Double diamond phase in pear-shaped nanoparticle systems with hard sphere solvent

The mechanisms behind the formation of bicontinuous nanogeometries, in particular in vivo, remain intriguing. Of particular interest are the many systems where more than one type or symmetry occurs, such as the Schwarz’ diamond surface and Schoen’s gyroid surface; a current example are the butterfly nanostructures often based on the gyroid, and the beetle nanostructures often based on the diamond surface. Here, we present a computational study of self-assembly of the bicontinuous Pn3m diamond phase in an equilibrium ensemble of pear-shaped particles when a small amount of a hard-sphere ‘solvent’ is added. Our results are based on previous work that showed the emergence of the gyroid Ia3d phase in a pure system of pear-shaped particles (Schönhöfer et al 2017 Interface Focus 7 20160161), in which the pear-shaped particles form an interdigitating bilayer reminiscent of a warped smectic structure. We here show that the addition of a small amount of hard spherical particles tends to drive the system towards the bicontinuous Pn3m double diamond phase, based on Schwarz diamond minimal surface. This result is consistent with the higher degree of spatial heterogeneity of the diamond minimal surface as compared to the gyroid minimal surface, with the hard-sphere ‘solvent’ acting as an agent to relieve packing frustration. However, the mechanism by which this relief is achieved is contrary to the corresponding mechanism in copolymeric systems; the spherical solvent tends to aggregate within the matrix phase, near the minimal surface, rather than within the labyrinthine channels. While it may relate to the specific form of the potential used to approximate the particle shape, this mechanism hints at an alternative way for particle systems to both release packing frustration and satisfy geometrical restrictions in double diamond configurations. Interestingly, the lattice parameters of the gyroid and the diamond phase appear to be commensurate with those of the isometric Bonnet transform

Circular Dichroism in Biological Photonic Crystals and Cubic Chiral Nets

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 butterfl

Bloch Modes and Evanescent Modes of Photonic Crystals: Weak Form Solutions Based on Accurate Interface Triangulation

We propose a new approach to calculate the complex photonic band structure, both purely dispersive and evanescent Bloch modes of a finite range, of arbitrary three-dimensional photonic crystals. Our method, based on a well-established plane wave expansion and the weak form solution of Maxwell’s equations, computes the Fourier components of periodic structures composed of distinct homogeneous material domains from a triangulated mesh representation of the inter-material interfaces; this allows substantially more accurate representations of the geometry of complex photonic crystals than the conventional representation by a cubic voxel grid. Our method works for general two-phase composite materials, consisting of bi-anisotropic materials with tensor-valued dielectric and magnetic permittivities ε and μ and coupling matrices ς. We demonstrate for the Bragg mirror and a simple cubic crystal closely related to the Kelvin foam that relatively small numbers of Fourier components are sufficient to yield good convergence of the eigenvalues, making this method viable, despite its computational complexity. As an application, we use the single gyroid crystal to demonstrate that the consideration of both conventional and evanescent Bloch modes is necessary to predict the key features of the reflectance spectrum by analysis of the band structure, in particular for light incident along the cubic [111] direction

Tangled (up in)cubes

The 'simplest' entanglements of the graph of edges of the cube are enumerated, forming two-cell {6,3} (hexagonal mesh) complexes on the genus-one two-dimensional torus. Five chiral pairs of knotted graphs are found. The examples contain non-trivial knotted and/or linked subgraphs [(2,2), (2,4) torus links and (3,2), (4,3) torus knots]

Fabrication and characterization of three-dimensional biomimetic chiral composites

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

Minimal surface scaffold designs for tissue engineering

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

Morphology and linear-elastic moduli of random network solids

The effective linear-elastic moduli of disordered network solids are analyzed by voxel-based finite element calculations. We analyze network solids given by Poisson-Voronoi processes and by the structure of collagen fiber networks imaged by confocal microscopy. The solid volume fraction φ is varied by adjusting the fiber radius, while keeping the structural mesh or pore size of the underlying network fixed. For intermediate φ, the bulk and shear modulus are approximated by empirical power-laws K(φ) α φn and G(φ) α φm with n≈ 1.4 and m≈ 1.7. The exponents for the collagen and the Poisson-Voronoi network solids are similar, and are close to the values n = 1.22 and m = 2.11 found in a previous voxel-based finite element study of Poisson-Voronoi systems with different boundary conditions. However, the exponents of these empirical power-laws are at odds with the analytic values of n = 1 and m= 2, valid for low-density cellular structures in the limit of thin beams. We propose a functional form for K(φ) that models the cross-over from a power-law at low densities to a porous solid at high densities; a fit of the data to this functional form yields the asymptotic exponent n≈ 1.00, as expected. Further, both the intensity of the Poisson-Voronoi process and the collagen concentration in the samples, both of which alter the typical pore or mesh size, affect the effective moduli only by the resulting change of the solid volume fraction. These findings suggest that a network solid with the structure of the collagen networks can be modeled in quantitative agreement by a Poisson-Voronoi process. 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

The Tricontinuous 3ths(5) Phase: A New Morphology in Copolymer Melts

Self-assembly remains the most efficient route to the formation of ordered nanostructures, including the double gyroid network phase in diblock copolymers based on two intergrown network domains. Here we use self-consistent field theory to show that a tricontinuous structure with monoclinic symmetry, called 3ths(5), based on the intergrowth of three distorted ths nets, is an equilibrium phase of triblock star-copolymer melts when an extended molecular core is introduced. The introduction of the core enhances the role of chain stretching by enforcing larger structural length scales, thus destabilizing the hexagonal columnar phase in favor of morphologies with less packing frustration. This study further demonstrates that the introduction of molecular cores is a general concept for tuning the relative importance of entropic and enthalpic free energy contributions, hence providing a tool to stabilize an extended repertoire of self-assembled nanostructured materials

The chiral structure of porous chitin within the wing-scales of Callophrys rubi

The structure of the porous three-dimensional reticulated pattern in the wing scales of the butterfly Callophrys rubi (the Green Hairstreak) is explored in detail, via scanning and transmission electron microscopy. A full 3D tomographic reconstruction o