Absence of circular polarisation in reflections of butterfly wing scales with chiral Gyroid structure - Saba, Matthias; Wilts, Bodo D.; Hielscher, Johannes; Schröder-Turk, Gerd (Materials Today, 2014)
The single Gyroid, a triply-periodic ordered chiral network of cubic symmetry, appears as a nanostructure in the green-colored wing scales of various butterflies. In lossless and perfectly ordered single Gyroid materials, the structural chirality leads to circularly polarized reflections from crystals oriented in the [100] direction. Here we report a circular polarisation study of the macroscopic reflections of the wing scales of Callophrys rubi and Teinopalpus imperialis that reveals no circular dichroism, that is, we find no significant difference in the reflectance values for left- and right-circularly polarized light. The reasons for the absence of circularly polarized reflections is likely to be a compound effect of various factors, including crystallite orientation, presence of both leftand right-handed single Gyroid enantiomers, and structural disorder. Each of these factors weakens, but does not fully extinguish, the circular polarisation signal. We further find a substantial amount of blue-absorbing pigment in those wing scales of C. rubi that are structured according to the single Gyroid. Numerical simulations demonstrate that absorption, while evidently reducing overall reflectance, does generally not reduce the circular dichroism strength. The experimental findings of this paper, however, clearly demonstrate that circular dichroism is absent from the reflections of the butterfly wing scale. Henceforth, the chiro-optical response of the idealised structure does not fulfil a biological photonic function.
EAM Import::2019-03-13

A Chiral Beamsplitter Inspired by Butterfly Nanostructures - Turner, Mark D.; Saba, Matthias; Zhang, Qiming; Cumming, Benjamin P.; Schröder-Turk, Gerd; Gu, Min (Frontiers in Optics 2013, 2013)
Here we present a chiral analog of the linearly polarizing beamsplitter. This microscopic device was built from a biomimetic chiral photonic crystal and separates left and right-handed circularly polarized light at near-infrared wavelengths.
EAM Import::2019-03-13

Adaptive optics enhanced direct laser writing of high refractive index gyroid photonic crystals in chalcogenide glass - Cumming, Benjamin P.; Turner, Mark D.; Schröder-Turk, Gerd; Debbarma, Sukanta; Luther-Davies, Barry; Gu, Min (Optics Express, 2014)
Chiral gyroid photonic crystals are fabricated in the high refractive index chalcogenide glass arsenic trisulfide with an adaptive optics enhanced direct laser writing system. The severe spherical aberration imparted when focusing into the arsenic trisulfide is mitigated with a defocus decoupled aberration compensation technique that reduces the level of aberration that must be compensated by over an order of magnitude. The fabricated gyroids are shown to have excellent uniformity after our adaptive optics method is employed, and the transmission spectra of the gyroids are shown to have good agreement with numerical simulations that are based on a uniform and diffraction limited fabrication resolution.
EAM Import::2019-03-13

Anomalous transport of a tracer on percolating clusters - Spanner, Markus; Hoefling, Felix; Schröder-Turk, Gerd; Mecke, Klaus; Franosch, Thomas (Journal of Physics: Condensed Matter, 2011)
We investigate the dynamics of a single tracer exploring a course of fixed obstacles in the vicinity of the percolation transition for particles confined to the infinite cluster. The mean-square displacement displays anomalous transport, which extends to infinite times precisely at the critical obstacle density. The slowing down of the diffusion coefficient exhibits power-law behavior for densities close to the critical point and we show that the mean-square displacement fulfills a scaling hypothesis. Furthermore, we calculate the dynamic conductivity as a response to an alternating electric field. Last, we discuss the non-Gaussian parameter as an indicator for heterogeneous dynamics.

Appearance of the single gyroid network phase in "nuclear pasta" matter - Schuetrumpf, Bastian; Klatt, Michael; Iida, K.; Schröder-Turk, Gerd; Maruhn, J. A.; Mecke, Klaus; Reinhard, Paul-Gerhard (Physical Review C - Nuclear Physics, 2015)
Nuclear matter under the conditions of a supernova explosion unfolds into a rich variety of spatially structured phases, called nuclear pasta. We investigate the role of periodic networklike structures with negatively curved interfaces in nuclear pasta structures, by static and dynamic Hartree-Fock simulations in periodic lattices. As the most prominent result, we identify for the first time the single gyroid network structure of cubic chiral I4(1)23 symmetry, a well-known configuration in nanostructured soft-matter systems, both as a dynamical state and as a cooled static solution. Single gyroid structures form spontaneously in the course of the dynamical simulations. Most of them are isomeric states. The very small energy differences from the ground state indicate its relevance for structures in nuclear pasta.

Beyond the percolation universality class: the vertex split model for tetravalent lattices - Nachtrab, Susan; Hoffmann, Matthias; Kapfer, Sebastian; Schröder-Turk, Gerd; Mecke, Klaus (New Journal of Physics, 2015)
We propose a statistical model defined on tetravalent three-dimensional lattices in general and the three-dimensional diamond network in particular where the splitting of randomly selected nodes leads to a spatially disordered network, with decreasing degree of connectivity. The terminal state, that is reached when all nodes have been split, is a dense configuration of self-avoiding walks on the diamond network. Starting from the crystallographic diamond network, each of the four-coordinated nodes is replaced with probability p by a pair of two edges, each connecting a pair of the adjacent vertices. For all values 0 <= p <= 1 the network percolates, yet the fraction f(p) of the system that belongs to a percolating cluster drops sharply at p(c) = 1 to a finite value f(p)(c). This transition is reminiscent of a percolation transition yet with distinct differences to standard percolation behaviour, including a finite mass f(p)(c) > 0 of the percolating clusters at the critical point. Application of finite size scaling approach for standard percolation yields scaling exponents for p -> p(c) that are different from the critical exponents of the second-order phase transition of standard percolation models. This transition significantly affects the mechanical properties of linear-elastic realizations (e.g. as custom-fabricated models for artificial bone scaffolds), obtained by replacing edges with solid circular struts to give an effective density phi. Finite element methods demonstrate that, as a low-density cellular structure, the bulk modulus K shows a cross-over from a compression-dominated behaviour, K(phi) alpha phi(k) with kappa approximate to 1, at p = 0 to a bending-dominated behaviour with kappa approximate to 2 at p = 1.
Keywords: percolation; networks in phase transitions; mechanical properties of solids

Bicontinuous minimal surface nanostructures for polymer blend solar cells - Kimber, Robin G. E.; Walker, Alison B.; Schröder-Turk, Gerd; Cleaver, Douglas J. (Physical Chemistry Chemical Physics, 2010)
This paper presents the first examination of the potential for bicontinuous structures such as the gyroid structure to produce high efficiency solar cells based on conjugated polymers. The solar cell characteristics are predicted by a simulation model that shows how the morphology influences device performance through integration of all the processes occurring in organic photocells in a specified morphology. In bicontinuous phases, the surface defining the interface between the electron and hole transporting phases divides the volume into two disjoint subvolumes. Exciton loss is reduced because the interface at which charge separation occurs permeates the device so excitons have only a short distance to reach the interface. As each of the component phases is connected, charges will be able to reach the electrodes more easily. In simulations of the current–voltage characteristics of organic cells with gyroid, disordered blend and vertical rod (rods normal to the electrodes) morphologies, we find that gyroids have a lower than anticipated performance advantage over disordered blends, and that vertical rods are superior. These results are explored thoroughly, with geminate recombination, i.e. recombination of charges originating from the same exciton, identified as the primary source of loss. Thus, if an appropriate materials choice could reduce geminate recombination, gyroids show great promise for future research and applications.
EAM Import::2019-03-13

Chiral Photonic Crystals with 4-fold Symmetry: Band Structure and S-Parameters of Eight-Fold Intergrown Gyroid Nets - Saba, Matthias; Turner, Mark D.; Gu, Min; Mecke, Klaus; Schröder-Turk, Gerd (Proceedings of SPIE, 2013)
The Single Gyroid, or srs, nanostructure has attracted interest as a circular-polarisation sensitive photonic material. We develop a group theoretical and scattering matrix method, applicable to any photonic crystal with symmetry I432, to demonstrate the remarkable chiral-optical properties of a generalised structure called 8-srs, obtained by intergrowth of eight equal-handed srs nets. Exploiting the presence of four-fold rotations, Bloch modes corresponding to the irreducible representations E- and E+ are identified as the sole and non-interacting transmission channels for right- and left-circularly polarised light, respectively. For plane waves incident on a finite slab of the 8-srs, the reflection rates for both circular polarisations are identical for all frequencies and transmission rates are identical up to a critical frequency below which scattering in the far field is restricted to zero grating order. Simulations show the optical activity of the lossless dielectric 8-srs to be large, comparable to metallic metamaterials, demonstrating its potential as a nanofabricated photonic material.
EAM Import::2019-03-13

Circular Dichroism in Biological Photonic Crystals and Cubic Chiral Nets - Saba, Matthias; Thiel, M.; Turner, M. D.; Hyde, S. T.; Gu, Min; Grosse-Brauckmann, K.; Neshev, D. N.; Mecke, Klaus; Schröder-Turk, Gerd (Physical Review Letters, 2011)
Nature provides impressive examples of chiral photonic crystals, with the notable example of the cubic so-called srs network (the label for the chiral degree-three network modeled on SrSi2) or gyroid structure realized in wing scales of several butterfly species. By a circular polarization analysis of the band structure of such networks, we demonstrate strong circular dichroism effects: The butterfly srs microstructure, of cubic I4(1)32 symmetry, shows significant circular dichroism for blue to ultraviolet light, that warrants a search for biological receptors sensitive to circular polarization. A derived synthetic structure based on four like-handed silicon srs nets exhibits a large circular polarization stop band of a width exceeding 30%. These findings offer design principles for chiral photonic devices.

Deformation of Platonic foam cells: Effect on growth rate - Evans, Myfanwy; Zirkelbach, Johannes; Schröder-Turk, Gerd; Kraynik, Andrew M.; Mecke, Klaus (Physical Review E, 2012)
The diffusive growth rate of a polyhedral cell in dry three-dimensional foams depends on details of shape beyond cell topology, in contrast to the situation in two dimensions, where, by von Neumann's law, the growth rate depends only on the number of cell edges. We analyze the dependence of the instantaneous growth rate on the shape of single foam cells surrounded by uniform pressure; this is accomplished by supporting the cell with films connected to a wire frame and inducing cell distortions by deforming the wire frame. We consider three foam cells with a very simple topology; these are the Platonic foam cells, which satisfy Plateau's laws and are based on the trivalent Platonic solids (tetrahedron, cube, and dodecahedron). The Surface Evolver is used to model cell deformations induced through extension, compression, shear, and torsion of the wire frames. The growth rate depends on the deformation mode and frame size and can increase or decrease with increasing cell distortion. The cells have negative growth rates, in general, but dodecahedral cells subjected to torsion in small wire frames can have positive growth rates. The deformation of cubic cells is demonstrated experimentally.

Direct relations between morphology and transport in Boolean models - Scholz, Christian; Klatt, Michael; Mecke, Klaus; Wirner, Frank; Hirneise, Daniel; Schröder-Turk, Gerd; Bechinger, Clemens (Physical Review E, 2015)
We study the relation of permeability and morphology for porous structures composed of randomly placed overlapping circular or elliptical grains, so-called Boolean models. Microfluidic experiments and lattice Boltzmann simulations allowus to evaluate a power-lawrelation between theEuler characteristic of the conducting phase and its permeability. Moreover, this relation is so far only directly applicable to structures composed of overlapping grains where the grain density is known a priori. We develop a generalization to arbitrary structures modeled by Boolean models and characterized by Minkowski functionals. This generalization works well for the permeability of the void phase in systems with overlapping grains, but systematic deviations are found if the grain phase is transporting the fluid. In the latter case our analysis reveals a significant dependence on the spatial discretization of the porous structure, in particular the occurrence of single isolated pixels. To link the results to percolation theory we performed Monte Carlo simulations of the Euler characteristic of the open cluster, which reveals different regimes of applicability for our permeability-morphology relations close to and far away from the percolation threshold.

Disordered spherical bead packs are anisotropic - Schröder-Turk, Gerd; Mickel, Walter; Schröter, Matthias; Delaney, Gary; Saadatfar, Mohammad; Senden, T.J.; Mecke, Klaus; Aste, T. (EPL - Europhysics Letters, 2010)
Investigating how tightly objects pack space is a long-standing problem, with relevance for many disciplines from discrete mathematics to the theory of glasses. Here we report on the fundamental yet so far overlooked geometric property that disordered mono-disperse spherical bead packs have significant local structural anisotropy manifest in the shape of the free space associated with each bead. Jammed disordered packings from several types of experiments and simulations reveal very similar values of the cell anisotropy, showing a linear decrease with packing fraction. Strong deviations from this trend are observed for unjammed configurations and for partially crystalline packings above 64%. These findings suggest an inherent geometrical reason why, in disordered packings, anisotropic shapes can fill space more efficiently than spheres, and have implications for packing effects in non-spherical liquid crystals, foams and structural glasses.

Elastic and Morphological Properties of Porous Biomaterials - Kapfer, Sebastian; Sporer, Susan; Hyde, Stephen T.; Mecke, Klaus; Schröder-Turk, Gerd (Biophysical Journal, 2010)

Erratum: Finding auxetic frameworks in periodic tessellations - Mitschke, Holger; Schwerdtfeger, Jan; Schury, Fabian; Stingl, Michael; Körner, Carolin; Singer, Robert; Robins, Vanessa; Mecke, Klaus; Schröder-Turk, Gerd (Advanced Materials, 2011)

Fabrication and characterization of three-dimensional biomimetic chiral composites - Turner, Mark D.; Schröder-Turk, Gerd; Gu, Min (Optics Express, 2011)
Here we show the fabrication and characterization of a novel class of biomimetic photonic chiral composites inspired by a recent finding in butterfly wing-scales. These three-dimensional networks have cubic symmetry, are fully interconnected, have robust mechanical strength and possess chirality which can be controlled through the composition of multiple chiral networks, providing an excellent platform for developing novel chiral materials. Using direct laser writing we have fabricated different types of chiral composites that can be engineered to form novel photonic devices. We experimentally show strong circular dichroism and compare with numerical simulations to illustrate the high quality of these three-dimensional photonic structures.
EAM Import::2019-03-14

Fiber Network Elasticity as Function of Crosslinker Density - Sporer, Susan; Kapfer, Sebastian; Arns, Christoph; Mecke, Klaus; Schröder-Turk, Gerd (Biophysical Journal, 2010)

Finding Auxetic Frameworks in Periodic Tessellations - Mitschke, Holger; Schwerdtfeger, Jan; Schury, Fabian; Stingl, Michael; Körner, Carolin; Singer, Robert; Robins, Vanessa; Mecke, Klaus; Schröder-Turk, Gerd (Advanced Materials, 2011)
It appears that most models for micro-structured materials with auxetic deformations were found by clever intuition, possibly combined with optimization tools, rather than by systematic searches of existing structure archives. Here we review our recent approach of finding micro-structured materials with auxetic mechanisms within the vast repositories of planar tessellations. This approach has produced two previously unknown auxetic mechanisms, which have Poisson's ratio v(ss) = -1 when realized as a skeletal structure of stiff incompressible struts pivoting freely at common vertices. One of these, baptized Triangle-Square Wheels, has been produced as a linear-elastic cellular structure from Ti-6Al-4V alloy by selective electron beam melting. Its linear-elastic properties were measured by tensile experiments and yield an effective Poisson's ratio v(LE) approximate to -0.75, also in agreement with finite element modeling. The similarity between the Poisson's ratios v(SS) of the skeletal structure and v(LE) of the linear-elastic cellular structure emphasizes the fundamental role of geometry for deformation behavior, regardless of the mechanical details of the system. The approach of exploiting structure archives as candidate geometries for auxetic materials also applies to spatial networks and tessellations and can aid the quest for inherently three-dimensional auxetic mechanisms.

Finite auxetic deformations of plane tessellations - Mitschke, Holger; Robins, Vanessa; Mecke, Klaus; Schröder-Turk, Gerd (Proceedings of the Royal Society A-Mathematical Physical and Engineering Sciences, 2013)
We systematically analyse the mechanical deformation behaviour, in particular Poisson's ratio, of floppy bar-and-joint frameworks based on periodic tessellations of the plane. For frameworks with more than one deformation mode, crystallographic symmetry constraints or minimization of an angular vertex energy functional are used to lift this ambiguity. Our analysis allows for systematic searches for auxetic mechanisms in archives of tessellations; applied to the class of one- or two-uniform tessellations by regular or star polygons, we find two auxetic structures of hexagonal symmetry and demonstrate that several other tessellations become auxetic when retaining symmetries during the deformation, in some cases with large negative Poisson ratios nu < -1 for a specific lattice direction. We often find a transition to negative Poisson ratios at finite deformations for several tessellations, even if the undeformed tessellation is infinitesimally non-auxetic. Our numerical scheme is based on a solution of the quadratic equations enforcing constant edge lengths by a Newton method, with periodicity enforced by boundary conditions.
Keywords: Poisson's ratio; strain amplification; tilings and tessellations; isostaticity; skeletal frameworks; cellular structures

Group theory of circular-polarization effects in chiral photonic crystals with four-fold rotation axes applied to the eight-fold intergrowth of gyroid nets - Saba, Matthias; Turner, Mark D.; Mecke, Klaus; Gu, Min; Schröder-Turk, Gerd (Physical Review B, 2013)
We use group or representation theory and scattering matrix calculations to derive analytical results for the band structure topology and the scattering parameters, applicable to any chiral photonic crystal with body-centered-cubic symmetry I432 for circularly polarized incident light. We demonstrate in particular that all bands along the cubic [100] direction can be identified with the irreducible representations E-+/-, A, and B of the C-4 point group. E+ and E- modes represent the only transmission channels for plane waves with wave vector along the Delta line, and E- and E+ are identified as noninteracting transmission channels for right-and left-circularly polarized light, respectively. Scattering matrix calculations provide explicit relationships for the transmission and reflectance amplitudes through a finite slab which guarantee equal transmission rates for both polarizations and vanishing ellipticity below a critical frequency, yet allowing for finite rotation of the polarization plane. All results are verified numerically for the so-called 8-srs geometry, consisting of eight interwoven equal-handed dielectric gyroid networks embedded in air. The combination of vanishing losses, vanishing ellipticity, near-perfect transmission, and optical activity comparable to that of metallic metamaterials makes this geometry an attractive design for nanofabricated photonic materials.

Jammed spheres: Minkowski tensors reveal onset of local crystallinity - Kapfer, Sebastian; Mickel, Walter; Mecke, Klaus; Schröder-Turk, Gerd (Physical Review E, 2012)
The local structure of disordered jammed packings of monodisperse spheres without friction, generated by the Lubachevsky-Stillinger algorithm, is studied for packing fractions above and below 64%. The structural similarity of the particle environments to fcc or hcp crystalline packings (local crystallinity) is quantified by order metrics based on rank-four Minkowski tensors. We find a critical packing fraction phi(c) approximate to 0.649, distinctly higher than previously reported values for the contested random close packing limit. At phi(c), the probability of finding local crystalline configurations first becomes finite and, for larger packing fractions, increases by several orders of magnitude. This provides quantitative evidence of an abrupt onset of local crystallinity at phi(c). We demonstrate that the identification of local crystallinity by the frequently used local bond-orientational order metric q(6) produces false positives and thus conceals the abrupt onset of local crystallinity. Since the critical packing fraction is significantly above results from mean-field analysis of the mechanical contacts for frictionless spheres, it is suggested that dynamic arrest due to isostaticity and the alleged geometric phase transition in the Edwards framework may be disconnected phenomena.

Local anisotropy of fluids using Minkowski tensors - Kapfer, Sebastian; Mickel, Walter; Schaller, Fabian; Spanner, Markus; Goll, Christian; Nogawa, T.; Ito, Nobuyasu; Mecke, Klaus; Schröder-Turk, Gerd (Journal of Statistical Mechanics-Theory and Experiment, 2010)
Statistics of the free volume available to individual particles have previously been studied for simple and complex fluids, granular matter, amorphous solids, and structural glasses. Minkowski tensors provide a set of shape measures that are based on strong mathematical theorems and easily computed for polygonal and polyhedral bodies such as free volume cells (Voronoi cells). They characterize the local structure beyond the two-point correlation function and are suitable to define indices 0 <= beta(a,b)(v) <= 1 of local anisotropy. Here, we analyze the statistics of Minkowski tensors for configurations of simple liquid models, including the ideal gas (Poisson point process), the hard disks and hard spheres ensemble, and the Lennard-Jones fluid. We show that Minkowski tensors provide a robust characterization of local anisotropy, which ranges from beta(a,b)(v) approximate to 0.3 for vapor phases to beta(a,b)(v) -> 1 for ordered solids. We find that for fluids, local anisotropy decreases monotonically with increasing free volume and randomness of particle positions. Furthermore, the local anisotropy indices beta(a,b)(v) are sensitive to structural transitions in these simple fluids, as has been previously shown in granular systems for the transition from loose to jammed bead packs.
Keywords: granular matter; random/ordered microstructures (theory); monodisperse fluids; mixtures and polydisperse fluids

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In particulate soft matter systems the average number of contacts extitZ of a particle is an important predictor of the mechanical properties of the system. Using x-ray tomography, we analyze packings of frictional, oblate ellipsoids of various aspect ratios α, prepared at different global volume fractions ϕ extitg

Miniature chiral beamsplitter based on gyroid photonic crystals - Turner, Mark D.; Saba, Matthias; Zhang, Qiming; Cumming, Benjamin P.; Schröder-Turk, Gerd; Gu, Min (Nature Photonics, 2013)
The linearly polarizing beamsplitter is a widely used optical component in photonics. It is typically built from a linearly birefringent crystal such as calcite, which has different critical reflection angles for s- and p-polarized light, leading to the transmission of one linear polarization and angled reflection of the other. However, the analogue for splitting circularly polarized light has yet to be demonstrated due to a lack of natural materials with sufficient circular birefringence. Here, we present a nano-engineered photonic-crystal chiral beamsplitter that fulfils this task. It consists of a prism featuring a nanoscale chiral gyroid network and can separate left- and right-handed circularly polarized light in the wavelength region around 1.615 µm. The structure is fabricated using a galvo-dithered direct laser writing method and could become a useful component for developing integrated photonic circuits that provide a new form of polarization control.
EAM Import::2019-03-12

Minimal surface scaffold designs for tissue engineering - Kapfer, Sebastian; Hyde, Stephen T.; Mecke, Klaus; Arns, Christoph H.; Schröder-Turk, Gerd (Biomaterials, 2011)
Triply-periodic minimal surfaces are shown to be a more versatile source of biomorphic scaffold designs than currently reported in the tissue engineering literature. A scaffold architecture with sheetlike morphology based on minimal surfaces is discussed, with significant structural and mechanical advantages over conventional designs. These sheet solids are porous solids obtained by inflation of cubic minimal surfaces to sheets of finite thickness, as opposed to the conventional network solids where the minimal surface forms the solid/void interface. Using a finite-element approach, the mechanical stiffness of sheet solids is shown to exceed that of conventional network solids for a wide range of volume fractions and material parameters. We further discuss structure property relationships for mechanical properties useful for custom-designed fabrication by rapid prototyping. Transport properties of the scaffolds are analyzed using Lattice-Boltzmann computations of the fluid permeability. The large number of different minimal surfaces, each of which can be realized as sheet or network solids and at different volume fractions, provides design flexibility essential for the optimization of competing design targets. (C) 2011 Elsevier Ltd. All rights reserved.
Keywords: Bone tissue engineering; Finite element analysis; Mechanical properties; Scaffold; Rapid prototyping

Minimal surfaces in nuclear pasta with the time-dependent hartree-fock approach - Schuetrumpf, Bastian; Klatt, Michael; Iida, K.; Schröder-Turk, Gerd; Maruhn, J. A.; Mecke, Klaus; Reinhard, Paul-Gerhard (Proceedings of Science, 2014)
In continuation to the studies of the whole variety of pasta shapes in [1], we present here calculations performed with the Hartree-Fock and time-dependent Hartree- Fock method concerning the mid-density range of pasta shapes: The slab-like, connected rod-like (p-surface) and the gyroidal shapes. On the one hand we present simulations of the dynamic formation of these shapes at finite temperature. On the other hand we calculate the binding energies of these shapes for varying simulation box lengths and mean densities. All of these shapes are found to be at least metastable. The slab shape has a slightly lower energy because of the lack of curvature, but among these three configurations the gyroidal shape is metastable for the widest range in mean density.
CRIS-Team Scopus Importer:2023-07-24

Minkowski Tensor Shape Analysis of Cellular, Granular and Porous Structures - Peukert, Wolfgang; Schröder-Turk, Gerd; Mickel, Walter; Kapfer, Sebastian; Klatt, Michael; Schaller, Fabian; Hoffmann, Matthias; Kleppmann, Nicola; Inayat, Amer; Armstrong, Patrick; Hug, Daniel; Reichelsdorfer, Martin; et al. (Advanced Materials, 2011)
Predicting physical properties of materials with spatially complex structures is one of the most challenging problems in material science. One key to a better understanding of such materials is the geometric characterization of their spatial structure. Minkowski tensors are tensorial shape indices that allow quantitative characterization of the anisotropy of complex materials and are particularly well suited for developing structure-property relationships for tensor-valued or orientation-dependent physical properties. They are fundamental shape indices, in some sense being the simplest generalization of the concepts of volume, surface and integral curvatures to tensor-valued quantities. Minkowski tensors are based on a solid mathematical foundation provided by integral and stochastic geometry, and are endowed with strong robustness and completeness theorems. The versatile definition of Minkowski tensors applies widely to different types of morphologies, including ordered and disordered structures. Fast linear-time algorithms are available for their computation. This article provides a practical overview of the different uses of Minkowski tensors to extract quantitative physically-relevant spatial structure information from experimental and simulated data, both in 2D and 3D. Applications are presented that quantify (a) alignment of co-polymer films by an electric field imaged by surface force microscopy; (b) local cell anisotropy of spherical bead pack models for granular matter and of closed-cell liquid foam models; (c) surface orientation in open-cell solid foams studied by X-ray tomography; and (d) defect densities and locations in molecular dynamics simulations of crystalline coppe
Keywords: geometric; statistic; spatial; anisotropy; shape; morphology; crystal

Minkowski tensors of anisotropic spatial structure - Schröder-Turk, Gerd; Mickel, W.; Kapfer, Sebastian; Schaller, Fabian; Breidenbach, B.; Hug, Daniel; Mecke, Klaus (New Journal of Physics, 2013)
This paper describes the theoretical foundation of and explicit algorithms for a novel approach to morphology and anisotropy analysis of complex spatial structure using tensor-valued Minkowski functionals, the so-called Minkowski tensors. Minkowski tensors are generalizations of the well-known scalar Minkowski functionals and are explicitly sensitive to anisotropic aspects of morphology, relevant for example for elastic moduli or permeability of microstructured materials. Here we derive explicit linear-time algorithms to compute these tensorial measures for three-dimensional shapes. These apply to representations of any object that can be represented by a triangulation of its bounding surface; their application is illustrated for the polyhedral Voronoi cellular complexes of jammed sphere configurations and for triangulations of a biopolymer fibre network obtained by confocal microscopy. The paper further bridges the substantial notational and conceptual gap between the different but equivalent approaches to scalar or tensorial Minkowski functionals in mathematics and in physics, hence making the mathematical measure theoretic formalism more readily accessible for future application in the physical sciences.

Monte Carlo simulations of confined hard sphere fluids - Schröder-Turk, Gerd; Kuczera, S.; Hoerndlein, D.; Krach, Florian; Hiester, T.; Mecke, Klaus (Physics Procedia, 2010)
We describe a grand-canonical Monte Carlo simulation of a hard sphere fluid confined to porous media with arbitrary pore shape. The pore geometry is given by a triangulated surface representation of the solid-void interface. This algorithm is developed to study the dependence of thermodynamic quantities, such as the average density and adsorption, on the shape of the pore. Particular pore shape geometries that are analysed are the labyrinthine domains bounded by so-called triply-periodic minimal surfaces.
EAM Import::2019-03-12

Morphology and Linear-Elastic Moduli of Random Network Solids - Nachtrab, Susan; Kapfer, Sebastian; Arns, Christoph H.; Madadi, Mahyar; Mecke, Klaus; Schröder-Turk, Gerd (Advanced Materials, 2011)
The dependence of linear-elastic properties on effective density is studied for porous network solids, by voxel-based finite element methods. The same dependence is found for solid structures derived from Poisson-Voronoi processes and from confocal microscopy images of collagen scaffolds. We recover the power-law for the bulk modulus for low densities and suggest a functional form for the cross-over to a high-density porous solid.

Networklike Propagation of Cell-Level Stress in Sheared Random Foams - Evans, Myfanwy; Kraynik, Andrew M.; Reinelt, Douglas A.; Mecke, Klaus; Schröder-Turk, Gerd (Physical Review Letters, 2013)
Quasistatic simple shearing flow of random monodisperse soap froth is investigated by analyzing SURFACE EVOLVER simulations of spatially periodic foams. Elastic-plastic behavior is caused by irreversible topological rearrangements (T1s) that occur when Plateau's laws are violated; the first T1 determines the elastic limit and frequent T1 avalanches sustain the yield-stress plateau at large strains. The stress and shape anisotropy of individual cells is quantified by Q, a scalar derived from an interface tensor that gauges the cell's contribution to the global stress. During each T1 avalanche, the connected set of cells with decreasing Q, called the stress release domain, is networklike and nonlocal. Geometrically, the networklike nature of the stress release domains is corroborated through morphological analysis using the Euler characteristic. The stress release domain is distinctly different from the set of cells that change topology during a T1 avalanche. Our results highlight the connection between the unique rheological behavior of foams and the complex large-scale cooperative rearrangements of foam cells that accompany distinctly local topological transitions.

Non-universal Voronoi cell shapes in amorphous ellipsoid packs - Schröter, Matthias; Schaller, Fabian; Kapfer, Sebastian; Hilton, James E.; Cleary, Paul W.; Mecke, Klaus; de Michele, Christiano; Schilling, Tanja; Saadatfar, Mohammad; Delaney, Gary; Schröder-Turk, Gerd (EPL - Europhysics Letters, 2015)
In particulate systems with short-range interactions, such as granular matter or simple fluids, local structure determines the macroscopic physical properties. We analyse local structure metrics derived from the Voronoi diagram of oblate ellipsoids, for various aspect ratios and global packing fractions φg. We focus on jammed static configurations of frictional ellipsoids, obtained by tomographic imaging and by discrete element method simulations. The rescaled distribution of local packing fractions φl, defined as the ratio of particle volume and its Voronoi cell volume, is found to be independent of the particle aspect ratio, and coincide with results for sphere packs. By contrast, the typical Voronoi cell shape, quantified by the Minkowski tensor anisotropy index β = β02,0, points towards a difference between random packings of spheres and those of oblate ellipsoids. While the average cell shape β of all cells with a given value of is similar in dense and loose jammed sphere packings, the structure of dense and loose ellipsoid packings differs substantially such that this does not hold true.

Observation of optical activity in dielectric biomimetic 8-srs networks - Turella, Fabio; Cumming, Benjamin P.; Schröder-Turk, Gerd; Gu, Min (Optics Letters, 2015)
We demonstrate the first experimental realization of the 8-srs network by direct laser writing in a dielectric medium. The optical properties of this photonic crystal are both numerically simulated and experimentally measured. The numerical simulations indicate a strong optical activity (OA) for regions of the spectrum close to the unit cell size (3 extgreekmm) and a high sensitivity to angular resolution. Experimentally, we observe 40° of OA for a region of the spectrum between 3 and 3.5 extgreekmm, confirming both simulation predictions.
EAM Import::2019-03-08

Pattern-fluid interpretation of chemical turbulence - Scholz, Christian; Schröder-Turk, Gerd; Mecke, Klaus (Physical Review E, 2015)
The spontaneous formation of heterogeneous patterns is a hallmark of many nonlinear systems, from biological tissue to evolutionary population dynamics. The standard model for pattern formation in general, and for Turing patterns in chemical reaction-diffusion systems in particular, are deterministic nonlinear partial differential equations where an unstable homogeneous solution gives way to a stable heterogeneous pattern. However, these models fail to fully explain the experimental observation of turbulent patterns with spatio-temporal disorder in chemical systems. Here we introduce a pattern-fluid model as a general concept where turbulence is interpreted as a weakly interacting ensemble obtained by random superposition of stationary solutions to the underlying reaction-diffusion system. The transition from turbulent to stationary patterns is then interpreted as a condensation phenomenon, where the nonlinearity forces one single mode to dominate the ensemble. This model leads to better reproduction of the experimental concentration profiles for the "stationary phases" and reproduces the turbulent chemical patterns observed by Q. Ouyang and H. L. Swinney [Chaos 1, 411 (1991)].

Permeability of Porous Materials Determined from the Euler Characteristic - Scholz, Christian; Wirner, Frank; Götz, Jan; Rüde, Ulrich; Schröder-Turk, Gerd; Mecke, Klaus; Bechinger, Clemens (Physical Review Letters, 2012)
We study the permeability of quasi-two-dimensional porous structures of randomly placed overlapping monodisperse circular and elliptical grains. Measurements in microfluidic devices and lattice Boltzmann simulations demonstrate that the permeability is determined by the Euler characteristic of the conducting phase. We obtain an expression for the permeability that is independent of the percolation threshold and shows agreement with experimental and simulated data over a wide range of porosities. Our approach suggests that the permeability explicitly depends on the overlapping probability of grains rather than their shape. © 2012 American Physical Society.
UnivIS-Import:2015-03-09:Pub.2012.tech.IMMD.lsinfs.permea

Robust Pore Size Analysis of Filamentous Networks from Three-Dimensional Confocal Microscopy - Mickel, Walter; Münster, Stefan; Jawerth, Louise M.; Vader, David A.; Weitz, David A.; Sheppard, Adrian P.; Mecke, Klaus; Fabry, Ben; Schröder-Turk, Gerd (Biophysical Journal, 2008)
We describe a robust method for determining morphological properties of. lamentous biopolymer networks, such as collagen or other connective tissue matrices, from confocal microscopy image stacks. Morphological properties including pore size distributions and percolation thresholds are important for transport processes, e. g., particle diffusion or cell migration through the extracellular matrix. The method is applied to fluorescently labeled fiber networks prepared from rat-tail tendon and calf-skin collagen, at concentrations of 1.2, 1.6, and 2.4 mg/ml. The collagen fibers form an entangled and branched network. The medial axes, or skeletons, representing the collagen fibers are extracted from the image stack by threshold intensity segmentation and distance-ordered homotopic thinning. The size of the fiuid pores as defined by the radii of largest spheres that fit into the cavities between the collagen fibers is derived from Euclidean distance maps and maximal covering radius transforms of the fluid phase. The size of the largest sphere that can traverse the fluid phase between the collagen fibers across the entire probe, called the percolation threshold, was computed for both horizontal and vertical directions. We demonstrate that by representing the fibers as the medial axis the derived morphological network properties are both robust against changes of the value of the segmentation threshold intensity and robust to problems associated with the point-spread function of the imaging system. We also provide empirical support for a recent claim that the percolation threshold of a fiber network is close to the fiber diameter for which the Euler index of the networks becomes zero.

Set Voronoi diagrams of 3D assemblies of aspherical particles - Schaller, Fabian M.; Kapfer, Sebastian C.; Evans, Myfanwy; Hoffmann, Matthias; Aste, Tomaso; Saadatfar, Mohammad; Mecke, Klaus; Delaney, Gary W.; Schröder-Turk, Gerd (TAYLOR & FRANCIS LTD, 2013)
Several approaches to quantitative local structure characterization for particulate assemblies, such as structural glasses or jammed packings, use the partition of space provided by the Voronoi diagram. The conventional construction for spherical mono-disperse particles, by which the Voronoi cell of a particle is that of its centre point, cannot be applied to configurations of aspherical or polydisperse particles. Here, we discuss the construction of a Set Voronoi diagram for configurations of aspherical particles in three-dimensional space. The Set Voronoi cell of a given particle is composed of all points in space that are closer to the surface (as opposed to the centre) of the given particle than to the surface of any other; this definition reduces to the conventional Voronoi diagram for the case of mono-disperse spheres. An algorithm for the computation of the Set Voronoi diagram for convex particles is described, as a special case of a Voronoi-based medial axis algorithm, based on a triangulation of the particles' bounding surfaces. This algorithm is further improved by a pre-processing step based on morphological erosion, which improves the quality of the approximation and circumvents the problems associated with small degrees of particle-particle overlap that may be caused by experimental noise or soft potentials. As an application, preliminary data for the volume distribution of disordered packings of mono-disperse oblate ellipsoids, obtained from tomographic imaging, is computed.
Keywords: Set Voronoi diagram; ellipsoid packings; random close packing; aspherical particles; medial axes and surfaces; Area Voronoi diagram; navigation map; skeleton by zone of influence

Shortcomings of the bond orientational order parameters for the analysis of disordered particulate matter - Mickel, Walter; Kapfer, Sebastian; Schröder-Turk, Gerd; Mecke, Klaus (Journal of Chemical Physics, 2013)
Local structure characterization with the bond-orientational order parameters q(4), q(6), ... introduced by Steinhardt et al. [Phys. Rev. B 28, 784 (1983)] has become a standard tool in condensed matter physics, with applications including glass, jamming, melting or crystallization transitions, and cluster formation. Here, we discuss two fundamental flaws in the definition of these parameters that significantly affect their interpretation for studies of disordered systems, and offer a remedy. First, the definition of the bond-orientational order parameters considers the geometrical arrangement of a set of nearest neighboring (NN) spheres, NN(p), around a given central particle p; we show that the choice of neighborhood definition can have a bigger influence on both the numerical values and qualitative trend of q(l) than a change of the physical parameters, such as packing fraction. Second, the discrete nature of neighborhood implies that NN(p) is not a continuous function of the particle coordinates; this discontinuity, inherited by q(l), leads to a lack of robustness of the q(l) as structure metrics. Both issues can be avoided by a morphometric approach leading to the robust Minkowski structure metrics q(l)'. These q(l)' are of a similar mathematical form as the conventional bond-orientational order parameters and are mathematically equivalent to the recently introduced Minkowski tensors [G. E. Schroder-Turk et al., Europhys. Lett. 90, 34001 (2010); S. Kapfer et al., Phys. Rev. E 85, 030301(R) (2012)]. (C) 2013 American Institute of Physics. [http://dx.doi.org/10.1063/1.4774084]

Structural similarity between dry and wet sphere packings - Weis, Simon; Schröder-Turk, Gerd; Schröter, Matthias (New Journal of Physics, 2019)
The mechanical properties of granular materials change significantly in the presence of a wetting liquid which creates capillary bridges between the particles. This results e.g. in a reduced value of the volume fraction when a packing is prepared with added liquid. Here we use x-ray tomography to demonstrate that this change in mechanical properties is not accompanied by structural differences between dry and wet sphere packings when compared at the same volume fraction. We characterize the structure of the packings by the average numbers of contacts of each sphere < Z > and the shape isotropy <beta(2,0)(0)> of the Voronoi cells of the particles. Additionally, we show that the number of liquid bridges per sphere < B > is approximately equal to < Z > + 2, independent of the volume fraction of the packing. These findings will be helpful in guiding the development of both particle-based models and continuum mechanical descriptions of wet granular matter.
CRIS-Team WoS Importer:2019-04-30

Symmetry detection of auxetic behaviour in 2D frameworks - Mitschke, Holger; Schröder-Turk, Gerd; Mecke, Klaus; Fowler, P. W.; Guest, S. D. (EPL - Europhysics Letters, 2013)
A symmetry-extended Maxwell treatment of the net mobility of periodic bar-and-joint frameworks is used to derive a sufficient condition for auxetic behaviour of a 2D material. The type of auxetic behaviour that can be detected by symmetry has Poisson's ratio -1, with equal expansion/contraction in all directions, and is here termed equiauxetic. A framework may have a symmetry-detectable equiauxetic mechanism if it belongs to a plane group that includes rotational axes of order n = 6, 4, or 3. If the reducible representation for the net mobility contains mechanisms that preserve full rotational symmetry (A modes), these are equiauxetic. In addition, for n = 6, mechanisms that halve rotational symmetry (B modes) are also equiauxetic. Copyright (c) EPLA, 2013

Tensorial Minkowski functionals and anisotropy measures for planar patterns - Schröder-Turk, Gerd; Kapfer, Sebastian; Breidenbach, B.; Beisbart, C.; Mecke, Klaus (Journal of Microscopy, 2010)
Quantitative measures for anisotropic characteristics of spatial structure are needed when relating the morphology of microstructured heterogeneous materials to tensorial physical properties such as elasticity, permeability and conductance. Tensor-valued Minkowski functionals, defined in the framework of integral geometry, provide a concise set of descriptors of anisotropic morphology. In this article, we describe the robust computation of these measures for microscopy images and polygonal shapes. We demonstrate their relevance for shape description, their versatility and their robustness by applying them to experimental data sets, specifically microscopy data sets of non-equilibrium stationary Turing patterns and the shapes of ice grains from Antarctic cores.
Keywords: Anisotropy; integral geometry; microstructured and cellular; materials; Minkowski functionals; morphology; strain and deformation; Turing patterns

Tensorial Minkowski functionals of triply periodic minimal surfaces - Mickel, Walter; Schröder-Turk, Gerd; Mecke, Klaus (Interface Focus, 2012)
A fundamental understanding of the formation and properties of a complex spatial structure relies on robust quantitative tools to characterize morphology. Asystematic approach to the characterization of average properties of anisotropic complex interfacial geometries is provided by integral geometry which furnishes a family of morphological descriptors known as tensorial Minkowski functionals. These functionals are curvature-weighted integrals of tensor products of position vectors and surface normal vectors over the interfacial surface. We here demonstrate their use by application to non-cubic triply periodic minimal surface model geometries, whose Weierstrass parametrizations allow for accurate numerical computation of the Minkowski tensors.
Keywords: morphology; anisotropy; lipid self-assembly; minimal surface; porous material; integral geometry

The microscopic structure of mono-disperse granular heaps and sediments of particles on inclined surfaces - Topic, Nikola; Schaller, Fabian; Schröder-Turk, Gerd; Pöschel, Thorsten (Soft Matter, 2016)
Granular heaps of particles created by deposition of mono-disperse particles raining from an extended source of finite size are characterized by a non-homogeneous field of density. It was speculated that this inhomogeneity is due to the transient shape of the sediment during the process of construction of the heap, thus reflecting the history of the creation of the heap. By comparison of structural characteristics of the heap with sediments created on top of inclined planes exploiting the method of Minkowski tensors, we provide further evidence to support this hypothesis. Moreover, for the case of sediments generated by homogeneous rain on surfaces, we provide relationships between the inclination of the surface and the Minkowski measures characterizing the isotropy of local particle environments.

Tomographic Analysis of Jammed Ellipsoid Packings - Schröter, Matthias; Schaller, Fabian; Neudecker, Max; Saadatfar, Mohammad; Delaney, Gary; Mecke, Klaus; Schröder-Turk, Gerd (American Institute of Physics, 2013)
Disordered packings of ellipsoidal particles are an important model for disordered granular matter. Here we report a way to determine the average contact number of ellipsoid packings from tomographic analysis. Tomographic images of jammed ellipsoid packings prepared by vertical shaking of loose configurations are recorded and the positions and orientations of the ellipsoids are reconstructed. The average contact number can be extracted from a contact number scaling (CNS) function. The size of the particles, that may vary due to production inaccuracies, can also be determined by this method.
Keywords: Granular matter; Ellipsoids; Tomography; Contact numbers

Tuning Elasticity of Open-Cell Solid Foams and Bone Scaffolds via Randomized Vertex Connectivity - Nachtrab, Susan; Kapfer, Sebastian; Rietzel, Dominik; Drummer, Dietmar; Madadi, Mahyar; Arns, Christoph; Kraynik, Andrew M.; Schröder-Turk, Gerd; Mecke, Klaus (Advanced Engineering Materials, 2012)
Tuning mechanical properties of and fluid flow through open-cell solid structures is a challenge for material science, in particular for the design of porous structures used as artificial bone scaffolds in tissue engineering. We present a method to tune the effective elastic properties of custom-designed open-cell solid foams and bone scaffold geometries by almost an order of magnitude while approximately preserving the pore space geometry and hence fluid transport properties. This strong response is achieved by a change of topology and node coordination of a network-like geometry underlying the scaffold design. Each node of a four-coordinated network is disconnected with probability p into two two-coordinated nodes, yielding network geometries that change continuously from foam- or network-like cellular structures to entangled fiber bundles. We demonstrate that increasing p leads to a strong, approximately exponential decay of mechanical stiffness while leaving the pore space geometry largely unchanged. This result is obtained by both voxel-based finite element methods and compression experiments on laser sintered models. The physical effects of randomizing network topology suggest a new design paradigm for solid foams, with adjustable mechanical properties. Copyright © 2012 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.
UnivIS-Import:2015-03-09:Pub.2012.tech.FT.FT-KT.tuning