Composite material in the sea urchin Cidaris rugosa: ordered and disordered micrometre-scale bicontinuous geometries.

The sponge-like biomineralized calcite materials found in echinoderm skeletons are of interest in terms of both structure formation and biological function. Despite their crystalline atomic structure, they exhibit curved interfaces that have been related to known triply periodic minimal surfaces. Here, we investigate the endoskeleton of the sea urchin Cidaris rugosa that has long been known to form a microstructure related to the Primitive surface. Using X-ray tomography, we find that the endoskeleton is organized as a composite material consisting of domains of bicontinuous microstructures with different structural properties. We describe, for the first time, the co-occurrence of ordered single Primitive and single Diamond structures and of a disordered structure within a single skeletal plate. We show that these structures can be distinguished by structural properties including solid volume fraction, trabeculae width and, to a lesser extent, interface area and mean curvature. In doing so, we present a robust method that extracts interface areas and curvature integrals from voxelized datasets using the Steiner polynomial for parallel body volumes. We discuss these very large-scale bicontinuous structures in the context of their function, formation and evolution.

Curvature in Biological Systems: Its Quantification, Emergence, and Implications across the Scales.

Surface curvature both emerges from, and influences the behavior of, living objects at length scales ranging from cell membranes to single cells to tissues and organs. The relevance of surface curvature in biology is supported by numerous experimental and theoretical investigations in recent years. In this review, first, a brief introduction to the key ideas of surface curvature in the context of biological systems is given and the challenges that arise when measuring surface curvature are discussed. Giving an overview of the emergence of curvature in biological systems, its significance at different length scales becomes apparent. On the other hand, summarizing current findings also shows that both single cells and entire cell sheets, tissues or organisms respond to curvature by modulating their shape and their migration behavior. Finally, the interplay between the distribution of morphogens or micro-organisms and the emergence of curvature across length scales is addressed with examples demonstrating these key mechanistic principles of morphogenesis. Overall, this review highlights that curved interfaces are not merely a passive by-product of the chemical, biological, and mechanical processes but that curvature acts also as a signal that co-determines these processes.

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.

Low-temperature statistical mechanics of the Quantizer problem: Fast quenching and equilibrium cooling of the three-dimensional Voronoi liquid.

The quantizer problem is a tessellation optimization problem where point configurations are identified such that the Voronoi cells minimize the second moment of the volume distribution. While the ground state (optimal state) in 3D is almost certainly the body-centered cubic lattice, disordered and effectively hyperuniform states with energies very close to the ground state exist that result as stable states in an evolution through the geometric Lloyd's algorithm [M. A. Klatt et al. Nat. Commun. 10, 811 (2019)]. When considered as a statistical mechanics problem at finite temperature, the same system has been termed the "Voronoi liquid" by Ruscher, Baschnagel, and Farago [Europhys. Lett. 112, 66003 (2015)]. Here, we investigate the cooling behavior of the Voronoi liquid with a particular view to the stability of the effectively hyperuniform disordered state. As a confirmation of the results by Ruscher et al., we observe, by both molecular dynamics and Monte Carlo simulations, that upon slow quasi-static equilibrium cooling, the Voronoi liquid crystallizes from a disordered configuration into the body-centered cubic configuration. By contrast, upon sufficiently fast non-equilibrium cooling (and not just in the limit of a maximally fast quench), the Voronoi liquid adopts similar states as the effectively hyperuniform inherent structures identified by Klatt et al. and prevents the ordering transition into a body-centered cubic ordered structure. This result is in line with the geometric intuition that the geometric Lloyd's algorithm corresponds to a type of fast quench.

Quo vadis biophotonics? Wearing serendipity and slow science as a badge of pride, and embracing biology.

This article is a reflection on the themes of the Faraday Discussion meeting on 'Biological and bio-inspired optics' held from 20 to 22 July 2020. It is a personal perspective on the nature of this field as a broad and interdisciplinary field that has led to a sound understanding of the material properties of biological nanostructured and optical materials. The article describes how the nature of the field and the themes of the conference are reflected in particular in work on the 3D bicontinuous biophotonic nanostructures known as single gyroids and in bicontinuous structures more broadly. Such single gyroid materials are found for example in the butterfly Thecla opisena, where the questions of biophotonic response, of bio-inspired optics, of the relationship between structure and function, and of the relationship between natural and synthetic realisations are closely interlinked. This multitude of facets of research on single gyroid structures reflects the beauty of the broader field of biophotonics, namely as a field that lives through embracing the serendipitous discovery of the biophotonic marvels that nature offers to us as seeds for in-depth analysis and understanding. The meandering nature of its discoveries, and the need to accept the slowness that comes from exploration of intellectually new or foreign territory, mean that the field shares some traits with biological evolution itself. Looking into the future, I consider that a closer engagement with living tissue and with the biological questions of function and formation, rather than with the materials science of biological materials, will help ensure the continuing great success of this field.

Self-assembly and entropic effects in pear-shaped colloid systems. I. Shape sensitivity of bilayer phases in colloidal pear-shaped particle systems.

The role of particle shape in self-assembly processes is a double-edged sword. On the one hand, particle shape and particle elongation are often considered the most fundamental determinants of soft matter structure formation. On the other hand, structure formation is often highly sensitive to details of shape. Here, we address the question of particle shape sensitivity for the self-assembly of hard pear-shaped particles by studying two models for this system: (a) the pear hard Gaussian overlap (PHGO) and (b) the hard pears of revolution (HPR) model. Hard pear-shaped particles, given by the PHGO model, are known to form a bicontinuous gyroid phase spontaneously. However, this model does not replicate an additive object perfectly and, hence, varies slightly in shape from a "true" pear-shape. Therefore, we investigate in the first part of this series the stability of the gyroid phase in pear-shaped particle systems. We show, based on the HPR phase diagram, that the gyroid phase does not form in pears with such a "true" hard pear-shaped potential. Moreover, we acquire first indications from the HPR and PHGO pair-correlation functions that the formation of the gyroid is probably attributed to the small non-additive properties of the PHGO potential.

Self-assembly and entropic effects in pear-shaped colloid systems. II. Depletion attraction of pear-shaped particles in a hard-sphere solvent.

We consider depletion effects of a pear-shaped colloidal particle in a hard-sphere solvent for two different model realizations of the pear-shaped colloidal particle. The two models are the pear hard Gaussian overlap (PHGO) particles and the hard pears of revolution (HPR). The motivation for this study is to provide a microscopic understanding for the substantially different mesoscopic self-assembly properties of these pear-shaped colloids, in dense suspensions, that have been reported in the previous studies. This is done by determining their differing depletion attractions via Monte Carlo simulations of PHGO and HPR particles in a pool of hard spheres and comparing them with excluded volume calculations of numerically obtained ideal configurations on the microscopic level. While the HPR model behaves as predicted by the analysis of excluded volumes, the PHGO model showcases a preference for splay between neighboring particles, which can be attributed to the special non-additive characteristics of the PHGO contact function. Lastly, we propose a potentially experimentally realizable pear-shaped particle model, the non-additive hard pear of revolution model, which is based on the HPR model but also features non-additive traits similar to those of PHGO particles to mimic their depletion behavior.

Geometric effects in random assemblies of ellipses.

Assemblies of anisotropic particles commonly appear in studies of active many-body systems. However, in two dimensions, the geometric ramifications of the finite density of such objects are not entirely understood. To fully characterize these effects, we perform an in-depth study of random assemblies generated by a slow compression of frictionless elliptical particles. The obtained configurations are then analysed using the Set Voronoi tessellation, which takes the particle shape into account. Not only do we analyse most scalar and vectorial morphological measures, which are commonly discussed in the literature or which have recently been addressed in experiments, but we also systematically explore the correlations between them. While in a limited range of parameters similarities with findings in 3D assemblies could be identified, important differences are found when a broad range of aspect ratios and packing fractions are considered. The data discussed in this study should thus provide a unique reference set such that geometric effects and differences from random assemblies could be clearly identified in more complex systems, including ones with soft and active particles that are typically found in biological systems.

Patchy particles by self-assembly of star copolymers on a spherical substrate: Thomson solutions in a geometric problem with a color constraint.

Confinement or geometric frustration is known to alter the structure of soft matter, including copolymeric melts, and can consequently be used to tune structure and properties. Here we investigate the self-assembly of ABC and ABB 3-miktoarm star copolymers confined to a spherical shell using coarse-grained dissipative particle dynamics simulations. In bulk and flat geometries the ABC stars form hexagonal tilings, but this is topologically prohibited in a spherical geometry which normally is alleviated by forming pentagonal tiles. However, the molecular architecture of the ABC stars implies an additional 'color constraint' which only allows even tilings (where all polygons have an even number of edges) and we study the effect of these simultaneous constraints. We find that both ABC and ABB systems form spherical tiling patterns, the type of which depends on the radius of the spherical substrate. For small spherical substrates, all solutions correspond to patterns solving the Thomson problem of placing mobile repulsive electric charges on a sphere. In ABC systems we find three coexisting, possibly different tilings, one in each color, each of them solving the Thomson problem simultaneously. For all except the smallest substrates, we find competing solutions with seemingly degenerate free energies that occur with different probabilities. Statistically, an observer who is blind to the differences between B and C can tell from the structure of the A domains if the system is an ABC or an ABB star copolymer system.

A Dynamic Optical Signal in a Nocturnal Moth.

The wings of butterflies and moths generate some of the most spectacular visual displays observed in nature [1-3]. Particularly striking effects are seen when light interferes with nanostructure materials in the wing scales, generating bright, directional colors that often serve as dynamic visual signals [4]. Structural coloration is not known in night-flying Lepidoptera, yet here we show a highly unusual form of wing coloration in a nocturnal, sexually dimorphic moth, Eudocima materna (Noctuidae). Males feature three dark wing patches on the dorsal forewings, and the apparent size of these patches strongly varies depending on the angle of the wing to the viewer. These optical special effects are generated using specialized wing scales that are tilted on the wing and behave like mirrors. At near-normal incidence of light, these "mirror scales" act as thin-film reflectors to produce a sparkly effect, but when light is incident at ∼20°-30° from normal, the reflectance spectrum is dominated by the diffuse scattering of the underlying, black melanin-containing scales, causing a shape-shifting effect. The strong sexual dimorphism in the arrangement and architecture of the scale nanostructures suggests that these patterns might function for sexual signaling. Flickering of the male's wings would yield a flashing, supernormal visual stimulus [5] to a viewer located 20°-30° away from the vertical, while being invisible to a viewer directly above the animal. Our findings reveal a novel use of structural coloration in nature that yields a dynamic, time-dependent achromatic optical signal that may be optimized for visual signaling in dim light.

Nature-Inspired Design and Application of Lipidic Lyotropic Liquid Crystals.

Amphiphilic lipids aggregate in aqueous solution into a variety of structural arrangements. Among the plethora of ordered structures that have been reported, many have also been observed in nature. In addition, due to their unique morphologies, the hydrophilic and hydrophobic domains, very high internal interfacial surface area, and the multitude of possible order-order transitions depending on environmental changes, very promising applications have been developed for these systems in recent years. These include crystallization in inverse bicontinuous cubic phases for membrane protein structure determination, generation of advanced materials, sustained release of bioactive molecules, and control of chemical reactions. The outstanding diverse functionalities of lyotropic liquid crystalline phases found in nature and industry are closely related to the topology, including how their nanoscopic domains are organized. This leads to notable examples of correlation between structure and macroscopic properties, which is itself central to the performance of materials in general. The physical origin of the formation of the known classes of lipidic lyotropic liquid crystalline phases, their structure, and their occurrence in nature are described, and their application in materials science and engineering, biology, medical, and pharmaceutical products, and food science and technology are exemplified.

Universal hidden order in amorphous cellular geometries.

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.

Density functional theory for hard uniaxial particles: Complex ordering of pear-shaped and spheroidal particles near a substrate.

We develop a density functional for hard particles with a smooth uniaxial shape (including non-inversion-symmetric particles) within the framework of fundamental measure theory. By applying it to a system of tapered, aspherical liquid-crystal formers, reminiscent of pears, we analyse their behaviour near a hard substrate. The theory predicts a complex orientational ordering close to the substrate, which can be directly related to the particle shape, in good agreement with our simulation results. Furthermore, the lack of particle inversion-symmetry implies the possibility of alternating orientations in subsequent layers as found in a smectic/lamellar phase of such particles. Both theory and Monte Carlo simulations confirm that such ordering occurs in our system. Our results are relevant for adsorption processes of asymmetric colloidal particles and molecules at hard interfaces and show once again that tapering strongly affects the properties of orientationally ordered phases.

Metallic gyroids with broadband circular dichroism.

Circular dichroism is a useful property for filtering or separating beams containing opposite spin angular momentum. Of the many geometries exhibiting circular dichroism, the gyroid has proven to be an excellent template for exploring circular dichroism in three dimensions. However, the bandwidth of the circular dichroism from dielectric gyroids is limited by its narrow circularly polarized stop band. Here we investigate conductive silver gyroid micro-structures using direct laser writing of polymeric templates followed by the electroless deposition of a uniform silver coating. We show that the transformation from dielectric to silver gyroid micro-structure can increase the circular dichroism bandwidth by close to a factor of 3.

Purely entropic self-assembly of the bicontinuous Ia3d gyroid phase in equilibrium hard-pear systems.

We investigate a model of hard pear-shaped particles which forms the bicontinuous Ia[Formula: see text]d structure by entropic self-assembly, extending the previous observations of Barmes et al. (2003 Phys. Rev. E68, 021708. (doi:10.1103/PhysRevE.68.021708)) and Ellison et al. (2006 Phys. Rev. Lett.97, 237801. (doi:10.1103/PhysRevLett.97.237801)). We specifically provide the complete phase diagram of this system, with global density and particle shape as the two variable parameters, incorporating the gyroid phase as well as disordered isotropic, smectic and nematic phases. The phase diagram is obtained by two methods, one being a compression-decompression study and the other being a continuous change of the particle shape parameter at constant density. Additionally, we probe the mechanism by which interdigitating sheets of pears in these systems create surfaces with negative Gauss curvature, which is needed to form the gyroid minimal surface. This is achieved by the use of Voronoi tessellation, whereby both the shape and volume of Voronoi cells can be assessed in regard to the local Gauss curvature of the gyroid minimal surface. Through this, we show that the mechanisms prevalent in this entropy-driven system differ from those found in systems which form gyroid structures in nature (lipid bilayers) and from synthesized materials (di-block copolymers) and where the formation of the gyroid is enthalpically driven. We further argue that the gyroid phase formed in these systems is a realization of a modulated splay-bend phase in which the conventional nematic has been predicted to be destabilized at the mesoscale due to molecular-scale coupling of polar and orientational degrees of freedom.

Butterfly gyroid nanostructures as a time-frozen glimpse of intracellular membrane development.

The formation of the biophotonic gyroid material in butterfly wing scales is an exceptional feat of evolutionary engineering of functional nanostructures. It is hypothesized that this nanostructure forms by chitin polymerization inside a convoluted membrane of corresponding shape in the endoplasmic reticulum. However, this dynamic formation process, including whether membrane folding and chitin expression are simultaneous or sequential processes, cannot yet be elucidated by in vivo imaging. We report an unusual hierarchical ultrastructure in the butterfly Thecla opisena that, as a solid material, allows high-resolution three-dimensional microscopy. Rather than the conventional polycrystalline space-filling arrangement, a gyroid occurs in isolated facetted crystallites with a pronounced size gradient. When interpreted as a sequence of time-frozen snapshots of the morphogenesis, this arrangement provides insight into the formation mechanisms of the nanoporous gyroid material as well as of the intracellular organelle membrane that acts as the template.

Mean-intercept anisotropy analysis of porous media. I. Analytic formulae for anisotropic Boolean models.

PURPOSE: Structure-property relations, which relate the shape of the microstructure to physical properties such as transport or mechanical properties, need sensitive measures of structure. What are suitable fabric tensors that quantify the shape of anisotropic heterogeneous materials? The mean intercept length is among the most commonly used characteristics of anisotropy in porous media, for example, of trabecular bone in medical physics. METHODS: We analyze the orientation-biased Boolean model, a versatile stochastic model that represents microstructures as overlapping grains with an orientation bias towards a preferred direction. This model is an extension of the isotropic Boolean model, which has been shown to truthfully reproduce multi-functional properties of isotropic porous media. We explain the close relationship between the concept of intersections with test lines to the elaborate mathematical theory of queues, and how explicit results from the latter can be directly applied to characterize microstructures. RESULTS: In this series of two papers, we provide analytic formulas for the anisotropic Boolean model and demonstrate often overlooked conceptual shortcomings of this approach. Queuing theory is used to derive simple and illustrative formulas for the mean intercept length. It separates into an intensity-dependent and an orientation-dependent factor. The global average of the mean intercept length can be expressed by local characteristics of a single grain alone. CONCLUSIONS: We thus identify which shape information about the random process the mean intercept length contains. The connection between global and local quantities helps to interpret observations and provides insights into the possibilities and limitations of the analysis. In the second paper of this series, we discuss, based on the findings in this paper, short-comings of the mean intercept analysis for (bone-)microstructure characterization. We will suggest alternative and better defined sensitive anisotropy measures from integral geometry.

Mean-intercept anisotropy analysis of porous media. II. Conceptual shortcomings of the MIL tensor definition and Minkowski tensors as an alternative.

PURPOSE: Structure-property relations, which relate the shape of the microstructure to physical properties such as transport or mechanical properties, need sensitive measures of structure. What are suitable fabric tensors to quantify the shape of anisotropic heterogeneous materials? The mean intercept length is among the most commonly used characteristics of anisotropy in porous media, e.g., of trabecular bone in medical physics. Yet, in this series of two papers we demonstrate that it has conceptual shortcomings that limit the validity of its results. METHODS: We test the validity of general assumptions regarding the properties of the mean-intercept length tensor using analytical formulas for the mean-intercept lengths in anisotropic Boolean models (derived in part I of this series), augmented by numerical simulations. We discuss in detail the functional form of the mean intercept length as a function of the test line orientations. RESULTS: As the most prominent result, we find that, at least for the example of overlapping grains modeling porous media, the polar plot of the mean intercept length is in general not an ellipse and hence not represented by a second-rank tensor. This is in stark contrast to the common understanding that for a large collection of grains the mean intercept length figure averages to an ellipse. The standard mean intercept length tensor defined by a least-square fit of an ellipse is based on a model mismatch, which causes an intrinsic lack of accuracy. CONCLUSIONS: Our analysis reveals several shortcomings of the mean intercept length tensor analysis that pose conceptual problems and limitations on the information content of this commonly used analysis method. We suggest the Minkowski tensors from integral geometry as alternative sensitive measures of anisotropy. The Minkowski tensors allow for a robust, comprehensive, and systematic approach to quantify various aspects of structural anisotropy. We show the Minkowski tensors to be more sensitive, in the sense, that they can quantify the remnant anisotropy of structures not captured by the mean intercept length analysis. If applied to porous tissue and microstructures, this improved structure characterization can yield new insights into the relationships between geometry and material properties.

Harmonic distortions enhance circular dichroism of dielectric single gyroids.

The departure from strict periodic order in two-phase dielectric materials can offer properties that are otherwise inaccessible to perfectly ordered photonic crystals. Herewith, we investigate the circular dichroism of the single gyroid photonic crystal in the presence of spatial distortions. FDTD simulations and microwave transmission measurements on 3D-printed replicas show that certain harmonic long-wavelength spatial distortions ("sinusoidal chirp") nearly doubles the imbalance of the circular polarisation reflectances, as well as significantly strengthens polarisation-incoherent reflectance. The observed changes are partially rationalised by comparison with simpler distortion models (linear chirp and tetragonal deformation) of the Gyroid.