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 project considers a particular test system, that of disordered cellular-like 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 analysis of microstructures, in many cases with visual inspection as a key analysis tool. As an illustrative example, Fig. (a) and Fig. (b) show disordered packings of oil droplets within an emulsion. There is often some structure within the disorder, and in Fig. (c), some chain-like structures have been selected and coloured, guided by the human eye. Meanwhile, our rigorous mathematical computations adequately capture such features.
Goals of the Project and Methodologies
In this project, we have access to data on a variety of different cellular systems, including systems that develop over time. These include foam, grain growth, and packing simulations, metallic foam experiments, in situ tomography, and stochastic random geometries. We 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 the physical properties of the systems is also examined.
There is a clear benefit to the physics and material sciences community in having more robust descriptive mathematical tools for the analysis of material microstructures. One of the large open questions of material science is the relationship between structure and function, and a fundamental step in this direction is a deeper understanding of structure itself. From the other perspective, this interdisciplinary field challenges mathematics to develop a descriptive interface with the sciences. This is certainly an innovative approach to a new application field for mathematics. In the case of aluminum (Al-alloy) based foams that are produced by inserting air bubbles (by blowing in air or by a chemical reaction) into a metallic liquid that is solidified over the process, the resulting material consists of a metallic framework that contains voids. In the physics literature, such wet foams are usually modelled by replacing each (typically non-convex) foam bubble with a round ball of the same volume for which then their Laguerre–Voronoi diagram is computed.
Achievements and Outlook
As a first test case, we considered disordered foam structures coming from Al-alloy foam samples. However, there is 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 behavior on multiple length scales.
In an approach to better capture the non-convexity of the foam bubbles, we rely on the combinatorial part of Plateau’s rules for dry foams that four bubbles jointly meet to form a vertex of a dual triangulation and three bubbles lie around each edge of the dual triangulation. In our modelling, we allow parallel edges as well as multiple triangles and tetrahedra as simplices of a generalized triangulation—dual to a simple decomposition of the foam that replaces the standard Laguerre–Voronoi diagram (joint work with Paul Kamm, Helmholtz-Zentrum Berlin, and Junichi Nakagawa, U Tokyo).
In the images below, we present some of the key images that resulted from our topological modelling and corresponding computations: First, a cluster of 1911 Al-alloy foam bubbles (Fig. d1) along with our combinatorial-topology representation of the foam (Fig. d2). Second, an individual elongated, non-convex bubble (Fig. e1) with its reconstruction (Fig. e2) in comparison to the standard Laguerre-Voronoi description (Fig. e3). Third, two multi-connected cells (Fig. f1) with their combinatorial-topology representation (Fig. f2). Fourth, a 2D reconstruction of a stingray skeleton (Fig. g1-g2, joint work with Daniel Baum, Zuse-Institute-Berlin). Fifth, a 3D reconstruction of a plant (Fig. h1-h3, joint work with Vira Raichenko, U Potsdam, and Kay Schneitz et. al., Technische Universität München).
The future of this research is to apply the geometric and topological tools developed in this project to the vast playground of further systems beyond the ones presented here to obtain structural descriptions that are 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.
(a) Experimental image of an emulsion.
(b) Bubble nucleation within a metal foam.
(c) Force chains within an emulsion, experimental image
(d1) A sample of Al-alloy foam with 1,911 bubbles.
(d2) A corresponding combinatorial-topology modelling for the foam in Fig. d1.
(e1) An elongated non-convex bubble in sample d1.
(e2) Our combinatorial-topology modelling of the bubble in Fig. e1.
(e3) The Laguerre-Voronoi cell associated with the bubble in Fig. e2.
(f1) Two multi-connected bubbles in sample d1.
(f2) Our combinatorial-topology modelling of the multi-connected pair of bubbles in Fig. f1.
(g1) A partial reconstruction (with defects) of a stingray bone.
(g2) Our combinatorial-topology completion of the partial reconstruction in Fig. g1.
(h1) A plant cell sample consisting of 1,683 cells.
(h2) An elongated non-convex cell in sample h1.
(h3) Our combinatorial-topology modelling of the cell in Fig. h2.
Please insert any kind of pictures (photos, diagramms, simulations, graphics) related to the project in the above right field (Image with Text), by choosing the green plus image on top of the text editor. (You will be directed to the media library where you can add new files.)
(We need pictures for a lot of purposes in different contexts, like posters, scientific reports, flyers, website,…
Please upload pictures that might be just nice to look at, illustrate, explain or summarize your work.)
As Title in the above form please add a copyright.
And please give a short description of the picture and the context in the above textbox.
Don’t forget to press the “Save changes” button at the bottom of the box.
If you want to add more pictures, please use the “clone”-button at the right top of the above grey box.