron (where a sphere is placed at the tetrahedron vertices) is the best solution with N = 4 spheres. 5 tetrahedra can share a common edge but a void necessarily remains between two triangular faces. This is due to the fact that the tetrahedron dihedral ang1e(%7O0) is not a submultiple of 27~. This is an example of geometrical frustration. This misfit angle manifests itself when one tries to propagate the tetrahedral local configuration and completely

surrounds a given vertex. An imperfect icosahedron is then obtained. Note that amorphous metal structure is well described by the so-called pseudo icosahedral

("compact or polytetrahedral") models. It is desirable to define an ideal mode1 in which the space can be perfectly tiled by tetrahedra. This is achieved using an S3 spherical space. This hypersphere can be embedded in the 4D euclidean space with equation :

Note that only 3 of the 4 coordinates are independent, S3 being a 3D (curved) mani- fold. In term of the tetrahedron edge length the radius of curvatureequals the golden ratio T = (1

+

/5)/2.

The perfect tetrahedral packing on S3 is called a "polytope" (the analogous of a polyhedron in higher dimension). This polytope is a finite structure (S3 is finite) and contains 120 vertices. Exactly5 tetrahedra share a common edge, and each vertex has 12 neighbours in a perfect icosahedral configuration. This polytope is called {3,3,5} using the standard Schlaffli notations and is well described in the books by Coxeter ( 6 )

.

This polytope is a structure which is completely free of frustration. In order to decrease the space curvature, it is necessary to add disclination lines. A

disclination line is a local concentration of curvature which allows a change of the mean curvature of the space. Negative disclination lines can be used to flatten a spherical space.

Disclinations are created by cutting the structure and adding (or removing) a wedge of material between the two lips of the cut. The symmetry operation is a rotation, while it is a translation foran usual dislocation. As dislocations introduce a strain field in the material, so do the disclinations. But in this case it is also possible to describe the induced deformation by a change of the space curvature. Adding a wedge of material to a structure defined in spherical space decreases the curvature, but structural defects appear along the edges and the faces of the wedge. Nevertheless, if the two faces of a wedge are equivalent by a rotation belonging to the structure symmetry group of the polytope, the defects are confined near the edge. Thus a perfect disclination line is created.

In 2-D and 3-D disclinations can change rings formed by bonds between atoms, or change the coordinationof atoms, depending on their relation with the position of atoms. In a {3,3,5} polytope, the local effect of a disclination running along some edges is to change the coordination number ( z = 14) of points lying on the disclination line.

The effect of a disclination on a structure being purely local, al1 other coordination polyhedra remain icosahedra.

A polytope is a perfect structure with orientational order. If a high density of disclination is added to such a structure, the orientational order disappears,and it remains only local order. Nevertheless, if disclination lines are introduced in a ~eriodic way, as in Frank and Kasper structures ( 7), the structure is crystalline.

IV

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PENROSE TILING

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C9-84 JOURNAL DE PHYSIQUE

There are different shapes of set of Penrose tiles : for example the kite and dart

described in M. Gardner reference paper ( 8 ) , but also the two rhombuses which appear

to be very important using the 5.D euclidean mappine, description. But al1 these structures can be obtained one from the other by decoration procedures.

One of the main properties of Penrose tesselations is the self similarity : a new

figure formed with smallest tiles Zan be obtained by decoration of the initial tessellation. A perfect orientational order is the consequence of the self-simila-

rity : two parallel edges of an initial tiling are separated by a large number of

new smaller tiles after a large number of iterations of the decoration procedure, and numerous edges of these small tiles are parallel to the two initial ones.

For example, on the fig. 4, which presents the dart and kite tiling we observe an

orientational order :

4 . : _{The Penrose tiling obtained from "Dart" and "Kitet'}_tiles.

_A

_matching

rule is imposed by two kinds of points.

Al1 bonds are parallel to five directions with a pentagonal symmetry. The self- similar decoration procedure is used to show the non-periodic character of the

tiling. Consider the decoration of dart (D) and kite (K) (fig. 5).

Fig. 5. : The two Penrose tiles : Dart and Kite in heavy line and how they are

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The decomposition at each iteration gives :

D + D + 2 K D + D + K

i

which can be written in a matrix formulation.with nK and n1 the number of kite and

dart tiles at the iteration i. D

U

matrix gives

-

= (1

+

/5)/2. This is an irrational number (the golden ratio 7 ) . na K

It proves the non periodicity.

It is striking when looking at the figure 6, showing a rhombus tiling,to see projection 2f cubes.

Fig. 6. : The Penrose rhombus tiling. Observe how it gives locally the impression of cube projections.

It has been shown by Kramer and Neri ( 9 ) and independently by Duneau and Katz (10)'

that Penrose tiling can be obtained £rom regular structure (cubic lattice) in space of high dimensions (5.D space) mapped on a plane. In order to have a local 5 fold symmetry the 2 D plane is orthogonal to the hyper-cube diagonal. The £ive unit vectors of the 5 D cubic lattice are mapped ont0 five vectors in the plane with a pentagonal symmetry. This description is very useful to explain why diffraction patterns by Penrose structure show spots.

\.Je use the 1 D equivalent of a Penrose tiling : a tiling with segment of length I and T of the line with an appropriate self-similarity. The fig. 7 explains how this tiling can be obtained by mapping of a subset of a square lattice points.

The subset is defined by a strip of irrational slope ( t g a = T ) with a width defined by the translation of the square unit cell.

The subset is a chain of points connected by vertical or horizontal edges. By orthogonal mapping on the line which borders the strip, these edges give two lengths of segment in the ratio T. This 1 D tiling is non-periodic since the line slope is irrational.

This way to describe Penrose structure can be used for 2 D and 3 D tiling, using cubic lattice in space of high dimensions : 5 D for 2 D Penrose tiling, 6 D for 3 D Penrose tiling.

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Fig. 7.

, , '

Fig. 8.

A 1 D Penrose tiling is obtained by projection of points of a square lattice if these poists are in a strip.

The diffraction pattern (Fig. 8) of the 1 D Penrose tiling is a section in the

2 D diffraction pattern of the points in the strip.

Optical Fourier transforms ( 11,12 ) seem to exhibit a discrete diffraction pattern.

The cubic lattice mapping description is a very useful method which allows an in-

dexation of the reflexionsusing 5 or 6 Miller indices. It also showswhatthe Fourier

transform looks like. For example, the 1 D diffraction pattern can be obtained from

the reciprocal lattice of the 2 D square lattice. The convolution of the reciprocal square lattice by the Fourier transform of the strip gives the Fourier transform of the chain of points inside the strip.

The Fourier transform of the subset of points orthogonally mapped on the straight

line is a section in the 2 D Fourier transform by a line parallel to the strip

(fig. 8).

Consequently, in a Penrose diffraction pattern, there is light everywhere, but on each point there is a delta function with an intensity completely different from

the intensity of an infinitely close point. In the 1

D

example, only points close

to a point of the reciprocal square lattice have an observable intensity.

The existence of such structuresin nature has been discovered by Shechtman et al (13)

in an Al Mn quenched alloy : the diffraction pattern can be perfectly indexed, using

6

the 6 D representation of a 3 D Penrose tiling.

This new kind of order (perfect orientational order, and non-periodic positional order) is probably occuring in numerous materials and willhelp renewing concepts in structure studies.

There is another description of Penrose tiling : a hierarchy of defects in a curved

structure (14).

A

curved structure is used to define a perfect local order and then

al1 local configurations which are different, are called defects. It has been shown that these defects are ordered like a Penrose structure with tiles larger than in tne initial one. Consequently, it is possible to define a hierarchy of defects.

V

-

HIERARCHY OF DISCLINATION LINES IN CURVED SPACE MODEL

Periodic arrangement of disclinations is a way to accomodate the local order and the topological constraint. But non-periodic solutions are also possible. We have described how to add one or two disclination lines in a {3,3,5] polytope, but it is difficult, due to the non-commutativity of symmetry operation associated with disclinations, to insert disclinations one by one in order to flatten the structure.

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try group contained in the symmetry group of the polytope,the above mentioned difficulty is bypassed. The required procedure which can be iterated infinitely uses the following geometrical objects :

-

the {5,3,3} polytope considered as a packing of dodecahedral cells.

-

the {3,3,5} polytope with atoms on its vertices. The description of this ~olytope shells by shells is the most appropriate in this case.

In fact,we really need only a part of this polytope which is within the second dodecahedral shell. This object contains 13 enclosed vertices and 20 vertices on its surface. Atoms are located on these vertices.

We need also al1 the geometric objects like dodecahedron or icosahedron, and objects obtained from them by a disclination procedure going through their centre. The idea consists in doing a suitable homothety in order to make the {3,3,5} dodecahedral shell coincide with the {5,3,3} dodecahedral cell.

We have now a new structure. Atoms within the {3,3,5} dodecahedral shell (13 in each {5,3,3} cell) keep their local icosahedral coordination polyhedron. Atoms lying on the {3,3,5} dodecahedral shell which are now located at {5,3,3} vertices, have a coordination polyhedron with 16 vertices. It is the polyhedron yet observed in Laves phase and enclosing a crossing point of 2 disclinations with a tetrahedral local symmetry. This is sufficient to explain why the structure can be understood as a medium with a local {3,3,5} configuration pierced by a network of disclinations formed by the edges of the {5,3,3} polytope (figure 9). There is an image helpful to "see" the underlying space of this structure : a strawberry. Far away, a strawberry looks like a sphere. This sphere plays the role of the {5,3,3) polytope. But on the surface of this sphere, there are some grains which play the role of the {3,3,5) parts filling the {5,3,3} hyper-surface.

New iteration needs to start from the dual of the above structure : it is a packing of dodecahedral cells, with disclinated dodecahedral cells ( 4 hexagonal faces and

12 pentagonal faces). The dodecahedral cells can be filled up with part of the {3,3,5} polytope like in the first iteration. The other cells are filled by part of disclinated {3,3,5} in order to keep the continuity of the disclination line. There is no difficulty to proceed like this an infinite number of timesfollowing the scheme :

Po (the {3,3,5} polytope )

d9a1

(the {5,3,3) polytope)

P (first iteration structure) + (filling with {3,3,5} parts)

1

Pi (ith iteration structure) du$1 (packkg of dodecahedra cells with disclinated cells)

Pi+l (filling with {3,3,5} parts or disclinated {3,3,5} parts)

The hierarchical polytope Pi presents a high degree of orientational order. Some of its edges are parallel to the P edges, but due to the distortion of the spherical space, this is not true for a11 edges.

In order to test the effect of this partial orientational order on the diffraction pattern of this structure, we have (15) mapped a small part of a P _{polytope on the}

2

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C9-88 JOURNAL DE PHYSIQUE

Fig. 9. : Two steps in the hierarchized network of disclination lines.

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-

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VI

-

CONCLUSIONS

The concept of orientational order has been introduced in the 2 D melting study. It is a way to describe States between the crystal and the liquid. More recently, it appears also very useful in 3 D systems. In this case, the concept of geometrical frustration seems to be very important. A compromise to solve this frustration can destroy the orientational order (in amorphous structure for example). Nevertheless, we can describe models with orientational order (3 D Penrose tiling or hierarchical polytopes), and there are real structures which show orientational order - the quasi-crystalline state.

In fact, hierarchical structure of intrinsic defect sets, self-similarity and orientational order are associated.

REFERENCES

/1/ Kosterlitz, J.M. and Thouless, D.J., J. Phys. C. 6 (1973) 1181 /2/ Nelson, D.R. and Halperin, B.I., Phys. Rev. B 19 (1979) 2457

/3/ A Review Paper by Bruckman W.F., Fisher, D.S. and Moncton, D.E., Sciences 217 (1982) 693

141 Steinhart, P.J., Nelson, D.R. and Ronchetti, M., Phys. Rev. Lett. 47 (1981) 1297; Phys. Rev. B 28 (1984) 784

151 Sadoc, J.F., J. of Non-Cryst. Sol. 44 (1981) 1

/6/ Coxeter, H.S.M., Regular polytopes - Dover Pub. N.Y. (1973)

171 Sadoc, J.F. and Citations here in. Lett. Journ. de Phys. (Paris) 44 (1983) L707 181 Gardner, M., Scientific American Jour. (1977)

/9/ Kramer, P., and Neri, P., Acta-Cryst. A 40 (1984) 580 /IO/ Duneau and Katz,Phys. Rev. Lett. 54 (1985) 2688 /Il/ MacKay, Physica 114A (1982) 609

1121 Mosseri, R., Sadoc, J.F., Proceeding on the Conf. on Structure of Non-Cryst. Mat.

-

Cambridge (1982)

1131 Shechtman, D., Blech, I., Gratias, D. ,Cahn

,

J.W., Phys. Rev. Lett. 53 (1984) 1951

1141 Sadoc, J.F., Mosseri, R., J. of Non-Cryst. Sol. 61-62 (1984) 487

1151 Mosseri, R., Sadoc, J.F., To appear in J. of Non Cryst. Sol. (Bloomfield 85, and to appear in J. de Phys. (Paris).J.de Phys.Lett.(Paris)45(1984)L-827