(Updated: 4th June 1999)

•  Summary
•  Introduction
•  Outline of proposed research
•  Programme
•  Research team and collaborators
•  External parties
•  Bibliography

Summary:

"The structure" of a condensed material, be it an atomic crystal, a lipid-water mixture in a living cell, a polymer melt, an oil- or melt-bearing rock, or the skeletal tissue of a sea-urchin, is a remarkably incoherent concept, that can evince as many answers as disciplines of those answering. Physicists and chemists seek structure at the atomic and molecular scales, materials and earth scientists slightly larger, engineers still larger. To understand structure/function fully, one must probe many length scales simultaneously. For example, we now know that correlations occur at many scales in rocks, governing transport of fluids within the rock, dramatically affecting oil recovery potential, the formation of ore deposits, and even the nucleation of earthquakes. A second example relates to the differences between biological and inorganic form, of central importance to the current debate over the possibility of extra-terrestrial life. Those perceived differences are becoming blurred, due to the recognition of common length scales in both domains. Biomineralisation, the meeting ground of bio and geo materials, is an exquisite interplay of molecular and atomic interactions. Our aim is to pursue studies of self-assembly in biological, geological and synthetic materials beyond the traditional domains of physics and chemistry to include all relevant length scales. Materials include the micron-sized porous networks in biological skeletons, in fractured rocks and in composite materials such as papers and coatings, multi-continuous molecular mixtures such as novel three- and four-continuous liquid crystals and pore morphologies of geological materials.

Introduction:

The Department of Applied Mathematics has been intersted in understanding of self-assembly of molecules - particularly amphiphilic systems - for many years.

That self-assembly work has naturally evolved into a more general study of molecular and atomic structuring in condensed crystalline materials, including microporous catalysts, such as zeolites and theoretical frameworks. En route a new "language of shape" has developed, based on the recognition of the ubiquity of hyperbolic geometries in condensed materials. That work is summarised in a book published in 1997 ("The Language of Shape", cf. bibliography), that has occasioned widespread debate (e.g. Nature, 387, p. 249, 1997). In tandem with those developments loosely grouped within the colloidal area, research has evolved to consider ever-longer length scales in both ordered and disordered media, including rocks, polymer blends, microemulsions, and biominerals, which conventionally lie within the rubric of "porous media". This is of immense interest to the petroleum industry, the paper industry, as well as to fundamental biological, geological and materials sciences.

Outline of research programme:

How can we distinguish living from non-living forms? That issue is central to current speculations concerning extra-terrestrial life (e.g. on Mars). Work in progress in Applied Mathematics on gel crystallisation, biomineralisation and templating is of direct relevance (figure 1). This is a very high profile area, likely to become increasingly so with planned manned NASA missions to Mars, and delivery of samples to Earth. Our contribution would be essentially that of devil's advocate: How broad is the spectrum of non-living forms, particularly at "biological" length scales (µm)?

  Figure 1: Which is biological and which is purely inorganic? Left: a "biological remnant" (?) from a Martian meteorite [D. Mackay et al., Science, August (1997)] Right: a carbonate crystal aggregate, grown in silica gel, scale bar = 1µm (Dougherty and Hyde, 1998).

How can we describe porous forms of arbitrary shape (figure 2)? Given modern imaging techniques, scientists are observing generic forms of extreme geometric complexity. To understand their genesis and generality, we must be able to quantify their forms. That process must involve tools from integral, statistical and differential geometry and topology. We aim to continue our contributions to that development (figures 2-4) [bibliography A]. The problem will be attacked from two directions. The first involves analytic generation of ordered surface morphologies and quantification of the effect of topology and symmetry on geometrical characteristics. The characteristics include intrinsic parameters, such as the distribution of curvatures over the surface, as well as extrinsic euclidean measures, including two- and higher-point correlations functions. The second approach will extend extant work on random morphologies generated from random vector fields, (e.g. Gaussian random field models), and explore the relations between randomness and resulting intrinsic and extrinsic geometrical characteristics.

Figure 2: (L to R) A termite nest, the calcitic skeleton of the sea urchin Riccio atomico, and the pore space of Berea sandstone (App. Maths/Vizlab ANUSF images).

Can we generate a complete inventory of forms of (particularly porous) networks? That (seemingly trivial) question is now amenable to solution, using novel techniques of statistical and non-euclidean geometry, developed in the past few years in-house. A definitive catalogue is a central issue for ab initio understanding of composite networks in general, found in rocks, liquid crystals and cells [bibliography B]. The work will focus on generation of ordered networks/graphs, produced from reticulations of triply periodic hyperbolic surfaces, and characterisation of those networks via topological measures, including coordination sequences (a series describing numbers of neighbours, next neighbours,… about any vertex in the net), ring sizes and topological density and geometric measures, including tortuosity and spatial correlations. We then plan to extend that inventory to cover disordered structures, using, for example, topological transformations developed by Nick Rivier and associates (Louis Pasteur University, Strasbourg) with reference to foam structures.

Figure 3: Modelling real materials: (L to R) A random sponge (cf. termite nest, fig. 2), a 3-periodic minimal surface (cf. sea urchin skeleton) an intersection-set of two random fields (cf. sandstone) (App. Maths/Vizlab images).

How does the property of a material depend on its structure? The characterisation and realistic modelling of random disordered materials as diverse as soils, sedimentary rocks, minerals, wood, bone, polymer composites, foams, catalysts, coatings, gels, concretes and ceramics has been a major problem for physicists, materials scientists, earth scientists and engineers for years. We aim to predict properties (e.g., diffusive, multi-phase flow, mechanical, optical, dielectric) of the materials from measures of the morphology and topology (figure 4).

Figure 4: Pore topology can be quantified by the pore graph. (L to R) Pore graphs of the termite nest (cf. fig. 2), the sea-urchin skeleton and the sandstone. The colouring of graph edges reflects the pore dimensions (App. Maths/Vizlab images).

What is the fundamental physics governing formation of porous materials, and what generic features govern formation of similar morphologies over a range of length scales (cf. figure 2)? Physical concepts developed for liquid crystals, such as curvature energy can be extended to generic materials. In addition, explication of the contribution of surface entropy to structure is needed. Both areas are being pioneered here [bibliography D].

Programme:

Theory is expected to focus squarely on:

• quantitative structural characterisation of random and ordered porous materials, invoking the tools developed and already used to characterise complex fluids and polymer and mineral composites;
• develop the physics underlying those structural formations, using the skills present in-house to characterise enthalpic and entropic and transport measures as a function of morphologies. A large component of computational work is expected to be involved, given the complex morphologies, and visualisation aspects of the programme. That work will be boosted by access to the Wedge, allowing manipulation and exploration of complex morphologies in real time.

Experimental efforts will be directed towards determination of mesostructured (and larger) morphologies of rocks, papers, biominerals and crystallite aggregates grown in gels. This work will involve two- and three-dimensional imaging techniques (X-ray CT, g-ray CT, transmission electron microscopy), as well as small angle X-ray and light scattering techniques and high-pressure/high temperature hot isostatic pressing and deformation studies at RSES. The imaging techniques are available at the ANU (Applied Maths, EM Unit), MicroAnalytical Research Centre, Physics, Melbourne University (proton tomographic probe), and our own high-resolution X-ray CT facility being built in-house.

Research team and collaborators:

• Prof. Stephen Hyde (App. Maths., RSPhysSE)
• Prof. Barry Ninham (App. Maths., RSPhysSE)
• Prof. Stjepan Marcelja (App. Maths., RSPhysSE)
• Dr. Mark Knackstedt (App. Maths., RSPhysSE)
• Prof. Stephen Cox (joint RSES & Dept. of Geology, The Faculties)
• Dr. P. Evans (Forestry, The Faculties)
• Dr. B. Gingold (ANU Supercomputer Facility)

External parties:

• Prof. W.V. Pinczewski (U.N.S.W.; Australian Petroleum C.R.C.)
• Dr. L. Paterson (C.S.I.R.O. for Petroleum Resources; Australian Petroleum C.R.C.)
• A. Prof. D. Jamieson (MARC Centre, Physics, Melbourne University)
• Dr. A.P. Roberts (currently Fulbright Fellow at Princeton University, from mid-1999 at the Centre for Microscopy, University of Queensland).
• Dr. F. Tiberg (Head, Forest Products Section, Y.K.I., Stockholm)
• Dr. J. Daicic (Project Area Manager Paper Coatings, Y.K.I., Stockholm)

Bibliography:

S.T. Hyde, S. Andersson, K. Larsson, Z. Blum, T. Landh, S. Lidin and B.W. Ninham, "The Language of Shape", Elsevier, (1997).

[A, works related to morphology quantification]:

1. D.F. Evans, D.J. Mitchell and B.W. Ninham, "Oil, water and surfactant: Properties and conjectured structure of simple microemulsions", J. Phys. Chem., 90, 2817, (1986).
2. M.A. Knackstedt and S.F. Cox, "Percolation and the pore geometry of crustal rocks", Phys. Rev. E (Rapid Comm.), 51, R5181 (1996).
3. H. Jinnai, T. Koga, Y. Nishikawa, T. Hashimoto and S.T. Hyde, "Curvature determination of interfaces in spinodally phase-separated structures of a polymer blend", Phys. Rev. Lett., 78, 2248-2251, (1997).
4. M.A. Knackstedt, A.P. Sheppard and W.V. Pinczewski, "Porosimetry on correlated grids: Evidence for extended heterogeneity at the pore scale in rocks", Phys. Rev. E (Rapid Comm.), 58, R6923, (1998).
5. S.T. Hyde, "Sponges", in Foams, Emulsions and Cellular Materials, N. Rivier and J.-F. Sadoc (eds.), NATO Advanced Study Series, Kluwer, 1999.

[B, cataloguing form]:

1. A.P. Roberts and M.A. Knackstedt, "Correlations in model composite materials", Phys. Rev E, 54, 2313, (1996).
2. A. Fogden and S.T. Hyde, "Parametrisation of triply periodic minimal surfaces", Acta Cryst., A48, 575, (1992).
3. S.T. Hyde, "The density of three-dimensional nets", Acta Cryst., A50, 753-759, (1994).
4. M. O'Keeffe and S.T. Hyde, "The asymptotic behaviour of coordination sequences for the 4-connected nets of zeolites and related structures", Z. Kristallogr., 211, 73-78, (1996).
5. S.T. Hyde and S. Ramsden, "Crystalline networks: Two-dimensional non-euclidean geometry and topology", in "Mathematical Chemistry", 6, D. Bonchev and D.H. Rouvray (eds.), (in press, 1999).

[C, structure-property relations]:

1. M.A. Knackstedt, B.W. Ninham and M. Monduzzi, "Diffusion in model disordered media", Phys. Rev Lett., 75, 653, (1995).
2. M.A. Knackstedt and A.P. Roberts, "Morphology and macroscopic properties of conducting polymer blends", Macromolecules, 29, 1369, (1996).
3. S.T. Hyde, "Swelling and structure. Analysis of the topology and geometry of lamellar and sponge lyotropic mesophases", Langmuir, 13, 842-851, (1997).

[D, physics of formation]:

1. S.T. Hyde, "Hyperbolic and elliptic layer warping in some sheet alumino-silicates", Phys. Chem. Minerals, 20, 190-200, (1993).
2. P. Pieruschka and S. Marcelja, "Monte Carlo simulation of curvature-elastic interfaces", Langmuir, 10, 345, (1994).
3. S.T. Hyde and M. O'Keeffe, "Elastic warping of graphitic carbon sheets: relative energies of some fullerenes, schwarzites and buckytubes", Phil. Trans. R. Soc. Lond. A, 354, 1999-, (1996).
4. S. Marcelja, "Entropy of phase separated structures", Physica A, (1996).