Myfanwy E. Evans

Polycontinuous geometries for inverse lipid phases with more than two aqueous network domains.

Inverse bicontinuous cubic phases with two aqueous network domains separated by a smooth bilayer are firmly established as equilibrium phases in lipid/water systems. The purpose of this article is to highlight the generalisations of these bicontinuous geometries to polycontinuous geometries, which could be realised as lipid mesophases with three or more network-like aqueous domains separated by a branched bilayer. An analysis of structural homogeneity in terms of bilayer width variations reveals that ordered polycontinuous geometries are likely candidates for lipid mesophase structures, with similar chain packing characteristics to the inverse micellar phases (that once were believed not to exist due to high packing frustration). The average molecular shape required by global geometry to form these multi-network phases is quantified by the surfactant shape parameter, v/(al); we find that it adopts values close to those of the known lipid phases. We specifically analyse the 3etc(187 193) structure of hexagonal symmetry P6(3) /mcm with three aqueous domains, the 3dia(24 220) structure of cubic symmetry I43d composed of three distorted diamond networks, the cubic chiral 4srs(24 208) with cubic symmetry P4232 and the achiral 4srs(5 133) structure of symmetry P42/nbc, each consisting of four intergrown undistorted copies of the srs net (the same net as in the QII(G) gyroid phase). Structural homogeneity is analysed by a medial surface approach assuming that the headgroup interfaces are constant mean curvature surfaces. To facilitate future experimental identification, we provide simulated SAXS scattering patterns that, for the 4srs(24 208) and 3dia(24 220) structures, bear remarkable similarity to those of bicontinuous QII(G)-gyroid and QII(D)-diamond phases, with comparable lattice parameters and only a single peak that cannot be indexed to the well-established structures. While polycontinuous lipid phases have, to date, not been reported, the likelihood of their formation is further indicated by the reported observation of a solid tricontinuous mesoporous silicate structure, termed IBN-9, which formed in the presence of surfactants [Han et al., Nat. Chem., 2009, 1, 123].

Hierarchical self-assembly of a striped gyroid formed by threaded chiral mesoscale networks

Significance Chirality and hierarchical ordering are two fundamental properties found in many of nature’s most complex self-assembled structures such as living cells. Simultaneous control over these properties in synthetic systems is vital to mimic or even surpass nature’s designs. Via numerical simulations, we describe a class of complex morphologies that afford radically new architectures for self-assembled shapes. Specifically, a mixture of two star block-copolymers are shown to form multiple interwoven 2D and 3D labyrinths—all chiral—and hierarchically ordered on two different length scales. Furthermore, we show that such intricate network morphologies forming at a confined, hyperbolic interface can be classified and modeled in terms of a much simpler isotropic model of packing based on tilings of the hyperbolic plane. Numerical simulations reveal a family of hierarchical and chiral multicontinuous network structures self-assembled from a melt blend of Y-shaped ABC and ABD three-miktoarm star terpolymers, constrained to have equal-sized A/B and C/D chains, respectively. The C and D majority domains within these patterns form a pair of chiral enantiomeric gyroid labyrinths (srs nets) over a broad range of compositions. The minority A and B components together define a hyperbolic film whose midsurface follows the gyroid minimal surface. A second level of assembly is found within the film, with the minority components also forming labyrinthine domains whose geometry and topology changes systematically as a function of composition. These smaller labyrinths are well described by a family of patterns that tile the hyperbolic plane by regular degree-three trees mapped onto the gyroid. The labyrinths within the gyroid film are densely packed and contain either graphitic hcb nets (chicken wire) or srs nets, forming convoluted intergrowths of multiple nets. Furthermore, each net is ideally a single chiral enantiomer, induced by the gyroid architecture. However, the numerical simulations result in defect-ridden achiral patterns, containing domains of either hand, due to the achiral terpolymeric starting molecules. These mesostructures are among the most topologically complex morphologies identified to date and represent an example of hierarchical ordering within a hyperbolic pattern, a unique mode of soft-matter self-assembly.

Set Voronoi diagrams of 3D assemblies of aspherical particles

Abstract 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.

Periodic entanglement I: Networks from hyperbolic reticulations

High-symmetry free tilings of the two-dimensional hyperbolic plane ({\bb H}^2) can be projected to genus-3 3-periodic minimal surfaces (TPMSs). The three-dimensional patterns that arise from this construction typically consist of multiple catenated nets. This paper presents a construction technique and limited catalogue of such entangled structures, that emerge from the simplest examples of regular ribbon tilings of the hyperbolic plane via projection onto four genus-3 TPMSs: the P, D, G(yroid) and H surfaces. The entanglements of these patterns are explored and partially characterized using tools from TOPOS, GAVROG and a new tightening algorithm.

Networklike propagation of cell-level stress in sheared random foams.

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 Plateaus 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 cells 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.

Trading spaces: building three-dimensional nets from two-dimensional tilings

We construct some examples of finite and infinite crystalline three-dimensional nets derived from symmetric reticulations of homogeneous two-dimensional spaces: elliptic (S2), Euclidean (E2) and hyperbolic (H2) space. Those reticulations are edges and vertices of simple spherical, planar and hyperbolic tilings. We show that various projections of the simplest symmetric tilings of those spaces into three-dimensional Euclidean space lead to topologically and geometrically complex patterns, including multiple interwoven nets and tangled nets that are otherwise difficult to generate ab initio in three dimensions.

From three-dimensional weavings to swollen corneocytes

A novel technique to generate three-dimensional Euclidean weavings, composed of close-packed, periodic arrays of one-dimensional fibres, is described. Some of these weavings are shown to dilate by simple shape changes of the constituent fibres (such as fibre straightening). The free volume within a chiral cubic example of a dilatant weaving, the ideal conformation of the G129 weaving related to the Σ+ rod packing, expands more than fivefold on filament straightening. This remarkable three-dimensional weaving, therefore, allows an unprecedented variation of packing density without loss of structural rigidity and is an attractive design target for materials. We propose that the G129 weaving (ideal Σ+ weaving) is formed by keratin fibres in the outermost layer of mammalian skin, probably templated by a folded membrane.

Ideal geometry of periodic entanglements

Three-dimensional entanglements, including knots, knotted graphs, periodic arrays of woven filaments and interpenetrating nets, form an integral part of structure analysis because they influence various physical properties. Ideal embeddings of these entanglements give insight into identification and classification of the geometry and physically relevant configurations in vivo. This paper introduces an algorithm for the tightening of finite, periodic and branched entanglements to a least energy form. Our algorithm draws inspiration from the Shrink-On-No-Overlaps (SONO) (Pieranski 1998 In Ideal knots (eds A Stasiak, V Katritch, LH Kauffman), vol. 19, pp. 20–41.) algorithm for the tightening of knots and links: we call it Periodic-Branched Shrink-On-No-Overlaps (PB-SONO). We reproduce published results for ideal configurations of knots using PB-SONO. We then examine ideal geometry for finite entangled graphs, including θ-graphs and entangled tetrahedron- and cube-graphs. Finally, we compute ideal conformations of periodic weavings and entangled nets. The resulting ideal geometry is intriguing: we see spontaneous symmetrisation in some cases, breaking of symmetry in others, as well as configurations reminiscent of biological and chemical structures in nature.

A geometric exploration of stress in deformed liquid foams

We explore an alternate way of looking at the rheological response of a yield stress fluid: using discrete geometry to probe the heterogeneous distribution of stress in soap froth. We present quasi-static, uniaxial, isochoric compression and extension of three-dimensional random monodisperse soap froth in periodic boundary conditions and examine the stress and geometry that result. The stress and shape anisotropy of individual cells is quantified by Q, a scalar measure derived from the interface tensor that gauges each cells contribution to the global stress. Cumulatively, the spatial distribution of highly deformed cells allows us to examine how stress is internally distributed. The topology of highly deformed cells, how they arrange relative to one another in space, gives insight into the heterogeneous distribution of stress.

Periodic entanglement III: tangled degree-3 finite and layer net intergrowths from rare forests.

Entanglements of two-dimensional honeycomb nets are constructed from free tilings of the hyperbolic plane (H2) on triply periodic minimal surfaces. The 2-periodic nets that comprise the structures are guaranteed by considering regular, rare free tilings in H2. This paper catalogues an array of entanglements that are both beautiful and challenging for current classification techniques, including examples that are realized in metal-organic materials. The compactification of these structures to the genus-3 torus is considered as a preliminary method for generating entanglements of finite θ-graphs, potentially useful for gaining insight into the entanglement of the periodic structure. This work builds on previous structural enumerations given in Periodic entanglement Parts I and II [Evans et al. (2013). Acta Cryst. A69, 241-261; Evans et al. (2013). Acta Cryst. A69, 262-275].