How can we cloak the Bragg peaks of a lattice with random perturbations and will traces of the underlying long-range order persist nevertheless? Our affirmative answer to both questions has been published as an Editors’ Suggestion:

The Editors of Physical Review E have suggested our paper as a Highlight: “When a regular lattice is perturbed, traces of the original lattice usually remain visible as Bragg peaks in the diffraction pattern. The authors of this paper consider uniformly randomized lattices and show that fine-tuned distributions of perturbations can hide the Bragg peaks. Interestingly, as the strength of the perturbations increases, long-range order oscillates and Bragg peaks appear and disappear.”

In our paper, published in PNAS, we show how to construct a foam-based photonic crystal with a substantial band gap using the famous Weaire-Phelan foam.

The Weaire-Phelan foam has the smallest surface area among all known tessellations with equal-volume cells. While it has only a slightly smaller surface area than the Kelvin foam. Its corresponding photonic network provides a distinctly larger band gap and a much smaller critical refractive index. To determine the latter, we have created a detailed map of gap sizes.

Among the promising prospects that phoamtonics offers for applications are multifunctional characteristics, the self-organization of large photonic networks, and a high degree of isotropy.

In this paper, we combine physics, microbiology and mathematics to study how bacteria attach to nanostructured surfaces, and how topography of nanostructured substrates can directly quantify the adhesion.

Aiming for 3D-structured materials for or against bio-adhesion, we studied mechanisms by which the bacteria adhere to different types of materials. Our insights help, for example, to prevent adhesion of pathogens (like Staphylococcus aureus) before the bacteria can begin to grow a biofilm, which protects individual bacteria from attack by antibiotics.

We found that the measured forces with which the individual bacterial cells adhered to nanostructured surfaces decrease with an increasing size of these nanostructures. In fact, with our morphometric analysis based on Minkowski functionals, we found that the strength of the adhesion force can be predicted from measurements of the surface area that is accessible to the tethering proteins with which the bacteria adhere to the surface. More precisely, we found that the ratio of the experimentally measured adhesion forces at the rough surface compared to a smooth surface agrees with the fraction of surface area that is accessible to the tethering proteins, which are about 50 nm long.

It has been my great pleasure to open the 15th International Congress for Stereology and Image Analysis in Aarhus, Denmark, with my keynote on:

How to find a hidden order among disorder

where I gave an introduction and overview on hyperuniformity. The study of this anomalous suppression of density fluctuations on large length scales has shed light on a variety of seemingly unrelated fields, from the eyes of chicken to exotic many-particle ensembles and random matrices. The unique properties of hyperuniform amorphous materials (with a hidden order such that the system remains macroscopically uniform, despite not being crystalline) have recently led to intense research in physics, mathematics, material science, and biology. Aiming for an intuitive understanding of the rigorous mathematical definitions, I presented both basic concepts and recent examples.

I thank the organizers for a great conference in the last week of May 2019 with many stimulating discussions across disciplines.

It has been my great pleasure to organize together with Lisa Manning, Gregory Grason, and Gerd Schröder-Turk a GSOFT Focus Session at the APS March Meeting in Boston this year:

"Hyperuniformity and Optimal Tessellations: Structure, Formation and Properties"

After recent
breakthroughs in the search for ordered optimal tessellations (for
example, including Frank-Kasper phases in copolymer melts), now
findings of the optimal properties of amorphous tessellations are
emerging, e.g., in biological tissues.

At the same time, there have been intensive studies of amorphous systems with an anomalous suppression of density fluctuations on large length scales, known as hyperuniformity. This geometric concept qualitatively and quantitatively characterizes a hidden-order in amorphous states that allows for unique physical properties – combining those of crystalline and disordered phases. Thus it offers candidates for optimal amorphous tessellations of space.

The two invited speakers of our session on Thursday, March 7, were Jasna Brujic and Salvatore Torquato.

In this paper, we analyze the evolution of random point patterns and their Voronoi diagrams as “quantizer” of space. Applying an iterative local optimization of their so-called Quantizer energy, we show that the patterns converge to the apparently same effectively hyperuniform state regardless of their initial conditions.

M.A. Klatt, J. Lovrić, D. Chen, S. C. Kapfer, F. M. Schaller, P. W. A. Schönhöfer, B. S. Gardiner, A.-S. Smith, G. E. Schröder-Turk, S. Torquato. Universal hidden order in amorphous cellular geometries. Nature Communications 10, 811 (2019)

Partitioning space into cells with certain extreme geometrical properties is a central problem in many fields of science and technology. Here we investigate the Quantizer problem, defined as the optimisation of the moment of inertia of Voronoi cells, i.e., similarly-sized ‘sphere-like’ polyhedra that tile space are preferred. We employ Lloyd’s centroidal Voronoi diagram algorithm to solve this problem and find that it converges to disordered states associated with deep local minima. These states are universal in the sense that their structure factors are characterised by a complete independence of a wide class of initial conditions they evolved from. They moreover exhibit an anomalous suppression of long-wavelength density fluctuations and quickly become effectively hyperuniform. Our findings warrant the search for novel amorphous hyperuniform phases and cellular materials with unique physical properties.

On Oct 1st 2018, I joined the Department of Physics at Princeton University as a Postdoctoral Research Associate, continuing to combine mathematics and physics in my research on amorphous geometries, material physics, and spatial statistics.

We propose a class of goodness-of-fit tests for complete spatial randomness (CSR). In contrast to standard tests, our procedure utilizes a transformation of the data to a binary image, which is then characterized by geometric functionals. Under a suitable limiting regime, we derive the asymptotic distribution of the test statistics under the null hypothesis and almost sure limits under certain alternatives. The new tests are computationally efficient, and simulations show that they are strong competitors to other tests of CSR. The tests are applied to a real data set in gamma-ray astronomy, and immediate extensions are presented to encourage further work.

How do unique geometrical features of heterogeneous materials bring forth unique physical properties? Here, we study the link between the structure of sphere packings and a variety of physical properties using analytic approximations and rigorous bounds. In particular, we are interested in the most disordered among all mechanically stable packings of hard spheres (that are monodisperse and frictionless).

This maximally random jammed (MRJ) state exhibits an anomalous suppression of large-scale density fluctuations, known as hyperuniformity. Although the system is isotropic and locally disordered, it appears uniform on large scales. We compare the diffusion in the pore-space of the MRJ spheres to that of an equilibrium hard-sphere liquid or non-interacting spheres. Moreover, we study flow properties and effective conductivity; the latter also for anisotropic packings of spheroids. Therefore, we provide a comprehensive overview of rigorous bounds that connect these seemingly unrelated physical properties.

The most surprising result is found for electromagnetic waves that propagate through the MRJ sphere packings, where the wavelengths are much larger than the radii of the spheres. Usually disorder causes dissipation, but because of the unique property of hyperuniformity, the MRJ state forms, to a very good approximation, a dissipationless isotropic heterogeneous medium. This is demonstrated using an analytic strong-contrast expansion. It holds for any phase dielectric contrast ratio. The anomalous suppression of density fluctuations also suppresses the scattering of the electromagnetic waves.

This attribute could be useful for the design of photonic materials with novel structural color characteristics or color-sensing capabilities. Additive manufacturing fabrication techniques offer a simple production of samples for experiments with microwaves. So, our analytic results call for an experimental testing of these predicted qualitative trends in the physical properties associated with the MRJ structure.

The “Kulturpreis Bayern 2017” a price for culture and science awarded on October 26th 2017 by the Bayernwerk and the Federal State of Bavaria (Press Release in German).