APPROACHING PHASE TRANSITIONS WITH NODAL SURFACES MODELS
Stefano Leoni, Reinhard Nesper
Laboratorium für
Anorganische Chemie, ETH Zürich, Universitätstrasse 6, CH-8092
Zürich
leoni@inorg.chem.ethz.ch
Keywords: Topological Phase Transitions, Periodic Nodal Surfaces, Hyperbolic Tesselations.
The topological organisation of periodically structured matter can be described and understood in a very effective and elegant way by using Periodic Nodal Surfaces (PNS) [1,2].
PNS are generated by short Fourier summations, involving only a small set of reflections in reciprocal space. The choice of reflections near the origin of reciprocal space and of characteristic intrinsic symmetry for a particular space group reduces the sometimes enormous amount of detail a structure may contain to its general organisation. The roots of the resulting density spaces generate a family of continuous, orientable, 3-periodic surfaces (closely related to minimal surfaces) partitioning euclidean space in non-intersecting subspaces, showing arrangements of interacting units and directions of principal interactions. For example, strong covalent networks like those in zeolites may develop along such curved 2-dimensional manifolds; electrostatic interactions are found to develop orthogonal to PNS between oppositely charged units placed in different subspaces separated by the surface. [1,2]
Working on the curved shape of such Periodic Nodal Surfaces, hyperbolic tilings corresponding to open framewoks can be generated. On the one hand such tilings are projections of covalent netwoks placed in the labyrinths, onto the surface: The network of analcime tiles the gyroide surface in a [6242] network, faujasite can be projected similarly on a diamond surface. On the other hand, graphitic networks can be generated, where rigid units, like hexagonal tiles, correspond to places of higher point symmetry, for example around 3-fold axes on planar portions of the surface, and more flexibly to saddle points, like four- and eight-rings. Tilings can also be related to each other, using transformations relating nodal surfaces. [4]
If the transformation between nodal surfaces is smooth, the topological variations in their labyrinths are more dramatic in appearance and consequence; the 3-connected graph embedded in the labyrinths of the gyroide becomes four-connected, at the end of a Bonnet-like transformation. Using the measure of the genus of the surfaces, chemical frameworks can be topologically related to each other. The transition can then be modelled by interpolating between limiting, network-embedding PNS. This is of particular utility in the case of reconstructive transitions where no mechanism is known and the frameworks are not related to each other in a simple way.
This approach was successfully employed in the description of a pathway relating cristobalite, tridymite and quartz; while the first step is basically a shearing mechanism, the transition from high-tridymite to quartz involves a 3-connected network as an intermediate, like that of boron oxide [3]. Other networks can be related to each other in the same phase system, for example low- to high-cristobalite and CaCl2-type to stichovite. [4]
Polyanionic networks, like those appearing in silicides, have been related to each other; the polymorphism of SrSi2 was understood on the basis of a diamond-like network, where a dumb-bell of silicon atoms is situated at each node; a concerted rotation of these transforms the SrSi2 structure (SG P4332) to the a-ThSi2 type (SG I41/amd). The same concept can be applied for transformations to other structures, for example to the orthorhombic BaSi2, or the trigonal CaSi2 [4]
Different modifications of diamond can be related to each
other; hexagonal and rhombohedral graphite can be made to
transform continuously into cubic or hexagonal diamond along a
continuous path where the nodes of the surface labyrinths always
represent the atomic positions.[4]