Medial Surface Analysis of Hyperbolic Structure

Hyperbolic interfaces and bicontinuous labyrinth structures are ubiquitous in condensed matter science. Examples are the void space channel system in a sandstone, isopotential surfaces of crystalline structures, bilayer interfaces in mesophases of liquid crystal self-assembly, membranes of complex cells, etc. Traditionally the analysis of hyperbolic surfaces relies on curvature properties of the interface surface or on topological aspects of the underlying graph network. This project aims at analysing hyperbolic structure from a slightly different perspective - their capability of tiling space with parallel sets of the surface. This approach has interesting implications for the notion of homogeneity of hyperbolic surfaces, and is important in understanding the physical origins of these complex structures. It is a numerical problem, since it relies on Voronoi diagram and medial axis methods, which are rarely accessible to analytic approaches.

Principal Investigator

Stephen Hyde
Applied Mathematics, RSPhysSE
Australian National University

Project

e99

Co-Investigator

Gerd Schroeder
Applied Mathematics, RSPhysSE
Australian National University

RFCD Codes

240202

Significant Achievements, Anticipated Outcomes and Future Work

We have extended the medial surface analysis of spatial structures from cubic to non-cubic infinite periodic minimal surfaces, whose geometry is generated using the Weierstrass-Enneper equations. These surfaces we have considered this period are one-parameter families of crystallogrpahic deformations (tetragonal and rhombohedral distortions of the cubic examples); of particular interest are those that exhibit cubic symmetry for singular values of the free parameter as they might serve as possible transition structures between the singular cubic members.

Our analysis of the rPD surface family has shed light on packing properties of this surface that serves as a pathway between the ubiquitous cubic members, the Primitive and the Diamond surface (see ref 2). We provide a robust definition for a channel diameter, and show that its fluctuations are minimal for the cubic Diamond surface and a horizontal inflection point for the Primitive surface. These fluctuations are linked to chain stretching contribution to the free energy of mesophases in liquid crystalline self-assembly.

Initial progress has also been made on the analysis of experimental data sets, such as computed X-ray or electron tomography. We are in the process of adapting and optimising our code to accommodate larger data sets, to extract smoothed surface dtaa from experimental voxel data and to handle the additional complication of noise. We have started to extract the spatial structure of the porous shell of sea urchins that shows an incommensurability between the microscopic crystal structure and the macroscopic porous structure.


 

Computational Techniques Used

This project uses a range of techniques from computational geometry, in particular Delaunay triangulations and Voronoi diagrams of 3D point sets and medial surface computations of triangulated data sets. The main computation, the Delaunay triangulation of large sets of surface vertices, has so far been done using Ernst Muecke's 'detri' code, but we are changing to a C/C++ implementation using CGAL, the Computational Geometry Algorithms Library. As 'detri' it allows for robust computation of Delaunay triangulations through the use of adaptive precision schemes, but is more efficient for very large data sets.

 

Publications, Awards and External Funding

External Funding and Awards

Australian-German Joint Research Co-operation Scheme, "Signatures of spatial morphology in ordered and disordered media" ($18k in toto).

Publications

G.E. Schroeder, S.J. Ramsden, A.G. Christy, S.T. Hyde, Medial surfaces of hyperbolic structures, Eur. Phys. J. B 35, 2003, 551-564
G.E. Schroeder, S.J. Ramsden, A. Fogden, S.T. Hyde, A rhombohedral family of minimal surfaces as a pathway between the P and D cubic mesophases, Physica A, 2004, in press
S.T. Hyde, G.E. Schroeder, Novel Surfactant mesostructural topologies: between lamellae and columnar (hexagonal) forms, Current Opinion in Colloid and Interface Science 8, 2003, 5-15