Myfanwy Evans, Francisco Garcia-Moreno, Frank Lutz, John Sullivan
Ihab Sabik (TU)
01.04.2020 – 30.09.2023
The macroscopic physical properties of materials, from foams and polymer melts to steel and ceramics, are to a large extent governed by their internal microstructure. Understanding this microstructure – in particular its geometric and topological features – is complicated. This proposal considers a particular test system, that of disordered cellular structures, with the aim of exploiting current methods in random and computational geometry and topology to gain further insight into these materials, and to explore the creative mathematical ideas that result.
The macroscopic physical properties of materials, from foams and polymer melts to steel and ceramics, are to a large extent governed by their internal microstructure. Until now, this microstructure has typically been considered from either the microscopic or the macroscopic perspective. At intermediate length scales, these materials are characterized by complicated geometric and topological features. We are motivated by recent fruitful interactions between material scientists, geometers and topologists, which have yielded deep insights into material structure using and developing the descriptive power of mathematics.
Current techniques in material science and physics utilise various, non-rigorous techniques for the description and anaylsis of microstructures, in many cases with visual inspection as a key analysis tool. As an illustrative example, The figure show a disordered packing of oil droplets within an emulsion. There is some structure within the disorder, and here some chain-like structures have been selected and coloured, guided by the human eye. We aim to develop rigorous mathematical characterization to adequately capture such features.
Goals of the Project and Methodologies
We currently have access to data on a variety of different cellular systems, including systems that develop over time, from distinct systems. These include foam, grain growth and packing simulations, metallic foam experiments, in situ tomography and stochastic random geometries. We will examine these test systems using the topological techniques of combinatorial roundness and topological data analysis, and the geometric techniques of Minkowski tensor analysis. The goal of this process is to extract descriptive information about the structures, comparing and contrasting between systems. The link of these structural descriptions to physical properties of the systems will also be examined.
There is a clear benefit to the physics and material sciences community in having more robust descriptive mathematical tools for analysis. One of the large open questions of material science is the relationship of structure and function, and a fundamental step in this direction is a deeper understanding of structure itself. From the other perspective, this interdisciplinary field will challenge mathematics to develop the descriptive interface with the sciences. This is certainly an innovative approach to a new application field for mathematics.
Future Research and New Horizons
The proposal here covers a first test case, that of disordered cellular structures, however there are a wealth of material systems where descriptive geometry and topology are essential. Systems such as polymeric materials, with their long tangled filaments, or biological materials containing cellular, filamentous and network-like components, are highly complicated and display distinct behaviour on multiple length scales. The future of this research is to develop the geometric and topological tools beyond the system presented here, opening a vast playground of structural description that is crucial to fields right across the breadth of the natural sciences and engineering.
F. García-Moreno, P. H. Kamm, T. R. Neu, and J. Banhart. Time-resolved in situ tomography for the analysis of evolving metal-foam granulates. J. Synchrotron Rad., 25:1505–1508, 2018.
A. Giustiniani, S. Weis, C. Poulard, P. H. Kamm, F. García-Moreno, M. Schröter, and W. Drenckhan. Skinny emulsions take on granular matter. Soft Matter, 14(36):7310–7323, 2018.
M. E. Evans, G. E. Schröder-Turk, and A. M. Kraynik. A geometric exploration of stress in deformed liquid foams. Journal of Physics: Condensed Matter, 29:124004, 2017.
F. H. Lutz, J. K. Mason, E. A. Lazar, and R. D. MacPherson. Roundness of grains in cellular microstructures. Phys. Rev. E, 96:023001, 2017.
S. Hilgenfeldt, A. M. Kraynik, D. A. Reinelt, and J. M. Sullivan. The structure of foam cells: isotropic Plateau polyhedra. Europhys. Lett., 67:484–490, 2004.
F. García-Moreno, P. H. Kamm, T. R. Neu, F. Bülk, R. Mokso, C. M. Schlepütz, M. Stampanoni and J. Banhart. Using X-ray tomoscopy to explore the dynamics of foaming metal. Nature Communiations, 10:3762, 2019.
Experimental image of an emulsion.
Bubble nucleation within a metal foam.
Force chains within an emulsion, experimental image
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