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Hierarchical Interlaced Networks of Disclination Lines in Non-Periodic Structures J.-F

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Hierarchical Interlaced Networks of Disclination Lines in Non-Periodic Structures J.-F 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 HAL Id: jpa-00210132 https://hal.archives-ouvertes.fr/jpa-00210132 Submitted on 1 Jan 1985 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. J. Physique 46 (1985) 1809-1826 NOVEMBRE 1985,1 1809 Classification Physics Abstracts 61.40D - 02.40 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. 1. Introduction. 3 coordinates are independent. The polytope model, or o Constant Curvature Idealization » (CCI) has Amorphous systems generally present an appreciable been extended to several other kinds of disordered amount of Short-Range Order (SRO). For example materials such as tetracoordinated covalent sys- amorphous metals can be well described by close tems [4]. A simple example is given by the packing of packing tetrahedra [1]. A regular tetrahedron is pentagonal dodecahedra which is forbidden in R3 the densest configuration for the packing of (as in the tetrahedral case) because of the polyhedron four equal spheres. The dense random pack- dihedral angle value. Packing these dodecahedra on ing of hard spheres problem can thus be mapped S3 leads to the regular polytope { 5, 3, 3 } which is on the tetrahedral packing problem. The dihedral dual of the above mentioned { 3, 3, 5 } and thus angle of a tetrahedron is not commensurable with 2 n, possesses the same symmetry group. However the consequently a perfect tiling of the Euclidean space values of the curvature associated with these two R3 is impossible with regular tetrahedra. Note that, polytopes are not identical (when scaled to the edge at this local level, the o frustration » (deviation to length) and this reflects the fact that the local angular perfectness) is of metrical rather than topological mismatch (in R3) are not equal. Polytope { 5, 3, 3 } nature. One of us (J.F.S.) has proposed to define an can be useful for modelling the « caged »-like tetra- ideal (unfrustrated) amorphous structure by allowing valent structures and those which are related like for curvature in the space in order for the local confi- amorphous ice for example. Several other CCI have ’ guration to propagate without defects throughout been proposed, like the regular honeycombs in the the whole space [2]. It is possible to pave a 3D manifold, 3D hyperbolic space H3 (with constant negative the hypersphere S3, by 600 regular tetrahedra arranged curvature) [2, 5] and the continuous double twisted by five around a common edge. The obtained geo- configuration of directors on S3 (as an ideal model metric object is called the polytope { 3, 3, 5 } using for the cholesteric blue phase) [6]. the standard notation [3]. Note that the underlying The idea is that the disordered material contains space S3 is 3 Dimensional although not Euclidean, « ordered >> regions where the local order can be put even if one often thinks of S3 as being imbedded in in correspondence with the ideal model, the comple- R4. Indeed S3 is the locus of points of R4 given by mentary regions being the locus of defects. We xi + x2 + x3 + x4 = R 2, which shows that only expect that a suitable map of the ideal model onto JOURNAL DE PHYSIQUE. - T. 46, ? 11, NOVEMBRE 1985 111 Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0198500460110180900 1810 R3 (minimizing the energy) will provide a realistic and its complexity is encoded in the information amorphous structure. The mapping introduces dis- content of the word. tortions and topological defects, among which the In a recent letter, hereafter referred to as 1 [14], we disclination lines play an important role. The final have shown how it is possible to bypass the above- structure consists in mixing of regions with positive mentioned difficulties and achieve the complete flat- curvature (where the local order is that of the poly- tening of the polytope. The key idea is, at each step, tope) and regions with negative curvature (the locus to introduce a disclination network (instead of a of defects), arranged such as to give zero curvature single disclination line) whose symmetry group is on the average. In this Corrugated Space Approach [7], contained in G. In the present paper we shall give a also called «variable curvature idealization », the complete description of how the method works. In local order is still perfect within coherence regions. order for this paper to be self-contained, we shall As has already been mentioned, disclination lines give a detailed description of the different tools which have proved to be natural defects for non-crystalline are needed (symmetry group, orthoscheme, bary- structure. S. N. Rivier [8] proposed that one type of centric transformation...). linear defects, characterized by oddness rather than In section 2, a simple 2-Dimensional example is intensity, is stable in real glasses as a result of the described, in which an icosidodecahedron is itera- double connectedness of the rotation group SO(3). tively flattened and gives rise to an asymptotic non- In the curved space approach, line defects can also periodic plane tiling. The main ideas that will be be classified using the homotopy theory of defects [9]. used later in the 3-Dimensional case are introduced Let us call Y the icosahedral group (subgroup of and visualized. SO(3)) and Y’ its lift in SU(2) (the covering group Section 3 contains the application of this method of SO(3)). The full symmetry group G* of polytope to the polytope { 3, 3, 5 } case. At each step a « defect » {3,3,5}, with 14 400 elements, is described in subnetwork is generated which has the same symme- Appendix B. G, the subgroup of G* containing only try properties as the structure itself. Also the matricial the direct symmetry operations (preserving orien- description of the iterative procedure is introduced. tation), is given by : G = Y’ x Y’/C2. C2 is the two- We show how to generate more disorder by combining element group. As shown by Nelson and Widom [10] two different iterative procedures which are compa- the defect lines that can be generated in the polytope tible since their associated defect structure share the { 3, 3, 5 } belong to the conjugacy classes of R = same symmetry group. In the last section (4) we ni (SO(4)/G) = Y’ x Y’. More recently Trebin [11] describe miscellaneous aspects and extensions of the has proposed a « coarser » classification for the line model. defects in polytope { 3, 3, 5 } by considering the first homology group of the order parameter space. Since 2. A 2D example : iterative flattening of an icosi- Y’ is a « perfect group » (it is isomorphic to its own dodecahedron. commutator subgroup), the homology group is trivial The icosidodecahedron (Fig. 1) is a quasi-regular which expresses the fact that any line can be transform- ed into any other by a suitable combination process. or Archimedean polyhedron [15]. It is noted 3 [3] One might hope to generate increasing numbers 5 of disclination lines in the { 3, 3, 5 } in order to achieve and shares the same symmetry group with the ico- sahedron and the dodecahedron a complete flattening of the polytope. We have { 3, 5 } { 5, 3 }. 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.
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