Order and Defects in Geometrically Frustrated Systems

In many physical (and perhaps biological) systems, there is a contradiction between the local order (local building rule) and the possibility for this local configuration to generate a perfect tiling of the euclidean space. It is the case, for instance, for amorphous metals where long range order is absent while the local configuration is rather well defined with the presence of five fold (icosahedral) symmetry1. Our aim is to provide a definition for the notions or “order” and “defects” in these materials. In a first step the underlying space geometry is modified in order for the local configuration to propagate freely (without “frustration”). This is often possible by allowing for curvature in space (either positive or negative). The obtained configuration in space is called the “Constant Curvature Idealization” of the disordered material. Let us take a simple example. In 2D a perfect pentagon tiling of the euclidean plane is impossible (fig. 1-a). It is however possible to tile a positively curved surface, a sphere, with pentagons leading to a pentagonal dodecahedron. Let us try now to densely pack spheres in 3D (this is an approximation to amorphous metals structures). Four spheres build a regular tetrahedron, but a perfect tetra-hedral packing is impossible because the tetrahedron dihedral angle is not a submultiple of 2π (fig. 1-b.). A perfect tetrahedral packing becomes possible Fig. 1geometrical frustration in 2 and 3 Dimensions if space is curved. if space is curved.