The Hard Core of Soft Matter: Kugelsysteme in der weichen Materie

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In a Material World. Hyperbolische Geometrie in biologischen Materialien

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Emergence and function of complex form in self-assembly and biological cells

This year, 2017, marks the centenary of the first publication of 'On Growth and Form', by the extraordinary Scottish classicist, biologist and part-time mathematician, D'Arcy Wentworth Thompson [1]..

Absence of circular polarisation in reflections of butterfly wing scales with chiral gyroid structure

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 left- and 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

The microscopic structure of mono-disperse granular heaps and sediments of particles on inclined surfaces

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

Deformation of Platonic foam cells: Effect on growth rate

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

Chiral photonic crystals with four-fold symmetry: Band structure and S-parameters of eight-fold intergrown Gyroid nets

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

SPIRE—a software tool for bicontinuous phase recognition: application for plastid cubic membranes

Bicontinuous membranes in cell organelles epitomize nature’s ability to create complex functional nanostructures. Like their synthetic counterparts, these membranes are characterized by continuous membrane sheets draped onto topologically complex saddle-shaped surfaces with a periodic network-like structure. Their structure sizes, (around 50–500 nm), and fluid nature make transmission electron microscopy (TEM) the analysis method of choice to decipher their nanostructural features. Here we present a tool, Surface Projection Image Recognition Environment (SPIRE), to identify bicontinuous structures from TEM sections through interactive identification by comparison to mathematical “nodal surface” models. The prolamellar body (PLB) of plant etioplasts is a bicontinuous membrane structure with a key physiological role in chloroplast biogenesis. However, the determination of its spatial structural features has been held back by the lack of tools enabling the identification and quantitative analysis of symmetric membrane conformations. Using our SPIRE tool, we achieved a robust identification of the bicontinuous diamond surface as the dominant PLB geometry in angiosperm etioplasts in contrast to earlier long-standing assertions in the literature. Our data also provide insights into membrane storage capacities of PLBs with different volume proportions and hint at the limited role of a plastid ribosome localization directly inside the PLB grid for its proper functioning. This represents an important step in understanding their as yet elusive structure–function relationship

Geometry: The leading parameter for the Poisson’s ratio of bending-dominated cellular solids

Control over the deformation behaviour that a cellular structure shows in response to imposed external forces is a requirement for the effective design of mechanical metamaterials, in particular those with negative Poisson’s ratio. This article sheds light on the old question of the relationship between geometric microstructure and mechanical response, by comparison of the deformation properties of bar-and-joint-frameworks with those of their realisation as a cellular solid made from linear-elastic material. For ordered planar tessellation models, we find a classification in terms of the number of degrees of freedom of the framework model: first, in cases where the geometry uniquely prescribes a single deformation mode of the framework model, the mechanical deformation and Poisson’s ratio of the linearly-elastic cellular solid closely follow those of the unique deformation mode; the result is a bending-dominated deformation with negligible dependence of the effective Poisson’s ratio on the underlying material’s Poisson’s ratio and small values of the effective Young’s modulus. Second, in the case of rigid structures or when geometric degeneracy prevents the bending-dominated deformation mode, the effective Poisson’s ratio is material-dependent and the Young’s modulus View the MathML sourceE˜cs large. All analysed structures of this type have positive values of the Poisson’s ratio and large values of View the MathML sourceE˜cs. Third, in the case, where the framework has multiple deformation modes, geometry alone does not suffice to determine the mechanical deformation. These results clarify the relationship between mechanical properties of a linear-elastic cellular solid and its corresponding bar-and-joint framework abstraction. They also raise the question if, in essence, auxetic behaviour is restricted to the geometry-guided class of bending-dominated structures corresponding to unique mechanisms, with inherently low values of the Young’s modulus